Power System Operation and Control - GitHub

POWERSYSTEMOPERATIONANDCONTROL

S.SivanagarajuAssociateProfessor

DepartmentofElectricalandElectronicsEngineering

UniversityCollegeofEngineeringJNTUKakinada

Kakinada,AndhraPradesh

G.SreenivasanAssociateProfessor

DepartmentofElectricalandElectronicsEngineering

INTELLEngineeringCollegeAnantapur,AndhraPradesh

Chennai•Delhi•Chandigarh

BriefContents

Chapter1EconomicAspects

Chapter2EconomicLoadDispatch-I

Chapter3EconomicLoadDispatch-II

Chapter4OptimalUnitCommitment

Chapter5OptimalPower-FlowProblem—SolutionTechnique

Chapter6Hydro-ThermalScheduling

Chapter7LoadFrequencyControl-I

Chapter8LoadFrequencyControl-II

Chapter9ReactivePowerCompensation

Chapter10VoltageControl

Chapter11ModelingofPrimeMoversandGenerators

Chapter12ModelingofSpeedGoverningandExcitationSystems

Chapter13PowerSystemSecurityandStateEstimation

Contents

Chapter1EconomicAspects

1.1Introduction

1.2LoadCurve

1.3Load–DurationCurve

1.4IntegratedLoad–DurationCurve

1.4.1UsesofIntegratedLoad–DurationCurve

1.5DefinitionofTermsandFactors

1.5.1ConnectedLoad

1.5.2MaximumDemand

1.5.3DemandFactor

1.5.4AverageLoad

1.5.5LoadFactor

1.5.6DiversityFactor

1.5.7PlantCapacity

1.5.8PlantCapacityFactor

1.5.9UtilizationFactor(orPlant-UseFactor)

1.5.10FirmPower

1.5.11PrimePower

1.5.12DumpPower

1.5.13SpillPower

1.5.14ColdReserve

1.5.15HotReserve

1.5.16SpinningReserve

1.6BaseLoadandPeakLoadonaPowerStation

1.7LoadForecasting

1.7.1PurposeofLoadForecasting

1.7.2ClassificationofLoadForecasting

1.7.3ForecastingProcedure

KeyNotes

ShortQuestionsandAnswers

Multiple-ChoiceQuestions

ReviewQuestions

Problems

Chapter2EconomicLoadDispatch-I

2.1Introduction

2.2CharacteristicsofPowerGeneration(Steam)Unit

2.3SystemVariables

2.3.1ControlVariables(P andQ )

2.3.2DisturbanceVariables(P andQ )

2.3.3StateVariables(Vandδ)

2.4ProblemofOptimumDispatch—Formulation

2.5Input–OutputCharacteristics

2.5.1UnitsofTurbineInput

2.6CostCurves

2.7IncrementalFuelCostCurve

2.8HeatRateCurve

2.9IncrementalEfficiency

2.10Non-SmoothCostFunctionswithMultivalveEffect

2.11Non-smoothCostFunctionswithMultipleFuels

2.12CharacteristicsofaHydro-PowerUnit

2.12.1EffectoftheWaterHeadonDischargeofWaterforaHydro-Unit

2.12.2IncrementalWaterRateCharacteristicsofHydro-Units

2.12.3IncrementalCostCharacteristicofaHydro-Unit

2.12.4ConstraintsofHydro-PowerPlants

2.13IncrementalProductionCosts

2.14ClassicalMethodsforEconomicOperationofSystemPlants

2.15OptimizationProblem—MathematicalFormulation(NeglectingtheTransmissionLosses)

2.15.1ObjectiveFunction

2.15.2ConstraintEquations

2.16MathematicalDeterminationofOptimalAllocationofTotalLoadAmongDifferentUnits

2.17ComputationalMethods

2.17.1AnalyticalMethod

2.17.2GraphicalMethod

2.17.3SolutionbyUsingaDigitalComputer

G G

D D

2.18EconomicDispatchNeglectingLossesandIncludingGeneratorLimits

2.19FlowchartforObtainingOptimalSchedulingofGeneratingUnitsbyNeglectingtheTransmissionLosses

2.20EconomicalLoadDispatch—InOtherUnits

2.20.1Nuclearunits

2.20.2Pumpedstoragehydro-units

2.20.3Hydro-plants

2.20.4Includingreactive-powerflows

KeyNotes



ReviewQuestions

Problems

Chapter3EconomicLoadDispatch-II

3.1Introduction

3.2OptimalGenerationSchedulingProblem:ConsiderationofTransmissionLosses

3.2.1Mathematicalmodeling

3.3TransmissionLossExpressioninTermsofReal-PowerGeneration—Derivation

3.4MathematicalDeterminationofOptimumAllocationofTotalLoadwhenTransmissionLossesareTakenintoConsideration

3.4.1DeterminationofITLformula

3.4.2PenaltyFactor

3.5FlowchartfortheSolutionofanOptimizationProblemwhenTransmissionLossesareConsidered

KeyNotes



ReviewQuestions

Problems

Chapter4OptimalUnitCommitment

4.1Introduction

4.2ComparisonwithEconomicLoadDispatch

4.3NeedforUC

4.4ConstraintsinUC

4.4.1SpinningReserve

4.4.2ThermalUnitConstraints

4.4.3Hydro-Constraints

4.4.4MustRun

4.4.5FuelConstraints

4.5CostFunctionFormulation

4.5.1Start-upCostConsideration

4.5.2Shut-downCostConsideration

4.6ConstraintsforPlantCommitmentSchedules

4.7UnitCommitment—SolutionMethods

4.7.1EnumerationScheme

4.7.2Priority-listMethod

4.7.3DynamicProgramming

4.8ConsiderationofReliabilityinOptimalUCProblem

4.8.1Patton’ssecurityfunction

4.9OptimalUCwithSecurityConstraint

4.9.1IllustrationofSecurityConstraintwithExample4.2

4.10Start-UpConsideration

KeyNotes



ReviewQuestions

Problems

Chapter5OptimalPower-FlowProblem—SolutionTechnique

5.1Introduction

5.2OptimalPower-FlowProblemwithoutInequalityConstraints

5.2.1AlgorithmforComputationalProcedure

5.3OptimalPower-FlowProblemwithInequalityConstraints

5.3.1InequalityConstraintsonControlVariables

5.3.2InequalityConstraintsonDependentVariables—PenaltyFunctionMethod

KeyNotes



ReviewQuestions

Chapter6Hydro-ThermalScheduling

6.1Introduction

6.2Hydro-ThermalCo-ordination

6.3SchedulingofHydro-UnitsinaHydro-ThermalSystem

6.4Co-ordinationofRun-offRiverPlantandSteamPlant

6.5Long-TermCo-ordination

6.6Short-TermCo-ordination

6.6.1ConstantHydro-GenerationMethod

6.6.2ConstantThermalGenerationMethod

6.6.3MaximumHydro-EfficiencyMethod

6.7GeneralMathematicalFormulationofLong-TermHydro-ThermalScheduling

6.7.1SolutionofProblem-DiscretizationPrinciple

6.7.2SolutionTechnique

6.7.3Algorithm

6.8SolutionofShort-TermHydro-ThermalSchedulingProblems—Kirchmayer’sMethod

6.9AdvantagesofOperationofHydro-ThermalCombinations

6.9.1Flexibility

6.9.2GreaterEconomy

6.9.3SecurityofSupply

6.9.4BetterEnergyConservation

6.9.5ReserveCapacityMaintenance

KeyNotes



ReviewQuestions

Problems

Chapter7LoadFrequencyControl-I

7.1Introduction

7.2NecessityofMaintainingFrequencyConstant

7.3LoadFrequencyControl

7.4GovernorCharacteristicsofaSingleGenerator

7.5AdjustmentofGovernorCharacteristicofParallelOperatingUnits

7.6LFC:(P–fControl)

7.7Q–VControl

7.8GeneratorControllers(P–fandQ–VControllers)

7.9P–fControlversusQ–VControl

7.10DynamicInteractionBetweenP–fandQ–VLoops

7.11Speed-GoverningSystem

7.11.1Speed-GoverningSystemModel

7.12TurbineModel

7.12.1Non-reheat-typeSteamTurbines

7.12.2IncrementalorSmallSignalforaTurbine-GovernorSystem

7.12.3ReheatTypeofSteamTurbines

7.13Generator-LoadModel

7.14ControlAreaConcept

7.15IncrementalPowerBalanceofControlArea

7.16SingleAreaIdentification

7.16.1BlockDiagramRepresentationofaSingleArea

7.17SingleArea—Steady-StateAnalysis

7.17.1Speed-ChangerPositionisConstant(UncontrolledCase)

7.17.2LoadDemandisConstant(ControlledCase)

7.17.3SpeedChangerandLoadDemandareVariables

7.18StaticLoadFrequencyCurves

7.19DynamicAnalysis

7.20RequirementsoftheControlStrategy

7.20.1IntegralControl

7.21AnalysisoftheIntegralControl

7.22RoleofIntegralControllerGain(K )Setting

7.23ControlofGeneratorUnitPowerOutput

KeyNotes



ReviewQuestions

Problems

Chapter8LoadFrequencyControl-II

8.1Introduction

8.2CompositeBlockDiagramofaTwo-AreaCase

8.3ResponseofaTwo-AreaSystem—UncontrolledCase

8.3.1StaticResponse

8.3.2DynamicResponse

8.4AreaControlError—Two-AreaCase

8.5CompositeBlockDiagramofaTwo-AreaSystem(ControlledCase)

8.5.1Tie-lineBiasControl

I

8.5.2Steady-StateResponse

8.5.3DynamicResponse

8.6OptimumParameterAdjustment

8.7LoadFrequencyandEconomicDispatchControls

8.8DesignofAutomaticGenerationControlUsingtheKalmanMethod

8.9Dynamic-State-VariableModel

8.9.1ModelofSingle-AreaDynamicSysteminaState-VariableForm

8.9.2OptimumControlIndex(I)

8.9.3OptimumControlProblemandStrategy

8.9.4DynamicEquationsofaTwo-AreaSystem

8.9.5State-VariableModelforaThree-AreaPowerSystem

8.9.6AdvantagesofState-VariableModel

KeyNotes



ReviewQuestions

Problems

Chapter9ReactivePowerCompensation

9.1Introduction

9.2ObjectivesofLoadCompensation

9.2.1P.f.Correction

9.2.2VoltageRegulationImprovement

9.2.3LoadBalancing

9.3IdealCompensator

9.4SpecificationsofLoadCompensation

9.5TheoryofLoadCompensation

9.5.1P.f.correction

9.5.2VoltageRegulation

9.6LoadBalancingandp.f.ImprovementofUnsymmetricalThree-PhaseLoads

9.6.1P.f.Correction

9.6.2LoadBalancing

9.7UncompensatedTransmissionLines

9.7.1FundamentalTransmissionLineEquation

9.7.2CharacteristicImpedance

9.7.3SurgeImpedanceorNaturalLoading

9.8UncompensatedLinewithOpen-Circuit

9.8.1VoltageandCurrentProfiles

9.8.2TheSymmetricalLineatno-Load

9.8.3UnderexcitedOperationofGeneratorsDuetoLine-Charging

9.9TheUncompensatedLineUnderLoad

9.9.1RadiallinewithfixedSending-endVoltage

9.9.2ReactivePowerRequirements

9.9.3TheUncompensatedLineUnderLoadwithConsiderationofMaximumPowerandStability

9.10CompensatedTransmissionLines

9.11Sub-SynchronousResonance

9.11.1EffectsofSeriesandShuntCompensationofLines

9.11.2ConceptofSSRinLines

9.12ShuntCompensator

9.12.1Thyristor-ControlledReactor

9.12.2Thyristor-SwitchedCapacitor

9.13SeriesCompensator

9.14UnifiedPower-FlowController

9.15BasicRelationshipforPower-FlowControl

9.15.1WithoutLineCompensation

9.15.2WithSeriesCapacitiveCompensation

9.15.3WithShuntCompensation

9.15.4WithPhaseAngleControl

9.16ComparisonofDifferentTypesofCompensatingEquipmentforTransmissionSystems

9.17VoltageStability—Whatisit?

9.17.1VoltageStability

9.17.2VoltageCollapse

9.18Voltage-StabilityAnalysis

9.18.1P–VCurves

9.18.2ConceptofVoltageCollapseProximateIndicator

9.18.3Voltage-StabilityAnalysis:Q–VCurves

9.19DerivationforVoltage-StabilityIndex

KeyNotes



ReviewQuestions

Problems

Chapter10VoltageControl

10.1Introduction

10.2NecessityofVoltageControl

10.3GenerationandAbsorptionofReactivePower

10.4LocationofVoltage-ControlEquipment

10.5MethodsofVoltageControl

10.5.1ExcitationControl

10.5.2ShuntCapacitorsandReactors

10.5.3SeriesCapacitors

10.5.4Tap-ChangingTransformers

10.5.5BoosterTransformers

10.5.6SynchronousCondensers

10.6RatingofSynchronousPhaseModifier

KeyNotes



ReviewQuestions

Problems

Chapter11ModelingofPrimeMoversandGenerators

11.1Introduction

11.2HydraulicTurbineSystem

11.2.1ModelingofHydraulicTurbine

11.3SteamTurbineModeling

11.3.1Non-reheatType

11.3.2Reheattype

11.4SynchronousMachines

11.4.1Salient-pole-typeRotor

11.4.2Non-salient-pole-typeRotor

11.5SimplifiedModelofSynchronousMachine(NeglectingSaliencyandChangesinFluxLinkages)

11.6EffectofSaliency

11.7GeneralEquationofSynchronousMachine

11.8DeterminationofSynchronousMachineInductances

11.8.1Assumptions

11.9RotorInductances

11.9.1RotorSelf-Inductance

11.9.2StatortoRotorMutualInductances

11.10StatorSelf-Inductances

11.11StatorMutualInductances

11.12DevelopmentofGeneralMachineEquations—MatrixForm

11.13Blondel’sTransformation(or)Park’sTransformationto‘dqo’Components

11.14InversePark’sTransformation

11.15Power-InvariantTransformationin‘f-d-q-o’Axes

11.16FluxLinkageEquations

11.17VoltageEquations

11.18PhysicalInterpretationofEquations(11.62)and(11.68)

11.19GeneralizedImpedanceMatrix(Voltage–CurrentRelations)

11.20TorqueEquation

11.21SynchronousMachine—Steady-stateAnalysis

11.21.1Salient-poleSynchronousMachine

11.21.2Non-salient-poleSynchronous(CylindricalRotor)Machine

11.22DynamicModelofSynchronousMachine

11.22.1Salient-poleSynchronousGenerator—Sub-TransientEffect

11.22.2DynamicModelofSynchronousMachineIncludingDamperWinding

11.22.3EquivalentCircuitofSynchronousGenerator—IncludingDamperWindingEffect

11.23ModelingofSynchronousMachine—SwingEquation

KeyNotes



ReviewQuestions

Chapter12ModelingofSpeedGoverningandExcitationSystems

12.1Introduction

12.2ModelingofSpeed-GoverningSystems

12.3ForSteamTurbines

12.3.1Mechanical–Hydraulic-ControlledSpeed-GoverningSystems

12.3.2Electro–Hydraulic-ControlledSpeed-GoverningSystems

12.3.3GeneralModelforSpeed-GoverningSystems

12.4ForHydro-Turbines

12.4.1Mechanical–Hydraulic-ControlledSpeed-GoverningSystems

12.4.2Electric–Hydraulic-ControlledSpeed-GoverningSystem

12.5ModelingwithLimits

12.5.1Wind-upLimiter

12.5.2Non-wind-upLimiter

12.6ModelingofaSteam-GovernorTurbineSystem

12.6.1ReheatSystemUnit

12.6.2BlockDiagramRepresentation

12.6.3TransferFunctionoftheSteam-GovernorTurbineModeling

12.7ModelingofaHydro-Turbine-SpeedGovernor

12.8ExcitationSystems

12.9EffectofVaryingExcitationofaSynchronousGenerator

12.9.1Explanation

12.9.2LimitationsofIncreaseinExcitation

12.10MethodsofProvidingExcitation

12.10.1CommonExcitationBusMethod

12.10.2IndividualExcitationMethod

12.10.3BlockDiagramRepresentationStructureofaGeneralExcitationSystem

12.11ExcitationControlScheme

12.12ExcitationSystems—Classification

12.12.1DCExcitationSystem

12.12.2ACExcitationSystem

12.12.3StaticExcitationSystem

12.13VariousComponentsandtheirTransferFunctionsofExcitationSystems

12.13.1PTandRectifier

12.13.2VoltageComparator

12.13.3Amplifiers

12.14Self-excitedExciterandAmplidyne

12.15DevelopmentofExcitationSystemBlockDiagram

12.15.1TransferFunctionoftheStabilizingTransformer

12.15.2TransferFunctionofSynchronousGenerator

12.15.3IEEEType-1ExcitationSystem

12.15.4TransferFunctionofOverallExcitationSystem

12.16GeneralFunctionalBlockDiagramofanExcitationSystem

12.16.1TerminalVoltageTransducerandLoadCompensation

12.16.2ExcitersandVoltageRegulators

12.16.3ExcitationSystemStabilizerandTransientGainReduction

12.16.4PowerSystemStabilizer

12.17StandardBlockDiagramRepresentationsofDifferentExcitationSystems

12.17.1DCExcitationSystem

12.17.2ACExcitationSystem

12.17.3StaticExcitationSystem

KeyNotes



ReviewQuestions

Chapter13PowerSystemSecurityandStateEstimation

13.1Introduction

13.2TheConceptofSystemSecurity

13.2.1Long-TermPlanning

13.2.2OperationalPlanning

13.2.3On-lineOperation

13.3SecurityAnalysis

13.3.1DigitalSimulation

13.3.2HybridComputerSimulation

13.3.3LyapunovMethods

13.3.4PatternRecognition

13.4SecurityEnhancement

13.5SSSAnalysis

13.5.1RequirementsofanSSSAssessor

13.6TransientSecurityAnalysis

13.6.1DigitalSimulation

13.6.2PatternRecognition

13.6.3LyapunovMethod

13.7StateEstimation

13.7.1StateEstimator

13.7.2Static-StateEstimation

13.7.3ModelingofUncertainty

13.7.4SomeBasicFactsofStateEstimation

13.7.5LeastSquaresEstimation

13.7.6ApplicationsofStateEstimation

KeyNotes



ReviewQuestions

AppendixA

Alsobythesameauthor

ElectricPowerTransmissionandDistributionisacomprehensivetextdesignedforundergraduatecourses.Apartoftheelectricalengineeringcurriculum,thisbookisdesignedtomeettherequirementsofstudentstakingelementarycoursesinelectricpowertransmissionanddistribution.Writteninasimple,easy-to-understandmanner,thisbookintroducesthereadertoelectrical,mechanicalandeconomicaspectsofthedesignandconstructionofpowertransmissionanddistributionsystems.

Toourparents

Preface

ThisbookentitledPowerSystemOperationandControlhasbeenintendedforusebyundergraduatestudentsinIndianuniversities.Withajudiciousmixofadvancedtopics,thebookmayalsobeusefulforsomeinstitutionsasafirstcourseforpostgraduates.Theorganizationofthisbookreflectsourdesiretoprovidethereaderwithathoroughunderstandingofthebasicprinciplesandtechniquesofpowersystemoperationandcontrol.Writtentoaddresstheneedforatextthatclearlypresentstheconceptofeconomicsystemoperationinamannerthatkindlesinterest,thetopicsaredealtwithusingalucidapproachthatmaybenefitbeginnersaswellasadvancedlearnersofthesubject.Ithasbeendesignedasafunctionalaidtohelpstudentslearnindependently.

Chapter1introducestheeconomicaspectsofpowersystemandprovidesdefinitionsforthevarioustermsusedinitsanalysis.Itexplainsreserverequirements,theimportanceofloadforecasting,anditsclassification.

Chapter2describessystemvariablesandtheirfunctions.Thecharacteristicsofthermalandhydro-powerunitsareillustratedinthisunit.Non-smoothcostfunctionswithmulti-valveeffectandwithmulti-fueleffectarebrieflydiscussed.Thischapterexplainsthemathematicalformulationofeconomicloaddispatchamongvariousunitsbyneglectingtransmissionlosses,anditalsogivesanoverviewoftheapplicationsofvariouscomputationalmethodstosolvetheoptimizationproblem.Theflowchartrequiredtoobtaintheoptimalschedulingofgeneratingunitsisalsodescribedhere.

Chapter3looksatthederivationoftheexpressionfortransmissionlossandexplainsthemathematical

determinationofeconomicloaddispatchtakingtransmissionlossintoconsideration.Thetheoryofincrementaltransmissionlossandpenaltyfactorisclearlydiscussed.Italsoanalyzestheoptimalschedulingofgeneratingunits,determinedwiththehelpofaflowchart.

Chapter4expoundsontheoptimalunitcommitmentproblemanditssolutionmethodsbytakingareliableexample.Reliabilityandstart-upconsiderationsinoptimalunitcommitmentproblemsareeffectivelydiscussed.

Chapter5explainstheoptimalpower-flowproblemanditssolutiontechniqueswithandwithoutinequalityconstraints.Inthischapter,inequalityconstraintsareconsideredfirstoncontrolvariables,andthenondependentvariables.Kuhn–Tuckerconditionsforthesolutionofanoptimalpowerflowarepresentedinthisunit.

Chapter6spellsouttheimportantprincipleofhydro-thermalschedulinganditsclassification.Itdiscussesthegeneralmathematicalformulationsandmethodsofsolvinglong-termandtheshort-termhydro-thermalschedulingproblems.

Chapter7dealswithsingle-arealoadfrequencycontrol.Itdescribesthecharacteristicsofthespeedgovernoranditsadjustmentincaseofparalleloperatingunits.Generatorcontrollers,namely,P–fandQ–Vcontrollers,thespeed-governingsystemmodel,theturbinemodel,andthegenerator–loadmodelandtheirblockdiagramrepresentationsareclearlydiscussed.Steady-anddynamic-stateanalysesofasingle-arealoadfrequencycontrolsystemarealsoexplained.Thechapteralsodiscussestheanalysisofintegralcontrolofasingle-arealoadfrequencycontrolsystem.

Chapter8dealswiththeresponseofatwo-arealoadfrequencycontrolforuncontrolledandcontrolledcases

veryeffectively.Adynamic-statevariablemodelforatwo-arealoadfrequencycontrolandforathree-arealoadfrequencycontrolsystemisderived.

Chapter9delineatesreactive-powercompensationalongwiththeobjectivesofloadcompensation.Thischapterdiscussesuncompensatedtransmissionlinesunderno-loadandloadconditions,andcompensatedtransmissionlineswiththeeffectsofseriesandshuntcompensationusingthyristor-controlledreactorsandcapacitors.ItalsoelucidatestheconceptofvoltagestabilityandmakesclearhowtheanalysisofvoltagestabilityiscarriedoutusingP–VcurvesandQ–Vcurves.

Therelationshipamongactivepower,reactivepower,andvoltageisderivedinChapter10.Thischapteralsospeaksaboutthemethodsofvoltagecontrolandthelocationofvoltage-controlequipments.

Chapter11dealswiththeprinciplesofmodelinghydro-turbinesandsteamturbines.Italsolooksatthemodelingofsynchronousmachinesincludingthesimplifiedmodelwiththeeffectofsaliency.Thedeterminationofself-inductanceandmutualinductance,andthedevelopmentofgeneralmachineequationsarediscussedinthischapter.Park’stransformationanditsinverse,thederivationsoffluxlinkageequationsandvoltageequationsofsynchronousmachines,andthesteady-stateanddynamic-statemodelanalysisareelucidated.

Chapter12offersaninsightintothemodelingofspeed-governingsystemsforsteam-andhydro-turbines.Mechanical–hydraulic-controlledspeed-governingsystems,electro–hydraulic-controlledspeed-governingsystems,andthegeneralmodelforspeed-governingsystemsforsteamturbinesareexplainedindetail.Itthrowslightonexcitationsystemmodelinginvariousaspectssuchasmethodsofprovidingexcitation,classificationofexcitationsystems,andvariouscomponentswiththeirtransferfunctions.Standard

blockdiagramrepresentationsforthedifferentexcitationsystemsareillustratedinthischapter.

Chapter13explainsthesteady-statesecurityanalysisandthetransientsecurityanalysisofapowersystem.Theconceptofstateestimationisdevelopedinthischapter,andthemethodofleastsquaresestimationofasystemstatehasbeenclearlyexplained.

1

EconomicAspects

OBJECTIVES

Afterreadingthischapter,youshouldbeableto

knowtheeconomicaspectsofpowersystems

analyzethevariousloadcurvesofeconomicpowergeneration

definethevarioustermsofeconomicpowergeneration

understandtheimportanceofloadforecasting

1.1INTRODUCTION

Apowersystemconsistsofseveralgeneratingstations,whereelectricalenergyisgenerated,andseveralconsumersforwhoseusetheelectricalenergyisgenerated.Theobjectiveofanypowersystemistogenerateelectricalenergyinsufficientquantitiesatthebest-suitedlocationsandtotransmitittothevariousloadcentersandthendistributeittothevariousconsumersmaintainingthequalityandreliabilityataneconomicprice.Qualityimpliesthatthefrequencybemaintainedconstantatthespecifiedvalue(50Hzinourcountry;though60-Hzsystemsarealsoprevailinginsomecountries)andthatthevoltagebemaintainedconstantatthespecifiedvalue.Further,theinterruptionstothesupplyofenergyshouldbeasminimumaspossible.

Oneimportantcharacteristicofelectricenergyisthatitshouldbeusedasitisgenerated;otherwiseitmaybestatedthattheenergygeneratedmustbesufficienttomeettherequirementsoftheconsumersatalltimes.Becauseofthediversifiednatureofactivitiesofthe

consumers(e.g.,domestic,industrial,agricultural,etc.),theloadonthesystemvariesfrominstanttoinstant.However,thegeneratingstationmustbeina‘stateofreadiness’tosupplytheloadwithoutanyintimationfromtheconsumer.This‘variableloadproblem’istobetackledeffectivelyeversincetheinceptionofapowersystem.Thisnecessitatesathoroughunderstandingofthenatureoftheloadtobesupplied,whichcanbereadilyobtainedfromtheloadcurve,load–durationcurve,etc.

1.2LOADCURVE

Aloadcurveisaplotoftheloaddemand(onthey-axis)versusthetime(onthex-axis)inthechronologicalorder.

Fromoutoftheloadconnected,aconsumerusesdifferentfractionsofthetotalloadatvarioustimesofthedayasperhis/herrequirements.Sinceapowersystemhastosupplyloadtoallsuchconsumers,theloadtobesuppliedvariescontinuouslywithtimeanddoesnotremainconstant.Iftheloadismeasured(inunitsofpower)atregularintervalsoftime,say,onceinanhour(orhalf-an-hour)andrecorded,wecandrawacurveknownastheloadcurve.

Atimeperiodofonly24hoursisconsidered,andtheresultingloadcurve,whichiscalleda‘Dailyloadcurve’,isshowninFig.1.1.However,topredicttheannualrequirementsofenergy,theoccurrenceofloadatdifferenthoursanddaysinayearandinthepowersupplyeconomics,‘Annualloadcurves’areused.

FIG.1.1Dailyloadcurve

Anannualloadcurveisaplotoftheloaddemandoftheconsumeragainsttimeinhoursoftheyear(1year=8,760hours).

Significance:FromthedailyloadcurveshowninFig.1.1,thefollowinginformationcanbeobtained:

Observethevariationofloadonthepowersystemduringdifferenthoursoftheday.Areaunderthiscurvegivesthenumberofunitsgeneratedinaday.Highestpointonthatcurveindicatesthemaximumdemandonthepowerstationonthatday.Theareaofthiscurvedividedby24hoursgivestheaverageloadonthepowerstationintheday.Ithelpsinselectionoftheratingandnumberofgeneratingunitsrequired.

1.3LOAD–DURATIONCURVE

Theload–durationcurveisaplotoftheloaddemands(inunitsofpower)arrangedinadescendingorderofmagnitude(onthey-axis)andthetimeinhours(onthex-axis).Theload–durationcurvecanbedrawnasshowninFig.1.2.

FIG.1.2Load–durationcurve

1.4INTEGRATEDLOAD–DURATIONCURVE

Theintegratedload–durationcurveisaplotofthecumulativenumberofunitsofelectricalenergy(onthex-axis)andtheloaddemand(onthey-axis).

Intheoperationofhydro-electricplants,itisnecessarytoknowtheamountofenergybetweendifferentloadlevels.Thisinformationcanbeobtainedfromtheload–durationcurve.Thus,letthedurationcurveofaparticularpowerstationbeasindicatedinFig.1.3(a);obviouslytheareaenclosedbytheload–durationcurverepresentsthedailyenergygenerated(inMWh).

Theminimumloadonthestationisd (MW).Theenergygeneratedduringthe24-hourperiodis24d(MWh),i.e.,theareaoftherectangleod b a .So,wecanassumethattheenergygeneratedvarieslinearlywiththeloaddemandfromzerotod tod MWasindicatedinFig.1.3(a).Astheloaddemandincreasesfromd todMW,thetotalenergygeneratedwillbelessthan24dMWh,sincetheloaddemandofd MWpersistsforadurationoflessthan24hours.Thetotalenergygeneratedisgivenbytheareaod b a .So,theenergygeneratedbetweentheloaddemandsofd andd is(area

1

1

1 1 1

1 2

1 2

2

2

2 2 1

2 1

od b a –areaod b a )=aread d b (showncross-latchedinFig.1.3(a)).

Now,ifthetotalnumberofunitsgeneratedwastobeplottedasabscissacorrespondingtoagivenload,weshallobtainwhatiscalledtheintegratedload–durationcurve.Thus,iftheareaod b a weredesignatedasc(MWh),thenpointphastheco-ordinates(e ,d )ontheintegratedload–durationcurveshowninFig.1.3(b).

Theintegratedload–durationcurveisalsotheplotofthecumulativeintegrationofareaundertheloadcurvestartingatzeroloadstotheparticularload.Itexhibitsanincreasingslopeuptothepeakload.

1.4.1Usesofintegratedload–durationcurve

1. Theamountofenergygeneratedbetweendifferentloadlevelscanbeobtained.

2. Fromacknowledgmentofthedailyenergyrequirements,theloadthatcanbecarriedonthebaseorpeakcanbeeasilydetermined.

FIG.1.3Integratedload–durationcurve

2 2 1 1 1 1 1 2 1

2 2 1 2

2 2


Tohaveaclearideaof‘base-load’and‘peakload’,letusconsiderapowersystem,thedailyloadcurveofwhichisdepictedinFig.1.4.

Inapowersystem,theremaybeseveraltypesofgeneratingstationssuchashydro-electricstations,fossil-fuel-firedstations,nuclearstations,andgas-turbine-drivengeneratingstations.Ofthesestations,someactasbase-loadstations,whileothersactaspeakloadstations.

Base-loadstationsrunat100%capacityona24-hourbasis.Nuclearreactorsareideallysuitedforthispurpose.

Intermediateorcontrolled-powergenerationstationsnormallyarenotfullyloaded.Hydro-electricstationsarethebestchoiceforthispurpose.

Peakloadstationsoperateduringthepeakloadhoursonly.Sincethegas-turbine-drivengeneratorscanpickuptheloadveryquickly,theyarebestsuitedtoserveaspeakloadstations.Whereavailable,pumped-storagehydro-electricplantscanbeoperatedaspeakloadstations.

Abase-loadstationoperatesatahigh-loadfactor,whereasthepeakloadplantoperatesatalow-load

factor.So,thebase-loadstationshouldhavelowoperatingcosts.

1.5DEFINITIONOFTERMSANDFACTORS

Severaltermsareusedinconnectionwithpowersupplytoanarea,whetheritbeforthefirsttime(asisthecasewhentheareaisbeingelectrifiedforthefirsttime)orsubsequently(duetotheloadgrowth).Thesetermsareexplainedbelow.

1.5.1Connectedload

Aconsumer,forexample,adomesticconsumer,mayhaveseveralappliancesratedatdifferentwattages.Thesumoftheseratingsishis/herconnectedload.

Connectedloadisthesumoftheratings(W,kW,orMW)oftheapparatusinstalledonaconsumer’spremises.

1.5.2Maximumdemand

Itisthemaximumloadusedbyaconsumeratanytime.Itcanbelessthanorequaltotheconnectedload.Ifallthedevicesconnectedintheconsumer’shouseruntotheirfullestextentsimultaneously,thenthemaximumdemandwillbeequaltotheconnectedload.Butgenerally,theactualmaximumdemandwillbelessthantheconnectedloadsincealltheappliancesareneverusedatfullloadatatime.

Themaximumdemandisusuallymeasuredinunitsofkilowatts(kW)ormegawatts(MW)byamaximumdemandindicator.(Usually,inthecaseofhigh-tensionconsumers,themaximumdemandismeasuredintermsofkVAorMVA.)

1.5.3Demandfactor

Theratioofthemaximumdemandtotheconnectedloadiscalledthe‘demandfactor’:

Note:Maximumdemandandtheconnectedloadaretobeexpressedinthesameunits(W,kW,orMW).

1.5.4Averageload

IfthenumberofkWhsuppliedbyastationinonedayisdividedby24hours,thenthevalueobtainedisknownasthedailyaverageload:

Dailyaverageload

Monthlyaverageload

Yearlyaverageload

1.5.5Loadfactor

Theratiooftheaveragedemandtothemaximumdemandiscalledtheloadfactor:

Loadfactor(LF)

IftheplantisinoperationforaperiodT,

Loadfactor

Theloadfactormaybeadailyloadfactor,amonthlyloadfactor,oranannualloadfactor,ifthetimeperiodisconsideredinadayoramonthorayear,respectively.Loadfactorisalwayslessthanonebecauseaverageloadissmallerthanthemaximumdemand.Itplaysakeyroleindeterminingtheoverallcostperunitgenerated.Highertheloadfactorofthepowerstation,lesserwillbethecostperunitgenerated.

1.5.6Diversityfactor

Diversityfactoristheratioofthesumofthemaximumdemandsofagroupofconsumerstothesimultaneousmaximumdemandofthegroupofconsumers:

Diversityfactor

Apowersystemsuppliesloadtovarioustypesofconsumerswhosemaximumdemandsgenerallydonotoccuratthesametime.Therefore,themaximumdemandonthepowersystemisalwayslessthanthesumofindividualmaximumdemandsoftheconsumers.

Ahighdiversityfactorimpliedthatwithasmallermaximumdemandonthestation,itispossibletocatertotheneedsofseveralconsumerswithvaryingmaximumdemandsoccurringatdifferenthoursoftheday.Thelesserthemaximumdemand,thelesserwillbethecapitalinvestmentonthegenerators.Thishelpsinreducingtheoverallcostoftheunits(kWh)generated.

Thus,ahigherdiversityfactorandahigherloadfactorarethedesirablecharacteristicsoftheloadonapowerstation.Theloadfactorcanbeimprovedbyencouraging

theconsumerstousepowerduringoff-peakhourswithcertainincentiveslikeofferingareductioninthecostofenergyconsumedduringoff-peakhours.

1.5.7Plantcapacity

Itisthecapacityorpowerforwhichaplantorstationisdesigned.Itshouldbeslightlymorethanthemaximumdemand.Itisequaltothesumoftheratingsofallthegeneratorsinapowerstation:

1.5.8Plantcapacityfactor

Itistheratiooftheaveragedemandonthestationtothemaximuminstalledcapacityofthestation.

Plantcapacityfactor

orcapacityfactor=(loadfactor)×(utilizationfactor).

Reservecapacity=plantcapacity−maximumdemand

1.5.9Utilizationfactor(orplant-usefactor)

ItistheratioofkWhgeneratedtotheproductoftheplantcapacityandthenumberofhoursforwhichtheplantwasinoperation:

Plant-usefactor

1.5.10Firmpower

Itisthepowerthatshouldalwaysbeavailableevenunderemergency.

1.5.11Primepower

Itisthemaximumpower(maybethermalorhydraulicormechanical)continuouslyavailableforconversion

intoelectricpower.

1.5.12Dumppower

Thisisthetermusuallyusedinhydro-electricplantsanditrepresentsthepowerinexcessoftheloadrequirements.Itismadeavailablebysurpluswater.

1.5.13Spillpower

Itisthepowerthatisproducedduringfloodsinahydro-powerstation.

1.5.14Coldreserve

Itisthereserve-generatingcapacitythatisnotinoperation,butcanbemadeavailableforservice.

1.5.15Hotreserve

Itisthereserve-generatingcapacitythatisinoperation,butnotinservice.

1.5.16Spinningreserve

Itisthereserve-generatingcapacitythatisconnectedtobusbarsandisreadytotaketheload.

1.6BASELOADANDPEAKLOADONAPOWERSTATION

Baseload:Itistheunvaryingloadthatoccursalmostduringthewholedayonthestation.

Peakload:Itisthevariouspeakdemandsofloadoverandabovethebaseloadofthestation.

Example1.1:Ageneratingstationhasamaximumdemandof35MWandhasaconnectedloadof60MW.Theannualgenerationofunitsis24×10 kWh.Calculatetheloadfactorandthedemandfactor.

Solution:

7

No.ofunitsgeneratedannually = 24×107kWh

No.ofhoursinayear(assuming365daysinayear)

= 365×24

= 8,760hours

∴Averageloadonthestation

∴LoadFactor

Demandfactor

Example1.2:Ageneratingstationsuppliesfourfeederswiththemaximumdemands(inMW)of16,10,12,and7MW.Theoverallmaximumdemandonthestationis20MWandtheannualloadfactoris45%.Calculatethediversityfactorandthenumberofunitsgeneratedannually.

Solution:

Sumofmaximumdemands=16+10+12+7=45MW

Simultaneousmaximumdemand=20MW

∴Diversityfactor

Averagedemand=(maximumdemand)×(loadfactor)

=20×0.45=9MW

∴No.ofunitsgeneratedannually=9×8,760=78,840MWh

Alternatively,

Annualloadfactor

i.e,

sothatthenumberofunitsgeneratedannually=0.45×20×8,760MWh

=78,840MWh

Example1.3:Theyearlyload–durationcurveofapowerplantisastraightline(Fig.1.5).Themaximumloadis30MWandtheminimumloadis20MW.Thecapacityoftheplantis35MW.Calculatetheplantcapacityfactor,theloadfactor,andtheutilizationfactor.

Solution:

No.ofunitsgeneratedperyear=AreaOACD=AreaOBCD+AreaBAC

∴Averageannualload

∴Loadfactor

Plantcapacityfactor

Utilizationfactor

Alternatively,

Utilizationfactor


Example1.4:Calculatethetotalannualenergygenerated,ifthemaximumdemandonapowerstationis120MWandtheannualloadfactoris50%.

Solution:

Maximumdemandonapowerstation=120MW

Annualloadfactor=50%

Loadfactor

∴Energygenerated/annum

= maximumdemand×LF×hoursinayear

= (120×103)×(0.5)×(24×365)kWh

= 525.6×106kWh

Example1.5:Determinethedemandfactorandtheloadfactorofageneratingstation,whichhasaconnectedloadof50MWandamaximumdemandof25MW,theunitsgeneratedbeing40×10 /annum.

Solution:

Connectedload = 50MW

Maximumdemand = 25MW

Unitsgenerated = 40×106/annum

Demandfactor

Averagedemand

Loadfactor

6

Example1.6:Calculatetheannualloadfactorofa120MWpowerstation,whichdelivers110MWfor4hours,60MWfor10hours,andisshutdownfortherestofeachday.Forgeneralmaintenance,itisshutdownfor60daysperannum.

Solution:

Capacityofpowerstation = 120MW

Powerdelivered = 110MWfor4hours

= 60MWfor10hours

= 0fortherestofeachday

Andforgeneralmaintenance,itisshutdownfor60daysperannum.

Energysuppliedin1day=(110×4)+(60×10)=1,040MWh

No.ofworkingdaysinayear=365−60=305

Energysuppliedperyear=1,040×305=3,17,200MWh

Annualloadfactor

Example1.7:customer-connectedloadsare10lampsof60Weachandtwoheatersof1,500Weach.His/hermaximumdemandis2kW.Onaverage,he/sheuses10lamps,7hoursaday,andeachheaterfor5hoursaday.

Determinehis/her:(i)averageload,(ii)monthlyenergyconsumption,and(iii)loadfactor.

Solution:

Maximumdemand=2kW

Connectedload=10×60+2×1,500=3,600W

Dailyenergyconsumption=numberoflampsused×wattageofeachlamp×workinghoursperday+numberofheaters×wattageofeachheater×workinghoursperday

= 10×60×7+2×1,500×5

= 19.2kWh

1. Averageload

2.

Monthlyenergyconsumption

= dailyenergyconsumption×no.ofdaysinamonth

= 19.2×30=576kWh

= 576kWh

3. Monthlyloadfactor

Example1.8:Themaximumdemandonageneratingstationis20MW,aloadfactorof75%,aplantcapacity

factorof50%,andaplant-usefactorof80%.Calculatethefollowing:

1. dailyenergygenerated,2. reservecapacityoftheplant,3. maximumenergythatcouldbeproduceddailyiftheplantwereinuse

allthetime.

Solution:

Maximumdemand,MD

= 20MW

Loadfactor,LF = 75%

Powercapacityfactor = 50%

Plant-usefactor = 80%

Averageload = MD×LF

= 20×0.75=15MW

1. Dailyenergygenerated=averageload×24=15×24=360MWh2. Powerstationinstalledcapacity=

Plantreservecapacity=installedcapacity−maximumdemand

=30−20

=10MW

3. Themaximumenergythatcanbeproduceddailyiftheplantisrunningallthetime

Example1.9:Acertainpowerstation’sannualload–durationcurveisastraightlinefrom25to5MW(Fig.1.6).Tomeetthisload,threeturbine-generatorunits,tworatedat15MWeachandoneratedat7.5MWareinstalled.Calculatethefollowing:

1. installedcapacity;2. plantfactor;3. unitsgeneratedperannum;4. utilizationfactor.

Solution:

1. Installedcapacity=2×15+7.5

=37.5MW

2. Fromtheload–durationcurveshowninFig.1.6,

Averagedemand

∴Plantfactor

3. Unitsgeneratedperannum=area(inkWh)underload–durationcurve

4. Utilizationfactor


Example1.10:Aconsumerhasaconnectedloadof12lampseachof100Wathis/herpremises.His/herloaddemandisasfollows:

Frommidnightto5A.M.:200W.

5A.M.to6P.M.:noload.

6P.M.to7P.M.:700W.

7P.M.to9P.M.:1,000W.

9P.M.tomidnight:500W.

Drawtheloadcurveandcalculatethe(i)energyconsumptionduring24hours,(ii)demandfactor,(iii)averageload,(iv)maximumdemand,and(v)loadfactor.

Solution:

FromFig.1.7,

1. Electricalenergyconsumptionduringtheday=areaofloadcurve

=200×5+700×1+1,000×2+500×3

=5,200Wh

=5.2kWh

2. Averageload

3. Demandfactor

4. Maximumdemand=1,000W5. Loadfactor

FIG.1.7Loadcurve

Example1.11:Calculatethediversityfactorandtheannualloadfactorofageneratingstation,whichsuppliesloadstovariousconsumersasfollows:

Industrialconsumer=2,000kW;

Commercialestablishment=1,000kW

Domesticpower=200kW;

Domesticlight=500kW

andassumethatthemaximumdemandonthestationis3,000kW,andthenumberofunitsproducedperyearis50×10 .

Solution:

5

Loadindustrialconsumer =2,000kW

Loadcommercialestablishment =1,000kW

Domesticpowerload =200kW

Domesticlightingload =500kW

Maximumdemandonthestation =3,000kW

NumberofkWhgeneratedperyear

=50×10

Diversityfactor

Averagedemand

Loadfactor

Example1.12:Calculatethereservecapacityofageneratingstation,whichhasamaximumdemandof20,000kW,theannualloadfactoris65%,andthecapacityfactoris45%.

Solution:

Maximumdemand = 20,000kW

Annualloadfactor = 65%

5

Capacityfactor = 45%

Energygenerated/annum

= maximumdemandLFhoursinayear

= (20,000)×(0.65)×(8,760)kWh=113.88×106kWh

Capacityfactor

0.45

∴Plantcapacity

Reservecapacity

= plantcapacity−maximumdemand

= 28,888.89−20,000=8,888.89kW

Example1.13:Themaximumdemandonapowerstationis600MW,theannualloadfactoris60%,andthecapacityfactoris45%.Findthereservecapacityoftheplant.

Solution:

Utilizationfactor

Plantcapacity

Reservecapacity

= plantcapacity−maximumdemand

= 800−600

= 200MW

Example1.14:Apowerstation’smaximumdemandis50MW,thecapacityfactoris0.6,andtheutilizationfactoris0.85.Calculatethefollowing:(i)reservecapacityand(ii)annualenergyproduced.

Solution:

Energygenerated/annum=maximumdemand×loadfactor×hoursinayear

=(50×LF×8,760)MWh

Loadfactor

Energygenerated/annum

= 50×0.706×8,760

= 3,09,228MWh=0.3×106MWh

Plantcapacity

Reservecapacity = plantcapacity−maximumdemand

= 58.82−50

= 8.82MW

Example1.15:Apowerstationistofeedfourregionsofloadwhosepeakloadsare12,7,10,and8MW.Thediversityfactoratthestationis1.4andtheaverageannualloadfactoris65%.Determinethefollowing:(i)maximumdemandonthestation,(ii)annualenergysuppliedbythestation,and(iii)suggesttheinstalledcapacity.

Solution:

1. Maximumdemandonstation

2. Unitsgenerated/annum=max.demand×LF×houseinayear

=(26.43×10 )×0.65×8,760kWh

=150.49×10 kMh

3. Theinstalledcapacityofthestationshouldbe15%to20%morethanthemaximumdemandinordertomeetthefuturegrowthofload.

Takingtheinstalledcapacitytobe20%morethanthemaximumdemand,

Installedcapacity=1.2×max.demand

=1.2×26.43

=31.716≅32MW

1.7LOADFORECASTING

Electricalenergycannotbestored.Ithastobegeneratedwheneverthereisademandforit.Itis,therefore,imperativefortheelectricpowerutilitiesthattheloadon

3

6

theirsystemsshouldbeestimatedinadvance.Thisestimationofloadinadvanceiscommonlyknownasloadforecasting.Itisnecessaryforpowersystemplanning.

Powersystemexpansionplanningstartswithaforecastofanticipatedfutureloadrequirements.Theestimationofbothdemandandenergyrequirementsiscrucialtoaneffectivesystemplanning.Demandpredictionsareusedfordeterminingthegenerationcapacity,transmission,anddistributionsystemadditions,etc.Loadforecastsarealsousedtoestablishprocurementpoliciesforconstructioncapitalenergyforecasts,whichareneededtodeterminefuturefuelrequirements.Thus,agoodforecast,reflectingthepresentandfuturetrends,isthekeytoallplanning.

Ingeneral,thetermforecastreferstoprojectedloadrequirementsdeterminedusingasystematicprocessofdefiningfutureloadsinsufficientquantitativedetailtopermitimportantsystemexpansiondecisionstobemade.Unfortunately,theconsumerloadisessentiallyuncontrollablealthoughminorvariationscanbeaffectedbyfrequencycontrolandmoredrasticallybyloadshedding.Thevariationinloaddoesexhibitcertaindailyandyearlypatternrepetitions,andananalysisoftheseformsthebasisofseveralload-predictiontechniques.

1.7.1Purposeofloadforecasting

1. Forproperplanningofpowersystem;2. Forproperplanningoftransmissionanddistributionfacilities;3. Forproperpowersystemoperation;4. Forproperfinancing;5. Forpropermanpowerdevelopment;6. Forpropergridformation;7. Forproperelectricalsales.

(i)ForProperPlanningofPowerSystem

Todeterminethepotentialneedforadditionalnewgeneratingfacilities;Todeterminethelocationofunits;

Todeterminethesizeofplants;Todeterminetheyearinwhichtheyarerequired;Todeterminethattheyshouldprovideprimarypeakingcapacityorenergyorboth;TodeterminewhethertheyshouldbeconstructedandownedbytheCentralGovernmentorStateGovernmentorElectricityBoardsorbysomeotherautonomouscorporations.

(ii)ForProperPlanningofTransmissionandDistributionFacilities

Forplanningthetransmissionanddistributionfacilities,theloadforecastingisneededsothattherightamountofpowerisavailableattherightplaceandattherighttime.Wastageduetomisplanninglikepurchaseofequipment,whichisnotimmediatelyrequired,canbeavoided.

(iii)ForProperPowerSystemOperation

Loadforecastbasedoncorrectvaluesofdemandanddiversityfactorwillpreventoverdesigningofconductorsize,etc.aswellasoverloadingofdistributiontransformersandfeeders.Thus,theyhelptocorrectvoltage,powerfactor,etc.andtoreducethelossesinthedistributionsystem.

(iv)ForProperFinancing

TheloadforecastshelptheBoardstoestimatethefutureexpenditure,earnings,andreturnsandtoscheduleitsfinancingprogramaccordingly.

(v)ForProperManpowerDevelopment

AccurateloadforecastingannuallyreviewedwillcometotheaidoftheBoardsintheirpersonnelandtechnicalmanpowerplanningonalong-termbasis.SucharealisticforecastwillreduceunnecessaryexpenditureandputtheBoards’financesonasoundandprofitablefooting.

(vi)ForProperGridFormation

Interconnectionsbetweenvariousstategridsarenowbecomingmoreandmorecommonandtheaimistohavefullyinterconnectedregionalgridsandultimately

evenasupergridforthewholecountry.Theseexpensivehigh-voltageinterconnectionsmustbebasedonreliableloaddata,otherwisethegeneratorsconnectedtothegridmayfrequentlyfalloutofstepcausingpowertobeshutdown.

(vii)ForProperElectricalSales

Incountries,wherespinningreservesaremore,properplanningandtheexecutionofelectricalsalesprogramareaidedbyproperloadforecasting.

1.7.2Classificationofloadforecasting

Theloadforecastingcanbeclassifiedas:(i)demandforecastand(ii)energyforecast.

(i)DemandForecast

Thisisusedtodeterminethecapacityofthegeneration,transmission,anddistributionsystemadditions.Futuredemandcanbepredictedonthebasisoffastrateofgrowthofdemandfrompasthistoryandgovernmentpolicy.Thiswillgivetheexpectedrateofgrowthofload.

(ii)EnergyForecast

Thisisusedtodeterminethetypeoffacilitiesrequired,i.e.,futurefuelrequirements.

1.7.3Forecastingprocedure

Dependingonthetimeperiodofinterest,aspecificforecastingproceduremaybeclassifiedas:

Short-term.Medium(intermediate)-term.Long-termtechnique.

(1)Short-TermForecast

Forday-to-dayoperation,coveringonedayoraweek,short-termforecastingisneededinordertocommitenoughgeneratingcapacityformattingtheforecastingdemandandformaintainingtherequiredspinning

reserve.Hence,itisusuallydone24hoursaheadwhentheweatherforecastforthefollowingdaybecomesavailablefromthemeteorologicaloffice.Thismostlyconsistsofestimatingtheweather-dependentcomponentandthatduetoanyspecialeventorfestivalbecausethebaseloadforthedayisalreadyknown.

Thepowersupplyauthoritiescanbuildupaweatherloadmodelofthesystemforthispurposeorcanconsultsometables.Thefinalestimateisobviouslydoneafteraccountingthetransmissionanddistributionlossesofthesystem.Inadditiontothepredictionofhourlyvalues,ashort-termloadforecasting(STLF)isalsoconcernedwithforecastingofdailypeak-systemload,systemloadatcertaintimesofaday,hourlyvaluesofsystemenergy,anddailyandweeklysystemenergy.

ApplicationsofSTLFaremainly:

Todrivetheschedulingfunctionsthatdecidethemosteconomiccommitmentofgenerationsources.Toaccessthepowersystemsecuritybasedontheinformationavailabletothedispatcherstopreparethenecessarycorrectiveactions.Toprovidethesystemdispatcherwiththelatestweatherpredictionssothatthesystemcanbeoperatedbotheconomicallyandreliably.

(2)Long-TermForecast

Thisisdonefor1–5yearsinadvanceinordertopreparemaintenanceschedulesofthegeneratingunits,planningfutureexpansionofthegeneratingcapacity,enterintoanagreementforenergyinterchangewiththeneighboringutilities,etc.Basically,twoapproachesareavailableforthispurposeandarediscussedasfollows.

(a)PeakLoadApproach

Inthiscase,thesimplestapproachistoextrapolatethetrendcurve,whichisobtainedbyplottingthepastvaluesofannualpeaksagainstyearsofoperation.Thefollowinganalyticalfunctionscanbeusedtodeterminethetrendcurve.

1. Straightline,Y=a+bx2. Parabola,Y=a+bx+cx3. S-curve,Y=a+bx+cx +dx4. Exponential,Y=ce5. Gompertz,log Y=a+ce

Intheabove,Yrepresentspeakloadsandxrepresentstimeinyears.Themostcommonmethodoffindingcoefficientsa,b,c,anddistheleastsquarescurve-fittingtechnique.

Theeffectofweatherconditionscanbeignoredonthebasisthatweatherconditions,asinthepast,aretobeexpectedduringtheperiodunderconsiderationbuttheeffectofthechangeintheeconomicconditionshouldbeaccommodatedbyincludinganeconomicvariablewhenextrapolatingthetrendcurve.Theeconomicvariablemaybethepredictednationalincome,grossdomesticproduct,etc.

(b)EnergyApproach

Anothermethodistoforecastannualenergysalestodifferentclassesofcustomerslikeresidential,commercial,industrial,etc.,whichcanthenbeconvertedtoannualpeakdemandusingtheannualloadfactor.Adetailedestimationoffactorssuchasrateofhousebuilding,saleofelectricalappliances,growthinindustrialandcommercialactivitiesarerequiredinthismethod.Forecastingtheannualloadfactoralsocontributescriticallytothesuccessofthemethod.Boththesemethods,however,havebeenusedbytheutilitiesinestimatingtheirlong-termsystemload.

KEYNOTES

Aloadcurveisaplotoftheloaddemand(onthey-axis)versusthetime(onthex-axis)inthechronologicalorder.Theload–durationcurveisaplotoftheloaddemands(inunitsofpower)arrangedinadescendingorderofmagnitude(onthey-axis)andthetimeinhours(onthex-axis).Intheoperationofhydro-electricplants,itisnecessarytoknowtheamountofenergybetweendifferentloadlevels.Thisinformationcan

e

2

2 3

dx

dx

beobtainedfromtheload–durationcurve.Theintegratedload–durationcurveisalsotheplotofthecumulativeintegrationofareaundertheloadcurvestartingatzeroloadstotheparticularload.Abase-loadstationoperatesatahigh-loadfactorwhilethepeakloadplantoperatesata-lowloadfactor.Demandfactoristheratioofthemaximumdemandtotheconnectedload.Loadfactoristheratiooftheaveragedemandtothemaximumdemand.Highertheloadfactorofthepowerstation,lesserwillbethecostperunitgenerated.Diversityfactoristheratioofthesumofthemaximumdemandsofagroupofconsumersandthesimultaneousmaximumdemandofthegroupofconsumers.Baseloadistheunvaryingloadthatoccursalmostthewholedayonthestation.Peakloadisthevariouspeakdemandsofloadoverandabovethebaseloadofthestation.

SHORTQUESTIONSANDANSWERS

1. Whatismeantbyconnectedload?

Itisthesumoftheratingsoftheapparatusinstalledonaconsumer’spremises.

2. Definethemaximumdemand.

Itisthemaximumloadusedbyaconsumeratanytime.

3. Definethedemandfactor.

Theratioofthemaximumdemandtotheconnectedloadiscalledthedemandfactor.

4. Definetheaverageload.

IfthenumberofkWhsuppliedbeastationinonedayisdividedby24hours,thenthevalueobtainedisknownasthedailyaverageload.

5. Definetheloadfactor.

Itistheratiooftheaveragedemandtothemaximumdemand.

6. Definethediversityfactor.

Itistheratioofthesumofthemaximumdemandsofagroupofconsumerstothesimultaneousmaximumdemandofthegroupofconsumers.

7. Definetheplantcapacity.

Itisthecapacityorpowerforwhichaplantorstationisdesigned.

8. Definetheutilizationfactor.

ItistheratioofkWhgeneratedtotheproductoftheplantcapacityandthenumberofhoursforwhichtheplantwasin

operation.

9. Whatismeantbybaseload?

Itistheunvaryingloadthatoccursalmostthewholedayonthestation.

10. Whatismeantbypeakload?

Itisthevariouspeakdemandsofloadoverandabovethebaseloadofthestation.

11. Whatismeantbyloadcurve?

Aloadcurveisaplotoftheloaddemandversusthetimeinthechronologicalorder.

12. Whatismeantbyload–durationcurve?

Theload–durationcurveisaplotoftheloaddemandsarrangedinadescendingorderofmagnitudeversusthetimeinhours.

MULTIPLE-CHOICEQUESTIONS

1. Inordertohavealowcostofelectricalgeneration,

1. Theloadfactoranddiversityfactorarehigh.2. Theloadfactorshouldbelowbutthediversityfactorshouldbehigh.3. Theloadfactorshouldbehighbutthediversityfactorshouldbelow.4. Theloadfactorandthediversityfactorshouldbelow.

2. Apowerplanthavingmaximumdemandmorethantheinstalledcapacitywillhaveutilizationfactor:

1. Lessthan100%.2. Equalto100%.3. Morethan100%.4. Noneofthese.

3. Thechoiceofnumberandsizeofunitsinastationaregovernedbybestcompromisebetween:

1. Aplantloadfactorandcapacityfactor.2. Plantcapacityfactorandplant-usefactor.3. Plantloadfactorandusefactor.4. Noneofthese.

4. Ifaplanthaszeroreservecapacity,theplantloadfactoralways:

1. Equalsplantcapacityfactor.2. Isgreaterthanplantcapacityfactor.3. Islessthanplantcapacityfactor.4. Noneofthese.

5. Ifsomereserveisavailableinapowerplant,

1. Itsusefactorisalwaysgreaterthanitscapacityfactor.2. Itsusefactorequalsthecapacityfactor.3. Itsusefactorisalwayslessthanitscapacityfactor.4. Noneofthese.

6. Ahigherloadfactormeans:

1. Costperunitisless.2. Lessvariationinload.3. Thenumberofunitsgeneratedaremore.4. Allofthese.

7. Themaximumdemandoftwopowerstationsisthesame.Ifthedailyloadfactorsofthestationsare10and20%,thentheunitsgeneratedbythemareintheratio:

1. 2:1.2. 1:2.3. 3:3.4. 1:4.

8. Aplanthadanaverageloadof20MWwhentheloadfactoris50%.Itsdiversityfactoris20%.Thesumofmax.demandsofallloadsamountsto:

1. 12MW.2. 8MW.3. 6MW.4. 4MW.

9. Apeakloadstation:

1. Shouldhavealowoperatingcost.2. Shouldhavealowcapitalcost.3. Canhaveaoperatingcosthigh.4. (a)and(c).5. (b)and(c).

10. TwoareasAandBhaveequalconnectedloads;howevertheloaddiversityinareaAismorethaninB,then:

1. Maximumdemandoftwoareasissmall.2. MaximumdemandofAisgreaterthanthemaximumdemandofB.3. ThemaximumdemandofBisgreaterthanthemaximumdemandofA.4. ThemaximumdemandofAmoreorlessthanthatofB.

11. Theareaunderthedailyloadcurvegives

1. Thenumberofunitsgeneratedintheday.2. Theaverageloadoftheday.3. Theloadfactoroftheday.4. Thenumberofunitsgeneratedintheyear.

12. Theannualpeakloadona60-MWpowerstationis50MW.Thepowerstationsuppliesloadshavingaveragedemandsof9,10,17,and20MW.Theannualloadfactoris60%.Theaverageloadontheplantis:

1. 4,000kW.2. 30,000kW.3. 2,000kW.4. 1,000kW.

13. Ageneratingstationhasaconnectedloadof40MWandamaximumdemandof20MW.Thedemandfactoris:

1. 0.7.2. 0.6.3. 0.59.4. 0.4.

14. A100MWpowerplanthasaloadfactorof0.5andautilizationfactorof0.2.Itsaveragedemandis:

1. 10MW.2. 5MW.3. 7MW.4. 6MW.

15. Thevalueofthedemandfactorisalways:

1. Lessthanone.2. Equaltoone.3. Greaterthanone.4. Noneofthese.

16. Ifcapacityfactor=loadfactor,then:

1. Utilizationfactoriszero.2. Utilizationcapacityisnon-zero.3. Utilizationfactorisequaltoone.4. Noneofthese.

17. Ifcapacityfactor=loadfactor,thentheplant’s

1. Reservecapacityismaximum.2. Reservecapacityiszero.3. Reservescapacityisless.4. Noneofthese.

18. Installedcapacityofpowerplantis:

1. Morethanthemaximumdemand.2. Lessthanthemaximumdemand.3. Equaltothemaximumdemand.4. Bothand.

19. Inaninterconnectedsystem,diversityfactordetermining:

1. Decreases.2. Increases.3. Zero.4. Noneofthese.

20. Theknowledgeofdiversityfactorhelpsindetermining:

1. Plantcapacity.2. Reservecapacity.3. Maximumdemand.4. Averagedemand.

21. Apowerstationhasaninstalledcapacityof300MW.Itscapacityfactoris50%anditsloadfactoris75%.Itsmaximumdemandis:

1. 100MW.2. 150MW.3. 200MW.4. 250MW.

22. Theconnectedloadofaconsumeris2kWandhis/hermaximumdemandis1.5kW.Theloadfactoroftheconsumeris:

1. 0.75.2. 0.375.3. 1.33.4. noneofthese.

23. Themaximumdemandofaconsumeris2kWandhis/herdailyenergyconsumptionis20units.His/herloadfactoris:

1. 10.15%.2. 41.6%.3. 50%.4. 52.6%.

24. Inapowerplant,areserve-generatingcapacity,whichisnotinservicebutinoperationisknownas:

1. Hotreserve.2. Spinningreserve.3. Coldreserve.4. Firmpower.

25. Thepowerintendedtobealwaysavailableisknownas:


26. Inapowerplant,areserve-generatingcapacity,whichisinservicebutnotinoperationis:


27. Whichofthefollowingisacorrectfactor?

1. Loadfactor=capacity×utilizationfactor.2. Utilizationfactor=capacityfactor×loadfactor.3. Utilizationfactor=loadfactor/utilizationfactor.4. Capacityfactor=loadfactor×utilizationfactor.

28. Iftheratedplantcapacityandmaximumloadofgeneratingstationareequal,then:

1. Loadfactoris1.2. Capacityfactoris1.3. Loadfactorandcapacityfactorareequal.4. Utilizationfactorispoor.

29. Thecapitalcostofplantdependson:

1. Totalinstalledcapacityonly.2. Totalnumberofunitsonly.3. Bothand.4. Noneofthese.

30. Thereservecapacityinasystemisgenerallyequalto:

1. Capacityofthelargestgeneratingunit.2. Capacityoftwolargestgeneratingunits.3. Thetotalgeneratingcapacity.4. Noneoftheabove.

31. Themaximumdemandofaconsumeris5kWandhis/herdailyenergyconsumptionis24units.His/her%loadfactoris:

1. 5.

2. 20.3. 24.4. 48.

32. Ifloadfactorispoor,then:

1. Electricenergyproducedissmall.2. ChargeperkWhishigh.3. FixedchargesperkWhishigh.4. Alloftheabove.

33. Ifageneratingstationhadmaximumloadsforadayat100kWandaloadfactorof0.2,itsgenerationinthatdaywas:

1. 8.64MWh.2. 21.6units.3. 21.6units.4. 2,160kWh.

34. Theknowledgeofmaximumdemandisimportantasithelpsindetermining:

1. Installedcapacityoftheplant.2. Connectedloadoftheplant.3. Averagedemandoftheplant.4. Either(a)or(b).

35. Apowerstationisconnectedto4.5and6kW.Itsdailyloadfactorwascalculatedas0.2,whereitsgenerationonthatdaywas24units.Calculatethedemandfactor.

1. 2.6.2. 3.1.3. 3.0.4. 0.476.

36. A50-MWpowerstationhadproduced24unitsinadaywhenitsmaximumdemandwas50Mw.Itsplantloadfactorandcapacityfactorthatdayin%were:

1. 1and2.2. 2and3.3. 2and2.4. 4and3.

37. Loadcurveofapowergenerationstationisalways:

1. Negative.2. Zeroslope.3. Positive.4. Anycombinationof(a),(b),and(c).

38. Loadcurvehelpsindecidingthe:

1. Totalinstalledcapacityoftheplant.2. Sizeofthegeneratingunits.3. Operatingscheduleofthegeneratingunits.4. Alloftheabove.

39. Theloadfactorfordomesticloadsmaybetaken:

1. About85%.2. 50−60%.3. 25−50%.

4. 20−15%.

REVIEWQUESTIONS

1. Explainthesignificanceofthedailyloadcurve.2. Discussthedifferencebetweentheloadcurveandtheload–

durationcurve.3. Explainthedifferencesinoperationsofpeakloadandbase-load

stations.4. Explainthesignificanceoftheloadfactorandthediversityfactor.5. Definethefollowing:

1. Loadfactor,2. Demandfactor,3. Diversityfactor,4. Plantcapacityfactor,and5. Utilizationfactor.

6. Explaintheloadforecastingprocedures.

PROBLEMS

1. Calculatediversityfactorandannualloadfactorofageneratingstationthatsuppliesloadstovariousconsumersasfollows:

Industrialconsumer=1,500kW;

Commercialestablishment=7,500kW

Domesticpower=100kW;

Domesticlight=400kW

Inaddition,assumethatthemaximumdemandonthestationis2,500kWandthenumberofunitsproducedperyearis40×10 kWh.

2. Apowerstationistofeedfourregionsofloadwhosepeakloadsare10,5,14,and6MW,respectively.Thediversityfactoratthestationis1.3andtheaverageannualloadfactoris60%.Determinethefollowing:(i)maximumdemandonthestation,(ii)annualenergysuppliedbythestation,and(iii)suggesttheinstalledcapacity.

3. Acertainpowerstation’sannualload–durationcurveisastraightlinefrom20to7MW.Tomeetthisload,threeturbine-generatorunits,tworatedat12MWeachandoneratedat8MWareinstalled.Calculatethefollowing:

1. Installedcapacity,2. Plantfactor,3. Unitsgeneratedperannum,4. Utilizationfactor.

5

2

EconomicLoadDispatch-I

OBJECTIVES

Afterreadingthischapter,youshouldbeableto:

studythedifferentcharacteristicsofsteamandhydro-powergenerationunits

knowthemeaningofeconomicalloaddispatch

developthemathematicalmodelforeconomicalloaddispatch

discussthedifferentcomputationalmethodsforoptimization

2.1INTRODUCTION

Powersystemsneedtobeoperatedeconomicallytomakeelectricalenergycost-effectivetotheconsumerinthefaceofconstantlyrisingpricesoffuel,wages,salaries,etc.Newgenerator-turbineunitsaddedtoasteampowerplantoperatemoreefficientlythanotherolderunits.Thecontributionofnewerunitstothegenerationofpowerwillhavetobemore.Intheoperationofpowersystems,thecontributionfromeachloadandfromeachunitwithinaplantmustbesuchthatthecostofelectricalenergyproducedisaminimum.

2.2CHARACTERISTICSOFPOWERGENERATION(STEAM)UNIT

Inanalyzingtheeconomicoperationofathermalunit,input–outputmodelingcharacteristicsaresignificant.Forthisfunction,considerasingleunitconsistingofaboiler,aturbine,andageneratorasshowninFig.2.1.Thisunithastosupplypowernotonlytotheloadconnectedtothepowersystembutalsotothelocalneedsfortheauxiliariesinthestation,whichmayvaryfrom2%

to5%.Thepowerrequirementsforstationauxiliariesarenecessarytodriveboilerfeedpumps,fansandcondensercirculatingwaterpumps,etc.

ThetotalinputtothethermalunitcouldbeBritishthermalunit(Btu)/hrorCal/hrintermsofheatsuppliedorRs./hrintermsofthecostoffuel(coalorgas).ThetotaloutputoftheunitatthegeneratorbuswillbeeitherkWorMW.

FIG.2.1Thermalgenerationsystem

2.3SYSTEMVARIABLES

Toanalyzethepowersystemnetwork,thereisaneedofknowingthesystemvariables.Theyare:

1. Controlvariables.2. Disturbancevariables.3. Statevariables.

2.3.1Controlvariables(P andQ )

Therealandreactive-powergenerationsarecalledcontrolvariablessincetheyareusedtocontrolthestateofthesystem.

2.3.2Disturbancevariables(P andQ )

Therealandreactive-powerdemandsarecalleddemandvariablessincetheyarebeyondthesystemcontrolandarehenceconsideredasuncontrolledordisturbancevariables.

G G

D D

2.3.3Statevariables(Vandδ)

ThebusvoltagemagnitudeVanditsphaseangleδdispatchthestateofthesystem.Thesearedependentvariablesthatarebeingcontrolledbythecontrolvariables.

2.4PROBLEMOFOPTIMUMDISPATCH—FORMULATION

Schedulingistheprocessofallocationofgenerationamongdifferentgeneratingunits.Economicschedulingisacost-effectivemodeofallocationofgenerationamongthedifferentunitsinsuchawaythattheoverallcostofgenerationshouldbeminimum.Thiscanalsobetermedasanoptimaldispatch.

Letthetotalloaddemandonthestation=P andthetotalnumberofgeneratingunits=n.

TheoptimizationproblemistoallocatethetotalloadP amongthese‘n’unitsinanoptimalwaytoreducetheoverallcostofgeneration.

LetP ,P ,P ,…,P bethepowergeneratedbyeach

individualunittosupplyaloaddemandofP .

Toformulatethisproblem,itisnecessarytoknowthe‘input–outputcharacteristicsofeachunit’.

2.5INPUT–OUTPUTCHARACTERISTICS

Theidealizedformofinput–outputcharacteristicsofasteamunitisshowninFig.2.2.Itestablishestherelationshipbetweentheenergyinputtotheturbineandtheenergyoutputfromtheelectricalgenerator.Theinputtotheturbineshownontheordinatemaybeeitherintermsoftheheatenergyrequirement,whichisgenerallymeasuredinBtu/hrorkCal/hrorintermsofthetotalcostoffuelperhourinRs./hr.TheoutputisnormallythenetelectricalpoweroutputofthatsteamunitinkWorMW.

D

D

Gi G2 G3 Gn

D

Inpractice,thecurvemaynotbeverysmooth,andfrompracticaldata,suchanidealizedcurvemaybeinterpolated.Thesteamturbine-generatingunitcurveconsistsofminimumandmaximumlimitsinoperation,whichdependuponthesteamcycleused,thermalcharacteristicsofmaterial,theoperatingtemperature,etc.

FIG.2.2Input–outputcharacteristicofasteamunit

2.5.1Unitsofturbineinput

Intermsofheat,theunitis10 kcal/hr(or)Btu/hrorintermsoftheamountoffuel,theunitistonsoffuel/hr,whichbecomesmillionsofkCal/hr.

2.6COSTCURVES

Toconverttheinput–outputcurvesintocostcurves,thefuelinputperhourismultipliedwiththecostofthefuel(expressedinRs./millionkCal).

i.e.,

= millionkCal/hr×Rs./millionkCal

6

= Rs./hr

2.7INCREMENTALFUELCOSTCURVE

Fromtheinput–outputcurves,theincrementalfuelcost(IFC)curvecanbeobtained.

TheIFCisdefinedastheratioofasmallchangeintheinputtothecorrespondingsmallchangeintheoutput.

Incrementalfuelcost

where∆representssmallchanges.

Asthe∆quantitiesbecomeprogressivelysmaller,itis

seenthattheIFCis andisexpressedin

Rs./MWh.AtypicalplotoftheIFCversusoutputpowerisshowninFig.2.3(a).

Theincrementalcostcurveisobtainedbyconsideringthechangeinthecostofgenerationtothechangeinreal-powergenerationatvariouspointsontheinput–outputcurves,i.e.,slopeoftheinput–outputcurveasshowninFig.2.3(b).

FIG.2.3(a)Incrementalcostcurve;(b)Incrementalfuelcostcharacteristicintermsoftheslopeoftheinput–outputcurve

TheIFCisnowobtainedas

(IC) =slopeofthefuelcostcurve

i.e.,tanβ

i

th

TheIFC(IC)ofthei thermalunitisdefined,foragivenpoweroutput,asthelimitoftheratiooftheincreasedcostoffuelinput(Rs./hr)tothecorrespondingincreaseinpoweroutput(MW),astheincreasingpoweroutputapproacheszero.

whereC isthecostoffuelofthei unitandP isthe

powergenerationoutputofthati unit.

Mathematically,theIFCcurveexpressioncanbeobtainedfromtheexpressionofthecostcurve.

Cost-curveexpression,

(Second-degreepolynomial)

TheIFC,

(linearapproximation)foralli=1,

2,3,…,n

where istheratioofincrementalfuelenergyinputin

BtutotheincrementalenergyoutputinkWh,whichiscalled‘theincrementalheatrate’.

Thefuelcostisthemajorcomponentandtheremainingcostssuchasmaintenance,salaries,etc.willbeofverysmallpercentageoffuelcost;hence,theIFCis

i Gi

th

th

th

verysignificantintheeconomicloadingofageneratingunit.

2.8HEATRATECURVE

TheheatratecharacteristicobtainedfromtheplotofthenetheatrateinBtu/kWhorkCal/kWhversuspoweroutputinkWisshowninFig.2.4.

FIG.2.4Heatratecurve

Thethermalunitismostefficientataminimumheatrate,whichcorrespondstoaparticulargenerationP .Thecurveindicatesanincreaseinheatrateatlowandhighpowerlimits.

Thermalefficiencyoftheunitisaffectedbythefollowingfactors:conditionofsteam,steamcycleused,re-heatstages,condenserpressure,etc.

2.9INCREMENTALEFFICIENCY

Thereciprocaloftheincrementalfuelrateorheatrate,whichisdefinedastheratioofoutputenergytoinputenergy,givesameasureoffuelefficiencyfortheinput.

i.e.,Incrementalefficiency

G

2.10NON-SMOOTHCOSTFUNCTIONSWITHMULTIVALVEEFFECT

Forlargesteamturbinegenerators,theinput–outputcharacteristicsareshowninFig.2.5(a).

Largesteamturbinegeneratorswillhaveanumberofsteamadmissionvalvesthatareopenedinsequencetoobtainanever-increasingoutputoftheunit.Figures2.5(a)and(b)showinput–outputandincrementalheatratecharacteristicsofaunitwithfourvalves.Astheunitloadingincreases,theinputtotheunitincreasesandtherebytheincrementalheatratedecreasesbetweentheopeningpointsforanytwovalves.However,whenavalveisfirstopened,thethrottlinglossesincreaserapidlyandtheincrementalheatraterisessuddenly.Thisgivesrisetothediscontinuoustypeofcharacteristicsinordertoschedulethesteamunit,althoughitisusuallynotdone.Thesetypesofinput–outputcharacteristicsarenon-convex;hence,theoptimizationtechniquethatrequiresconvexcharacteristicsmaynotbeusedwithimpunity.

FIG.2.5Characteristicsofasteamgeneratorunitwithmultivalveeffect:(a)Input–outputcharacteristicand(b)incrementalheatratecharacteristic

2.11NON-SMOOTHCOSTFUNCTIONSWITHMULTIPLEFUELS

Generally,apiece-wisequadraticfunctionisusedtorepresenttheinput–outputcurveofageneratorwithmultiplefuels.Figure2.6representstheincrementalheatratecharacteristicsofageneratorwithmultiplefuels.

2.12CHARACTERISTICSOFAHYDRO-POWERUNIT

Asimplehydro-powerplantisshowninFig.2.7(a).

Theinput–outputcharacteristicsofahydro-powerunitasshowninFig.2.7(b)canbeobtainedinthesamewayasforthesteamunitsassumingthewaterheadtobeconstant.Theordinatesarewaterinputordischarge(m /s)versusoutputpower(kWorMW).

3

FIG.2.6Incrementalheat-ratecharacteristicsofasteamgeneratorwithmultiplefuels

FIG.2.7(a)Atypicalsystemofahydro-powerplant;(b)Input–outputcharacteristicsofahydro-unit;(c)Effectofwaterheadonwaterdischarge;(d)Incrementalwaterratecharacteristicofahydro-unit;(e)Incremental

costcharacteristicofahydro-unit

FromFig.2.7(b),itisobservedthatthereisalinearwaterrequirementuptotheratedloadandafterthat

greaterdischargeisneededtomeettheincreasedloaddemandsuchthattheefficiencyoftheunitdecreases.

2.12.1Effectofthewaterheadondischargeofwaterforahydro-unit

Figure2.7(c)showstheeffectofthewaterheadonwaterdischarge.Itisobservedthatwhentheheadofthewaterfalls,theinput–outputcharacteristicofahydro-powerplantmovesverticallyupwards,suchthatahigherdischargeofwaterisneededforthesamepowergeneration.Thereversewillhappenwhentheheadrises.

2.12.2incrementalwaterratecharacteristicsofhydro-units

AtypicalincrementalwaterratecharacteristicisshowninFig.2.7(d).Itcanbeobtainedfromtheinput–outputcharacteristicofahydro-unitasshowninFig.2.7(b).

FromFig.2.7(d),itisseenthatthecurveisastraighthorizontallineuptotheratedloadindicatingaconstantslopeandafterthatitrisesrapidly.Whentheloadincreasesmorethantherated,moreunitswillbeputintooperation(service).

2.12.3Incrementalcostcharacteristicofahydro-unit

Actually,theinputofahydro-plantisnotdependentonthecost.Buttheinputwaterflowcostsareduetothecapacityofstorage,requirementofwaterfortheagriculturalpurpose,andrunningoftheplantduringoffseason(dryseason).Theartificialstoragerequirement(i.e.,costofconstructionofdams,canals,conduits,gates,penstocks,etc.)imposesacostonthewaterinputtotheturbineaswellasthecostofcontrolonthewateroutputfromtheturbineduetoagriculturalneed.

TheincrementalcostcharacteristiccanbeobtainedfromtheincrementalwaterratecharacteristicbymultiplyingitwithcostofwaterinRs./m .

3

Incrementalcost

= (Incrementalwaterrate)×costofwaterinRs./m3

= m3/MWh×Rs./m3

= Rs./MWh

Theincrementalcostcharacteristic(or)incrementalproductioncostcharacteristicisshowninFig.2.7(e).

Theanalyticalexpressionofanincrementalcostcharacteristicis

(IC) = C ,(0≤P ≤P )

= mP +C ,(P ≤P ≤P )

whereP isthepowergenerationofahydro-unitandmistheslopeofcharacteristicbetweenP andP .

2.12.4Constraintsofhydro-powerplants

Thefollowingconstraintsaregenerallyusedinhydro-powerplants.

(i)Waterstorageconstraints

Letγ bethestoragevolumeattheendofintervalj,γ≤γ ≤γ .

(ii)Waterspillageconstraints

Eventhoughtheremaybecircumstanceswhereallowingwaterspillage(S )>0forsomeintervalj,prohibitionofspillageisassumedsothatallS =0mightreducethecostofoperationofathermalplant.

H1 GH GH1

GH 1 GH1 GH GH2

GH

GH1 GH2

j min

j max

Pj

Pj

(iii)Waterdischargeflowconstraints

Thedischargeflowmaybeconstrainedbothinrateandintotalas

2.13INCREMENTALPRODUCTIONCOSTS

TheincrementalproductioncostofagivenunitismadeupoftheIFCplustheincrementalcostofitemssuchaslabor,supplies,maintenance,andwater.

Itisnecessaryforarigorousanalysistobeabletoexpressthecostsoftheseproductionitemsasafunctionofoutput.However,nomethodsarepresentlyavailableforexpressingthecostoflabor,supplies,ormaintenanceaccuratelyasafunctionofoutput.

Arbitrarymethodsofdeterminingtheincrementalcostsoflabor,supplies,andmaintenanceareused,thecommonestofwhichistoassumethesecoststobeafixedpercentageoftheIFCs.

Inmanysystems,forpurposesofschedulinggeneration,theincrementalproductioncostisassumedtobeequaltotheIFC.

2.14CLASSICALMETHODSFORECONOMICOPERATIONOFSYSTEMPLANTS

Previously,thefollowingthumbruleswereadoptedforschedulingthegenerationamongvariousunitsofgeneratorsinapowerstation:

1. Baseloadingtocapacity:Theturbo-generatorsweresuccessivelyloadedtotheirratedcapacitiesintheorderoftheirefficiencies.

2. Baseloadingtomostefficientload:Theturbo-generatorunitsweresuccessivelyloadedtotheirmostefficientloadsintheincreasingorderoftheirheatrates.

3. Proportionalloadingtocapacity:Theturbo-generatorsetswereloadedinproportiontotheirratedcapacitieswithoutconsiderationtotheirperformancecharacteristics.

Iftheincrementalgenerationcostsaresubstantiallyconstantovertherangeofoperation,thenwithoutconsideringreserveandtransmissionlinelimitations,themosteconomicwayofschedulinggenerationistoloadeachunitinthesystemtoitsratedcapacityintheorderofthehighestincrementalefficiency.Thismethod,knownasthemeritorderapproachtoeconomicloaddispatching,requiresthepreparationoftheorderofmerittablesbasedupontheincrementalefficiencies,whichshouldbeupdatedregularlytoreflectthechangesinfuelcosts,plantcycleefficiency,plantavailability,etc.Activepowerschedulingtheninvolveslookingintothetableswithouttheneedforanycalculations.

2.15OPTIMIZATIONPROBLEM—MATHEMATICALFORMULATION(NEGLECTINGTHETRANSMISSIONLOSSES)

Anoptimizationproblemconsistsof:

1. Objectivefunction.2. Constraintequations.

2.15.1Objectivefunction

Theobjectivefunctionistominimizetheoverallcostofproductionofpowergeneration.

Costinthermalandnuclearstationsisnothingbutthecostoffuel.LetnbethenumberofunitsinthesystemandC thecostofpowergenerationofunit‘i’:

∴TotalcostC=C +C +C +…+C

i.e.,

Thecostofgenerationofeachunitinthermalpowerplantsismainlyafuelcost.Thegenerationcostdependsontheamountofrealpowergenerated,sincethereal-powergenerationisincreasedbyincreasingthefuelinput.

i

1 2 3 n

Thegenerationofreactivepowerhasnegligibleinfluenceonthecostofgeneration,sinceitiscontrolledbythefieldcurrent.

Therefore,thegenerationcostofthei unitisafunctionofreal-powergenerationofthatunitandhencethetotalcostisexpressedas

i.e.,C=C (P )+C (P )+C (P )+…+C (P )

Thisobjectivefunctionconsistsofthesummationofthetermsinwhicheachtermisafunctionofseparateindependentvariables.Thistypeofobjectivefunctioniscalledaseparableobjectivefunction.

Theoptimizationproblemistoallocatethetotalloaddemand(P )amongthevariousgeneratingunits,suchthatthecostofgenerationisminimizedandsatisfiesthefollowingconstraints.

2.15.2Constraintequations

Theeconomicpowersystemoperationneedstosatisfythefollowingtypesofconstraints.

(1)Equalityconstraints

Thesumofreal-powergenerationofallthevariousunitsmustalwaysbeequaltothetotalreal-powerdemandonthesystem.

i.e.,

or

1 G1 2 G2 3 G3 n Gn

D

th

where totalreal-powergenerationandP isthe

totalreal-powerdemand.Equation(2.2)isknownasthereal-powerbalanceequationwhenlossesareneglected.

(2)Inequalityconstraints

Theseconstraintsareconsideredinaneconomicpowersystemoperationduetothephysicalandoperationallimitationsoftheunitsandcomponents.

Theinequalityconstraintsareclassifiedas:

(a)Accordingtothenature

Accordingtonature,theinequalityconstraintsareclassifiedfurtherintothefollowingconstraints:

1. Hard-typeconstraints:Theseconstraintsaredefiniteandspecificinnature.Noflexibilitywilltakeplaceinviolatingthesetypesofconstraints.

e.g.,:Therangeoftappingofanon-loadtap-changingtransformer.

2. Soft-typeconstraints:Theseconstraintshavesomeflexibilitywiththeminviolating.

e.g.,:Magnitudesofnodevoltagesandthephaseanglebetweenthem.

Somepenaltiesareintroducedfortheviolationsofthesetypesofconstraints.

(b)Accordingtopowersystemparameters

Accordingtopowersystemparameters,inequalityconstraintsareclassifiedfurtherintothefollowingcategories.

1. Outputpowerconstraints:Eachgeneratingunitshouldnotoperateaboveitsratingorbelowsomeminimumgeneration.Thisminimumvalueofreal-powergenerationisdeterminedfromthetechnicalfeasibility.

P ≤P ≤P (2.3a)

Similarly,thelimitsmayalsohavetobeconsideredovertherangeofreactive-powercapabilitiesofthegeneratorunitrequiringthat:

D

Gi(min) Gi Gi(max)

Q ≤Q ≤Q fori=1,2,3,…,n(2.3b)

andtheconstraintP +Q ≤(S ) mustbesatisfied,whereS

istheratingofthegeneratingunitforlimitingtheoverheatingofstator.

2. Voltagemagnitudeandphase-angleconstraints:Formaintainingbettervoltageprofileandlimitingoverloadings,itisessentialthatthebusvoltagemagnitudesandphaseanglesatvariousbusesshouldvarywithinthelimits.Thesecanbeillustratedbyimposingtheinequalityconstraintsonbusvoltagemagnitudesandtheirphaseangles.

V ≤V ≤V fori=1,2,…,n

δ ≤δ ≤δ fori=1,2,…,n

wherej=1,2,…,m,j≠i,nisthenumberofunits,andmthenumberofloadsconnectedtoeachunit.

3. Dynamicconstraints:Theseconstantsmayconsiderwhenfastchangesingenerationarerequiredforpickingupthesheddingdownorincreasingofloaddemand.Theseconstraintsareoftheform:

Inaddition,intermsofreactive-powergeneration,

4. Sparecapacityconstraints:Theseconstraintsarerequiredtomeetthefollowingcriteria:

1. Errorsinloadprediction.2. Theunexpectedandfastchangesinloaddemand.3. Unplannedlossofscheduledgeneration,i.e.,theforcedoutagesofoneormore

unitsonthesystem.

Thetotalpowergenerationatanytimemustbemorethanthetotalloaddemandandsystemlossesbyanamountnotlessthanaspecifiedminimumsparecapacity(P )

i.e.,P ≥(P +P )+P

whereP isthetotalpowergeneration,P +P isthetotalload

demandandsystemlosses,andP isthespecifiedminimumspare

power.

5. Branchtransfercapacityconstraints:Thermalconsiderationsmayrequirethatthetransmissionlinesbesubjectedtobranchtransfercapacityconstraints:

Gi(min) Gi Gi(max)

Gi Gi irated i

i (min) i i (max)

ij (min) ij ij (max)

SP

G D L SP

G D L

SP

2 2 2

S ≤S ≤S fori=1,2,…,n

wheren isthenumberofbranchesandS thei branchtransfer

capacityinMVA.

6. Transformertapposition/settingsconstraints:Thetappositions(or)settingsofatransformer(T)mustliewithintheavailablerange:

T ≤T≤T

Foranautotransformer,thetapsettingconstraintsare:

0≤T≤1

wheretheminimumtapsettingiszeroandthemaximumtapsettingis1.

Fora2-windingtransformer,tapsettingconstraintsare0≤T≤K,whereKisthetransformation(turns)ratio.

Foraphase-shiftingtransformer,theconstraintsareofthetype:

θ ≤θ ≤θ

whereθ isthephaseshiftobtainedfromthei transformer.

7. Transmissionlineconstraints:Theactiveandreactivepowerflowingthroughthetransmissionlineislimitedbythethermalcapabilityofthecircuit.

TC ≤TC

whereTC isthemaximumloadingcapacityofthei line.

8. Securityconstraints:Powersystemsecurityandpowerflowsbetweencertainimportantbusesarealsoconsideredforthesolutionofanoptimizationproblem.

Ifthesystemisoperatingsatisfactorily,thereisanoutagethatmaybescheduledorforced,butsomeoftheconstraintsarenaturallyviolated.Itmaybementionedthatconsiderationofeachandeverypossiblebranchforanoutagewillnotbeafeasibleproportion.Whenalargesystemisunderstudy,thenetworksecurityismaintainedsuchthatcomputationistobemadewiththeoutageofonebranchatonetimeandthenthecomputationofagroupofbranchesorunitsatanothertime.

So,theoptimizationproblemwasstatedearlierasminimizingthecostfunction(C)givenbyEquation(2.1),whichissubjectedtotheequalityandinequalityconstraint(Equations(2.2)and(2.3)).

i (min) bi i (max) b

b bi

(min) (max)

i (min) i i (max)

i

i i (max)

i (max)

th

th

th

2.16MATHEMATICALDETERMINATIONOFOPTIMALALLOCATIONOFTOTALLOADAMONGDIFFERENTUNITS

Considerapowerstationhaving‘n’numberofunits.Letusassumethateachunitdoesnotviolatetheinequalityconstraintsandletthetransmissionlossesbeneglected.

Thecostofproductionofelectricalenergy

whereC isthecostfunctionofthei unit.

Thiscostistobeminimizedsubjecttotheequalityconstraintgivenby

whereP isthereal-powergenerationofthei unit.

Thisisaconstrainedoptimizationproblem.

Togetthesolutionfortheoptimizationproblem,wewilldefineanobjectivefunctionbyaugmentingEquation(2.4)withanequalityconstraint(Equation(2.5))throughtheLagrangianmultiplier(λ)as

Theconditionforoptimalityofsuchanaugmentedobjectivefunctionis

i

Gi

th

th

FromEquation(2.6),

SinceP isaconstantandisanuncontrolledvariable,

SincetheexpressionofCisinavariableseparableform,i.e.,theoverallcostisthesummationofcostofeachgeneratingunit,whichisafunctionofreal-powergenerationofthatunitonly:

D

InEquation(2.8),eachofthesederivativesrepresentstheindividualincrementalcostofeveryunit.Hence,theconditionfortheoptimalallocationofthetotalloadamongthevariousunits,whenneglectingthetransmissionlosses,isthattheincrementalcostsoftheindividualunitsareequal.Itacalledaco-ordinationequation.

Assumethatoneunitisoperatingatahigherincrementalcostthantheotherunits.Iftheoutputpowerofthatunitisreducedandtransferredtounitswithlowerincrementaloperatingcosts,thenthetotaloperatingcostdecreases.Thatis,reducingtheoutputoftheunitwiththehigherincrementalcostresultsinamoredecreaseincostthantheincreaseincostofaddingthesameoutputreductiontounitswithlowerincrementalcosts.Therefore,allunitsmustrunwithsameincrementaloperatingcosts.

Aftergettingtheoptimalsolution,inthecasethatthegenerationofanyoneunitisbelowitsminimumcapacityoraboveitsmaximumcapacity,thenitsgenerationbecomesthecorrespondinglimit.Forexample,ifthegenerationofanyunitviolatestheminimumlimit,thenthegenerationofthatunitissetatitsminimumspecifiedlimitandviceversa.Then,theremainingdemandis

allocatedamongtheremainingunitsasfortheabovecriteria.

Inthesolutionofanoptimizationproblemwithoutconsideringthetransmissionlosses,wemakeuseofequalincrementalcosts,i.e.,themachinesaresoloadedthattheincrementalcostofproductionofeachmachineisthesame.

Itcanbeseenthatthismethoddoesnotsensethelocationofchangesintheloads.Aslongasthetotalloadisfixed,irrespectiveofthelocationofloads,thesolutionwillalwaysbethesameand,infact,forthisreasonthesolutionmaybefeasibleinthesensethattheloadvoltagesmaynotbewithinspecifiedlimits.Thereactive-powergenerationrequiredmayalsonotbewithinlimits.

2.17COMPUTATIONALMETHODS

Differenttypesofcomputationalmethodsforsolvingtheaboveoptimizationproblemareasfollows:

1. Analyticalmethod.2. Graphicalmethod.3. Usingadigitalcomputer.

Themethodtobeadopteddependsonthefollowing:

1. ThemathematicalequationrepresentingtheIFCofeachunit,whichcanbedeterminedfromthecostofgenerationofthatunit.

Thecostofthei unitisgivenby

∴TheIFCofthei unit

(IC) =a P +b (Linearmodel)(2.10)

wherea istheslopeoftheIFCcurveandb theinterceptoftheIFC

curve.

2. Numberofunits(n).3. Needtorepresentthediscontinuities(ifanyduetosteamvalve

opening)intheIFCcurve.

i i Gi i

i i

th

th

2.17.1Analyticalmethod

Whenthenumberofunitsaresmall(either2or3),incrementalcostcurvesareapproximatedasalinearorquadraticvariationandnodiscontinuitiesarepresentintheincrementalcostcurves.

WeknowthattheIFCofthei unit

Foranoptimalsolution,theIFCofalltheunitsmustbethesame(neglectingthetransmissionlosses):

Theanalyticalmethodconsistsofthefollowingsteps:

1. Chooseaparticularvalueofλ.

i.e.,λ=a P +b

2. Compute

3. Findtotalreal-powergeneration foralli=1,2,…,n.

4. Repeattheprocedurefromstep(ii)fordifferentvaluesofλ.5. Plotagraphbetweentotalpowergenerationandλ.6. Foragivenpowerdemand(P ),estimatethevalueofλfromFig.2.8.

Thatvalueofλwillbetheoptimalsolutionforoptimizationproblem.

2.17.2Graphicalmethod

Forobtainingthesolutioninthismethod,thefollowingprocedureisrequired:

i G1 i

D

th

1. (i)Considertheincrementalcostcurvesofallunits:

i.e.,(IC) =a P +b foralli=1,2,…,n

andthetotalloaddemandP isgiven.

FIG.2.8Estimationofoptimumvalueofλ—analyticalmethod

FIG.2.9Graphicalmethod

2. Foreachunit,drawagraphbetweenP and(IC)asshowninFig.2.9.

3. Chooseaparticularvalueofλand∆λ.4. Determinethecorrespondingreal-powergenerationsofallunits:

i.e.,P ,P ,…,P

i i Gi i

D

G

G1 G2 Gn

5. Computethetotalreal-powergeneration

6. Checkthereal-powerbalanceofEquation(2)asfollows:

1. If ,thentheλchosenwillbetheoptimalsolutionand

incrementalcostsofallunitsbecomeequal.

2. If ,increaseλby∆λandrepeattheprocedurefromstep(iv).

3. If ,decreaseλby∆λandrepeattheprocedurefromstep(iv).

7. Thisprocessisrepeateduntil iswithinaspecified

tolerance(ε),say1MW.

2.17.3Solutionbyusingadigitalcomputer

Formorenumberofunits,theλ-iterativemethodismoreaccurateandincrementalcostcurvesofallunitsaretobestoredinmemory.

informationabouttheIFCcurvesisgivenforallunits:

i.e.,λ=(IC) =a P +b

or (whenlossesareneglected)

i i Gi i

andsoon.

∴P =α +β (IC) +γ (IC ) +…(2.14)

fori=1,2,…,n

Thenumberoftermsincludeddependsonthedegreeofaccuracyrequiredandcoefficientsα ,β ,andγ aretobetakenasinput.

Algorithmforλ–IterativeMethod

1. Guesstheinitialvalueofλ withtheuseofcost-curveequations.2. CalculateP ,accordingtoEquation(2.14),i.e.,P =α +β (λ ) +

γ (λ ) +…

3. Calculate

4. Checkwhether :

5. If setanewvalueforλ,i.e.,λ′=λ +∆λandrepeatfrom

step(ii)tillthetolerancevalueissatisfied.

6. If setanewvalueforλ,i.e.,λ′=λ –∆λandrepeatfrom

step(ii)tillthetolerancevalueissatisfied.7. Stop.

Example2.1:ThefuelcostfunctionsinRs./hrforthreethermalplantsaregivenby

C =400+8.4P +0.006P

C =600+8.93P +0.0042P

Gi i i i i i

i i i

G1 G1 i i i

i i

1 1 1

2 2 2

2

o

o o o

o 2

o

o

2

2

2

C =650+6.78P +0.004P

whereP ,P ,andP areinMW.Neglectinglinelossesandgeneratorlimits,determinetheoptimalschedulingofgenerationofeachloadingusingtheiterativemethod.

1. P =550MW.

2. P =820MW.

3. P =1,500MW.

Solution:

For(i)P =550MW:

3 3 3

1 2 3

D

D

D

D

2

For(ii)P =820MW:

For(iii)P =1,500MW:

GENERATINGUNIT OPTIMALGENERATION(MW)

D

D

2.18ECONOMICDISPATCHNEGLECTINGLOSSESANDINCLUDINGGENERATORLIMITS

Theoutputpowerofanygeneratorshouldneitherexceeditsratingnorshoulditbebelowthatnecessaryforthestableoperationofaboiler.Thus,thegenerationsarerestrictedtoliewithingivenminimumandmaximumlimits.Theproblemistofindtheactivepowergenerationofeachplantsuchthattheobjectivefunction(i.e.,totalproductioncost)isminimum,subjecttotheequalityconstraint,andtheinequalityconstraintsare

respectively.

ThesolutionalgorithmforthiscaseisthesameasdiscussedinSection2.17.3withminormodifications.Ifanygeneratingunitviolatestheaboveinequalityconstraints,setitsgenerationatitsrespectivelimitasgivenbelow.Inaddition,thebalanceoftheloadisthensharedbetweentheremainingunitsonthebasisofequalincrementalcost.

Thenecessaryconditionsforoptimaldispatchwhenlossesareneglected:

forP ≤P ≤PGi(min) Gi Gi(max)

forP =≤P

for≤P =≤P

Example2.2:ThefuelcostfunctionsinRs./hr.forthreethermalplantsaregivenby

C =400+8.4P , 100≤P ≤600

C =600+8.93P , 60≤P ≤300

C =650+6.78P , 300≤P ≤650

whereP ,P ,andP areinMW.Neglectinglinelossesandincludinggeneratorlimits,determinetheoptimalschedulingofgenerationofeachloadingusingtheiterativemethod.

1. P =550MW.

2. P =820MW.

3. P =1,500MW.

Solution:

For(i)P =550MW:

Gi Gi(max)

Gi Gi(min)

1 1 1

2 2 2

3 3 3

1 2 3

D

D

D

D

2

2

2

Resultsfor(i)P =550MW:

Resultsfor(ii)P =820MW:

d

d

Resultsfor(iii)P =1,500MW:d

2.19FLOWCHARTFOROBTAININGOPTIMALSCHEDULINGOFGENERATINGUNITSBYNEGLECTINGTHETRANSMISSIONLOSSES

TheoptimalschedulingofgeneratingunitsisrepresentedbytheflowchartasshowninFig.2.10.

2.20ECONOMICALLOADDISPATCH—INOTHERUNITS

Theeconomicalloaddispatchproblemhasbeensolvedforapowersystemareaconsistingoffossilfuelunits.Foranareaconsistingofamixofdifferenttypesofunits,i.e.—fossilfuelunits,nuclearunits,pumpedstoragehydro-units,hydro-units,etc.—solvingtheeconomicalloaddispatchproblemwillbecomedifferent.

2.20.1Nuclearunits

Fortheseunits,thefixedcostishighandoperatingcostsarelow.Assuch,nuclearunitsaregenerallybaseloadplantsattheirratedoutputs,i.e.,thereferencepowersettingofturbinegovernorsfornuclearunitsisheldconstantattheratedoutput.Therefore,theseunitsdonotparticipateineconomicalloaddispatch.

2.20.2Pumpedstoragehydro-units

Theseunitsareoperatedassynchronousmotorstopumpwaterduringoff-peakhours.Duringpeakloadhours,thewaterisreleasedandtheunitsareoperatedassynchronousgeneratorstosupplypower.Theeconomic

operationoftheareaisdonebypumpingduringoff-peakhourswhentheareaincrementalcost(λ)islow,andbygeneratingduringpeakloadhourswhenλishigh.Sometechniquesareavailableforincorporatingpumpedstoragehydro-unitsintotheeconomicdispatchoffossilunits.

2.20.3Hydro-plants

Foranareaconsistingofhydro-plantslocatedalongariver,theobjectiveoftheeconomicdispatchproblembecomesmaximizingthepowergenerationovertheyearlywatercycleratherthanminimizingthetotaloperatingcosts.Forthesetypesofplants,reservoirsareprovidedtostorethewaterduringrainyseasons.Therearesomeconstraintsonthelevelofwatersuchasflowofriver,irrigation,etc.Optimalstrategiesareavailableforco-ordinatingtheoutputsofsuchplantsalongariver.Therearealsosomeeconomicdispatchstrategiesavailableforthemixoffossilfuelandhydro-systems.

FIG.2.10Flowchart

2.20.4Includingreactive-powerflows

Inthiscase,bothactiveandreactivepowersareselectedtominimizetheoperatingcosts.Inparticular,reactive-powerinjectionsfromgenerators,switchedcapacitorbanks,andstaticVArsystemsalongwithtransformersettingscanbeselectedtominimizethetransmissionlosses.

Example2.3:AsystemconsistsoftwounitstomeetthedailyloadcycleasshowninFig.2.11.

Thecostcurvesofthetwounitsare:

C =0.15P +60P +135Rs./hr

C =0.25P +40P +110Rs./hr

Themaximumandminimumloadsonaunitaretobe220and30MW,respectively.

Findout:

1. Theeconomicaldistributionofaloadduringthelight-loadperiodof7hrandduringtheheavy-loadperiodsof17hr.Inaddition,findtheoperationcostforthis24-hrperiodoperationoftwounits.

2. Theoperationcostwhenremovingoneoftheunitsfromserviceduring7hroflight-loadperiod.AssumethatacostofRs.525isincurredintakingaunitoffthelineandreturningittoserviceafter7hr.

3. Commentontheresults.

Solution:

(i)Whenbothunitsareoperatingthroughouta24-hrperiod,

Totaltime=24hr

1 G1 G1

1 G2 G2

2

2

FIG.2.11Dailyloadcycle

Totalload=84MWfor7hr+400MWfor17hr

(from4A.M.to11A.M.)(from11A.M.to4A.M.)

Foraheavyloadof400MW:

Heavy-loadperiod,t =17hr

load,P =400MW

Wehavetofindtheoptimalschedulingoftwounitswiththisload.

Wehavethecostcurvesoftwounits:

ForUnit1:

C =0.15P +60P +135Rs./hr

Incrementalfuelcost,

=0.3P +60Rs./MWh

ForUnit2:

C =0.25P +40P +110Rs./hr

h

Dh

1 G1 G1

G1

2 G2 G2

2

2

0.25×2P +40

=0.5P +40Rs./MWh

Fortheoptimaldistributionofaload,

0.3P +60=0.5P +40

0.3P −0.5P =−20(2.15)

P +P =400(given)(2.16)

FromEquations(2.15)and(2.16),wehave

SubstitutingtheP valueinEquation(2.16),weget

Here,P =225MWandP >P ;hence,setP atits

maximumgenerationlimit

i.e.,P =220MW

∴P =400-220=180MW

Theoperationcostforaheavy-loadperiod(i.e.,from11A.M.to4A.M.)withthisoptimaldistributionis

C = (C +C )×t

G2

G2

G1 G2

G1 G2

G1 G2

G1

G1 G1 Gmax G1

G1

G2

1 2 h

2 2

= [(0.15×220 +60×220+135)+(0.25×180 +40×180+110)]×17

= Rs.6,12,085

Foralightloadof84MW:

Period,t =7hr

load,P =84MW

Foroptimalloadsharing,

i.e.,0.3P +60=0.5P +40

0.3P −0.5P =−20(2.17)

P +P =84(2.18)

BysolvingEquations(2.17)and(2.18),weget

P =27.5MW;P =56.5MW

Here,P =27.5MW<P =30MW

Therefore,theloadsharedbyUnit-1issettoP =30

MWandP =84-30=54MW.

Theoperationcostforalight-loadperiod(i.e.,from4A.M.to11A.M.)withthisoptimaldistribution:

=[(0.15)×(30) +60×30+135)+(0.25×54 +40×54+110)]×7

=Rs.35,483

Hence,thetotalfuelcostwhenboththeunitsareoperatingthroughoutthe24-hrperiod

=Rs.(6,12,085+35,483)

=Rs.6,47,568

(ii)Ifonlyoneoftheunitsisrunduringthelight-loadperiod,

i.e.,Period,t =7hr

l

D

G1 G2

G1 G2

G1 G2

G1 G2

G1 Gmin

G1

G2

l

2 2

2 2

load,P =84MW

WhenUnit-1istoberun,

Costofoperation = C ×t

= [0.15×84 +60×84+135]×7

= Rs43,633.80

WhenUnit-2istoberun,

Costofoperation = C ×t

= [0.15×84 +40×84+110]×7

= Rs36,638

Fromtheabove,itisverifiedthatitiseconomicaltorunUnit-2duringalight-loadperiodandtoputoffUnit-1fromservice.

TheoperatingcostwithonlyUnit-2inoperation=Rs.36,638

Theoperatingcostfortheoperationofbothunitsinaheavy-loadperiodandUnit-2onlyinalight-loadperiod=Rs.(6,47,568+36,638)=Rs.6,48,723

Inaddition,giventhatacostofRs.525isincurredintakingaunitoffthelineandreturningittoserviceafter7hr,

Totaloperatingcost=operatingcost+start-upcost=Rs.(6,48,723+525)=Rs.6,49,248.

(iii)Totaloperatingcostfor(i)=Rs.6,47,568

D

1 1

2 1

2

2

Totaloperatingcostfor(ii)=Rs.6,49,248

Itisconcludedthatthetotaloperatingcostwhenbothunitsrunningthroughout24-hrperiodsislessthantheoperatingcostwhenoneoftheunitsisputofffromthelineandreturnedtotheserviceafteralight-loadperiod.Hence,itiseconomicaltorunbothunitsthroughout24hr.

Example2.4:Aconstantloadof400MWissuppliedbytwo210-MWgenerators1and2,forwhichthefuelcostcharacteristicsaregivenasbelow:

C =0.05P +20P +30.0Rs./hr

C =0.06P +15P +40.0Rs./hr

Thereal-powergenerationsofunitsP andP arein

MW.Determine:(i)themosteconomicalloadsharingbetweenthegenerators.(ii)ThesavinginRs./daytherebyobtainedcomparedtotheequalloadsharingbetweentwogenerators.

Solution:

TheIFCsare

=0.10P +20.0

=0.12P +15.0

(i)Foroptimalsharingofload,theconditionis

0.10P +20.0=0.12P +15.0

1 G1 G1

2 G2 G1

G1 G2

G1

G2

G1 G1

2

2

or 0.10P −0.12P =15.0−20.0

or 0.10P −0.12P =−5.0 (12.19)

Given:P +P =400 (12.20)

SolvingEquations(2.19)and(2.20),wehave

SubstitutingP =195.45MWinEquation(2.20),weget

P =400–195.45=204.55MW

Theloadof400MWiseconomicallysharedbythetwogeneratorswithP =195.45MWandP =204.55MW.

(ii)Whentheloadissharedbetweenthegeneratorsequally,then

P =200MWandP =200MW

Withthisequalsharingofload,theP valueis

increasedfrom195.45witheconomicalsharingto200MW.

∴Increaseinoperationcostofgenerator1

G1 G1

G1 G1

G1 G2

G1

G2

G1 G2

G1 G2

G1

TheP valueisdecreasedfrom204.55to200MW.

∴DecreaseinoperationcostofGenerator2

∴Savingincost=180.96–178.69=2.27Rs./hr

Thesavingincostperday=2.27×24

=56.75Rs./day

Example2.5:ConsiderthefollowingthreeICcurves:

P = −100+50(IC )+2(IC )

P = −150+60(IC )−2(IC )

P = −80+40(IC )−1.8(IC )

whereIC’sareinRs./MWhandP ’sareinMW.

Thetotalloadatacertainhourofthedayis400MW.Neglecttransmissionlossesanddevelopacomputerprogramforoptimumgenerationschedulingwithintheaccuracyof±0.05MW.

Note:AllP ’smustberealandpositive.

Solution:

α =−100, β =50, γ =2(∵Assumed ,d ,d are

neglected)

G2

G1 1 1

G2 2 2

G3 3 3

G

G

1 1 1 1 2 3

2

2

2

α =−150, β =60, γ =−2.5

α =−80, β =40, γ =−1.8

α =

∴a =0.02;a =0.0166;a =0.025

b =2;b =2.49;b =2

Foroptimalloaddistributionamongthevariousunits,

0.02P +2=0.0166P +2.49

⇒0.02P −0.0166P =0.49 (2.21)

2 2 2

3 3 3

i

1 2 3

1 2 3

G1 G2

G1 G2

0.0166P +2.49=0.025P +2

0.0166P −0.025P =−0.49 (2.22)

0.02P +2=0.025P +2

0.02P −0.025P =0(2.23)

Given:P +P +P =400(2.24)

orP +P =400−P (2.25)


0.0166P +0.0416P =7.13(2.26)

SolvingEquations(2.23)and(2.26),weget

G2 G3

G2 G3

G1 G3

G1 G3

G1 G2 G3

G2 G3 G1

G1 G3

SubstitutingP = 14.35MWin(2.26),weget

P = 142.9375MW

SubstitutingP = 142.9375MWandP =114.35MW

in(2.25),weget

P = 142.93175MW

Therefore,foroptimalgeneration,thethreeunitsmustshareatotalloadof400MWasfollows:

Costofgenerationof142.9375MWbyUnit-1

(C )= (142.9375) +2(142.9375)

C =490.186Rs./MWh

Similarly,

andC = ×0.025×(114.35) +(2×114.35)

=392.149Rs./MWh

Totalcostofgenerationof400MWwitheconomicalloadsharing

C=C +C +C

=490.186+525.359+392.149

=1,407.694Rs./MWh

G3

G1

G1 G3

G2

1

1

3

1 2 3

2

2

Costperday=1,407.694×24

=Rs.33,784.656/day

Totalcostperdaywithanequaldistributionofload

=1,412.838×24

=Rs.33,908.112/day

∴Savingincost=Rs.33,908.112–33,784.856=Rs.123.256/day

Example2.6:Thefuelcostoftwounitsaregivenby

C =C (P )=1.0+25P +0.2P Rs/hr

C =C (P )=1.5+35P +0.2P Rs/hr

Ifthetotaldemandonthegeneratorsis200MW,findtheeconomicloadschedulingofthetwounits.

Solution:

Theconditionforeconomicloadschedulingwhenneglectingthetransmissionlossesis

Foreconomicalloaddispatch,

1 1 G1 G1 G1

2 2 G2 G2 G2

2

2

25+0.4P =35+0.4P

or0.4P −0.4P =10MW (2.27)

andP +P =200MW(2.28)

MultiplyingbothsidesofEquation(2.28)by0.4,weget

0.4P +0.4P =80(2.29)

ByaddingEquations(2.27)and(2.29),weget

SubstitutingP =112.5MWinEquation(2.28)

112.5+P =200

∴P =87.5MW

Example2.7:Theincrementalcostcurvesofthreeunitsareexpressedintheformofpolynomials:

P = −150+50(IC )−2(IC )

P = −100+50(IC )−2(IC )

P = −150+50(IC )−2(IC )

Thetotaldemandatacertainhourofthedayequals200MW.Developacomputerprogramthatwillrendera

G1 G2

G1 G2

G1 G2

G1 G2

G1

G2

G2

G1 1 1

G2 2 2

G3 3 3

2

2

2

solutionfortheoptimumallocationofloadamongthreeunits.

Solution:

Step1:Assumeλ =10.

Step2:ComputeP correspondingtoλ ,i=1,2,3.

P =−150+50λ −2(λ ) =−150+50(10)−2(100)=

150MW

P =−100+50λ −2(λ ) =−100+50(10)−2(100)=

200MW

P =−150+50λ −2(λ ) =−150+50(10)−2(100)=

150MW

Step3:Compute :

i.e.,P +P +P =500MW

Step4:Checkif :

Wefind

i.e.,500>200

Step5:ReduceλbyΔλ=3:

i.e.,λ′=λ –Δλ=10-3=7

Step6:NowfindthegenerationP ,andP

Step7:Gotostep4.

Byrepeatingtheaboveprocedure,thefollowingresultsareobtainedandtheaboveequationsconvergeatλ=5

G1

G1

G2

G3

G1 G2 G3

G ,P1 G2G3

o

(o) o

(0) o o 2

(0) o o 2

(0) o o 2

o o o

o

11

1

Example2.8:Twounitseachof200MWinathermalpowerplantareoperatingallthetimethroughouttheyear.Themaximumandminimumloadoneachunitis200and50MW,respectively.Theincrementalfuelcharacteristicsforthetwounitsaregivenas

Ifthetotalloadvariesbetween100and400MW,findtheIFCandallocationofloadbetweentwounitsforminimumfuelcostforvarioustotalloads.

Solution:

Fortheminimumloadof100MW,

P ==50MW,P =50MWG1 G2

FromEquations(2.30)and(2.31),itisnotedthatatatotalminimumloadof100MW,Unit-1isoperatingatahigherIFCthanUnit-2.Therefore,additionalloadonUnit-2shouldbeincreasedtill(IC) =λ=19andatthatpoint,

13+0.1P =19

∴P =60

Hence,thetotalloadbeingdeliveredatequalincrementalcostsof19Rs./MWhis110MW,i.e.,P =

50andP =60.

Goonincreasingtheloadoneachunitsothattheunitsoperateatthesameincrementalcost,andtheseoperatingconditionsarefoundbyassumingvariousvaluesofλandbycalculatingtheoutputforeachunit.

Example2.9:DeterminethesavinginfuelcostinRs./hrfortheeconomicdistributionofthetotalloadof100MWbetweentwounitsoftheplantasgiveninExample2.8.Comparewithequaldistributionofthesametotalload.

Solution:

2

G2

G2

G1

G2

Fortheoptimaldistributionofthetotalloadbetweenthetwounits,

∴0.08P +15=0.1P +13

or0.08P −0.1P =13−15=−2(2.32)

GivenP +P =1100.08P −0.1P =13−15=−2

(2.33)


0.08P −0.1P =−2

Equation(2.33)×0.1⇒

orP =50MW

SubstitutingP inEquation(2.32),weget

P =60MW

OperatingcostofUnit-1,

OperatingcostofUnit-2,

TheoperatingcostsofUnit-1andUnit-2are

C =0.04(50) +15(50)=850Rs./hr

G1 G2

G1 G2

G1 G2 G1 G2

G1 G2

G1

G1

G2

1

2

2

C =0.05(60) +13(60)=960Rs./hr

Fortheequaldistributionofload⇒P =55MWandP

=55MW.

TheoperatingcostsofUnit-1andUnit-2are

C =0.04(55) +15(55)=946Rs./hr

C =0.05(55) +13(55)=866.25Rs./hr

TheincreaseincostforUnit-1whenthedeliveringpowerincreasesfrom50to55MWis946–850=96Rs./hrandforUnit-2decreasesincostduetodecreaseinpowergenerationfrom60to55MWis960–866.25=–93.75Rs./hr.

∴Savingincost=96–93.75=3.75Rs./hr.

Example2.10:Threepowerplantsofatotalcapacityof500MWarescheduledforoperationtosupplyatotalsystemloadof350MW.Findtheoptimumloadschedulingiftheplantshavethefollowingincrementalcostcharacteristicsandthegeneratorconstraints:

Solution:

Foreconomicloadschedulingamongthepowerplants,thenecessaryconditionis

2

G1 G2

1

2

2

2

2

Forthreeplants,

Giventotalload=P +P +P =350MW

(2.34)

40+0.25P =50+0.30P =20+0.20P =λ

(2.35)

⇒40+0.25P =50+0.30P

or0.25P −0.30P =50−40=10(2.36)

and40+0.25P =20+0.2P

or0.25P −0.2P =20−40=−20(2.37)

FromEquation(2.36),wehave

SubstitutingEquation(2.38)inEquation(2.34)

P +0.833P −33.33+P =350

or1.833P +P =383.33(2.39)

SolvingEquations(2.37)and(2.39)

G1 G2 G3

G1 G2 G3

G1 G2

G1 G1

G1 G1

G1 G1

G1 G1 G3

G1 G3

SubstitutingthevalueofP inEquation(2.39),

1.833×91.98+P =383.33G3

orP =214.73MW

SubstitutingthevaluesofP andP inEquation(2.34),

weget

91.98+P +214.73=350

orP =43.29MW

∴Foreconomicschedulingoftheload,thegenerationsofthreeplantsmustbe

P =91.98MW,P =43.29MW,andP =214.73MW

Example2.11:Thefuelcostoftwounitsaregivenby

C =0.1P =25P +1.6Rs./hr

C =0.1P +32P +2.1Rs./hr

Ifthetotaldemandonthegeneratorsis250MW,findtheeconomicalloaddistributionofthetwounits.

Solution:

Given

G1

G3

G3

G1 G2

G2

G2

G1 G2 G3

1 G1 G1

2 G2 G2

2

2

Giventhetotalload,P =250MW.Foreconomicaldistributionoftotalload,theconditionis

0.2P +25=0.2P +32

or0.2P −0.2P =7(2.40)

andP +P =250(Given)(2.41)


2P =285

orP =142.5MW


P =250−P =107.5MW

Example2.12:Aplanthastwogeneratorssupplyingtheplantbus,andneitheristooperatebelow20orabove125MW.Incrementalcostsofthetwounitsare

Foreconomicdispatch,findtheplantcostofthereceivedpowerinRs./MWh(λ)whenP +P equals:

(a)40MW,(b)100MW,and(c)225MW.

Solution:

Foreconomicoperation,

D

G1 G2

G1 G2

G1 G2

G1

G1

G1

G2 G1

G1 G2

(a)WhenP +P =40MW2(2.42)

0.15P −20=0.225P +17.5

or0.15P −0.225P =−2.5(2.43)

Equation(2.42)×0.15⇒0.15P +0.15P =6.0

(2.44)


−0.375P =−8.5

P =22.66MW

SubstitutingP =22.66MWinEquation(2.42)

P =40−22.666

=17.34MW

0.225P +17.5=λ

or0.225(226)+17.5=λ

∴=22.59Rs./MWh

(b)WhenP +P =100MW(2.45)

Equation(2.45)×0.15⇒0.15P +0.15P =15

(2.46)


G1 G2

G1 G2

G1 G2

G1 G2

G2

G2

G2

G1

G2

G1 G2

G1 G1


P =53.34MW

∴0.15P +20=λor0.225P +17.5=λ

0.15(53.34)+20=λorλ=0.225(46.66)+17.5

⇒λ=28Rs./MWh;λ=28Rs./MWh

(c)WhenP +P =225MW(2.47)

Equation(2.47)×0.15⇒−015P +015P =3375

(2.48)



P =128.34MW

∴λ=0.255P +17.5

=0.225(96.66)+17.5

=39.24Rs./MWh

Example2.13:Thecostcurvesoftwogeneratorsmaybeapproximatedbysecond-degreepolynomials:

C =0.1P +20P +α

G2

G1

G1 G2

G1 G2

G1 G2

G2

G1

G2

1 G1 G1 1

2

2

C =0.1P +30P +α

whereα andα areconstants

Ifthetotaldemandonthegeneratorsis200MW,findtheoptimumgeneratorsettings.Howmanyrupeesperhourwouldyouloseifthegeneratorswereoperatedabout15%oftheoptimumsettings?

Solution:


0.2P +20=0.2P +30

or0.2P +0.2P =10

orP −P =50(2.49)

andgiventhatP +P =200(2.50)


2P =250

orP =125MW


P =200–125=75MW

Ifthegeneratorswereoperatedabout15%oftheoptimumsettings,

P =125−125× =125−18.75=106.25MW

andP =75− =75−11.25=63.75MW

ThedecreaseincostforGenerator-1is

2 G2 G2 2

1 2

G1 G2

G1 G2

G1 G2

G1 G2

G1

G1

G1

G2

G1

G2

2

ThedecreaseincostforGenerator-2is

Thelossofamount = ΔC −ΔC

= −58.59−(−181.40625)

= −122.81Rs./hr

Example2.14:DeterminethesavinginfuelcostinRs./hrfortheeconomicdistributionofatotalloadof225MWbetweenthetwounitswithIFCs:

Comparewithequaldistributionofthesametotalload.

1 2

Solution:

Given:P +P =225MW(2.51)

Foroptimaloperation:

⇒0.075P +15=0.085P +12

or0.075P −0.085P =−3(2.52)

Equation(2.51)×0.085⇒0.085P +0.285P =225×

0.085=19.125(2.53)


0.16P =16.125

orP =100.78MW


P =225–100.78=124.218MW

Withequaldistributionofthetotalload,

⇒P =112.5MWandP =112.5MW

TheincreaseincostforUnit-1is

ForUnit-2,

G1 G2

G1 G2

G1 G2

G1 G2

G1

G1

G1

G2

G1 G2

Thenegativesignindicatesadecreaseincost.

∴Savinginfuelcost=Rs.269.53656−258.505

=11.03156Rs./hr

Example2.15:Threeplantsofatotalcapacityof500MWarescheduledforoperationtosupplyatotalsystemloadof310MW.Evaluatetheoptimumloadschedulingiftheplantshavethefollowingcostcharacteristicsandthelimitation:

C =0.06P +30P +10,30≤P ≤150

C =0.10P +40P +15,20≤P ≤100

C =0.075P +10P +20,50≤P ≤250

Solution:

TheIFCsofthethreeplantsare

Foroptimumschedulingofunits,

1 G1 G1 G1

2 G2 G2 G2

3 G3 G3 G3

2

2

2

0.12P +30=0.20P +40=0.15P +10

⇒0.12P +30=0.15P +10

orP −0.15P =−20(2.54)

andgiventhatP +P =310−P (2.55)


or0.27P +0.15P =26.5(2.56)

and

0.12P +30=0.2P +40

0.12P −0.2P =10(2.57)



G1 G2 G3

G1 G3

G1 G2

G1 G3 G2

G1 G2

G1 G2

G1 G2

G1

0.12(94.444)−0.15P =−20

11.33−0.15P =−20

31.33=0.15P

orP =208.86MW

SubstitutingtheP andP valuesinEquation(2.55),we

get

94.44+208.86+P =310

∴P =6.7MW

Theoptimalpowergenerationis

P =94.44MW

P =6.7MW

andP =208.86MW

Itisobservedthatthereal-powergenerationofUnit-2is6.7MWanditisviolatingitsminimumgenerationlimit.Hence,wehavetofixitsvalueatitsminimumgeneration,i.e.,P =20MW.

Given:P +P +P =310MW

P +P =310−20=290MW

Theremainingloadof290MWistobedistributedoptimallybetweenUnit-1andUnit-3asfollows:

0.12P +30=0.15P +10

or0.12P −0.15P =−20(2.58)

andP +P =290(2.59)

SolvingEquations(2.58)and(2.59),weget:

1

G3

G3

G3

G3

G1 G3

G2

G2

G1

G2

G3

G2

G1 G2 G3

1 G3

G1 G3

G1 G2

G1 G3


P =290−67.14=202.96MW

Thetotalloadof310MWisdistributedoptimallyamongtheunitsas

P =87.03MW

P =20MW

andP =202.96MW

Example2.16:Theincrementalcostcharacteristicsoftwothermalplantsaregivenby

Calculatethesharingofaloadof200MWformosteconomicoperations.Iftheplantsarerated150and250MW,respectively,whatwillbethesavingincostinRs./hrincomparisontotheloadinginthesameproportiontorating.

Solution:


G1

G3

G1

G2

G3

0.2P +60=0.3P +40

or0.2P −0.3P =−20(2.60)

orP +P =200(given)(2.61)


∴P =80MW

SubstitutingtheP valueinEquation(2.61),P =120

MW.Iftheplantsareloadedinthesameproportiontotherating,

i.e.,P =150MW,P =250MW

IncreaseintheoperationcostforPlant-1is

IncreaseintheoperationcostforPlant-2is

∴Savinginoperationcost=12,415−5,810=66Rs./hr

G1 G3

G1 G2

G1 G2

G1

G1 G2

G1 G2

Example2.17:TheIFCsoftwounitsinageneratingstationareasfollows:

Assumingcontinuousrunningwithatotalloadof150MW,calculatethesavingperhourobtainedbyusingthemosteconomicaldivisionofloadbetweentheunitsascomparedwithloadingeachequally.Themaximumandminimumoperationalloadingsarethesameforeachunitandare125and20MW,respectively.

Solution:

Given:

Totalload=P +P =150MW(2.62)

Foroptimality,

0.15P +35=0.20P +28

or0.15P −0.20P =−7(2.63)


G1 G2

G1 G2

G1 G2


Withanequalsharingofload,P =75MWandP =75

MW.

Withanequaldistributionofload,theloadonPlant-1isincreasedfrom65.714to75MW.

TheincreaseincostofoperationforPlant-1is

TheloadonPlant-2isdecreasedfrom84.286to75MW.

∴Thesolvingincost=423.01−407.921

=15.089Rs./hr

G1

G1 G2

Example2.18:Iftwoplantshavingcostcharacteristicsasgiven

C =0.1P +60P +135Rs./hr

C =0.15P +40P +100Rs./hr

havetomeetthefollowingdailyloadcycle:

0to6hrs–7MW

18to24hrs–70MW

findtheeconomicscheduleforthedifferentloadconditions.IfacostofRs.450isinvolvedintakingeitherplantoutofservicesortoreturntoservice,findwhetheritismoreeconomicaltokeepbothplantsinserviceforthewholedayortoremoveoneofthemduringlight-loadservice.

Solution:

For0–6hr:Totalload=7MW

i.e.,P +P =7MW(2.64)

Theconditionfortheoptimaldistributionofloadis

0.2P +60=0.3P +40

0.2P −0.3P =−40(2.65)


1 G1 G1

1 G1 G2

G1 G2

G1 G2

G1 G2

2

2

Sincethereal-powergenerationofPlant-1isP =—

35.8MW,itviolatestheminimumgenerationlimit.Hence,tomeettheloaddemandof7MW,itisnecessarytorunUnit-2onlywithgenerationof7MW.

OperationcostofUnit-2during0–6hris

C =0.15(7) +40(7)+100

=7.35+280+100

=387.35Rs./hr

For18–24hr:

Totalload=70MW

i.e.,P +P =70MW(2.66)


ThecostofoperationofPlant-1with2-MWgenerationis

C =0.1P +60P +135

=0.1(2) +60(2)+135=255.4Rs./hr

ThecostofoperationofPant-2with68-MWgenerationis

C =0.15(68) +40(68)+100=3,513.6Rs./hr

G1

2

G1 G2

1 G1 G1

2

2

2

2

Theoperatingcostduring18–24hr=255.4+3,513.6=3,769Rs./hr

Thetotaloperatingcostduringanentire24-hrperiodis

387.35×6+3,769×6=Rs.24,938.10

AcostofRs.450isincurredasthestart-upcost.

∴Totaloperatingcost=24,938.1+450=Rs.25,388.10

Example2.19:TheIFCsinrupeesperMWhforaplantconsistingoftwounitsare

CalculatetheextracostincreasedinRs./hr,ifaloadof220MWisscheduledasP =P =110M.

Solution:

Foroptimalschedulingofunits,

0.20P +40.0=0.25P +30

or0.20P −0.25P =10(2.67)

Given:P +P =220(2.68)


G1 G2

G1 G2

G1 G2

G1 G2


∴P =220−P =120MW

Foranequaldistributionofload,P =110MWandP

=110MW.TheoperationcostofUnit-1isincreasedastheloadsharedbyitisincreasedfrom100to110MW.

∴IncreaseinoperationcostofUnit-1

TheoperationcostofUnit-2isdecreasedastheloadsharedbyitisdecreasedfrom120to110MW.

∴DecreaseinoperationcostofUnit-2

TheextracostincurredinRs./hriftheloadisequallysharedbyUnit-1andUnit-2is

610–587.5=22.5Rs./hr

G1

G2 G1

G1 G2

Example2.20:Thefuelcostcharacteristicsoftwogeneratorsareobtainedasunder:

C (P )=1,000+50P +0.01P Rs./hr

C (P )=2,500+45P +0.005P Rs./hr

Ifthetotalloadsuppliedis1,000MW,findtheoptimalloaddivisionbetweentwogenerators.

Solution:

C (P )=1,000+50P +0.01P Rs./hr

C (P )=2,500+45P +0.005P Rs./hr

TheIFCcharacteristicsare

Theconditionforoptimalloaddivisionis

50+0.02P =45+0.01P

or0.02P +P =−5.0(2.69)

P +P =1,000(given)(2.70)


1 G1 G1 G1

2 G2 G2 G2

1 G1 G1 G1

2 G2 G2 G1

G1 G2

G1 G2

G1 G2

2

2

2

2


P =833MW

SubstitutingtheP andP valuesin

equation,weget

λ=53.33Rs./MWh

∴Thetotalloadof1,000MWoptimallydividedinbetweenthetwogeneratorsis

P =166MW

P =833MW

AndIFC,λ=53.33Rs./MWh

Example2.21:Determinetheeconomicoperationpointforthethreethermalunitswhendeliveringatotalof1,000MW:

UnitA: P =600MW,P =150MW

C =500+7P +0.0015P

UnitB: P =500MW,P =125MW

C =300+7.88P +0.002P

G1

G2

G1 G1

G1

G2

max min

A GA GA

max min

B GB GB

2

2

UnitC: P =300MW,P =75MW

C =80+7.99P +0.05P

Fuelcosts:

UnitA=1.1unitofprice/MBtu

UnitB=1.0unitofprice/MBtu

UnitC=1.0unitofprice/MBtu

FindthevaluesofP ,P andP foroptimaloperation.

Solution:

Costcurvesare:

C (P )=H ×1.1=550+7.7P +0.00165P

C (P )=H ×1.0=300+7.88P +0.002P

C (P )=H ×1.0=80+7.799P +0.005P

NowIFCsare:

Foraneconomicsystemoperation,

max min

C GC GC

GA GB GC

A GA A A A

B GB B B B

C GC C C C

2

2

2

2

7.7+0.0033P =7.99+0.001P

or0.0033P −0.01P =0.29(2.71)

P +P +P =1,000(given)

orP =1,000−(P +P )(2.72)

SubstitutingP fromEquation(2.72)inEquation(2.71),

weget

7.88+0.004P =7.99+0.01P

or0.004P +0.0133P =0.11(2.74)


0.0033(366.16)+0.0133P =3.01

orP =135.464MW

SubstitutingP andP valuesinEquation(2.72),we

get

P =498.376MW

GA G1

GA GC

GA GB GC

GA GB GC

GA

GB GC

GB GC

GB

GC

GC

GB GC

GA

Foratotalloadof1,000MW,theeconomicschedulingofthreeunitsare:

P =498.376MW (150MW<P <600

MW)

P =366.16MW (125MW<P <500

MW)

andP =135.464MW (75MW<P <300MW)

Example2.22:Thefuelcostcurveoftwogeneratorsaregivenas:

C (P )=800+45P +0.01P

C (P )=200+43P +0.003P

andifthetotalloadsuppliedis700MW,findtheoptimaldispatchwithandwithoutconsideringthegeneratorlimitswherethelimitshavebeenexpressedas:

50MW≤P ≤200MW

50MW≤P ≤600MW

Comparethesystem’sincrementatcostwithandwithoutgeneratorlimitsconsidered.

Solution:

Foreconomicoperation,I =I =λ

GA GA

GB GB

GC GC

A GA GA GA

B GB GB GB

GA

GB

CA CB

Consideringalongwiththegivenconstraintequations:

λ = 45+0.02P

λ = 43+0.02P

P +P = 700MW

Solvingtheseequations,

λ=46.7

P =84.6MW

P =615.4MW

Intheaboveillustration,generatorlimitshavenotbeenincluded.Iftheselimitsarenowincluded,itmaybeseenthatGenerator-Bhasviolatedthelimit.Fixingitattheuppermostlimits,let

P = 600MW

Andobviouslybysothat P = 100MW(sinceP +P =

700MW)

∴ λ = 45+0.02×100=47

λ = 43+0.006×600=46.6

Hence,itisobservedthatλ ≠λ ,i.e.,economicoperationisnotstrictlymaintainedinthisparticularcondition;incrementalcostofUnit-Aisnowmarginally

GA

GB

GA GB

GA

GB

G

B

G

A

GA GB

A

B

A B

morethanthatofUnit-B.However,inpractice,thisdifferenceofλ andλ isnotmuch;hence,thesystemoperationisjustifiedunderthiscondition.

Example2.23:Thefuelcostcurveoftwogeneratorsaregivenas

C =625+35P +0.06P

C =175+30P +0.005P

ifthetotalloadsuppliedis550MW,findtheoptimaldispatchwithandwithoutconsideringthegeneratorlimits:

35MW≤P ≤175MW

35MW≤P ≤600MW

andalsocommentabouttheincrementalcostofbothcases.

Solution:

Giventhattotalload=P +P =550MW(2.75)

Costoffirstunit,C =625+35P +0.06P

TheIFCoffirstunit,

Costofsecondunit,C =175+30P +0.005P

TheIFCofsecondunit,

Case-I:Withoutconsideringgeneratorlimits:

Foroptimaldispatchofload,thenecessaryconditionis

A B

1 G1 G1

2 G2 G1

G1

G2

G1 G2

1 G1 G1

2 G2 G2

2

2

2

2

0.12P +35=0.01P +30

0.12P +0.01P =−5(2.76)



P =550−3.846=546.154MW

Theaboveresultsareforthecasewithoutconsideringthegeneratorlimits.

TheIFCsare

TheIFC,λ=35.46Rs./MWh

Case-II:Consideringthegeneratorlimits:

35MW≤P ≤175MW

30MW≤P ≤600MW

FromCase-I,theobtainedpowergenerationsare

P =3.846MW

P =546.154MW

Itisobservedthatthereal-powergenerationofUnit-1isviolatingtheminimumgenerationlimit.Toachievetheoptimumoperation,fixupthegenerationofthefirst

G1 G2

G2 G2

G1

G1

G1

G2

G1

G2

unitatitsminimumgeneration,i.e.,P =35MW.Hence,

fortheloadof550MW,P =35MWandP =550-35=

515MW.

Then,theIFCsare

Hence,itisobservedthatλ ≠λ ,i.e.,economicoperationisnotstrictlymaintainedinthisparticularcondition.

Commentontheresults:Whenthegeneratorlimitsarenotconsidered,theeconomicoperationofgeneratingunitsisobtainedatanIFCof33.45Rs./MWh.Theireconomicoperationisnotobtainedwhenconsideringthegenerationlimits,sincetheIFCofthefirstunitissomewhatmarginallygreaterthanthatofthesecondunit.

KEYNOTES

Economicoperationofapowersystemisimportantinordertomaintainthecostofelectricalenergysuppliedtoaconsumeratareasonablevalue.Inanalyzingtheeconomicoperationofathermalunit,input–outputmodelingcharacteristicsareofgreatsignificance.Foroperationalplanning,dailyoperation,andforeconomicscheduling,thedatanormallyrequiredareasfollows:

Foreachgenerator

Maximumandminimumoutputcapacities.Fixedandincrementalheatrate.Minimumshutdowntime.Minimumstableoutput.Maximumrun-upandrun-downrates.

Foreachstation

Costandcalorificvalueofthefuel.

G1

G1 G2

1 2

Factorsreflectingrecentoperationalperformanceofthestation.Minimumtimebetweenloadingandunloading.

Forthesystem

Loadcycle.Specifiedconstraintsimposedontransmissionsystemcapability.Sparecapacityrequirement.Transmissionsystemparametersincludingmaximumcapacitiesandreliabilityfactors.Toanalyzethepowersystemnetwork,thereisaneedofknowingthesystemvariables.Theyare:

1. Controlvariables.2. Disturbancevariables.3. Statevariables.

Schedulingistheprocessofallocationofgenerationamongdifferentgeneratingunits.Economicschedulingisthecost-effectivemodeofallocationofgenerationamongthedifferentunitsinsuchawaythattheoverallcostofgenerationshouldbeminimum.Input–outputcharacteristicsestablishtherelationshipbetweentheenergyinputtotheturbineandtheenergyoutputfromtheelectricalgenerator.Incrementalfuelcostisdefinedastheratioofasmallchangeintheinputtothecorrespondingsmallchangeintheoutput.Incrementalefficiencyisdefinedasthereciprocalofincrementalfuelrate.Theinput–outputcharacteristicsofhydro-powerunitco-ordinatesarewaterinputordischarge(m /s)versusoutputpower(kWorMW).

ConstraintEquations

Theeconomicpowersystemoperationneedstosatisfythefollowingtypesofconstraints:

1. Equalityconstraints.2. Inequalityconstraints.

(a)Accordingtothenature:

1. Hard-typeconstraints.2. Soft-typeconstraints.

(b)Accordingtothepowersystemparameters:

1. Outputpowerconstraints.2. Voltagemagnitudeandphase-angleconstraints.3. Transformertapposition/settingsconstraints.4. Transmissionlineconstraints.

3


1. Justifytheproductioncostbeingconsideredasafunctionofreal-powergeneration.

Theproductioncostinthecaseofthermalandnuclearpowerstationsisafunctionoffuelinput.Thereal-powergenerationisafunctionoffuelinput.Hence,theproductioncostwouldbeafunctionofreal-powergeneration.

2. Givetheexpressionfortheobjectivefunctionusedforoptimizationofpowersystemoperation.

3. Statetheequalityandinequalityconstraintsontheoptimizationofproductcostofapowerstation.

Theequalityconstraintisthesumofreal-powergenerationofallthevariousunitsthatmustalwaysbeequaltothetotalreal-powerdemandonthesystem.

Theinequalityconstraintineachgeneratingunitshouldnotbeoperatingaboveitsratingorbelowsomeminimumgeneration.

i.e.,P ≤P ≤P ,

fori=1,2,3,…,n

4. Whatisanincrementalfuelcostandwhatareitsunits?

Incrementalfuelcostisthecostoftherateofincreaseoffuelinputwiththeincreaseinpowerinput.ItsunitisRs./MWh.

5. Howistheinequalityconstraintconsideredinthedeterminationofoptimumallocation?

Ifoneorseveralgeneratorsreachtheirlimitvalues,thebalancereal-powerdemand,whichisequaltothedifferencebetweenthetotaldemandandthesumofthelimitvalue,isoptimallydistributedamongtheremainingunitsbyapplyingtheequalincrementalfuelcostrule.

6. Onwhatfactorsdoesthechoiceofacomputationmethoddependonthedeterminationofoptimumdistributionofloadamongthe

Gi(min) Gi Gi (max)

units?

Thefactorsdependuponthefollowing:

1. Numberofgeneratingunits.2. ThedegreeofpolynomialrepresentingtheICcurve.3. ThepresenceofdiscontinuitiesintheICcurves.

7. Whatdoestheproductioncostofapowerplantcorrespondto?

Theproductioncostofapowerplantcorrespondstotheleastofminimumoroptimumproductioncostsofvariouscombinationsofunits,whichcansupplyagivenreal-powerdemandonthestation.

8. Togetthesolutiontoanoptimizationproblem,whatwillwedefineanobjective’sfunction?

Minimizethecostofproduction,minC′=minC(P )

9. Writetheconditionforoptimalityinallocatingthetotalloaddemandamongthevariousunits.

Theconditionforoptimalityistheincrementalfuelcost,

10. Writetheseparableobjectivefunctionandwhyitiscalledso.

Theaboveobjectivefunctionconsistsofasummationoftermsinwhicheachtermisafunctionofaseparateindependentvariable.Hence,itiscalledseparableobjectivefunction.

11. Brieflydiscusstheoptimizationproblem.

Minimizetheoverallcostofproduction,whichissubjectedtoequalityconstraintsandinequalityconstraints.

Equalityconstraintis:

Inequalityconstraintis

P ≤P ≤P

12. Whatisthereliableindicatorofacountry’sorstate’sdevelopment?

Gn

Gi(min) Gi Gi(max)

Itisthepercapitaconsumptionofelectricalenergy.

13. Stateinwordstheconditionforminimumfuelcostinapowersystemwhenlossesareneglected.

Theminimumfuelcostisobtainedwhentheincrementalfuelcostforallthestationsisthesameinthepowersystem.

14. Whatistheneedofsystemvariablesandwhatarethevariables?

Toanalyzethepowersystemnetwork,thereisaneedofknowingthesystemvariables.Theyare:

1. Controlvariables—P andQ

2. Disturbancevariables—P andQ

3. Statevariables—Vandδ.

15. Definethecontrolvariables.

Therealandreactive-powergenerationsarecalledcontrolvariablessincetheyareusedtocontrolthestateofthesystem.

16. Definethedisturbancevariables.

Therealandreactive-powerdemandsarecalleddemandvariablesandtheyarebeyondsystemcontrolandarehencecalleduncontrolledordisturbancevariables.

17. (Definethestatevariables.

ThebusvoltagemagnitudeVanditsphaseangleδdispatchthestateofthesystem.Theyaredependentvariablesthatarebeingcontrolledbythecontrolvariables.

18. Whatistheneedofinput–outputcharacteristicsofasteamunit?

Itestablishestherelationshipbetweentheenergyinputtotheturbineandtheenergyoutputfromtheelectricalgenerator.

19. Definetheincrementalfuelorheatratecurve.

Itisdefinedastheratioofasmallchangeintheinputtothecorrespondingsmallchangeintheoutput.

Incrementalfuelrate

20. Howdoyougetincrementalcostcurve?

Theincrementalcostcurveisobtainedbyconsideringatvariouspoints,thechangeincostofgenerationtothechangeinreal-powergeneration,i.e.,slopeoftheinput–outputcurve.

G G.

D D.

21. Howyougettheheatratecharacteristic?

TheheatratecharacteristicisobtainedfromtheplotofnetheatrateinkCal/kWhversuspoweroutputinkW.

22. Definetheincrementalefficiency.

Itisdefinedasthereciprocalofincrementalfuelrateandisgivenby

23. Whatarehard-typeconstraints?Giveexamples.

Hard-typeconstraintsaredefiniteandspecificinnature.Noflexibilitywillbetakenplaceinviolatingthesetypesofconstraints.

E.g.,Thetappingrangeofanon-loadtap-changingtransformer.

24. Whataresoft-typeconstraints?Giveexamples.

Soft-typeconstraintshavesomeflexibilitywiththeminviolatingthesetypeofconstraints.

E.g.,Magnitudesofnodevoltagesandthephaseanglebetweenthem.

25. Whatistheneedofsparecapacityconstraints?

Theseconstraintsarerequiredtomeet:

1. Errorsinloadprediction.2. Theunexpectedandfastchangesinloaddemand.3. Unplannedlossofscheduledgeneration,i.e.,theforcedoutagesofoneor

morealternatorsonthesystem.


1. Inathermal-electricgeneratingplant,theoverallefficiencyisimprovedwhen:

1. Boilerpressureisincreased.2. Thedifferencebetweeninitialpressureandtemperatureandexhaust

pressureandtemperatureareheldatamaximum.3. Loadontheunitsisincreased.4. Itsoperatingtimeisincreased.

2. Whenloadonathermalunitisincreased,fuelinput:

1. Increases.2. Doesnotchange.3. Decreases.

4. Noneofthese.

3. Incrementalheatratecurves,forthermalgeneratingunits,areusedtodeterminethe:

1. Fuelcostinrupeesperhour.2. Valuesatwhichtheunitsshouldbeloadedtoresultinminimumfuelcosts.3. Costperunitofelectricaloutput.4. Heatproducedperhour.

4. Whengeneratingunitsareloadedtoequalincrementalcosts,itresultsin:

1. Minimumfuelcosts.2. Fuelcostsareatamaximum.3. Fuelcostsarenotaffected.4. Maximumloadingofgeneratingunits.

5. Oneadvantageofcomputercontrolofgeneratingunitsisthat:

1. Netoutputoftheunitsisminimized.2. Allunitsunderthecontrolofthecomputerwillbeloadedtothesameload.3. Loadingoftheunitswillbefrequentlyadjustedtomaintainthematequal

incrementalcosts.4. Both(b)and(c).

6. Ifthefuelcostofoneunit,operatinginparallelwithotherunits,isincreasedanditisdesiredtomaintainaveragefuelcost,theloadontheunitwillbe:

1. Increased.2. Heldconstant.3. Decreased.4. Noneofthese.

7. Inapowersystemusingbothhydro-andthermal-generation,theproportionofhydro-generationcanbeincreasedby:

1. Increasingtheprice(gamma)ofwater.2. Reducingthepriceofwater.3. Increasingthefieldcurrentsofthehydro-generators.4. Noneofthese.

8. Economicoperationofpowersystemis:

1. Unitcommitment.2. Loadscheduling.3. Controllingofvoltageanditsmagnitude.4. Both(a)and(b).

9. Lagrangianmultipliermethodconvertsanon-linearconstrainedoptimizationprobleminto_____non-linearoptimizationproblem.

1. Gradient.2. Linear.3. Unconstrained.4. Allofthese.

10. Unitofheatratecurveis_____.

1. MillionkCal/hr.2. Rs.-hr.3. Rs./MWh.

4. Rs./hr.

11. Powerbalanceequationis_____constraint.

1. Equality.2. Inequality.3. Securityconstraints.4. Branchtransfercapacityconstraint.

12. Optimizationproblemswithonlyobjectivefunctionandwithoutconstraintsisa_____function.

1. Single-valued.2. Multi-valued.3. Both(a)and(b).4. Either(a)or(b).

13. Unitofλis_____.

1. Rs./hr.2. Rs./MW.3. Rs./MWh.4. MW/Rs.

14. Whichofthefollowinghasanegligibleeffectontheproductioncost?

1. Generationofrealpower.2. Realandreactive-powerdemands.3. Systemvoltageandangle.4. Generationofreactivepower.

1. (i)and(ii).2. Except(iii).3. (ii)and(iv).4. Allofthese.

15. Ananalyticalmethodofgettingthesolutiontooptimizationproblem,thefollowinggraphistobedrawn:

1. Totalreal-powerdemandversusλ.2. Totalreal-powergenerationversustotalreal-powerdemand.3. Totalreal-powergenerationversusλ.4. Totalreal-powergenerationversusfuelinput.

16. Thecontrolvariablesare:

1. P andQ

2. P andQ

3. Vandδ

4. Qandδ

17. P andQ are:

1. Controlvariables.2. Statevariables.3. Disturbancevariables.4. Constants.

18. P andQ are:

1. Disturbancevariables.2. Demandvariables.3. Uncontrollablevariables.

D D.

G G.

.

.

D D

D D

4. Allofthese.

19. Costcurvescanbeobtainedby:

1. Multiplythefuelinputwithcostoffuel.2. Subtractthefuelinputwithcostoffuel.3. Addthefuelinputwithcostoffuel.4. Noneofthese.

20. Costcurvesareexpressedas:

1. Rs./millioncal.2. Millioncal/hr×Rs./millioncal.3. Rs./hr.4. (b)and(c).

21. Thecurveobtainedbyconsideringthechangeincostofgenerationtochangeinreal-powergenerationatvariouspointsis:

1. Fuelcostcurve.2. Input–outputcurve.3. Incrementalcostcurve.4. Allofthese.

22. Incrementalfuelcost,I isgivenby:

1. Rs./MWh.2. Slopeofthefuelcostcurve.3. Tanβ=∆C/∆P

4. ∆i/p/∆o/p.

1. (i)and(ii).2. (ii)and(iii).3. Allexcept(iv).4. Allofthese.

23. Incrementalproductioncostofagivenunitismadeupof:

1. IC-incrementalcostoflabor,supplies,maintenance,etc.2. IC+incrementalcostoflabor,supplies,maintenance,etc.3. IC×incrementalcostoflabor,supplies,maintenance,etc.4. IC%incrementalcostoflabor,supplies,maintenance,etc.

24. Theoptimizationproblemis:

1. Toallocatetotalloaddemandamongvariousunitssuchthatthecostofgenerationismaintainedconstant.

2. Toallocatetotalloaddemandamongvariousunitssuchthatthecostofgenerationisminimized.

3. Toallocatetotalloaddemandamongvariousunitssuchthatthecostofgenerationisenormouslyincreased.

4. Toallocatetotalloaddemandamongvariousunitssuchthatthereisnoeffectwithcostofgeneration.

25. Themethodadoptedtogetanoptimalsolutiontooptimalschedulingproblemdependson:

1. ThemathematicalequationrepresentingI .

2. No.ofunits.3. Needtorepresentanydiscontinuityinincrementalcostcurve.4. Changeinlocation.

1. Only(i).2. Only(ii).

C

G.

C

3. Allexpect(iv).4. Allexpect(ii).

26. Inadigitalcomputermethodofgettingthesolutiontoanoptimizationproblem,

1. Thenumberoftermsincludedinexpressionfor dependsonthedegree

ofaccuracy.2. α,β,γcoefficientsaretobetaken

asoutput.3. α,β,γcoefficientsaretobetakenasinput.

1. Both(i)and(ii).2. Both(i)and(iii).3. Only(i).4. Only(iii).

27. Ifthereal-powerinequalityconstraintsareviolatedforanygenerator,then:

1. Itistiedtothecorrespondinglimitandtherestoftheloadiseconomicallydistributedamongtheremainingunits.

2. Itistiedtothecorrespondinglimitandthetotalloadiseconomicallydistributedamongalltheunits.

3. Itisnotconsideredandthetotalloadiseconomicallydistributedamongalltheunits.

4. Anyoftheabovemethods.

28. Themethodofgettingthesolutiontoanoptimizationproblemwithneglectedtransmissionlosses:

1. Doessensethechangesintheloads.2. Doesnotsensethelocationofthechangesintheload.3. Doessensethechangesintheloadandthelocationofchangesintheloads.4. Doesnotsenseboththelocationandthechangesintheload.

1. (i)and(ii).2. Either(iii)or(iv).3. Only(iv).4. Only(iii).

29. Togetanoptimalsolutiontoanoptimizationproblem,wewilldefineanobjectivefunctionas:

1.

2.

3.

4.

30. Theconditionforoptimalityis:

1. .

2. .

3. .4. (d)Both(a)and(b).

31. Whichofthefollowingistherealindicatorofthestateofdevelopmentofacountry?

1. Population.2. Facilities.3. Politics.4. Percapitaconsumptionofelectricity.

32. Equalityandinequalityconstraintsare

1.

2.

3.

4. (d)Noneoftheabove.

33. Inamathematicaldetermination,theoptimizationproblemshouldbemodifiedas:

1. Constrainedoptimizationproblem.2. Normalizedoptimizationproblem.3. Conditionaloptimizationproblem.4. Alltheabove.

REVIEWQUESTIONS

1. Explaintheimportantcharacteristicsofasteamunit.2. Describetheneedofeconomicdispatch.3. Explainwhytheproductioncostofelectricalenergyistreatedas

afunctionofreal-powergeneration.4. Obtaintheconditionforoptimumoperationofapowersystem

with‘n’plants.5. Bringoutthedifferencebetweenoptimaloperationofgenerators

inthermalstationsandoptimalschedulingofhydro-thermalsystems.

6. Explainhowtheincrementalproductioncostofathermalpowerstationcanbedetermined.

7. Explainthevariousfactorstobeconsideredinallocatinggenerationtodifferentpowerstationsforoptimumoperation.

8. Explainthesignificanceofequalityandinequalityconstraintsintheeconomicallocationofgenerationamongdifferentplantsinasystem.

PROBLEMS

1. Threepowerplantsofatotalcapacityof425MWarescheduledforoperationtosupplyatotalsystemloadof300MW.Findtheoptimumloadschedulingiftheplantshavethefollowingincrementalcostcharacteristicsandthegeneratorconstraints.

2. Aplantconsistsoftwounits.Theincrementalfuelcharacteristicsforthetwounitsaregivenas:

Findtheoptimalloadsharingoftwounitswhenatotalloadof300MWisconnectedtothesystem.AlsocalculatetheextracostincreasedinRs./hrifthetotalloadissharedequallybetweenthem.

3. Thecostcurvesofthethreeplantsaregivenasfollows:

C =0.04P +20P +230Rs./hr

C =0.06P +18P +200Rs./hr

C =0.05P +15P +180Rs./hr

Determinetheoptimumsharingofatotalloadof180MWforwhicheachplantwouldtakeupforminimuminputcostofreceivedpowerinRs/MWh.

4. TheincrementalfuelcostsinrupeesperMWhforaplantconsistingoftwounitsare:

1 G1 G1

2 G2 G2

3 G3 G3

2

2

2

CalculatetheextracostincreasedinRs./hr,ifaloadof210MWisscheduledasP =P =105MW.

G1 G2

3

EconomicLoadDispatch-II

OBJECTIVES


developthemathematicalmodelforeconomicalloaddispatchwhenlossesareconsideredderivetransmissionlossexpressionstudytheoptimalallocationoftotalloadamongtheunitsdevelopaflowchartforthesolutionofoptimizationproblem

3.1INTRODUCTION

Incaseofanurbanareawheretheloaddensityisveryhighandthetransmissiondistancesareverysmall,thetransmissionlosscouldbeneglectedandtheoptimumstrategyofgenerationcouldbebasedontheequalincrementalproductioncost.Iftheenergyistobetransportedoverrelativelylargerdistanceswithlowloaddensity,thetransmissionlosses,insomecases,mayamounttoabout20–30%ofthetotalload;hence,itisessentialtotaketheselossesintoaccountwhenformulatinganeconomicloaddispatchproblem.

3.2OPTIMALGENERATIONSCHEDULINGPROBLEM:CONSIDERATIONOFTRANSMISSIONLOSSES

Inapracticalsystem,alargeamountofpowerisbeingtransmittedthroughthetransmissionnetwork,whichcausespowerlossesinthenetwork(P )asshowninFig.3.1.

Infindinganoptimalsolutionforeconomicschedulingproblem(allocationoftotalloadamongthegeneratingunits),itismorerealistictoconsiderthetransmission

L

linelosses,whichareabout5–15%ofthetotalgeneration.

Ingeneral,theconditionforoptimality,whenlossesareconsidered,isdifferent.Equalincrementalfuelcosts(IFCs)forallgeneratingunitswillnotgiveanoptimalsolution.

3.2.1Mathematicalmodeling

Considertheobjectivefunction:

MinimizeEquation(3.1)subjecttothefollowingequalityandinequalityconstraints:

FIG.3.1Transmissionnetwork

(i)Equalityconstraint

Thereal-powerbalanceequation,i.e.,totalreal-powergenerationsminusthetotallossesshouldbeequaltothereal-powerdemand:

i.e.,

(or)

whereP isthetotaltransmissionlosses(MW),P thetotalreal-powerdemand(MW),andp thereal-powergenerationatthei unit(MW).

(ii)Inequalityconstraints

Alwaystherewillbeupperandlowerlimitsforrealandreactive-powergenerationateachofthestations.Theinequalityconstraintsarerepresented:

1. Intermsofreal-powergenerationas

P ≤P ≤P (3.3)

2. Intermsofreactive-powergenerationas

Q ≤Q ≤Q (3.4)

Thereactive-powerconstraintsaretobeconsideredsincethetransmissionlineresultsinlossisafunctionofrealandreactive-powergenerationsandalsothevoltageatthestationbus.

3. Inaddition,thevoltageateachofthestationsshouldbemaintainedwithincertainlimits:

i.e.,V ≤V ≤V (3.5)

TheoptimalsolutionshouldbeobtainedbyminimizingthecostfunctionsatisfyingconstraintEquations(3.2)–(3.5).

3.3TRANSMISSIONLOSSEXPRESSIONINTERMSOFREAL-POWERGENERATION–DERIVATION

TransmissionlossP isexpressedwithoutlossofaccuracyasafunctionofreal-powergenerations.ThepowerlossisexpressedusingB-coefficientsorlosscoefficients.

L D

Gi

Gi (min) Gi Gi (max)

Gi (min) Gi Gi (max)

i (min) i i (max)

L

th

TheexpressionfortransmissionpowerlossisderivedusingKron’smethodofreducingasystemtoanequivalentsystemwithasinglehypotheticalload.

Theexpressionisbasedonseveralassumptionsasfollows:

1. Allthelinesinthesystemhavethesame ratio.

2. Alltheloadcurrentshavethesamephaseangle.3. Alltheloadcurrentsmaintainaconstantratiotothetotalcurrent.4. Themagnitudeandphaseangleofbusvoltagesateachstationremain

constant.5. Powerfactorateachstationbusremainsconstant.

Wewillderiveanexpressionforthepowerlossofasystem,havingtwogeneratingstations,supplyinganarbitrarynumberofloadsthroughatransmissionnetworkasshowninFig.3.2(a).

Todeterminethecurrentinanyline,sayk line,applythesuperpositionprincipleanddeterminethecurrentpassingthroughtheline,I .

Thecurrentdistributionfactorofatransmissionlinew.r.t.apowersourceistheratioofthecurrentitwouldcarrytothecurrentthatthesourcewouldcarrywhenallothersourcesarerenderedinactive,i.e.,sourcesthatarenotsupplyinganycurrent.

Letusassumethattheentireloadcurrentissuppliedbygeneratingstation-1onlyasshowninFig.3.2(b).

Currentinthek line=I

Currentdistributionfactor,

IfweassumethattheentireloadissuppliedbythesecondgeneratingstationonlyasshowninFig.3.2(c):

Thecurrentflowingthroughthek line=I

k

k1

k1

th

th

th

Currentdistributionfactor,

FIG.3.2(a)Transmissionnetworkwithtwogeneratingstations;(b)loadsuppliedbygeneratingstation-1only;(c)loadsuppliedbygenerating

station-2only;(d)loadsuppliedbytwogeneratingstationssimultaneously;(e)sourcecurrentswithrespecttoreference;(f)currentink line

Becauseofassumptions(i)and(ii),thecurrentdistributionfactorswillberealnumbersratherthancomplexnumbers.

AndalsoassumingthatthetotalloadisbeingsuppliedbyboththestationsasshowninFig.3.2(d):

Thecurrentinthek line=I

∴I =I′ +I′

Fromtherelations,

Althoughthecurrentdistributionfactorsarerealnumbers,thevarioussourcecurrentssupplyingtotalloadwillnotbeinphase,i.e.,I andI arenotinphase.

LetthesourcecurrentsI andI beexpressedasI ∠σandI ∠σ asshowninFig.3.2(e).

Then,I =N I ∠σ +N I ∠σ withaphasedifferenceof(σ –σ )asshowninFig.3.2(f).

ByaddingI andI phasors,wehave

k

k k1 k2

1 2

1 2 1 1

2 2

k k1 1 1 k2 2 2

2 1

k1 k2

th

th

2 2 2

∣I ∣ =(N I ) +(N I ) +2(N I )(N I )cos(σ −σ )(3.7)

Thestationcurrentsarerelatedas

and

Thepowerlossinthek linecanbecalculatedas3|I | R

i.e.,

powerloss

Ifthereare‘l’numberoflinesinthesystem,totalpowerlossinthesystemcanbecalculatedas

i.e.,

k k1 1 k2 2 k1 1 k2 2 2 1

k k

2 2 2

th

2

Thisaboveexpressioncanbewrittenas

P =B P +B P +2B P P (3.8)

where

and

Equation(3.8)expressesthetotallossasafunctionofreal-powergenerations,P andP .

ThecoefficientsB ,B ,andB arecalledlosscoefficients(or)B-coefficientsandtheunitis(MW)andisalsoconsideredtobeaconstantinviewoftheassumptionsmade.

Thesameprocedurecanbeextendedforsystemshavingmorenumberofstations.Ifthesystemhas‘n’numberofstations,supplyingthetotalloadthroughtransmissionlines,thetransmissionlinelossisgivenby

whenn=2,

L 11 G1 22 G2 12 G1 G2

G1 G2

11 22 12

2 2

–1

Similarlyforn=3,

Sincethetransmissionlinesaresymmetrical,losscoefficientsB andB areequal,i.e.,B =B .

TheB coefficientsarelosscoefficientsandcanberepresentedinmatrixformofann-generatorsystemas

Thediagonalelementsofthesecoefficientsareallpositiveandstrong(sincegeneratingstationsareinterconnected)ascomparedwiththeoff-diagonalelements,whichmostlyarenegativeandarerelativelyweaker.

Thesecoefficientsaredeterminedforalargesystembyanelaboratedigitalcomputerprogramstartingfromtheassemblyoftheopen-circuitimpedancematrixofthetransmissionline,whichisquitelengthyandtime-consuming.Besides,theformulationsofB-coefficientsarebasedonseveralassumptionsanddonottakeintoaccounttheactualconditionsofthesystem;thesolution

pq qp pq qp

pq

fortheplantgenerationscannotbeexpectedtobethebestforminimumcostofgeneration.

3.4MATHEMATICALDETERMINATIONOFOPTIMUMALLOCATIONOFTOTALLOADWHENTRANSMISSIONLOSSESARETAKENINTO

CONSIDERATION

Considerapowerstationhaving‘n’numberofunits.Letusassumethateachunitdoesnotviolatetheinequalityconstraintsandletthetransmissionlossesbeconsidered.

Assumingthattheinequalityconstraintissatisfied,theobjectivefunctionisredefinedbyaugmentingEquation(3.1)withequalityconstraint(Equation(3.2))usingLagrangianmultiplier(λ)andisgivenby

Thisaugmentedobjectivefunctioniscalledconstrainedobjectivefunction.

Intheaboveobjectivefunction,thereal-powergenerationsarethecontrolvariablesandthecondition

foroptimalitybecomes

i.e.,

(or)

(or)

where representsthevariationoftotaltransmission

losswithrespecttoreal-powergenerationoftheistationandiscalledincrementaltransmissionloss(ITL)ofthei station.Equation(3.14)canbewrittenas

or

where andiscalledthepenaltyfactorof

thei station.Equation(3.15)canbeutilizedtoobtaintheoptimalcostofoperation.

TheconditionforoptimalitywhenthetransmissionlossesareconsideredisthattheIFCofeachplantmultipliedbyitspenaltyfactormustbethesameforalltheplants:

th

th

th

i.e.,

Equation(3.12)isasetofnequationswith(n+1)unknowns.Here,thepowersofngeneratorsareunknownandλisalsounknown.Theseequationsareknownasexactco-ordinationequationsbecausetheyco-ordinatetheITLwithIFC.

Thefollowingpointsshouldbekeptinmindforthesolutionofeconomicloaddispatchproblemswhentransmissionlossesareincludedandco-ordinated:

Althoughincrementalproductioncostofaplantisalwayspositive,ITLcaneitherbepositiveornegative.Theindividualunitswilloperateatdifferentincrementalproductioncosts.ThegenerationwiththehighestpositiveITLwilloperateatthelowestincrementalproductioncost.

ForasmallincreaseinreceivedloadbyΔP ,thei plantgenerationisonlychangedby∆P andthegenerations

oftheremainingunitsareunaffected.LetΔP bethechangeintransmissionloss,thepowerbalanceequationbecomes∆P −∆P =∆P .

Thus,

when istheincrementalcostofthereceived

powerofthei plantandthepenaltyfactor .This

D

G1

L

Gi L D

th

th

meansthatas∆P incrementhasalargerproportion

dissipatedasloss, approachesunityandthepenalty

factor‘L ’increaseswithoutbound.Thus,foralargerpenaltyfactor‘L ’,unit‘i’shouldbeoperatedatlowincrementalcostimplyingalowpoweroutput.

3.4.1DeterminationofITLformula

Whenasystemconsistsofthreegeneratingunits,i.e.,n=3,thetransmissionlossis

ITLofGenerator-1isobtainedas

Ingeneral,

WeknowthattheIFCofthei unitis

SubstituteEquations(3.17)and(3.18)inEquation(3.9);weget

Gi

i

i

th

Dividingtheaboveequationbyλ,weget

Tosolvethisallocationproblem,solvetheco-ordinationEquation(3.19)foraparticularvalueofλiterativelystartingwithaninitialsetofvaluesofP

(suchasallP settominimumvalues)andgetthe

solutionwithinaspecifiedtolerancetillallP ’s

converge,thencheckforpowerbalanceandifitistobesatisfied,thenitistheoptimalsolution.Ifthepowerbalanceequationisnotsatisfied,modifythevalueofλtoasuitablevalueandsolvetheco-ordinationequation.

3.4.2Penaltyfactor

ConsiderEquation(3.12):

Gi

Gi

Gi

Theaboveexpressioncanbewrittenas

or

where

iscalledthepenaltyfactorofthei station

Thepenaltyfactorofanyunitisdefinedastheratioofasmallchangeinpoweratthatunittothesmallchange

th

inreceivedpowerwhenonlythatunitsuppliesthissmallchangeinreceivedpower.

3.5FLOWCHARTFORTHESOLUTIONOFANOPTIMIZATIONPROBLEMWHENTRANSMISSIONLOSSESARECONSIDERED

Whentransmissionlossesaretakenintoaccount,thesolutionofanoptimizationproblemisrepresentedbythefollowingflowchart(Fig.3.3).

Example3.1:ThefuelcostfunctionsinRs./hrfortwothermalplantsaregivenby

C =420+9.2P +0.004P

C =350+8.5P +0.0029P

whereP ,P areinMW.Determinetheoptimalschedulingofgenerationifthetotalloadis640.82MW.Estimatevalueofλ=12Rs./MWh.Thetransmissionpowerlossisgivenbytheexpression

P =0.0346P +0.00643P

Solution:

1 1 2

2 2 2

1 2

L(pu) 1(pu) 2(pu)

2

2

2 2

FIG.3.3Flowchart

FINALOUTPUTOFMATLABPROGRAMdispatch3.m

lambda=12.1034


1.0000 177.2999

2.0000 489.8232

INCREMENTALFUELCOSTSANDPENALTYFACTORSARE:

UNITNO. IFC L

1.0000 10.6184 1.1398

2.0000 11.3410 1.0672

CHECKLAMBDA=IFC*L

UNITNO. LAMBDA

1.0000 12.1034

2.0000 12.1034

TOTALGENERATIONCOST(Rs./hr)=7386.20

Example3.2:ThefuelcostfunctionsinRs./hrfortwothermalplantsaregivenby

C =420+9.2P +0.004P ,100≤p ≤200

C =350+8.5P +0.0029P ,150≤P ≤500

whereP ,P ,P areinMWandplantoutputsaresubjectedtothefollowinglimits.Determinetheoptimalschedulingofgenerationifthetotalloadis640.82MW.Estimatethevalueofλ=12Rs./MWh.

P =0.0346P +0.00643P

Solution:

1 1 1 2

2 2 1

1 2 3

L(pu) 1(pu) 2(pu)

2

2 3

2 2

Results:

GENERATIONISWITHINTHELIMITS

FINALOUTPUTOFMATLABPROGRAMdispatch3.m

lambda=12.1034


1.0000 177.3001

2.0000 489.8236

INCREMENTALFUELCOSTSANDPENALTYFACTORSARE:

UNITNO. IFC L

1.0000 10.6184 1.1399

2.0000 11.3410 1.0672

CHECKLAMBDA=IFC*L

UNITNO. LAMBDA

1.0000 12.1034

2.0000 12.1034

TOTALGENERATIONCOST(Rs./hr)=7386.20

Example3.3TheIFCfortwoplantsare

Thelosscoefficientsaregivenas

B =0.0015/MW,B =−0.0004/MW,andB =0.0032/MWforλ=25Rs./MWh.Findthereal-powergenerations,totalloaddemand,andthetransmissionpowerloss.

Solution:

P +P =P +P

Andtransmissionloss,

Fornumberofplants,n=2,wehave

TheITLofPlant-1is

11 12 22

G1 G2 D L

PenaltyfactorofPlant-1:

TheITLofPlant-2is

andpenaltyfactorofplant-2is

Conditionforoptimumoperationis

or0.15P −0.02P =7(3.21)

and

G1 G2

or0.02P −0.24P =−9(3.22)

SolvingEquations(3.21)and(3.22),

Equation(3.21)×0.24⇒0.036P −0.0048P =1.68

Equation(3.22)×0.02⇒0.0004P −0.0048P =

−0.18

0.0356P =1.86

∴P =52.247MW


0.15(52.247)−0.02P =7

∴P =41.852MW

Transmissionloss,P

= 0.0015(52.247) −0.0008(52.247)(41.852)+0.0032(41.852)

= 7.95MW

Totalload,P

= P +P −P

= 52.247+41.852−7.95=86.149MW

Example3.4:Thecostcurvesoftwoplantsare

G1 G2

G1 G2

G1 G2

G1

G1

G1

G1

G2

L

D

G1 G2 L

2

2

C =0.05P +20P +150Rs./hr

C =(0.05P )+15P +180Rs./hr

ThelosscoefficientfortheabovesystemisgivenasB =0.0015/MW,B =B =–0.0004/MW,andB =0.0032/MW.Determinetheeconomicalgenerationschedulingcorrespondingtoλ=25Rs./MWhandthecorrespondingsystemloadthatcanbemetwith.Ifthetotalloadconnectedtothesystemis120MWtaking4%changeinthevalueofλ,whatshouldbethevalueofλinthenextiteration?

Solution:

Giventhatthecostcurvesoftwoplantsare

C =0.05P +20P +150Rs./hr

C =(0.05P )+15P +180Rs./hr

theincrementalcostsare

Transmissionloss,

Fortwoplants,n=2andwehave

1 G1 G1

2 G2 G1

11

12 21 22

1 G1 G1

2 G2 G1

2

2

2

2

TheITLofPlant-1is

TheITLofPlant-2is

ThepenaltyfactorofPlant-1is

andthepenaltyfactorofPlant-2is

Theconditionforoptimumoperationis

or1.09P −0.024P =10(3.23)

and

or0.024P −0.292P =−15(3.24)


Equation(3.23)×0.024⇒0.02616P −0.000576P =

0.24

Equation(3.24)×1.09⇒0.02616P −0.31828P =

−16.36

0.3177P =16.6

∴P =52.25MW


1.09P −0.024(52.25)=10

∴P =10.325MW

Transmissionloss,P = 0.0015(10.325) −0.0008(10.325)(52.25)+0.0032(52.25)

G1 G2

G1 G2

G1 G2

G1 G2

G1

G2

G1

G1

G1

L

2

2

= 8.465MW

Thecorrespondingsystemload,P

= P +P −P

= 10.325+52.25−8.465=54.11MW

For4%changeinvalueofλ,Δλ=4%of30=1.2Rs./MWh

Newloadconnectedtosystem,P =120MW

Changeinload,ΔP =120–54.11=65.89MW

Here,changeinload,ΔP >0;hence,togetanoptimumdispatchdecrementλbyΔλ,

Newvalueofλ=λ′=λ–Δλ=30–1.2=28.8Rs./MWh.

Example3.5:Asystemconsistsoftwopowerplantsconnectedbyatransmissionline.ThetotalloadlocatedatPlant-2isasshowninFig.3.4.Dataofevaluatinglosscoefficientsconsistofinformationthatapowertransferof100MWfromStation-1toStation-2resultsinatotallossof8MW.FindtherequiredgenerationateachstationandpowerreceivedbytheloadwhenλofthesysteminRs.100/MWh.TheIFCsofthetwoplantsaregivenby

Solution:

Totallossis

D

G1 G2 L

D

D

D

FIG.3.4IllustrationforExample3.5

Sincen=2,wehave

Sincepowertransferof100MWfromPlant-1toPlant-2(i.e.,P =100MW),P ,P ,B =0

∴P =B P

Given:P =8MW

∴8=B (100)

⇒B =8×10 MW

∴P =8×10 P

andthepenaltyfactorofPlant-1is

G1 G2 21 22

L 11 G1

L

11

11

L G1

2

2

−4 −1

−4 2

AndthepenaltyfactorofPlant-2is

Now,theconditionforoptimalityis

Forλ=100Rs./MWh

or0.12P +65=100(1−16×10 P )

or0.12P +0.16P =100−65

0.25P =35

amd0.25P +75=100

Powerreceivedbytheload = (P +P )−losses

= 125+100−8×10 ×P

G1 G1

G1 G1

G1

G2

G1 G2

G1

−4

−4 2

−4 2

= 125+100−8×10 ×125

= 225−12.5

= 212.5MW

Example3.6:ForExample.3.5,with212.5MWreceivedbytheload,findthesavingsinRs./hrobtainedbyco-ordinatingthetransmissionlossesratherthanneglectingindeterminingtheloaddivisionbetweentheplants.

Solution:

Byco-ordinatingthelossestosupplyaloadof212.5MW,thereal-powergenerationsatPlants1and2shouldbe125and100MW,respectively.

Whenlossesareneglected,totalload=212.5MWistobedistributedbetweenthetwoplantsmosteconomically.Conditionforoptimality,

Sincethelossesarenotco-ordinatedbutneglected,wehave

212.5P +0.49P −40−8×10 PG1 G1 G1

−4 2

−4 2

or8×10 P −1.48P 252.5=0

Bysolving,weget

P = 1,659.8MWand190.15MW

P = 1,659.8MW⇒isnottoberequiredtoovercomethatpowerdemandP

∴P = 190.15istoberequired

andP = 0.48×190.15−40

= 51.27MW

∴Powergeneration:withlossesarebeingco-ordinated,P =125MW,P =100MWwithlossesare

notbeingco-ordinated,P =190.15MW,P =51.27

MW.

IncreaseincostofPlant-1whenlossesareco-ordinated:

IncreaseincostofPlant-2,becauseincreaseingeneration:

G1 G1

G1

G1D

G1

G2

G1 G2

G1 G2

−4 2

SavingsinRs./hrbyco-ordinatingthelosses=5,466.67–4,576.17=890.50Rs./hr.

Example3.7:Onasystemconsistingoftwogeneratingplants,theincrementalcostsinRs./MWhwithP and

P inMWare

ThesystemisoperatingoneconomicdispatchwithP

=P =500MWamd Findthepenaltyfactorof

Plant-1.

Solution:

Giventhatthesystemoperatesoneconomicdispatchwithp =p =500MW,theconditionforthisoptimal

operationwhenconsideringthetransmissionlossis

andalsogiventhatITLofPlant-2,

G1

G2

G1

G2

G1 G2

ThepenaltyfactorofPlant-2,

∴Foroptimalcondition,

or

[(0.008×500)+80]L =(0.012×500+9.0)1.25

or

12L =18.75

or

L =1.5625

∴PenaltyfactorofPlant-1=1.5625.

Example3.8:DeterminetheincrementalcostofreceivedpowerandthepenaltyfactoroftheplantshowninFig.3.5iftheincrementalcostofproductionis

Solution:

Thepenaltyfactor

1

1

1

∴Costofreceivedpower


Example3.9:A2-bussystemconsistsoftwopowerplantsconnectedbyatransmissionlineasshowninFig.3.6.

Thecost-curvecharacteristicsofthetwoplantsare:

C =0.015P +18P +20Rs./hr

C =0.03P +33P +40Rs./hr

Whenapowerof120MWistransmittedfromPlant-1totheload,alossof16.425MWisincurred.DeterminetheoptimalschedulingofplantsandtheloaddemandifthecostofreceivedpowerisRs.26/MWh.Solvetheproblemusingco-ordinationequationsandthepenaltyfactormethodapproach.

Solution:

1 G1 G1

1 G2 G2

2

2

Fortwounits,P =p B p +2p B p +p B p

.

SincetheloadislocatedatBus-2alone,thelossesinthetransmissionlinewillnotbeaffectedbythegeneratorofPlant-2.

i.e.,B =B =0andB =0

∴P =B P (3.25)

16.425=B ×120

B =0.00114MW

Usingtheco-ordinationequationmethod:

Theco-ordinationequationforPlant-1is


PL=0.00114P

SubstituteEquations(3.27)and(3.28)inEquation(3.26);thentheequationforPlant-1becomes

L G1 11 G1 G1 12 G2 G2 21 G1

12 21 22

L 11 G1

11

11

G1

2

2

−1

2

Theco-ordinationequationforPlant-2is

∴Equation(3.29)becomes

∴Thetransmissionloss,P = B P

= 0.00114×(89.6)

= 9.15MW

∴Theload,P = P +P −P

= 89.6+66.67−9.15=147.12MW

L 11 G1

D G1 G2 L

2

2

Penaltyfactormethod:


Nowtheconditionforoptimalityis


Foroptimality,

Thetransmissionloss,P = B P

= 0.00114×(89.6)

= 9.15MW

L 11 G1

2

2

∴Theload,P = P +P −P

= 89.6+66.67−9.15=147.12MW.

Example3.10:AssumethatthefuelinputinBritishthermalunit(Btu)perhourforUnits1and2aregivenby

C =(8P +0.024P +80)10

C =(6P +0.024P +120)10

Themaximumandminimumloadsontheunitsare100and10MW,respectively.DeterminetheminimumcostofgenerationwhenthefollowingloadissuppliedasshowninFig.3.7.ThecostoffuelisRs.2permillionBtu.


Solution:

D G1 G2 L

1 G1 G1

2 G1 G2

2 6

2 6

1. Whentheloadis50MW:

Conditionfortheeconomicschedulingis

0.048P +8=0.08P +6

0.048P −0.08P =−2(3.30)

andP +P =50(3.31)


p =15625MW

p =34375MW

∴C =210.868millionBtu/h

C =373.5millionBtu/h

2. Whentheloadis150MW,

FromEquation(3.30),0.048P −0.08P =−2(3.32)

P +P 150(3.33)


p =78126MW

p =71874MW

andC =851.496millionBtu/hr

C =757.87millionBtu/hr

∴Totalcost=Rs.(210.868+373.5+851.496+757.87)×2

=Rs.52,649.61/hr.

G1 G2

G1 G2

G1 G2

G1

G2

1

2

G1 G2

G1 G2

G1

G2

1

2

Example3.11:TwopowerplantsareconnectedtogetherbyatransmissionlineandloadatPlant-2asshowninFig.3.8.When100MWistransmittedfromPlant-1,thetransmissionlossis10MW.Thecostcharacteristicsoftwoplantsare

C =0.05P +13P Rs./hr

C =0.06P +12P Rs./hr

Findtheoptimumgenerationforλ=22,λ=25,andλ=30.

Solution:

C =0.05P +13P

C =0.06P +12P

TheIFCcharacteristicsare

Thetransmissionpowerloss,

Heren=2,

P =B P +B P +2B P P

SincetheloadisconnectedatBus-2andthepowertransferisfromPlant-1only,B =0andB =0.

∴P =B P

10=B (100)

1 G1 G1

2 G2 G2

1 G1 G1

2 G2 G2

L 11 G1 22 G2 12 G1 G2

22 12

L 11 G1

11

2

2

2

2

2 2

2

2

FIG.3.8Singlelinediagramrepresentingtwopowerplantsconnectedbyatransmissionline

TheITLofPlant-1is


(∵thetransmissionpowerlossisnotthe

functionofP )


Foroptimality,whentransmissionlossesareconsidered,theconditionis

G2

or0.0144P =9

∴P =62.5MW,

and(0.12P +12)1=22

∴P =83.33MW

Similarly,wehave

Forλ=25,

p =80MW,p =108.33MW

Forλ=30,

p =106.25MW,p =150MW

Example3.12:ForExample3.11,thedataforthelossequationsconsistoftheinformationthat200MWtransmittedfromPlant-1totheloadresultsinatransmissionlossof20MW.Findtheoptimumgenerationscheduleconsideringtransmissionlossestosupplyaloadof204.41MW.Alsoevaluatetheamountoffinanciallossthatmaybeincurredifatthetimeofschedulingtransmissionlossesarenotco-ordinated.AssumethattheIFCcharacteristicsofplantsaregivenby

Solution:

G1

G1

G2

G2

G1 G2

G1 G2

2 2

P =B P +2B P P +B P

∴TheloadisatPlant-2,henceB =B =0

∴P =B P

AndgiventhatP =200MW,P =20M



Foroptimality,

⇒0.025P =(1−0.001P )(0.05P +16)

⇒0.025P =(0.05P −0.00005P P +16−0.016P )

0.041P −0.05P +0.00005P P =16(3.34)

L 11 G1 12 G1 G2 22 G2

2 22

L 11 G1

G1 L

G1 G1 G2

G1 G2 G1 G2 G1

G1 G2 G1 G2

2 2

2

2

andP +P =P +P

P +P =204.41+0.0005P

⇒P +P −0.0005P =204.41(3.35)


p =133.3MWandp =80MW

Ifthetransmissionlossesarenotco-ordinated,wehave

Whileinthesystem,apowerbalanceequationalwaysholdsgood.

P +P −P −P =0

P +P −204.41−0.0005P (3.37)


p =172.91MW

andp =46.45MWandP =00005p =1495MW

Fromtheresults,itisclearthatifthetransmissionlossesareco-ordinated,theloadonPlant-1isincreasedfrom133.3to172.91MW.

IncreaseinfuelcostofPlant-1is

G1 G2 D L

G1 G2 G1

G1 G2 G1

G1 G2

G1 G2 D L

G1 G2 G1

G1

G2 L G1

2

2

2

2

TheloadonPlant-2isdecreasedfrom80to46.45MW.ThedecreaseinthefuelcostofPlant-2is

Thenetfinancialloss = ∆C −∆C =706.15−642.70

= 63.45Rs./hr

Example3.13:ForthesystemshowninFig.3.9,withBus-1asthereferencebuswithavoltageof1.0∟0°p.u.,findthelossformula(B )coefficientsifthebranchcurrentsandimpedancesare:

I =(1.00–j0.15)p.u.;Z =0.02+j0.15p.u.

I =(0.50–j0.05)p.u.;Z =0.03+j0.15p.u.

I =(0.20–j0.05)p.u.;Z =0.02+j0.25p.u.

Ifthebaseis100MVA,whatwillbethemagnitudesofB coefficientsinreciprocalMW?

Solution:

Theassumptionindevelopingtheexpressionfortransmissionlossisthatallloadcurrentsmaintainaconstantratiotothetotalcurrent:

i.e.,

1 2

pq

a a

b b

c c

pq

∴Thecurrentdistributionfactors:

whereI isthecurrentinbranch‘a’whenPlant-1isin

operation;andI thecurrentwhenPlant-2isin

operation.

N =0.6649;N =0.6649

N =0.3353;N =0.3359

N =–0.3353;N =0.6649


VoltageatBus-1=V =1.0∠0°p.u.=(1.0+j0.0)p.u.

ThebusvoltageatBus-2is

V =V +I Z

=(1.00+j0.0)+(0.20–j0.05)(0.02+j0.25)

=1.0165+j0.049=1.0176∠2.76Theplantcurrentare

I =

a1

a2

a1 a2

b1 b2

c1 c2

1

2 1 c c

G1

I −I =(1.00−j0.15)−(0.20−j0.05)=0.80−j0.10=

0.8062∠−7.11°

I = I +I =(0.50−j0.10)+(0.20−j0.05)

= (0.70−j15)=0.7519∠−12.09°

Theplantcurrentsareintheformof

I =I ∠σ

I =I ∠σ

∴σ =−7.1°andσ =−12.09°

cos(σ −σ )=cos4.99°=0.996

Theplantp.f.sare

cosφ =cos7.1°=0.9923

cosφ =cos(anglebetweenV andI )

=cos(2.76°+12.09°)=0.9666

or

a c

G2 b c

1 1 1

2 2 2

1 2

2 1

1

2 2 G2

or

or

Example3.14:ForExample3.13,findtheITLattheoperatingconditionsconsidered.

Solution:

Transmissionpowerloss,P =B P +B P +

2B P PL 11 G1 22 G2

12 G1 G2

2 2

TheITLofPlant-1is


Example3.15:ForthesystemshowninFig.3.10,withBus-3asthereferencebuswithavoltageof1.2∠0 p.u.,findthelossformula(B )coefficientsofthesystemin

pq

o

p.u.andinactualunits,ifthebranchcurrentsandimpedancesare

I =2.5–j1.0p.u.;Z =0.02+j0.08p.u.

I =1.8–j0.6p.u.;Z =0.03+j0.09p.u.

I =1.5–j0.5p.u.;Z =0.013+j0.05p.u.

I =3.0–j1.0p.u.;Z =0.015+j0.06p.u.

Considerthatthebasehas100MVA.

Solution:

Theassumptionindevelopingtheexpressionfortransmissionlossisthatallloadcurrentsmaintainaconstantratiotothetotalcurrent.

∴Currentdistributionfactorsare

N =1; N =0

N =0.575; N =0.425

N =−0.425; N =0.575

N =0.425; N =0.575

ThebusvoltageatreferenceBus‘3’=1.2+j0.0p.u.

ThebusvoltageatPlant-1, V = V +I Z

a a

b b

c c

d d

a1 a2

b1 b2

c1 c2

d1 d2

1ref a a

= 1.2+j0.0+(2.2−j1)(0.02+j0.08)

= 1.342∠7.7°p.u.

ThebusvoltageatPlant-2,

V = V +I Z

= 1.2+(1.5+j0.5)(0.013−j0.05)

= 1.246∠3.15°p.u.

Thecurrentphaseanglesattheplants=(I =I ,I =I +I )

Theplantpowerfactorsare

cosφ =cos(7.7 +21.8 )=0.87

cosφ =cos(3.5 +18.43 )=0.928

Thelosscoefficientsare

1

2ref c c

1 a 2 d

c

1

2

o o

o o

Forobtainingthelosscoefficientvaluesinreciprocalmegawatts,thelosscoefficientsinp.u.mustbedividedbythebasevalue(i.e.,100MVA):

Example3.16:Asystemconsistsoftwogeneratingplantswithfuelcostsof

C = 0.05P +20P +1.5

and C = 0.075P +22.5P +1.6

1 G1 G1

2 G2 G2

2

2

Thesystemoperatesoneconomicaldispatchwith100MWofpowergenerationbyeachplant.TheITLofPlant-2is0.2.FindthepenaltyfactorofPlant-1.

Solution:

Given

C = 0.05P +20P +1.5

C = 0.075P +22.5P +1.6

P = P =100MW

and


IncrementalfuelcostofPlant-1

IncrementalfuelcostofPlant-2

Foroptimality,theconditionis

1 G1 G1

2 G2 G2

G1 G2

2

2

⇒(0.1P +20)L = (0.15P +22.5)1.25

(0.1×100+20)L = (0.15×100+22.5)1.25

= (37.5)(1.25)

∴30L = 46.875

orL = 1.5625

i.e.,thepenaltyfactorofPlant-1=L =1.5625.

Example3.17:TwothermalplantsareinterconnectedandsupplypowertoaloadasshowninFig.3.11.

Thefollowingaretheincrementalproductioncostsoftheplants:

wherep andp areexpressedinp.u.in100-MVAbase.

Thetransmissionlossisgivenby

P =0.1P +0.2P +0.1P P p.u.

Iftheincrementalcostofreceivedpoweris50Rs./MWh,findtheoptimalgeneration.

G1 1 G2

1

1

1

1

G1 G2

L G1 G2 G1 G2

2 2

FIG.3.11IllustrationorExample3.17

Solution:

Given:

and P = 0.1P +0.2P +0.1P P p.u. (3.38)

Generally,

P = B P +B P +2B P P (3.39)

ComparingthecoefficientsofEquations(3.38)and(3.39),weget

Incrementalcostofreceivedpower,λ=50Rs./MWh

Theconditionforoptimumallocationoftotalloadwhentransmissionlossesareconsideredis

LG1 G2 G1 G2

L11 G1 22 G2 12 G1 G2

2 2

2 2

TheITLofPlant-1is

TheITLofPlant-2is



∴Foroptimumoperation:

or 20+10P =50−10P −5P

or 10P +10P +5P =30

20P +5P =30 (3.40)


∴P =0.95652p.u.

Substitutingthep valueinEquation(3.40),weget

20p +5(0.95652)=30

or20p +4.7826=30

G1 G1 G2

G1 G2 G2

G1 G2

G1

G2

G1

G1

∴p =1.26087p.u.

Substitutingthep andp valuesinEquation(3.38),we

have

P = 0.1(1.26087) +0.2(0.95652) +0.1(1.26087)(0.95652)

= 0.158979+0.182986+0.12060

= 0.46256p.u.

= 0.46256×100=46.256MW

P inMW=p.u.valur×baseMVA

P = 1.2608×100

= 126.08MW

and P = 95.652MW

Example3.18:Apowersystemoperatesaneconomicloaddispatchwithasystemλof60Rs./MWh.IfraisingtheoutputofPlant-2by100kW(whiletheotheroutputiskeptconstant)resultsinincreasedpowerlossesof12kWforthesystem,whatistheapproximateadditionalcostperhouriftheoutputofthisplantisincreasedby1MW?

Solution:

Foreconomicoperation:

G1

G1 G2

L

G1

G1

G2

2 2

IfthePlant-2outputisincreasedby1MW,i.e.,∂P =

1MW,theadditionalcost,∂C =?

Given:

λ=60Rs./MWh,∂P =100kW,and∂P =12kW:


Thefuelcostwhentheoutputisincreasedby1MWis

∂C =52.817×dP =52.817×1=52.817Rs./hr

Example3.19:Apowersystemissuppliedbyonlytwoplants,bothofwhichoperateoneconomicaldispatch.AtthebusofPlant-1,theincrementalcostis55Rs./MWhandatPlant-2is50Rs./MWh.Whichplanthasthehigherpenaltyfactor?WhatisthepenaltyfactorofPlant-1ifthecostperhourofincreasingtheloadonsystemby1MWis75Rs./hr?

Solution:

G2

2

G2 L

2 G2

Given

ThecostinRs./hrtoincreasethetotalsystemloadby1MWiscalledsystemλ:

λ=25Rs./MWh

or

∂P =1MWandRs./hr=75(given)

Foreconomicaloperation,bothplantsoperatingatcommonλ,i.e.,λ=75Rs./MWh

and

Therefore,L isgreaterthanL .

KEYNOTES

Whentheenergyistransportedoverrelativelylargerdistanceswithlowloaddensity,thetransmissionlossesinsomecasesmayamounttoabout20–30%ofthetotalload.Hence,itbecomesveryessentialtotaketheselossesintoaccountwhenformulatinganeconomicdispatchproblem.

G1

2 1

Considertheobjectivefunction:

Minimizetheabovefunctionsubjecttotheequalityandinequalityconstraints.

Equalityconstraints

Thereal-powerbalanceequation,i.e.,totalreal-powergenerationsminusthetotallossesshouldbeequaltoreal-powerdemand:

Inequalityconstraints

Theinequalityconstraintsarerepresentedas:

1. Intermsofreal-powergenerationas

P ≤P ≤P

2. Intermsofreactive-powergenerationas

Q ≤Q ≤Q

3. Inaddition,thevoltageateachofthestationsshouldbemaintainedwithincertainlimits.

i.e.,V ≤V ≤V

Currentdistributionfactorofatransmissionlinew.r.tapowersourceistheratioofthecurrentitwouldcarrytothecurrentthatthesourcewouldcarrywhenallothersourcesarerenderedinactivei.e.,thesourcesthatdonotsupplyanycurrent.Ifthesystemhas‘n’numberofstations,supplyingthetotalloadthroughtransmissionlines,thetransmissionlinelossisgivenby

ThecoefficientsB ,B andB arecalledlosscoefficientsorB-

coefficientsandareexpressedin(MW) .Thetransmissionlossisexpressedasafunctionofreal-power

G (min)i Gi G (max)i


i(min) i i(max)

11 12 22 −1

generations.

Theincrementaltransmissionlossisexpressedas .

Thepenaltyfactorofanyunitisdefinedastheratioofasmallchangeinpoweratthatunittothesmallchangeinreceivedpowerwhenonlythatunitsuppliesthissmallchangeinreceivedpowerandisexpressedas

Theconditionforoptimalitywhentransmissionlossesareconsideredis


1. Stateinwordstheconditionforminimumfuelcostinapowersystemwhenlossesareconsidered.

Theminimumfuelcostisobtainedwhentheincrementalfuelcostofeachstationmultipliedbyitspenaltyfactoristhesameforallthestationsinthepowersystem.

2. Definethecurrentdistributionfactor.

Thecurrentdistributionfactorofatransmissionlinewithrespecttoapowersourceistheratioofthecurrentitwouldcarrytothecurrentthatthesourcewouldcarrywhenallothersourcesarerenderedinactive,i.e.,thesourcesthatarenotsupplyinganycurrent.

3. Writetheexpressionforthetotaltransmissionlossintermsofreal-powergenerationswhenn=2.

Forn=2,

4. Inthestudyofanoptimumallocationproblem,whataretheconsiderationsthatyouwillnoticeregardingequalityandinequalityconstraintsinthecaseoftransmissionloss

considerationandwhyarereactive-powerconstraintstaken?

Equalityconstraints,

Inequalityconstraints,

P ≤P ≤P and

Q ≤Q ≤Q

V ≤V ≤V

Reactive-powerconstraintsaretobetakensincethetransmissionlossesarefunctionsofrealandreactive-powergenerationsandalsothevoltageateachbus.

5. Whataretheassumptionsconsideredinderivingthetransmissionlossexpression?

Thefollowingassumptionsaretobeconsideredforderivingthetransmissionlossexpression:

1. Alllinesinthesystemhavethesame ratio.

2. Alltheloadcurrentshavethesamephaseangle.3. Alltheloadcurrentsmaintainaconstantratiotothetotalcurrent.4. Themagnitudeandphaseangleofbusvoltagesateachstationremain

constant.

6. Writethetransmissionlossexpressionforthek line,iftherearetwogeneratingstationsintermsofstationvoltages,real-powergenerations,andtheirpowerfactors.

7. Asimpletwo-plantsystemhastheIC’sthatare

dC /dp =0.01p +2.0

dC /dp =0.01p +1.5andthetotalloadonthesystemis

distributedoptimallybetweentwostationsasp =60MVand

p =110MW,correspondingtoλ=2.6andthelosscoefficients

ofthesystemaregivenas

p q B



(min)i i (max)i

1 G1 G1

2 G2 G2

G1

G2

pq

th

1 1 0.0015

1 2 –0.0015

2 2 0.0025

Determinethetransmissionloss.

Transmissionloss

= B P +2B P P +B P

= (0.0015)(60) +2(−0.0015)

×(60×110)+(0.0025)

×(110) =25.75MW

8. Whatisyouranalysisbyconsideringtheoptimizationproblemwithandwithouttransmissionlossconsideration?

Togetthesolutiontooptimizationproblem,i.e.,toallocatethetotalloadamongvariousunits:

Whentransmissionlossesareneglected,theconditionis

i.e.,theICofalltheunitsmustbethesame.

Whentransmissionlossesareconsidered,theconditionis

i.e.,theproductofICofanyunitanditspenaltyfactorgivestheoptimumsolution.

9. FindthepenaltyfactoroftheplantshowninFig.3.12.

= 59MW

11 G1 12 G1 G2 22 G2

2 2

2

2

Here,

p

P = 19MW

FIG.3.12IllustrationforQuestionnumber9

Penaltyfactor,

10. WritetheexpressionfortransmissionlossintermsofB

coefficientswhentherearethreegeneratingstations.

11. Writetheconditionforoptimalitywhenlossesaretakenintoconsideration.

i.e.,

G1

D

min

12. FindthepenaltyfactorsofboththeplantsshowninFig.3.13.

Given:p =125MW

p =75MW,B =0.0015

SincetheloadisatStation-2,thetransferofpowertotheloadisfromonlyStation-1andhenceB =B =B =0

P =B P +2B P P +B P

∴P =B P


=(0.0015)(125)2

=0.0015×P

PenaltyfactorofStation-1,

PenaltyfactorofStation-2,

G1

G2 11

12 22 21

L 11 G1 22 G1 G2 22 G2

L 11 G1

G1

2 2

2

2

13. Definetheoptimizationproblemwhentransmissionlossesareconsidered.

14. WhatdoyoumeanbyITLandpenaltyfactorofthesystem?Writeexpressionsforthem.

ITL=Incrementaltransmissionloss=

Itisdefinedastheratioofthechangeinreal-powerlosstothechangeinreal-powergeneration.

Penaltyfactor

15. Whyarethereactive-powerconstraintstobeconsideredasinequalityconstraintsinsolvinganoptimizationproblemwhentransmissionlossesareconsidered?

Thetransmissionlossisafunctionofrealandreactive-powergenerationssincereactivepowerisproportionaltothesquareofthevoltage.

16. IfthefuelcostinRs./hrofapowerstationisrelatedtothepowergeneratedinMWbyC =0.0002P +0.06P +300,whatis

theincrementalfuelcostatP =200MW?

17. Whatarethepointsthatshouldbekeptinmindforthesolutionofeconomicloaddispatchproblemswhentransmissionlossesareincludedandco-ordinated?

Thefollowingpointsshouldbekeptinmind:

1. Althoughtheincrementalproductioncostofaplantisalwayspositive,ITLcanbeeitherpositiveornegative.

2. Theindividualunitswilloperateatdifferentincrementalproductioncosts.3. ThegenerationwithhighestpositiveITLwilloperateatthelowest

incrementalproductioncost.


1. Intheeconomicoperationofapowersystem,theeffectof

1 G G

G

3 2

increasedpenaltyfactorbetweenageneratingplantandsystemloadcenteristo:

1. Decreasetheloadonthegeneratingplant.2. Increasetheloadontheplant.3. Holdtheplantloadconstant.4. Decreasetheloadfirstandthenincrease.

2. Inapowersysteminwhichgeneratingplantsareremotefromtheloadcenter,minimumfuelcostsoccurwhen:

1. Equalincrementalcostsaremaintainedatthegeneratingstationbuses.2. Equalincrementalcostsarereferredtosystemloadcenter.3. Equalunitsareoperatedatthesameload.4. Alltheabove.

3. Unitofpenaltyfactoris:

1. Rs.

2. MW .3. Rs./MWh.4. Nounits.

4. Unitcommitmentofmorenumberofgeneratingunitsisdoneusing:

1. Gradientmethod.2. Non-linearprogrammingmethod.3. Dynamicprogramming.4. Alltheabove.

5. Economicdispatchisdonefirstby___________andthenby___________.

1. Unitcommitmentandthenloadscheduling.2. Loadschedulingandthenunitcommitment.3. Either(a)or(b).4. Unitcommitmentandloadfrequencycontrol.

6. Transmissiionlossesareabout:

1. 50%ofthetotalgeneration.2. 100%ofthetotalgeneration.3. 5–15%ofthetotalgeneration.4. Noneofthese.

7. Inoptimalschedulingofhydro-thermalunits,theobjectiveis:

1. Waterdischargeminimization.2. Storageofwater.3. Both(a)and(b).4. Noneofthese.

8. Inoptimalgenerationscheduling,theco-ordinationequationforall‘i’valuesis:

1. IC =λ .

2. IC =λ L .

3. IC =λ /L .

4. IC =λ +L .

9. TransmissionlossbyB-coefficientsisPL=__________

i i

i i i

i i i

i i i

–1

T

1. P BP.

2. P B.3. BP.4. All.

10. Theconditionforoptimalitywithconsiderationoftransmissionlossis:

1. TheincrementalfuelcostsinRs./hrofalltheunitsmustbethesame.2. TheincrementalfuelcostsinRs./hrofalltheunitsmustbethesame.3. TheincrementaltransmissionlossesinRs./MWhofalltheunitsmustbe

thesame.4. Theincrementalfuelcostofeachmultipliedbyitspenaltyfactormustbe

thesameforallplants.

11. Expressionfortransmissionlossisderivedusing______________method.

1. Kron’s.2. Penaltyfunction.3. Kirchmayer’s.4. Kuhn-Tucker.

12.

1.

2.

3.

4.

13. Theequalityconstraint,whenthetransmissionlinelossesareconsidered,is:

1.

2.

3.

4.

14. Transmissionlossis:

T

T

1. Afunctionofreal-powergeneration.2. Independentofreal-powergeneration.3. Afunctionofreactive-powergeneration.4. Afunctionofbusvoltagemagnitudeanditsangle.

15. InKron’smethod,

1. Reducethesystemtoanequivalentsystemwithasinglehypotheticalload.2. Reducethesystemtoanequivalentsystemwithoutanyload.3. Reducethesystemtoanequivalentsystemwithalargenumberofloads.4. Enhancethesystemtoanequivalentsystemwithnopowerloss.

16. Thederivationoftransmissionlinelossisnotbasedonwhichassumption?

1. Alltheloadcurrentsmaintainaconstantratio.

2. Allthelinesinthesystemhavedifferent ratios.

3. Alltheloadcurrentshavesamephaseangle.4. Thepowerfactorateachstationremainsconstant.

17. ThelosscoefficientB isgivenby:

1.

2.

3.

4.

18. Whichofthefollowingiscorrect?

1.

2.

3.

4.

12

th

19. Thepenaltyfactorofthei stationis:

1.

2.

3.

4.

20. approximatepenaltyfactorithplantisexpressedas:

1.

2.

3.

4.

21. Theincrementaltransmissionlossis:

1.

2.

3.

th

4.

22. iscalledtheco-ordinationequationbecause:

1. Itco-ordinatesITLwithIC.2. Itco-ordinatesITLwithpenaltyfactor.3. Itco-ordinatesreal-powergenerationwithreactive-powergeneration.4. Itco-ordinatesbusvoltagemagnitudewithIC.

23. TheincrementalcostofreceivedpowerinRs./MWhoftheiplantis:

1.

2.

3.

4.

24. Insolvingoptimizationproblemwithtransmissionlossconsideration,theconditionforoptimalityisobtainedas:

1. TheICofalltheplantsmustbethesame.2. TheICofeachplantmultipliedwithitspenaltyfactormustbethesamefor

alltheplants.3. TheICofeachplantdividedbyitspenaltyfactormustbethesameforall

theplants.4. TheICofeachplantsubtractedfromitspenaltyfactormustbethesame

foralltheplants.

25. Thematrixformoftransmissionlossexpressionis:

1.

2.

3.

4.

26. Theexactco-ordinationequationofthei plantis:

th

th

1.

2.

3.

4.

27. Theoptimizationproblemissolvedbythecomputationalmethodwiththeexpressionforp whichisgivenas:

1.

2.

3.

4.

28. ThepenaltyfactoroftheplantshowninFig.3.14is:

1. 5.2. 5.25.3. 1.254. 12.5


Gi

29. Theincrementalcostofreceivedpowerfortheaboveplantif

Rs./MWhis:

1. 1.25.2. 16.82.3. 16.00.4. 12.80.

30. ForFig.3.15,whatisthepenaltyfactorofthesecondplantifapowerof125MWistransmittedfromthefirstplanttoloadwithanincurredlossof15.625MW?

1. 24.2. 1.25.3. Zero.4. 1.

31. Toderivethetransmissionlossexpression,whichofthefollowingassumptionsaretobetakenintoconsideration?

1. AllthelinesinthesystemhavethesameR/Xratio.2. P.f.ateachstationremainsconstant.3. Alltheloadcurrentsmaintainconstantratiotothetotalcurrent.4. Alltheloadcurrentshavetheirowndifferentphaseangles.

1. (i)and(ii).2. (ii)and(iii).3. Allexcept(iv).4. Allofthese.

FIG.3.15

32. Inderivingtheexpressionfortransmissionpowerloss,whichofthefollowingprinciplesareused?

1. Thevinin’stheorem.2. Kron’smethod.3. Max.power-transfertheorem.4. Superpositiontheorem.

1. (i)only.2. (ii)and(iii)only.3. (ii)and(iv)only.4. Allexcept(i).

33. Thetransmissionlossisexpressedas:

1.

2.

3.

4.

34. Infindingtheoptimalsolution,theobjectivefunctionisredefinedasconstrainedobjectivefunctionandisgivenby:

1.

2.

3.

4.

REVIEWQUESTIONS

1. Derivethetransmissionlossformulaandstatetheassumptionsmadeinit.

2. Obtaintheconditionforoptimumoperationofapowersystemwith‘n’plantswhenlossesareconsidered.

3. Brieflyexplainabouttheexactco-ordinationequationandderivethepenaltyfactor.

4. WhatareB-coefficients?Derivethem.5. Explaineconomicdispatchofthermalplantsco-ordinatingthe

systemtransmissionlinelosses.Deriverelevantequationsandstatethesignificanceandroleofpenaltyfactor.

6. Giveastep-by-stepprocedureforcomputingeconomicallocationofpowergenerationinathermalsystemwhentransmissionlinelossesareconsidered.

PROBLEMS

1. Asystemconsistsoftwogeneratingplantswithfuelcostsof:

C = 0.03P +15P +1.01 G1 G1

2

and

C = 0.04P +21P +1.4

Thesystemoperatesoneconomicaldispatchwith120MWofpowergenerationbyeachplant.TheincrementaltransmissionlossofPlant-2is0.15.FindthepenaltyfactorofPlant-1.

2. Asystemconsistsoftwogeneratingplants.TheincrementalcostsinRs./MWhwithp andp inMWare:

Thesystemoperatesoneconomicdispatchwithp =p =400

MWand .FindthepenaltyfactorofPlant-1.

3. Thecostcurvesoftwoplantsareasfollows:

C =0.04P +25P +120

C =0.035P +10P +160

ThelosscoefficientfortheabovesystemisgivenasB =

0.001/MW,B =B =–0.0002/MWandB =0.003/MW.

Determinetheeconomicalgenerationschedulingcorrespondingtoλ=20Rs./MWhandcorrespondingsystemloadthatcanbemetwith.Ifthetotalloadconnectedtothesystemis110MWtaking3.5%changeinthevalueofλ,whatshouldbethevalueofλinthenextiteration?

2 G2 G2

G1 G2

G1 G2

1 G1 G2

2 G2 G1

11

12 21 22

2

2

2

4

OptimalUnitCommitment

OBJECTIVES


knowtheneedofoptimalunitcommitment(UC)studythesolutionmethodsforUCsolvetheUCproblembydynamicprogramming(DP)approachpreparetheUCtablewithreliabilityandstart-upcostconsiderations

4.1INTRODUCTION

Thetotalloadofthepowersystemisnotconstantbutvariesthroughoutthedayandreachesadifferentpeakvaluefromonedaytoanother.Itfollowsaparticularhourlyloadcycleoveraday.TherewillbedifferentdiscreteloadlevelsateachperiodasshowninFig.4.1.

Duetotheabovereason,itisnotadvisabletorunallavailableunitsallthetime,anditisnecessarytodecideinadvancewhichgeneratorsaretostartup,whentoconnectthemtothenetwork,thesequenceinwhichtheoperatingunitsshouldbeshutdown,andforhowlong.Thecomputationalprocedureformakingsuchdecisionsiscalledunitcommitment(UC),andaunitwhenscheduledforconnectiontothesystemissaidtobecommitted.

FIG.4.1Discretelevelsofsystemloadofdailyloadcycle

TheproblemofUCisnothingbuttodeterminetheunitsthatshouldoperateforaparticularload.To‘commit’ageneratingunitisto‘turniton’,i.e.,tobringituptospeed,synchronizeittothesystem,andconnectit,sothatitcandeliverpowertothenetwork.

4.2COMPARISONWITHECONOMICLOADDISPATCH

Economicdispatcheconomicallydistributestheactualsystemloadasitrisestothevariousunitsthatarealreadyon-line.However,theUCproblemplansforthebestsetofunitstobeavailabletosupplythepredictedorforecastloadofthesystemoverafuturetimeperiod.

4.3NEEDFORUC

Theplantcommitmentandunit-orderingschedulesextendtheperiodofoptimizationfromafewminutestoseveralhours.Weeklypatternscanbedevelopedfromdailyschedules.Likewise,monthly,seasonal,andannualschedulescanbepreparedbytakingintoconsiderationtherepetitivenatureoftheloaddemandandseasonalvariations.Agreatdealofmoneycanbesavedbyturningofftheunitswhentheyarenotneededforthetime.Iftheoperationofthesystemistobe

optimized,theUCschedulesarerequiredforeconomicallycommittingunitsinplanttoservicewiththetimeatwhichindividualunitsshouldbetakenoutfromorreturnedtoservice.Thisproblemisofimportanceforschedulingthermalunitsinathermalplant;asforothertypesofgenerationsuchashydro,theiraggregatecosts(suchasstart-upcosts,operatingfuelcosts,andshut-downcosts)arenegligiblesothattheiron-offstatusisnotimportant.

4.4CONSTRAINTSINUC

TherearemanyconstraintstobeconsideredinsolvingtheUCproblem.

4.4.1Spinningreserve

Itisthetermusedtodescribethetotalamountofgenerationavailablefromallsynchronizedunitsonthesystemminusthepresentloadandlossesbeingsupplied.Here,thesynchronizedunitsonthesystemmaybenamedunitsspinningonthesystem.

LetP bethespinningreserve, thepowergeneration

ofthei synchronizedunit,P thetotalloadonthesystem,andp thetotallossofthesystem:

Thespinningreservemustbemaintainedsothatthefailureofoneormoreunitsdoesnotcausetoofaradropinsystemfrequency.Simply,ifoneunitfails,theremustbeanamplereserveontheotherunitstomakeupforthelossinaspecifiedtimeperiod.

Thespinningreservemustbeagivenapercentageofforecastedpeakloaddemand,oritmustbecapableof

Gsp

D

L

th

takingupthelossofthemostheavilyloadedunitinagivenperiodoftime.

Itcanalsobecalculatedasafunctionoftheprobabilityofnothavingsufficientgenerationtomeettheload.

Thereservesmustbeproperlyallocatedamongfast-respondingunitsandslow-respondingunitssuchthatthisallowstheautomaticgenerationcontrolsystemtorestorefrequencyandquicklyinterchangethetimeofoutageofageneratingunit.

Beyondthespinningreserve,theUCproblemmayconsidervariousclassesof‘scheduledreserves’oroff-linereserves.Theseincludequick-startdieselorgas-turbineunitsaswellasmosthydro-unitsandpumpedstoragehydro-unitsthatcanbebroughton-line,synchronized,andbroughtuptomaximumcapacityquickly.Assuch,theseunitscanbecountedintheoverallreserveassessmentaslongastheirtimetocomeuptomaximumcapacityistakenintoconsideration.Reservesshouldbespreadwellaroundtheentirepowersystemtoavoidtransmissionsystemlimitations(oftencalled‘bottling’ofreserves)andtoallowdifferentpartsofthesystemtorunas‘islands’,shouldtheybecomeelectricallydisconnected.

4.4.2Thermalunitconstraints

Athermalunitcanundergoonlygradualtemperaturechangesandthistranslatesintoatimeperiod(ofsomehours)requiredtobringtheunitontheline.Duetosuchlimitationsintheoperationofathermalplant,thefollowingconstraintsaretobeconsidered.

1. Minimumup-time:Duringtheminimumup-time,oncetheunitisoperating(upstate),itshouldnotbeturnedoffimmediately.

2. Minimumdown-time:Theminimumdown-timeistheminimumtimeduringwhichtheunitisin‘down’state,i.e.,oncetheunitisdecommitted,thereisaminimumtimebeforeitcanberecommitted.

3. Crewconstraints:Ifaplantconsistsoftwoormoreunits,theycannotbothbeturnedonatthesametimesincetherearenotenoughcrewmemberstoattendtobothunitswhilestartingup.

Start-upcost

Inadditiontotheaboveconstraints,becausethetemperatureandthepressureofthethermalunitmustbemovedslowly,acertainamountofenergymustbe

expendedtobringtheuniton-lineandisbroughtintotheUCproblemasastart-upcost.

Thestart-upcostmayvaryfromamaximum‘cold-start’valuetoaverysmallvalueiftheunitwasonlyturnedoffrecently,anditisstillrelativelyclosetotheoperatingtemperature.

Twoapproachestotreatingathermalunitduringits‘down’state:

Thefirstapproach(cooling)allowstheunit’sboilertocooldownandthenheatbackuptoaoperatingtemperatureintimeforascheduledturn-on.Thesecondapproach(banking)requiresthatsufficientenergybeinputtotheboilertojustmaintaintheoperatingtemperature.

FIG.4.2Time-dependentstart-upcosts

Thebestapproachcanbechosenbycomparingthecostsfortheabovetwoapproaches.

LetC bethecold-startcost(MBtu),Cthefuelcost,Cthefixedcost(includescrewexpensesandmaintainableexpenses),αthethermaltimeconstantfortheunit,Cthecostofmaintainingunitatoperatingtemperature(MBtu/hr),andtthetimetheunitwascooled(hr).

Start-upcostwhencooling=C (1–e )C+C ;

C F

t

c F

-t/α

Start-upcostwhenbanking=C ×t×C+C .

Uptoacertainnumberofhours,thecostofbanking<costofcoolingisshowninFig.4.2.

ThecapacitylimitsofthermalunitsmaychangefrequentlyduetomaintenanceorunscheduledoutagesofvariousequipmentsintheplantandthismustalsobetakenintoconsiderationintheUCproblem.

Theotherconstraintsareasfollows

4.4.3Hydro-constraints

AspointedoutalreadythattheUCproblemisofmuchimportancefortheschedulingofthermalunits,itisnotthemeaningofUCthatcannotbecompletelyseparatedfromtheschedulingofahydro-unit.

Thehydro-thermalschedulingwillbeexplainedasseparatedfromtheUCproblem.Operationofasystemhavingbothhydroandthermalplantsis,however,farmorecomplexashydro-plantshavenegligibleoperationcosts,butarerequiredtooperateunderconstraintsofwateravailableforhydro-generationinagivenperiodoftime.

Theproblemofminimizingtheoperatingcostofahydro-thermalsystemcanbeviewedasoneofminimizingthefuelcostofthermalplantsundertheconstraintofwateravailabilityforhydro-generationoveragivenperiodofoperation.

4.4.4Mustrun

Itisnecessarytogiveamust-runreorganizationtosomeunitsoftheplantduringcertaineventsoftheyear,bywhichweyieldthevoltagesupportonthetransmissionnetworkorforsuchpurposeassupplyofsteamforusesoutsidethesteamplantitself.

4.4.5Fuelconstraints

t F

AsysteminwhichsomeunitshavelimitedfuelorelsehaveconstraintsthatrequirethemtoburnaspecifiedamountoffuelinagiventimepresentsamostchallengingUCproblem.

4.5COSTFUNCTIONFORMULATION

LetF bethecostofoperationofthei unit,P the

outputofthei unit,andC therunningcostoftheiunit.Then,

F =C P

C mayvarydependingontheloadingcondition.

LetC bethevariablecostcoefficientforthei unitwhenoperatingatthej loadforwhichthecorrespondingactivepowerisP .

Sincethelevelofoperationisafunctionoftime,thecostefficiencymaybedescribedwithyetanotherindextodenotethetimeofoperation,sothatitbecomesC for

thesub-interval‘t’correspondingtoapoweroutputof.

Ifeachunitiscapableofoperationatkdiscretelevels,thentherunningcostF ofthei unitinthetime

intervaltisgivenby

Iftherearenunitsavailableforoperationinthetimeinterval‘t’,thenthetotalrunningcostofnunitsduringthetimeinterval‘t’is

i Gi

i

i i Gi

i

ij

Gij

ijt

it

th

th th

th

th

th

Fortheentiretimeperiodofoptimization,havingTsub-intervalsoftime,theoverallrunningcostforalltheunitsmaybecome

4.5.1Start-upcostconsideration

Supposethatforaplanttobebroughtintoservice,anadditionalexpenditureC hastobeincurredinaddition

totherunningcost(i.e.,start-upcostofthei unit),thecostofstarting‘x’numberofunitsduringanysub-intervaltisgivenby

whereδ =1,ifthei unitisstartedinsub-interval‘t’

andotherwiseδ =0.

4.5.2Shut-downcostconsideration

Similarly,ifaplantistakenoutofserviceduringtheschedulingperiod,itisnecessarytoconsidertheshut-downcost.

If‘y’numberofunitsarebetoshutdownduringthesub-interval‘t’,theshut-downcostmayberepresentedas

whereσ =1,whenthei unitisthrownoutofservicein

sub-interval‘t’;otherwiseσ =0.

si

it

it

it

it

th

th

th

OverthecompleteschedulingperiodofTsub-intervals,thestart-upcostisgivenby

andtheshut-downcostis

Now,thetotalexpressionforthecostfunctionincludingtherunningcost,thestart-upcost,andtheshut-downcostiswrittenintheform:

Foreachsub-intervaloftimet,thenumberofgeneratingunitstobecommittedtoservice,thegeneratorstobeshutdown,andthequantizedpowerloadinglevelsthatminimizethetotalcosthavetobedetermined.

4.6CONSTRAINTSFORPLANTCOMMITMENTSCHEDULES

Asintheoptimalpointgenerationscheduling,theoutputofeachgeneratormustbewithintheminimumandmaximumvalueofcapacity:

i.e.,

Theoptimumschedulesofgenerationarepreparedfromtheknowledgeofthetotalavailableplantcapacity,whichmustbeinexcessoftheplant-generatingcapacityrequiredinmeetingthepredictedloaddemandin

satisfyingtherequirementsforminimumrunningreservecapacityduringtheentireperiodofscheduling:

whereS isthetotalavailablecapacityinanysub-interval‘t’,S theminimumrunningreservecapacity,

α =1,ifthei unitisinoperationduringsub-interval

‘t’;otherwiseα =0

Inaddition,forapredictedloaddemandP ,thetotalgenerationoutputinsub-interval‘t’mustbeinexcessoftheloaddemandbyanamountnotlessthantheminimumrunningreservecapacityS .

(withoutconsideringthe

transmissionlosses)

Incaseofconsiderationoftransmissionlosses,theaboveequationbecomes

Thegeneratorstart-upandshut-downlogicindicatorsδ andσ ,respectively,shouldbeunityduringthe

correspondingsub-intervalsofoperation

4.7UNITCOMMITMENT—SOLUTIONMETHODS

ThemostimportanttechniquesforthesolutionofaUCproblemare:

TAC

rmin

it

it

D

rmin

it it

th

1. Priority-listschemes.2. Dynamicprogramming(DP)method.3. Lagrange’srelaxation(LR)method.

Now,wewillexplainthepriority-listschemeandtheDPmethod.

Asimpleshut-downruleorpriority-listschemecouldbeobtainedafteranexhaustiveenumerationofallunitcombinationsateachloadlevel.

4.7.1Enumerationscheme

Astraightforwardbuthighlytime-consumingwayoffindingthemosteconomicalcombinationofunitstomeetaparticularloaddemandistotryallpossiblecombinationsofunitsthatcansupplythisload.Thisloadisdividedoptimallyamongtheunitsofeachcombinationbytheuseofco-ordinationequationssoastofindthemosteconomicaloperatingcostofthecombination.Then,thecombinationthathastheleastoperatingcostamongalltheseisdetermined.

SomecombinationswillbeinfeasibleifthesumofallmaximumMWfortheunitscommittedislessthantheloadorifthesumofallminimumMWfortheunitscommittedisgreaterthantheload.

Example4.1:Letusconsideraplanthavingthreeunits.Thecostcharacteristicsandminimumandmaximumlimitsofpowergeneration(MW)ofeachunitareasfollows:

Unit-1,

C =0.002842P +8.46P +600.0Rs./hr,200≤

P ≤650

Unit-2,

C =0.002936P +8.32P +420.0Rs./hr,150≤

P ≤450

Unit-3,

1 G1 G1

G1

2 G2 G2

G2

2

2

2

C =0.006449P +9.884P +110.0Rs./hr,100≤

P ≤300

Tosupplyatotalloadof600MWmosteconomically,thecombinationsofunitsandtheirgenerationstatusaretabulatedinTable4.1.

Numberofcombinations=2 =2 =8

TABLE4.1Combinationsoftheunitsandtheirstatusforthedispatchofa600-MWload

Note:Theleastexpensivewasnottosupplythegenerationwithallthreeunitsrunningorevenanycombinationinvolvingtwounits.Rather,theoptimumcommitmentwastorunonlyunit-1,themosteconomicunit.Byonlyrunningit,theloadcanbesuppliedbythatunitoperatingclosertoitsbestefficiency.Ifanotherunitiscommitted,bothUnit-1andtheotherunitwillbeloadedfurtherfromtheirbestefficiencypointssuchthatthenetcostisgreaterthanunit-1alone.

3 G3 G3

G3

2

n 3

FIG.4.3Simplepeak–valleyloadpattern

4.7.1.1UCoperationofsimplepeak–valleyloadpattern:shut-downrule

Letusassumethattheloadfollowsasimple‘peak–valley’patternasshowninFig.4.3.

Tooptimizethesystemoperation,someunitsmustbeshutdownastheloaddecreasesandisthenrecommitted(putintoservice)asitgoesbackup.

Oneapproachcalledthe‘shut-downrule’mustbeusedtoknowwhichunitstodropandwhentodropthem.Asimplepriority-listschemeistobedevelopedfromthe‘shut-downrule’.

Considertheexample,withtheloadvaryingfromapeakof1,400MWtoavalleyof600MW(Table4.2).Toobtaina‘shut-downrule’,wesimplyuseabrute-forcetechniquewhereinallcombinationsofunitswillbetriedforeachloadleveltakeninstepsofsomeMW(here50MW).

TABLE4.2Shut-downrulederivation

Fromtheabovetable,wecanobservethatfortheloadabove1,100MW,runningallthethreeunitsiseconomical;between1,100and700MWrunningthefirstandsecondunitsiseconomical.Forbelow700MW,runningofonlyUnit-1iseconomicalasshowninFig.4.4.

FIG.4.4UCscheduleusingtheshut-downrule

TABLE4.3Priorityorderingofunits

TABLE4.4Prioritylistforsupplyof1,400MW

Combinationofunits

ForcombinationP ForcombinationP

2,1,and3 50 1,400

2and1 350 1,100

2 150 450

4.7.2Priority-listmethod

Asimplebutsub-optimalapproachtotheproblemistoimposepriorityordering,whereinthemostefficientunit

Gmin Gmax

isloadedfirsttobefollowedbythelessefficientunitsinorderastheloadincreases.

Inthismethod,firstwecomputethefull-loadaverageproductioncostofeachunit.Then,intheorderofascendingcosts,theunitsarearrangedtocommittheloaddemand.

ForExample4.1,weconstructaprioritylistasfollows:

First,thefull-loadaverageproductioncostwillbecalculated.

Thefull-loadaverageproductioncostofUnit-1=9.79Rs./MWh.



Apriorityorderoftheseunitsbasedontheaverageproductionisasfollows(Table4.3):

Byneglectingminimumup-ordown-time,start-upcosts,etc.theloaddemandcanbemetbythepossiblecombinationsasfollows(Table4.4):

4.7.2.1Priority-listschemeversusshut-downsequence

Inshut-downsequence,Unit-2wasshutdownat700MWleavingUnit-1.Withthepriority-listscheme,bothunitswouldbeheldONuntiltheloadhadreached450MWandthenUnit-1wouldbedropped.

Manypriority-listschemesaremadeaccordingtoasimpleshut-downalgorithm,suchthattheywouldhavestepsforshuttingdownaunitasfollows:

1. Duringthedroppingofload,attheendofeachhour,determinewhetherthenextunitontheprioritylistwillhavesufficientgenerationcapacitytomeettheloaddemandandtosatisfytherequirementofthespinningreserve.Ifyesgotothenextstepandotherwisecontinuetheoperationwiththeunitasitis.

2. Determinethetimeinnumberofhours‘h’beforethedropped

unit(inStep1)willbeneededagainforservice.3. Ifthenumberofhours(h)islessthanminimumshut-downtime

fortheunit,thenkeepthecommitmentoftheunitasitisandgotoStep5;ifnot,gotothenextstep.

4. Now,calculatethefirstcost,whichisthesumofhourlyproductioncostsforthenext‘h’hourswiththeunitin‘up’state.Then,recalculatethesamesumassecondcostfortheunit‘down’stateandinthestart-upcostforeithercoolingtheunitorbankingit,whicheverislessexpensive.Iftherearesufficientsavingsfromshuttingdowntheunit,itshouldbeshutdown,otherwisekeepiton.

5. Repeattheaboveprocedureforthenextunitontheprioritylistandcontinueforthesubsequentunit.

Thevariousimprovementstothepriority-listschemescanbemadebygroupingofunitstoensurethatvariousconstraintsaremet.

4.7.3Dynamicprogramming

DynamicprogrammingisbasedontheprincipleofoptimalityexplainedbyBellmanin1957.Itstatesthat‘anoptimalpolicyhastheproperty,that,whatevertheinitialstateandtheinitialdecisionsare,theremainingdecisionsmustconstituteanoptimalpolicywithregardtothestateresultingfromthefirstdecision’.

Thismethodcanbeusedtosolveproblemsinwhichmanysequentialdecisionsarerequiredtobetakenindefiningtheoptimumoperationofasystem,whichconsistsofadistinctnumberofstages.However,itissuitableonlywhenthedecisionsatthelaterstagesdonotaffecttheoperationattheearlierstages.

4.7.3.1SolutionofanoptimalUCproblemwithDPmethod

Dynamicprogramminghasmanyadvantagesovertheenumerationscheme,themainadvantagebeingareductioninthesizeoftheproblem.

Theimpositionofaprioritylistarrangedinorderofthefull-loadaveragecostratewouldresultinacorrectdispatchandcommitmentonlyif

1. No–loadcostsarezero.2. Unitinput–outputcharacteristicsarelinearbetweenzerooutputand

fullload.3. Therearenootherlimitations.4. Start-upcostsareafixedamount.

IntheDPapproach,weassumethat:

1. Astateconsistsofanarrayofunitswithspecifiedoperatingunitsandtherestareatoff-line.

2. Thestart-upcostofaunitisindependentofthetimeifithasbeenoff-line.

3. Therearenocostsforshuttingdownaunit.4. Thereisastrictpriorityorderandineachintervalaspecified

minimumamountofcapacitymustbeoperating.

Afeasiblestateisoneatwhichthecommittedunitscansupplytherequiredloadandthatmeetstheminimumamountofcapacityineachperiod.

Practically,aUCtableistobemadeforthecompleteloadcycle.TheDPmethodismoreefficientforpreparingtheUCtableiftheavailableloaddemandisassumedtoincreaseinsmallbutfinitesizesteps.InDPitisnotnecessarytosolveco-ordinateequations,whileatthesametimetheunitcombinationsaretobetried.

ConsiderablecomputationalsavingcanbeachievedbyusingthebranchandboundtechniqueoraDPmethodforcomparingtheeconomicsofcombinationsascertaincombinationsneednotbetriedatall.

Thetotalnumberofunitsavailable,theirindividualcostcharacteristics,andtheloadcycleonthestationareassumedtobeknownapriori.Further,itshallbeassumedthattheloadoneachunitorcombinationofunitschangesinsuitablysmallbutuniformstepsofsize∆MW(say1MW).

ProcedureforpreparingtheUCtableusingtheDPapproach:

Step1: Startarbitrarilywithconsiderationofanytwounits.

Step2: Arrangethecombinedoutputofthetwounitsintheformofdiscreteloadlevels.

Step3: Determinethemosteconomicalcombinationofthetwounitsforalltheloadlevels.Itistobeobservedthatateachloadlevel,theeconomicoperationmaybetoruneitheraunitorbothunitswithacertainloadsharingbetweenthetwounits.

Step4: Obtainthemosteconomicalcostcurveindiscreteformforthetwounitsandthatcanbetreatedasthecostcurveofasingleequivalentunit.

Step5: Addthethirdunitandrepeattheproceduretofindthecostcurveofthethreecombinedunits.Itmaybenotedthatbythisprocedure,theoperatingcombinationsofthethirdandfirstandthirdandsecondunitsarenotrequiredtobeworkedoutresultinginconsiderablesavingincomputation.

Step6: Repeattheprocesstillallavailableunitsareexhausted.

ThemainadvantageofthisDPmethodofapproachisthathavingobtainedtheoptimalwayofloading‘K’units,itisquiteeasytodeterminetheoptimalwayofloading(K+1)units.

Mathematicalrepresentation

LetacostfunctionF (x)betheminimumcostinRs./hrofgenerationof‘x’MWbyNnumberofunits,f (y)thecostofgenerationof‘y’MWbytheN unit,andF (x−y)theminimumcostofgenerationof(x−y)MWbyremaining(N−1)units.

ThefollowingrecursiverelationwillresultwiththeapplicationofDP:

N

N

N −1

th

Themostefficienteconomicalcombinationofunitscanefficientlybedeterminedbytheuseoftheaboverelation.Herethemosteconomicalcombinationofunitsissuchthatityieldstheminimumoperatingcost,fordiscreteloadlevelsrangesfromtheminimumpermissibleloadofthesmallestunittothesumofthecapacitiesofallavailableunits.

Inthisprocess,thetotalminimumoperatingcostandtheloadsharedbyeachunitoftheoptimalcombinationareautomaticallydeterminedforeachloadlevel.

Example4.2:Apowersystemnetworkwithathermalpowerplantisoperatingbyfourgeneratingunits.Determinethemosteconomicalunittobecommittedtoaloaddemandof8MW.Also,preparetheUCtablefortheloadchangesinstepsof1MWstartingfromtheminimumtothemaximumload.Theminimumandmaximumgeneratingcapacitiesandcost-curveparametersoftheunitslistedinatabularformaregiveninTable4.5.

Solution:

Weknowthat:

Thecostfunction,


Thetotalload=P =8MW(given)

Bycomparingthecost-curveparameters,wecometoknowthatthecostcharacteristicsofthefirstunitarethelowest.Ifonlyonesingleunitistobecommitted,Unit-1istobeemployed.

D

Now,findoutthecostofgenerationofpowerbythefirstunitstartingfromminimumtomaximumgeneratingcapacityofthatunit.

Let,

f (1)=themaincostinRs./hrforthegenerationof1MWbythefirstunit




…..…..…..…..………………………


TABLE4.5Capacitiesandcost-curveparametersoftheunits

1

1

1

1

1

ForthecommitmentofUnit-1only

Whenonlyoneunitistobecommittedtomeetaparticularloaddemand,i.e.,Unit-1inthiscaseduetoitslesscostparameters,thenF (x)=f (x).

where:

F (x)istheminimumcostofgenerationof‘x’MWbyonlyoneunit

f (x)istheminimumcostofgenerationof‘x’MWbyUnit-1

∴F (1)=f (1)=(0.37×1+22.9)1=23.27

F (2)=f (2)=(0.37×2+22.9)2=47.28

F (3)=f (3)=(0.37×3+22.9)3=72.03

Similarly,

F (4)=f (4)=97.52

F (5)=f (5)=123.75

F (6)=f (6)=150.72

F (7)=f (7)=178.43

F (8)=f (8)=206.88

WhenUnit-1istobecommittedtomeetaloaddemandof8MW,thecostofgenerationbecomes206.88Rs./hr.

Forthesecondunit

f (1)=min.costinRs./hrforthegenerationof1MWbythesecondunitonly

=(0.78P +25.9)P

1 1

1

1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

2

G2 G2

=(0.78×1+25.9)1=26.68

Similarly,

f (2)=54.92

f (3)=84.72

f (4)=116.08

f (5)=149.0

f (6)=183.48

f (7)=219.52

f (8)=257.12

Byobservingf (8)andf (8),itisconcludedthatf (8)<f (8),i.e.,thecostofgenerationof8MWbyUnit-1isminimumthanthatbyUnit-2.

ForcommitmentofUnit-1andUnit-2combination

F (8)=Minimumcostofgenerationof8MWbythesimultaneousoperationoftwounits

i.e.,Units-1and2.

Inotherwords,theminimumcostofgenerationof8MWbythecombinationofUnit-1andUnit-2is205.11Rs./hrandforthisoptimalcost,Unit-1supplies7MWandUnit-2supplies1MW.

2

2

2

2

2

2

2

1 2 1

2

2

i.e.,theminimumcostofgenerationof7MWwiththecombinationofUnit-1(by6-MWsupply)andUnit-2(by1-MWsupply)is177.4Rs./hr.

F (2) = min[f (0)+F (2),f (1)+F (1),f (2)+F (0)]

= min[47.28,49.95,54.92]

∴F (2) = 47.28Rs./hr

F (1) = min[f (0)+F (1),f (1)+F (0)]

= min[23.27,26.68]

∴F (1) = 23.27Rs./hr

Now,thecostofgenerationbyUnit-3onlyis

f (0)=0; f (5)=169.625

f (1)=29.985; f (6)=209.46

f (2)=61.94; f (7)=251.265

f (3)=95.865; f (8)=295.04

f (4)=131.76;

2 2 1 2 1 2 1

2

2 2 1 2 1

2

3 3

3 3

3 3

3 3

3

ForcommitmentofUnit-1,Unit-2,andUnit-3combination

F (8)=Theminimumcostofgenerationof8MWbythethreeunits,i.e.,Unit-1,Unit-2,andUnit-3

i.e.,forthegenerationof8MWbythreeunits,Unit-1andUnit-2willcommittomeettheloadof8MWwithUnit-1supplying7MW,Unit-2supplying1MW,andUnit-3isinanoff-statecondition.

3

F (2) = min[f (0)+F (2),f (1)+F (1),f (2)+F (0)]

= min[47.28,53.255,61.94]

∴F (2) = 47.28Rs./hr

F (1) = min[f (0)+F (1),f (1)+F (0)]

= min[23.27,29.958]

∴F (1) = 23.27Rs./hr

Costofgenerationbythefourthunit

3 3 2 3 2 3 2

3

3 3 2 3 2

3

f (0)=0

f (1)=31.88Rs./hr

f (2)=65.12Rs./hr

f (3)=99.72Rs./hr

f (4)=135.68Rs./hr

f (5)=173.0Rs./hr

f (6)=211.68Rs./hr

f (7)=251.72Rs./hr

f (8)=293.12Rs./hr

Minimumcostofgenerationbyfourunits,i.e.,Unit-1,Unit-2,Unit-3,andUnit-4

F (8)=Theminimumcostofgenerationof8MWbyfourunits

i.e.,forthegenerationof8MWbyfourunits,Unit-1andUnit-2willcommittomeettheloadof8MWwithUnit-1supplying7MW,Unit-2supplying1MW,andUnit-3aswellasUnit-4areinanoff-statecondition:

4

4

4

4

4

4

4

4

4

4

F (1) = min[f (0)+F (1),f (1)+F (0)]

= min[23.2731.88]

∴F (1) = 23.27Rs./hr

Fromtheabovecriteria,itisobservedthatforthegenerationof8MW,thecommitmentofunitsisasfollows:

f (8) = F (8)=theminimumcostofgenerationof8MWin

Rs./hrbyUnit-1only

= 206.88Rs./hr

F (8) = theminimumcostofgenerationof8MWbytwounitswithUnit-1supplying7MWandUnit-2supplying1MW

= 205.11Rs./hr

F (8) = theminimumcostofgenerationof8MWbythreeunitswithUnit-1supplying7MW,Unit-2supplying1MW,andUnit-3isinanoff-statecondition.

4 4 3 4 3

4

1 1

2

3

= 205.11Rs./hr

F (8) = minimumcostofgenerationof8MWbyfourunitswithUnit-1supplying7MW,Unit-2supplying1MW,andUnit-3andUnit-4areinanoff-statecondition.

= 205.11Rs./hr

ByexaminingthecostsF (8),F (8),F (8),andF (8),wehaveconcludedthatformeetingtheloaddemandof8MW,theoptimalcombinationofunitstobecommittedisUnit-1with7MWandUnit-2with1MW,respectively,atanoperatingcostof205.11Rs./hr

ForpreparingtheUCtable,theorderingofunitsisnotacriterion.Foranyorder,wegetthesamesolutionthatisindependentofnumberingunits.

Togetahigheraccuracy,thestepsizeoftheloadistobereduced,whichresultsinaconsiderableincreaseintimeofcomputationandrequiredstoragecapacity.

Status1ofanyunitindicatesunitrunningorunitcommittingandstatus0ofanyunitindicatesthattheunitisnotrunning.

TheUCtableispreparedonceandforallforagivensetofunits(Table4.6).Astheloadcycleonthestationchanges,itwouldonlymeanchangesinstartingandstoppingofunitswithoutchangingthebasicUCtable.

TheUCtableisusedingivingtheinformationofwhichunitsaretobecommittedtosupplyaparticularloaddemand.Theexactloadsharingbetweentheunitscommittedistobeobtainedbysolvingtheco-ordinationequationsasbelow.

Totalload,

4

1 2 3 4

P =P =8MW(given) (4.5)

⇒P =8−P

TABLE4.6TheUCtablefortheabove-consideredsystem

Foranoptimalloadsharing,

G1 G2

G2 G1

i.e.,loadsharedbythefirstunit,P =6.73MW

andP −8−P =8−6.73=1.27MW

i.e.,loadsharedbythesecondunit,P =1.27MW

Lagrangianmultiplier,λ = 0.74P +22.9=1.56P +

25.9

= 27.88Rs./MWh

G1

G2 G1

G2

G1 G2

ThetotalminimumoperatingcostwithanoptimalcombinationofUnit-1andUnit-2is

f +f =205.11Rs./hr

TopreparetheUCtable,theloadistovaryinstepsof1MWstartingfromaminimumgeneratingcapacitytoamaximumgeneratingcapacityofastationinsuitablesteps.

4.8CONSIDERATIONOFRELIABILITYINOPTIMALUCPROBLEM

Inadditiontotheeconomyofpowergeneration,thereliabilityorcontinuityofpowersupplyisalsoanotherimportantconsideration.Anysupplyundertakinghasassuredallitsconsumerstoprovidereliableandqualityofserviceintermsofthespecifiedrangeofvoltageandfrequency.

Theaspectofreliabilityinadditiontoeconomyistobeproperlyco-ordinatedinpreparingtheUCtableforagivensystem.

TheoptimalUCtableistobemodifiedtoincludethereliabilityconsiderations.

Sometimes,thereisanoccurrenceofthefailureofgeneratorsortheirderatingconditionsduetosmallandminordefects.Underthatcontingencyofforcedoutage,inordertomeettheloaddemand,‘staticreservecapacity’isalwaysmaintainedatageneratingstationsuchthatthetotalinstalledcapacityexceedstheyearlypeakdemandbyacertainmargin.Thisisaplanningproblem.

InarrivingattheeconomicUCdecisionatanyparticularperiod,theconstrainttakenintoconsiderationwasmerelyafactthatthetotalgeneratingcapacityon-linewasatleastequaltothetotalloaddemand.Iftherewasanymarginbetweenthecapacityofunitscommittedandtheloaddemand,itwasincidental.Underactualoperation,oneormorenumberofunitshadfailedrandomly;itmaynotbepossibletomeettheload

1 2

demandforacertainperiodoftime.Tostartthespare(standby)thermalunitandtobringitonthelinetotakeuptheloadwillinvolvelongperiodsoftimeusuallyfrom2to8hrandalsosomestartingcost.Incaseofahydro-generatingunit,itcouldbebroughton-lineinafewminutestotakeuptheload.

Hence,toensurecontinuityofsupplytomeetrandomfailures,thetotalgeneratingcapacityon-linemusthaveadefinitemarginovertheloadrequirementsatanypointoftime.Thismarginiscalledthespinningreserve,whichensurescontinuallybymeetingthedemanduptoacertainextentofprobablelossofgeneratingcapacity.Whilerulesofthumbhavebeenused,basedonpastexperiencetodeterminethesystemspinningreserveatanytime,arecentbetterapproachcalledPatton’sanalyticalapproachisthemostpowerfulapproachtosolvethisproblem.

ConsiderthefollowingpointsintheaspectofreliabilityconsiderationintheUCproblem:

1. Theprobabilityofoutageofanyunitthatincreaseswithitsoperatingtimeandaunit,whichistoprovideaspinningreserveatanyparticulartime,hastobestartedseveralhourslater.Hence,thesecurityofsupplyproblemhastobetreatedintotalityoveraperiodofoneday.

2. Theloadsareneverknownwithcompletecertainty.3. Thespinningreservehastobefacilitatedatsuitablegenerating

stationsofthesystemandnotnecessarilyateachgeneratingstation.

Aunit’susefullifespanundergoesalternateperiodsofoperationandrepairasshowninFig.4.5.

FIG.4.5Randomoutagephenomenaofageneratingunitexcludingthescheduledoutages

Aunitoperatingtimeisalsocalledunit‘up-time’(t )anditsrepairtimeasits‘down-time’(t ).

Thelengthsofindividualoperatingandrepairperiodsarearandomphenomenonwithmuchlongerperiodsofoperationcomparedtorepairperiods.

Thisrandomphenomenonwithalongeroperatingperiodofaunitisdescribedbyusingthefollowingparameters.

Meantimetofailure(mean‘up’time):

Meantimetorepair(mean‘down’time):

∴Meancycletime=

TherateoffailureandtherateofrepaircanbedefinedbyinversingEquations(4.6)and(4.7)as

up

down

Rateoffailure failures/year

Rateofrepair repairs/year

Thefailureandrepairratesaretobeestimatedfromthepastdataofunitsorothersimilarunitselsewhere.

Theratesoffailureareaffectedbyrelativemaintenanceandtheratesofrepairareaffectedbythesize,composition,andskillofrepairteams.

Bymakinguseoftheratiodefinitionofgeneratingunits,theprobabilityofaunitbeinginan‘up’stateand‘down’statecanbeexpressedas

Probabilityoftheunitinthe‘up’stateis

Theprobabilityoftheunitinthe‘down’stateis

Obviously,P +P =1(4.10)

P andP arealsoknownasavailabilityandunavailabilityoftheunit.

up down

up down

Inanysystemwithknumberofunits,theprobabilityofthesystemstatechanges,i.e.,whenkunitsarepresentinasystem,thesystemstatechangesduetorandomoutages.

Therandomoutage(failure)ofaunitcanbeconsideredasaneventindependentofthestateoftheotherunit.

Letaparticularsystemstate‘i’,inwhichx unitsareinthe‘down’stateandy unitsareinthe‘up’state:

i.e.,x +y =k

Theprobabilityofthesystembeinginstate‘i’isexpressedas

Πindicatesprobabilitymultiplicationofthesystemstate.

4.8.1Patton’ssecurityfunction

Someintolerableorundesirableconditionofsystemoperationistermedasa‘breachofsystemsecurity’.

InanoptimalUCproblem,theonlybreachofsecurityconsideredistheinsufficientgeneratingcapacityofthesystemataparticularinstantoftime.

TheprobabilitythattheavailablegeneratingcapacityataparticulartimeislessthanthetotalloaddemandonthesystematthattimeiscomplicativelyestimatedbyonefunctionknownasPatton’ssecurityfunction.

Patton’ssecurityfunctionisdefinedas

i

i

i i

whereP istheprobabilityofthesystembeingintheistateandr istheprobabilitythatthesystemstateicausesabreachofsystemsecurity.

Inconsideringallpossiblesystemstatestodeterminethesecurityfunction,fromthepracticalpointofview,thissumistobetakenoverthestatesinwhichnotmorethantwounitsareonforcedoutage,i.e.,stateswithmorethantwounitsoutmaybeneglectedastheprobabilityoftheiroccurrencewillbetoosmall.

r =1,iftheavailablegeneratingcapacity(sumofcapacitiesofunitscommitted)islessthanthetotalload

demand,i.e., .Otherwiser=0.

ThesecurityfunctionSgivesaquantitativeestimationofsysteminsecurity.

4.9OPTIMALUCWITHSECURITYCONSTRAINT

Fromapurelyeconomicalpointofview,aUCtableispreparedfromwhichweknowwhichunitsarecommittedforagivenloadonthesystem.

Foreachperiod,wewillestimatethesecurityfunction

Foranysystem,wewilldefinemaximumtolerableinsecuritylevel(MTIL).Thisisamanagementdecisionandthevalueisbasedonpastexperience.

WheneverthesecurityfunctionexceedsMTIL(S>MTIL),itisnecessarytomodifytheUCtabletoincludetheaspectsofsecurity.Itisnormallyachievedbycommittingthenextmosteconomicalunittosupplytheload.Withthenewunitbeingcommitted,wewillestimatethesecurityfunctionandcheckwhetheritisS<MTIL.

i

i

i

i

th

TheprocedureofcommittingthenextmosteconomicalunitiscontinueduptoS<MTIL.IfS=MTIL,thesystemdoesnothaveproperreliability.Addingunitsgoesuptoonesteponlybecauseforanother,itisnotnecessarytoaddthenextunitsmorethanoneunitsincethereisapresenceofspinningreserve.

4.9.1IllustrationofsecurityconstraintwithExample4.2

ReconsiderExample4.2andthedailyloadcurvefortheabovesystemasgiveninFig.4.6.

TheeconomicallyoptimalUCforthisloadcurveisobtainedbytheuseoftheUCTable4.6(whichwaspreviouslyprepared)(Table4.7).

ConsideringperiodE,inwhichtheminimumloadis5MWandUnit-1isbeingcommittedtomeettheload.Wewillcheckforthisperiodwhetherthesystemissecureornot.

Assumetherateofrepair,µ=99repairs/year

Andrateoffailure,λ=1failure/yearforallfourunits

AndalsoassumethatMTIL=0.005

WehavetoestimatethesecurityfunctionSforthisperiodE:

Valueofr dependingonwhetherthereisabreachofsecurityornot.

TherearetwopossiblestatesforUnit-1:

operatingstate(or)‘up’state

(or)

forcedoutagestate(or)‘down’state

i

TheprobabilityofUnit-1beinginthe‘up’state,


TABLE4.7EconomicallyoptimalUCtableforloadcurveshowninFig.4.4

r =0,sincethegenerationofUnit-1(max.capacity)isgreaterthantheload(i.e.,14MW>5MW).

ThereisnobreachofsecuritywhentheUnit-1isinthe‘up’state.

1

TheprobabilityofUnit-1beinginthe‘down’state:

r =1,sinceUnit-1isinthedownstate(P =0),the

loaddemandof5MWcannotbemet.

ThereisabreachofsecuritywhenUnit-1isinthe‘down’state.Now,findthevalueofthesecurityfunction.

whereirepresentsthestateofUnit-1.

Ifnisthenumberofunits,numberofstates=2

Forn=1,states=2 =2(i.e.,upanddownstates)

∴S=P r +P r

=P r +P r

=0.99×0+0.01×1

=0.01

Itisobservedthat0.01>0.005,i.e.,S>MTIL

Sinceinthiscase,S>MTILrepresentssysteminsecurity.Therefore,itisnecessarytocommitthenextmosteconomicalunit,i.e.,unit-2,toimprovethesecurity.WhenbothUnits-1and2areoperating,estimatethesecurityfunctionasfollows:

Here,numberofunits,n=2

∴Numberofstates=2 =2 =4

r =0,representsnobreachofsecurityand

r =1,representsbreachofsecurity

2 G1

1 1 2 2

1up 1 1down 2

i

i

n

1

n 2

TABLE4.8Representationofbreachofsecurityforthepossiblecombinations

Whentakingeitherupdownupcombinationsofstates,

downupup

thereisnobreachofsecurity,sincer=0

Forthecombinationdown

down,

thereisabreachofsecurity(Table4.8).

Itisobservedthat0.001<0.005

Therefore,thecombinationofUnit-1andUnit-2doesmeettheMTILof0.005.

Forallotherperiodsofaloadcycle,checkwhetherthesecurityfunctionislessthanMTIL.ItisalsofoundthatforallotherperiodsexceptE,thesecurityfunctionislessthanMTIL.Now,wewillobtaintheoptimalandsecurity-constrainedUCtableforExample4.2(Table4.9).

4.10START-UPCONSIDERATION

i

FromtheoptimalandsecuredUCtablegiveninTable4.9,dependingontheloadinaparticularperiod,itisobservedthatsomeunitsaretobedecommittedandrestartedinthenextperiod.Wheneveraunitistoberestarted,itinvolvessomecostaswellassometimebeforetheunitisputon-line.Forthermalunits,itisnecessarytobuildupcertaintemperatureandpressuregraduallybeforetheunitcansupplyanyloaddemand.ThecostinvolvedinrestartinganyunitafterthedecommittingperiodisknownasSTART-UPcost.

TABLE4.9OptimalandsecureUCtableforExample4.2

*Unitiscommittedfromthepointofviewofsecurityconsiderations.

Dependingontheconditionoftheunit,thestart-upcostswillbedifferent.Iftheunitistobestartedfromacoldconditionandbroughtuptonormaltemperatureandpressure,thestart-upcostswillbemaximumsincesomeenergyisrequiredtobuilduptherequiredpressureandtemperatureofthesteam.Sometimes,theunitmaybeswitchedoffandthetemperatureofsteammaynotbeinacoldcondition.Thisparticularconditioniscalledthebankingcondition.

FromtheUCtablegiveninTable4.7,itisobservedthatduringPeriodB,Unit-3isoperatingandduring

PeriodC,itisdecommitted.ItisrestartedduringPeriodD.

InPeriodC,checkwhetheritiseconomicaltorunonlytwounitsorallowallthethreeunits(Units-1,2,and3)tocontinuetorunsuchthatthestart-upcostsareeliminated.

Letusassumethatthestart-upcostofeachunit=Rs.500.

CaseA:Unit-3isnotinoperationinPeriodC,i.e.,onlytwoUnits-1and2areoperating.

ForPeriodBorD,totalload=15MW

Thisistobesharedbythreeunits,i.e.,P +P +P

=15

SubtractingEquation(4.14)fromEquation(4.13),weget

0.74P −1.56P =3(4.16)

SubtractingEquation(4.15)fromEquation(4.13),weget

0.74P −1.97P =6.1(4.17)

or0.74P −1.97(15−P −P )=6.1

or2.71P +1.97P =35.65(4.18)

G1 G2 G3

G1 G2

G1 G3

G1 G1 G2

G1 G2

BysolvingEquations(4.13)and(4.16),wehave

P =10.8MW,P =3.2MW

P =15−P −P =15−108−3.2=1MW

C =(0.37P +22.9)P =290.48Rs./hr

C =(0.78P +25.9)P =90.87Rs./hr

C =(0.985P +29)P =29.98Rs./hr

ForPeriodB,theoperatingtimeis4hr.

∴Totalcost,C=[C +C +C ]t

=[290.48+90.87+29.98]×4

=Rs.1,645.34

TotaloperatingcostduringPeriodBisRs.1,645.34.

InPeriodC,10MWofloadistobesharedbyUnits-1and2

i.e.,P +P =10MW(4.19)


P =8.086MWandP =1.913MW

TotaloperatingcostforPeriodC

=[(0.37P +22.9)P +(0.78P +25.9)P ]×4

=Rs.1,047.05.

ForperiodD,thetotaloperatingcostisthesameasthatofPeriodB=Rs.1,645.34.

Therefore,thetotaloperatingcostforPeriodsB,C,andDis

=Rs.[1,645.34+1,047.05+1,645.34]

=Rs.4,337.73.

InPeriodD,Unit-3isrestartedtocommittheload,hencethestart-upcostofUnit-3isaddedtothetotaloperatingcostforperiodsB,C,andD:

G1 G2

G3 G1 G2

1 G1 G1

2 G2 G2

3 G3 G3

1 2 3

G1 G2

G1 G2

G1 G1 G2 G2

Start-upcostforUnit-3=Rs.500(given)

∴TotalcostofoperatingofunitsduringperiodB,C,andDis

=4,337.73+500

=Rs.4,837.73

CaseB:Unit-3isallowedtoruninPeriodC.

Hence,10-MWloadistobesharedbyunits1,2,and3.

i.e.,P +P +P =10(4.20)

SubstitutingP fromEquation(4.20)inEquation

(4.17),weget

0.74P −1.97(10−P −P )=6.1(4.21)

or2.71P +1.97P =25.8(4.22)


P =8.1MW,P =1.9MW,andP =0MW

Fromtheabovepowers,itisobservedthatP violates

theminimumgenerationcapacity(i.e.,0<1).

Hence,setthegenerationcapacityofUnit-3atminimumcapacity,i.e.,P =1MW.

Thentheremaining9MWisoptimallysharedbyUnit-1andUnit-2as

P =7.4MW,P =1.6MW,andP =1MW

TheoperatingcostatPeriodC

=[(0.37P +22.9)P +(0.78P +25.9)P +

(0.985P +29)P ]×4hr

=Rs.1,048.57

TotalcostforPeriodsB,C,andD=Rs.1,645.34+Rs.1,048.57+Rs.1,645.34

=Rs.4,339.25.

G1 G2 G3

G3

G1 G1 G2

G1 G2

G1 G2 G3

G3

G3

G1 G2 G3

G1 G1 G2 G2

G3 G3

Rs.4,339.25<Rs.4,837.73

∴ItisconcludedthattorunUnit-3inPeriodCistheeconomicalway.

Now,theoptimalUCtableismodifiedas

*Unitiscommittedfromthepointofsecurityconsideration.

**Unitiscommittedfromthepointofstart-upconsiderations.

Hence,itiseconomicaltoallowallthethreeunitstocontinuetoruninPeriodsB,C,andD,i.e.,inPeriodCcontinuationofUnit-3iseconomical.

Example4.3:Apowersystemnetworkwithathermalpowerplantisoperatingbyfourgeneratingunits.Determinethemosteconomicalunitstobecommittedtoaloaddemandof10MW.AlsopreparetheUCtablefortheloadchangesinstepsof1MWstartingfromtheminimumtothemaximumload.Theminimumandmaximumgeneratingcapacitiesandcost-curveparametersoftheunitslistedinatabularformareasgiveninTable4.10.

Solution:

Weknow:

Thecostfunction,


Thetotalload=P =10MW(given)

Bycomparingthecost-curveparameters,wecometoknowthatthecostcharacteristicsofthefirstunitarethelowest.Ifonlyonesingleunitistobecommitted,unit-1istobeemployed.

TABLE4.10Capacitiesandcost-curveparametersoftheunits

Now,findthecost-of-generationofpowerbythefirstunitstartingfromtheminimumtothemaximumgeneratingcapacityofthatunit.

Let


D

1




…..…..…..…..…………………..…..…..……


ForthecommitmentofUnit-1only

Whenonlyoneunitistobecommittedtomeetaparticularloaddemand,i.e.,Unit-1,inthiscase,duetoitslowcostparameters,thenF (x)=f (x).

where

F (x)istheminimumcostofgenerationof‘x’MWbyonlyoneunit

f (x)istheminimumcostofgenerationof‘x’MWbyUnit-1

∴F (1)=f (1)=(0.34×1+22.8)1+823=846.14

F (2)=f (2)=(0.34×2+22.8)2+823=869.96

F (3)=f (3)=(0.34×3+22.8)3+823=894.46

1

1

1

1

1 1

1

1

1 1

1 1

1 1

Similarly,

F (4)=f (4)=916.64

F (5)=f (5)=945.50

F (6)=f (6)=972.04

F (7)=f (7)=996.26

F (8)=f (8)=1,027.16

F (9)=f (9)=1,055.74

F (10)=f (10)=1,085.00

WhenUnit-1istobecommittedtomeetaloaddemandof10MW,thecostofgenerationbecomes1,085Rs./hr.

Forthesecondunit:

f (1) = minimumcostinRs./hrforthegenerationof1MWbythesecondunitonly

= (0.765P +25.9)P +120

= (0.765×1+25.9)1+120=146.665

Similarly,

f (2)=174.860

f (3)=204.585

f (4)=235.840

f (5)=268.625

f (6)=302.940

f (7)=338.785

f (8)=376.160

1 1

1 1

1 1

1 1

1 1

1 1

1 1

2

G2 G2

2

2

2

2

2

2

2

f (9)=415.065

f (10)=455.500

Byobservingf (10)andf (10),itisconcludedthatf (10)<f (10),i.e.,thecostofgenerationof10MWbyunit-1isminimumthanthatbyUnit-2.

Forcommitmentofunit-1andUnit-2combination

F (10)=Minimumcostofgenerationof10MWbythesimultaneousoperationoftwounits,i.e.,Units-1and2

Inotherwords,theminimumcostofgenerationof10MWbythecombinationofUnit-1andUnit-2is455.5Rs./hrandforthisoptimalcost,Unit-1supplies0MWandUnit-2supplies10MW.

2

2

1 2

1 2

2

i.e.,theminimumcostofgenerationof9MWwiththecombinationofUnit-1(by0-MWsupply)andUnit-2(by9-MWsupply)is415.065Rs./hr.

Similarly,

∴F (8)=376.16Rs./hr

∴F (7)=338.785Rs./hr

∴F (6)=338.785Rs./hr

∴F (5)=268.625Rs./hr

∴F (4)=235.84Rs./hr

F (3)=min[894.46,1,016.6251,0221204.585]

∴F (3)=204.585Rs./hr

2

2

2

2

2

2

2

F (2)=min[869.96,992.805174.86]

∴F (2)=174.86Rs./hr

F (1)=min[846.14146.665]

∴F (1)=146.665Rs./hr

Now,thecostofgenerationbyUnit-3only:

f (0)=0; f (5)=649.75;f (9)=821.19

f (1)=509.99; f (6)=689.64;f (10)=869.00

f (2)=541.96; f (7)=731.51

f (3)=575.91; f (8)=775.36

f (4)=611.84;

ForcommitmentofUnit-1,Unit-2,andunit-3combination:

F (10)=Theminimumcostofgenerationof10MWbythethreeunitsi.e.,Unit-1,Unit-2,andUnit-3

2

2

2

2

2

3 3 3

3 3 3

3 3

3 3

3

3

i.e.,forthegenerationof10MWbythreeUnits,unit-2alonewillcommittomeettheloadof10MWandUnits-1and3areinanoff-statecondition:

F (3)=min[204.585,684.85,688.625,575.91]

∴F (3)=204.585Rs./hr

F (2)=min[174.86,565.655,541.96]

∴F (2)=174.86Rs./hr

F (1)=min[509.99,146.665]

∴F (1)=146.665Rs./hr

Costofgenerationbythefourthunit

f (0)=0

f (1)=531.115Rs./hr

f (2)=564.46Rs./hr

f (3)=600.035Rs./hr

f (4)=637.84Rs./hr

f (5)=677.875Rs./hr

f (6)=720.14Rs./hr

f (7)=764.635Rs./hr

3

3

3

3

3

3

4

4

4

4

4

4

4

4

f (8)=811.36Rs./hr

f (9)=860.315Rs./hr

f (10)=911.5Rs./hr

Minimumcostofgenerationbyfourunits,i.e.,Unit-1,Unit-2,Unit-3,andUnit-4:

F (10)=Theminimumcostofgenerationof10MWbyfourunits

i.e.,forthegenerationof10MWbyfourunits,Unit-2willcommittomeettheloadof10MW,andUnit-1,Unit-3,andUnit4areinanoff-statecondition:

4

4

4

4

F (3)=min[204.585,705.975,711.125,600.035]

∴F (3)=204.585Rs./hr

F (2)=min[46.96,554.255,564.46]

∴F (2)=46.96Rs./hr

F (1)=min[23.14,531.115]

∴F (1)=23.14Rs./hr

Fromtheabovecriteria,itisobservedthatforthegenerationof10MW,thecommitmentofunitsisasfollows:

f (10)=F (10)=theminimumcostofgenerationof10MWinRs./hrbyUnit-1only

=1085Rs./hr

F (10)=theminimumcostofgenerationof10MWbytwounitswithUnit-1supplying0MWandUnit-2supplying10MW

4

4

4

4

4

4

1 1

2

=455.5Rs./hr

F (10)=theminimumcostofgenerationof10MWbythreeunitswithUnit-2supplying10MW,Unit-1andUnit-3isinanoff-statecondition

=455.5Rs./hr

F (10)=theminimumcostofgenerationof10MWbyfourunitswithUnit-2supplying10MW,andUnit-1,Unit-3andUnit-4areinanoff-statecondition

=455.5Rs./hr

ByexaminingthecostsF (10),F (10),F (10),andF (10),wehaveconcludedthatformeetingtheloaddemandof10MW,theoptimalcombinationofunitstobecommittedisUnit-1,Unit-3,andUnit-4inanoff-stateconditionandUnit-2supplyinga10-MWloadatanoperatingcostof455.5Rs./hr.

ForpreparingtheUCtable,theorderingofunitsisnotacriterion.Foranyorder,wegetthesamesolutionthatisindependentofnumberingunits.

TABLE4.11TheUCtablefortheabove-consideredsystem

Togetahigheraccuracy,thestepsizeoftheloadistobereduced,whichresultsinconsiderableincreaseintimeofcomputationandrequiredstoragecapacity.

Status1ofanyunitindicatesunitrunningorunitcommittingandStatus0ofanyunitindicatesunitnotrunning.

TheUCtableispreparedonceandforallforagivensetofunits(Table4.11).Astheloadcycleonthestation

3

4

1 2 3

4

changes,itwouldonlymeanchangesinstartingandstoppingofunitswithoutchangingthebasicUCtable.

KEYNOTES

Unitcommitmentisaproblemofdeterminingtheunitsthatshouldoperateforaparticularload.To‘commit’ageneratingunitisto‘turniton’.Theconstraintsconsideredforunitcommitmentare:

1. Spinningreserve.2. Thermalunitconstraints.3. Hydro-constraints.4. Must-runconstraints.5. Fuelconstraints.

ThesolutionmethodstoaUCproblemare:

1. Priority-listscheme.2. Dynamicprogrammingmethod(DP).3. Lagrange’srelaxationmethod(LR).

Inthepriorityorderingmethod,themostefficientunitisloadedfirsttobefollowedbythelessefficientunitsinorderastheloadincreases.ThemainadvantageoftheDPmethodisresolutioninthedimensionalityofproblems,i.e.,havingobtainedtheoptimalwayofloadingKnumberofunits,itisquiteeasytodeterminetheoptimalwayofloading(K+1)numberofunits.


1. Duetotheloadvariation,itisnotadvisableto:

1. Runallavailableunitsatallthetimes.2. Runonlyoneunitateachdiscreteloadlevel.3. Both(a)and(b).4. Noneofthese.

2. Aunitwhenscheduledforconnectiontothesystemissaidtobe:

1. Loaded.2. Disconnected.3. Committed.4. Noneofthese.

3. Todeterminetheunitsthatshouldoperateforaparticularloadistheproblemof:

1. Unitcommitment.2. Optimalloadscheduling.3. Either(a)or(b).4. Noneofthese.

4. Tocommitageneratingunitis:

1. Tobringituptospeed.2. Tosynchronizeittothesystem.3. Toconnectitsothatitcandeliverpowertothenetwork.4. Allofthese.

5. Economicdispatchproblemisapplicabletovariousunits,Whichofthefollowingissuitable?

1. Theunitsarealreadyon-line.2. Tosupplythepredictedorforecastloadofthesystemoverafuturetime

period.3. Both(a)and(b).4. Noneofthese.

6. Unitcommitmentproblemplansforthebestsetofunitstobeavailable.Whichofthefollowingissuitable?

1. Theunitsarealreadyon-line.2. Tosupplythepredictedorforecastloadofthesystemoverafuturetime

period.3. Both(a)and(b).4. Noneofthese.

7. Spinningreserveisdefinedas:

1.

2.

3.

4. Noneofthese.

8. Spinningreservemustbe:

1. Maintainedsothatthefailureofoneormoreunitsdoesnotcausetoofaradropinsystemfrequency.

2. Capableoftakingupthelossofmostheavilyloadedunitinagivenperiodoftime.

3. Calculatedasafunctionoftheprobabilityofnothavingsufficientgenerationtomeettheload.

4. Allofthese.

9. Becauseoftemperatureandpressureofthermalunitthatmustbemovedslowly,acertainamountofenergymustbeexpendedtobringtheuniton-lineandisbroughtintotheUCproblemasa:

1. Runningcost.

2. Fixedcost.3. Fuelcost.4. Start-upcost.

10. Unitcommitmentproblemisofmuchimportancefor:

1. Schedulingofthermalunits.2. Schedulingofhydro-units.3. Schedulingofboththermalandhydro-units.4. Noneofthese.

11. ThermalunitconstraintsconsideredinaUCproblemare:

1. Minimumupandminimumdowntimes.2. Crewconstraints.3. Start-upcosts.4. Allofthese.

12. Thestart-upcostmayvaryfromamaximumcold-startvaluetoaverysmallvalueifthethermalunit:

1. Wasonlyturnedoffrecently.2. Isstillrelativelyclosetotheoperatingtemperature.3. Isstilloperatingatnormaltemperature.4. Both(a)and(b).

13. Unitcommitmentproblemis:

1. Ofmuchimportanceforschedulingofthermalunits.2. Cannotbecompletelyseparatedfromtheschedulingofhydro-units.3. Usedforhydro-thermalscheduling.4. Both(a)and(b).

14. TheconstraintsconsideredinaUCproblemare:

1. Thermalunitandhydro-unitconstraints.2. Spinningreserve.3. Must-runandfuelconstraints.4. Alltheabove.

15. ThemethodusedforobtainingthesolutiontoaUCproblemis:

1. Priority-listscheme.2. Dynamicprogrammingmethod.3. Lagrange’srelaxationmethod.4. Alltheabove.

16. Astraightforwardbuthighlytime-consumingwayoffindingthemosteconomicalcombinationofunitstomeetaparticularloaddemandis:

1. Enumerationscheme.2. Priority-listscheme.3. DPmethod.4. Allofthese.

17. Whichiscorrectregardingtheshut-downrule?

1. Toknowwhichunitstodropandwhen.2. Fromwhichasimplepriority-listschemeisdeveloped.3. Both(a)and(b).4. Toknowwhichunitstostartfromshut-downcondition.

18. Inthepriority-listmethodofsolvinganoptimalUCproblem:

1. Mostefficientunitisloadedfirsttobefollowedbythelessefficientunitinorderasloadincreases.

2. Lessefficientunitisloadedfirsttobefollowedbythemostefficientunitinorderasloadincreases.

3. Mostefficientunitisloadedfirsttobefollowedbythelessefficientunitinorderasloaddecreases.

4. Either(a)or(b).

19. Inthepriority-listmethod,theunitsarearrangedtocommittheloaddemandintheorderof:

1. Ascendingcostsofunits.2. Descendingcostsofunits.3. Either(a)or(b).4. Independentofcostsofunits.

20. ThechiefadvantageoftheDPmethodovertheenumerateschemeis:

1. Reductionintimeofcomputation.2. Reductioninthedimensionalityoftheproblem.3. Reductioninthenumberofunits.4. Allofthese.

21. IntheDPmethod,thecostfunctionF (x)represents:

1. MinimumcostinRs/hrofgenerationofNMWbyxnumberofunits.2. MinimumcostinRs/hrofgenerationofxMWbyNnumberofunits.

3. MinimumcostinRs/hrofgenerationofNMWbythex unit.

4. MinimumcostinRs/hrofgenerationofxMWbytheN unit.

22. IntheDPmethod,thecostfunctionF (y)represents:

1. CostofgenerationofNMWbyynumberofunits.2. CostofgenerationofyMWbyNnumberofunits.

3. CostofgenerationofNMWbythey unit.

4. CostofgenerationofyMWbytheN unit.

23. TherecursiverelationresultswiththeapplicationoftheDPmethodofsolvingtheUCproblemis:

1.

2.

3.

4.

24. ForpreparingtheUCtable,whichofthefollowingisnotacriterion?

1. Orderingofunits.2. Orderingofcostsofunits.3. Orderingofrangeofload.4. Allofthese.

25. InaUCtable,unitrunningorunitcommittingisindicatedby:

1. Status0.2. Status1.3. Status+.

N

N

th

th

th

th

4. Status>.

26. InaUCtable,thestatusoftheunitnotrunningisindicatedby:

1. Status0.2. Status0.3. Status+.4. Status−.

27. Theunscheduledormaintenanceoutagesofvariousequipmentsofathermalplantmustbetakenintoaccountin:

1. Optimalschedulingproblem.2. UCproblem.3. Loadfrequencycontrollingproblem.4. Allofthese.

28. Unitup-timeisnothingbut:

1. Aunitoperatingtime.2. Aunitrepairtime.3. Aunittotallifetime.4. Aunitdesigningtime.

29. Unitdown-timeisnothingbut:

1. Aunitoperatingtime.2. Aunitrepairtime.3. Aunittotallifetime.4. Aunitdesigningtime.

30. InreliabilityaspectsofaUCproblem,thelengthsofanindividualoperatingandrepairperiodsofaunitconsideredatarandomphenomenonwith:

1. Muchlongerperiodsofoperationcomparedtorepairperiods.2. Muchlongerperiodsofrepaircomparedtooperationperiods.3. Equalperiodsofoperationandrepair.4. Either(a)or(b).

31. Meanup-timeofaunit is:

1. Meantimetofailure.2. Meantimetorepair.3. Meanoffailureandrepairtimes.4. Meanoftotaltime.

32. Meandown-timeofaunit is:

1. Meantimetofailure.2. Meantimetorepair.3. Meanoffailureandrepairtimes.4. Meanoftotaltime.

33. Meancycletimeofaunitis:

1.

2.

3.

4.

34. Rateoffailureofaunitisexpressedas:

1.

2.

3.

4.

35. Rateofrepairofaunitisexpressedas:

1.

2.

3.

4.

36. Therateoffailureofaunitaffectedby:

1. Relativemaintenance.2. Size,compositionofrepairteam.3. Skillofrepairteam.4. Allofthese.

37. Therateofrepairofaunitisaffectedby:

1. Relativemaintenance.2. Size,compositionofrepairteam.3. Skillofrepairteam.4. Both(b)and(c).

38. P andP ofanyunitrepresent:

1. Unavailabilityandavailabilityofaunit.2. Availabilityandunavailabilityofaunit.3. Either(a)or(b).4. Both(a)and(b).

39. P +P =

1. Zero.2. 1.3. −1.4. Infinite.

up down

up down

40. AbreachofsystemsecurityconsideredinoptimalUCproblemis:

1. Sufficientgeneratingcapacityofthesystemataparticularinstantoftime.2. Insufficientgeneratingcapacityofthesystemataparticularinstantof

time.3. Insufficientgeneratingcapacityofthesystematalltimes.4. Either(a)or(b).

41. UseofPatton’ssecurityfunctionintheUCproblemistheestimationoftheprobabilitythattheavailablegeneratingcapacityataparticulartimeis:

1. Lessthanthetotalloaddemand.2. Morethanthetotalloaddemand.3. Equaltothetotalloaddemand.4. Independentofthetotalloaddemand.

42. Patton’ssecurityfunctionSgivesaquantitativeestimationof:

1. Systemsecurity.2. Systeminsecurity.3. Systemstability.4. Systemvariables.

43. ItisnecessarytomodifytheUCtabletoincludesecurityaspectsbycommittingthenextmosteconomicalunittosupplytheloadwhen:

1. S<MTIL.2. S>MTIL.3. S=MTIL.4. S=MTIL/2.

44. Theprocedureofcommittingamosteconomicalunit,toincludesecurityaspectintheUCtable,iscontinuedupto:

1. S<MTIL.2. S>MTIL.3. S=MTIL.4. S=MTIL/2.

45. Systeminsecurityisrepresentedby:

1. S<MTIL.2. S>MTIL.3. S=MTIL.4. S=MITL/2.

46. Iftheunitistobestartedfromacoldconditionandbroughtuptonormaltemperatureandpressure,thestart-upcostwillbe:

1. Minimum.2. Maximum.3. Havingnoeffect.4. Noneofthese.

47. WhenanyunitisintheUPstate,thereis:

1. Breachofsecurity.2. Nobreachofsecurity.3. Stability.4. Allofthese.

48. If theprobabilitythatthesystemstate‘i’causesabreachofsystemsecuritybecomes:

1. r =1.

2. r =0.

3. r =−1.

4. r =∞.

49. If theprobabilitythatthesystemstate‘i’causesabreachofsystemsecuritybecomes:

1. r =1.

2. r =0.

3. r =−1.

4. r =∞.


1. WhatisaUCproblem?

Itisnotadvisabletorunallavailableunitsatalltimesduetothevariationofload.Itisnecessarytodecideinadvance:

1. Whichgeneratorstostartup.2. Whentoconnectthemtothenetwork.3. Thesequenceinwhichtheoperatingunitsshouldbeshutdownandfor

howlong.

ThecomputationalprocedureformakingtheabovesuchdecisionsiscalledtheproblemofUC.

2. Whatdoyoumeanbycommitmentofaunit?

TocommitageneratingunitistoturnitON,i.e.,tobringituptospeed,synchronizeittothesystem,andconnectit,sothatitcandeliverpowertothenetwork.

3. WhyistheUCproblemimportantforschedulingthermalunits?

Asforothertypesofgenerationsuchashydro,theaggregatecostssuchasstart-upcosts,operatingfuelcosts,andshut-downcostsarenegligiblesothattheirON–OFFstatusisnotimportant.

4. ComparetheUCproblemwitheconomicloaddispatch.

Economicloaddispatcheconomicallydistributestheactualsystemloadasitrisestothevariousunitsalreadyon-line.ButtheUCproblemplansforthebestsetofunitstobeavailabletosupplythepredictedorforecastloadofthesystemoverfuturetimeperiods.

5. Whatarethedifferentconstraintsthatcanbeplacedontheucproblem?

1. Spinningreserve.2. Thermalunitconstraints.3. Hydro-constraints.4. Must-runconstraints.5. Fuelconstraints.

i

i

i

i

i

i

i

i

6. WhatarethethermalunitconstraintsconsideredintheUCproblem?

ThethermalunitconstraintsconsideredintheUCproblemare:

1. Minimumup-time.2. Minimumdown-time.3. Crewconstraints.4. Start-upcost.

7. Whymustthespinningreservebemaintained?

Spinningreservemustbemaintainedsothatfailureofoneormoreunitsdoesnotcausetoofaradropinsystemfrequency,i.e.,ifoneunitfails,theremustbeamplereserveontheotherunitstomakeupforthelossinaspecifiedtimeperiod.

8. WhyarethermalunitconstraintsconsideredinaUCtable?

Athermalunitcanundergoonlygradualtemperaturechangesandthistranslatesintoatimeperiodofsomehoursrequiredtobringtheunitontheline.Duetosuchlimitationsintheoperationofathermalplant,thethermalunitconstraintsaretobeconsideredintheUCproblem.

9. Whatisastart-upcostandwhatisitssignificance?

Becauseoftemperatureandpressureofathermalunitthatmustbemovedslowly,acertainamountofenergymustbemovedslowly,acertainamountofenergymustbeexpendedtobringtheuniton-line,anditisbroughtintotheUCproblemasastart-upcost.

Thestart-upcostmayvaryfromamaximumcold-startvaluetoaverysmallvalueiftheunitwasonlyturnedoffrecentlyandisstillrelativelyclosetotheoperatingtemperature.

10. Writetheexpressionsofastart-upcostwhencoolingandwhenbanking.

Start-upcostwhencooling=C (1–e )C+C

Start-upcostwhenbanking=C ×t×C+C

whereC isthecold-startcost(MBtu),Cisthefuelcost,C isthe

fixedcost(includescrewexpensesandmaintainableexpenses),αisthethermaltimeconstantfortheunit,C isthecostof

maintainingaunitatoperatingtemperature(MBtu/hr),andtisthetimetheunitwascooled(hr).

11. WhatarethetechniquesusedforgettingthesolutiontotheUCproblem?

1. Priority-listscheme.2. Dynamicprogramming(DP)method.3. Lagrange’srelaxation(LR)method.

12. Whatarethestepsofanenumerationschemeoffindingthemosteconomicalcombinationofunitstomeetaloaddemand?

1. Totryallpossiblecombinationsofunitsthatcansupplytheload.

c F

t F

C F

t

–t/α

2. Todividethisloadoptimallyamongtheunitsofeachcombinationbytheuseofco-ordinationequations,soastofindthemosteconomicaloperatingcostofthecombination.

3. Thentodeterminethecombinationthathastheleastoperatingcostamongallthese.

13. Whatisashut-downruleoftheUCoperation?

Iftheoperationofthesystemistobeoptimized,unitsmustbeshutdownastheloadgoesdownandthenrecommittedasitgoesbackup.Toknowwhichunitstodropandwhen,oneapproachcalledtheshut-downrulemustbeusedfromwhichasimplepriority-listschemeisdeveloped.

14. Whatisapriority-listmethodofsolvingaUCproblem?

Inthismethod,firstthefull-loadaverageproductioncostofeachunit,whichissimplythenetheatrateatfullloadmultipliedbythefuelcost,iscomputed.Then,intheorderofascendingcosts,theunitsarearrangedtocommittheloaddemand.

15. Inapriority-listmethod,whichunitisloadedfirstandtobefollowedbywhichunits?

Themostefficientunitisloadedfirst,tobefollowedbythelessefficientunitsintheorderasloadincreases.

16. WhatisthechiefadvantageoftheDPmethodoverothermethodsinsolvingtheUCproblem?

Resolutioninthedimensionalityofproblems,i.e.,havingobtainedtheoptimalwayofloadingKnumberofunits,itisquiteeasytodeterminetheoptimalwayofloading(K+1)numberofunits.

17. Whatisthethermalconstraintminimumup-time?

Minimumup-timeisthetimeduringwhichiftheunitisrunning,itshouldnotbeturnedoffimmediately.

18. Whatisminimumdown-time?

Iftheunitisstopped,thereisacertainminimumtimerequiredtostartitandputitontheline.

19. Whatisspinningreserve?

Toensurethecontinuityofsupplytomeetrandomfailures,thetotalgeneratingcapacityon-linemusthaveadefinitemarginovertheloadrequirementsatanypointoftime.Thismarginiscalledspinningreserve,whichensurescontinuationbymeetingthedemanduptoacertainextentofprobablelossofgeneratingcapacity.

20. Whatdoyoumeanbyabreachofsystemsecurity?

Someintolerableorundesirableconditionsofsystemoperationistermedasabreachofsystemsecurity.

21. InanoptimalUCproblem,whatisconsideredasabreachofsecurity?

Insufficientgeneratingcapacityofthesystemataparticularinstantoftime.

22. WhatisPatton’ssecurityfunction?Giveitsexpression.

Patton’ssecurityfunctionestimatestheprobabilitythattheavailablegeneratingcapacityataparticulartimeislessthanthetotalloaddemandonthesystematthattime.

Itisexpressedas

whereP istheprobabilityofthesystembeinginthei stateand

r theprobabilitythatthesystemstate‘i’causesabreachof

systemsecurity.

23. HowtheoptimalUCtableismodifiedwithconsiderationofsecurityconstraints?

WheneverthesecurityfunctionexceedsMTIL(S>MTIL),theUCtableismodifiedbycommittingthenextmosteconomicunittosupplytheloads.Withthenewunitbeingcommitted,thesecurityfunctionisthenestimatedandcheckedwhetheritisS<MTILornot.

24. Whatisthesignificanceofmust-runconstraintsconsideredinpreparingtheUCtable?

Someunitsaregivenamust-runrecognizationduringcertaintimesoftheyearforthereasonofvoltagesupportonthetransmissionnetworkorforsuchpurposesassupplyofsteamforusesoutsidethesteamplantitself.

REVIEWQUESTIONS

1. UsingtheDPmethod,howdoyoufindthemosteconomicalcombinationoftheunitstomeetaparticularloaddemand?

2. ExplainthedifferentconstraintsconsideredinsolvingaUCproblem.

3. CompareanoptimalUCproblemwithaneconomicalloaddispatchproblem.

4. ExplaintheneedofanoptimalUCproblem.5. DescribethereliabilityconsiderationinanoptimalUCproblem.6. Describethestart-upcostconsiderationinanoptimalUC

problem.

PROBLEMS

1. Apowersystemnetworkwithathermalpowerplantisoperatingbyfourgeneratingunits.Determinethemosteconomicalunitto

i

i

th

becommittedtoaloaddemandof10MW.Preparetheunitcommitmenttablefortheloadchangesinstepsof1MWstartingfromminimumloadtomaximumload.Theminimumandmaximumgeneratingcapacitiesandcost-curveparametersofunitslistedinatabularformaregiveninthefollowingtable.

2. PreparetheunitcommitmenttablewiththeapplicationofDPapproachforthesystemhavingfourthermalgeneratingunits,whichhavethefollowingcharacteristicparameters.Alsoobtainthemosteconomicalstationoperatingcostforthecompleterangeofstationcapacity.

Generatingunitparameters

3. ForthepowerplantofProblem2,andforthedailyloadcyclegiveninthefigure,preparethereliabilityconstrainedoptimalunitcommitmenttable.Alsoincludethestart-upconsiderationfromthepointofviewofoveralleconomywiththestart-upcostofanyunitbeingRs.75.

Dailyloadcurve

5

OptimalPower-FlowProblem—SolutionTechnique

OBJECTIVES


knowtheoptimalpowerflowproblemconceptstudythemajorstepsforoptimalpowerflowsolutiontechniquesformulatethemathematicalmodelingforoptimalpowerflowproblemwithandwithoutinequalityconstraintsdevelopalgorithmsforoptimalpowerflowproblemswithandwithoutinequalityconstraints

5.1INTRODUCTION

Theproblemofoptimizingtheperformanceofapowersystemnetworkisformulatedasageneraloptimizationproblem.Itisrequiredtostatefromwhichaspecttheperformanceofthepowersystemnetworkisoptimized.

Inoptimizationproblem,theobjectivefunctionbecomes‘tominimizetheoverallcostofgenerationineconomicschedulingandunitcommitmentproblem’:

Itisbasedonallocatingthetotalloadonastationamongvariousunitsinanoptimalwaywithcasesbeingtakenintoconsiderationinaload-schedulingproblem.Itisbasedonallocatingthetotalloadonthesystemamongthevariousgeneratingstations.

Theoptimalpowerflowproblem:

referstotheloadflowthatgivesmaximumsystemsecuritybyminimizingtheoverloads,aimsatminimumoperatingcostandminimumlosses,shouldbebasedonoperationalconstraints,andisastaticoptimizationproblemwiththecostfunctionasascalarobjectivefunction.

ThesolutiontechniqueforanoptimalpowerflowproblemwasfirstproposedbyDommelandTinneyandhasfollowingthreemajorsteps:

1. ItisbasedontheloadflowsolutionbytheNewton–Raphson(N–R)method.

2. Afirst-ordergradientmethodconsistsofanalgorithmthatadjuststhegradientforminimizingtheobjectivefunction.

3. Useofpenaltyfunctionstoaccountforinequalityconstraintsondependentvariables.

Theoptimizationproblemofminimizingtheinstantaneousoperatingcostsintermsofrealandreactive-powerflowsisstudiedinthisunit.

Theoptimalpowerflowproblemwithoutconsideringconstraints,i.e.,unconstrainedoptimalpowerflowproblem,isfirststudiedandthentheoptimalpowerflowproblemwithinequalityconstraintsisstudied.Theinequalityconstraintsareintroducedfirstoncontrolvariablesandthenondependentvariables.

5.2OPTIMALPOWER-FLOWPROBLEMWITHOUTINEQUALITYCONSTRAINTS

Theprimaryobjectiveoftheoptimalpowerflowsolutionistominimizetheoverallcostofgeneration.Thisisrepresentedbyanobjectivefunction(or)costfunctionas:

subjecttopower(load)flowconstraints:

and

whereV =|V |∠δ isthevoltageatbus‘i’,V =|V |∠δthevoltageatbus‘j’,Y =|Y |∠θ themutualadmittancebetweenthei andj buses,P thespecifiedreal-poweratbusi,Q thespecifiedreactivepoweratbusi,thenetrealpowerinjectedintothesystematthei bus,P =P

–P ,andthenetreactivepowerinjectedintothesystem

atthei bus,Q =Q –Q .

Forloadbuses(or)P–Qbuses,PandQarespecifiedandhenceEquations(5.2)and(5.3)formtheequalityconstraints.

ForP–Vbuses,Pand|V|arespecifiedasthefunctionofsomevectorsandarerepresentedas:

f(x,y)=0

wherexisavectorofdependentvariablesandisrepresentedas

andyisavectorofindependentvariablesandisrepresentedas

i i i j j j

ij ij ij

i

i

i Gi

Di

i Gi Di

th th

th

th

Outoftheseindependentvariables(Equation(5.6)),certainvariablesarechosenascontrolvariables,whicharetobevariedtoyieldanoptimalvalueoftheobjectivefunction.Theremainingindependentvariablesarecalledfixedordisturbanceoruncontrollableparameters.

Let‘u’bethevectorofcontrolvariablesand‘p’thevectoroffixedordisturbancevariables.

Hence,thevectorofindependentvariablescanberepresentedasthecombinationofvectorofcontrolvariables‘u’andvectoroffixedordisturbanceoruncontrollablevariables‘p’andisexpressedas

Thechoiceof‘u’and‘p’dependsonwhataspectofpowersystemistobeoptimized.Thecontrolparametersmaybe:

1. voltagemagnitudeatP–Vbuses,2. P atgeneratorbuseswithcontrollablepower,

3. slackbusvoltageandregulatingtransformertapsettingasadditionalcontrolvariables,and

4. inthecaseofbuseswithreactive-powercontrol,Q istakenasa

controlvariable.

Now,theoptimalpowerflowproblemcanbestatedas

minC=C(x,u)(5.8)

Gi

Gi

subjecttoequalityconstraints:

f(x,u,p)=0(5.9)

DefinethecorrespondingLagrangianfunction byaugmentingtheequalityconstrainttotheobjectivefunctionthroughaLagrangianmultiplierλas

min· =C(x,u)+λ [f(x,u,p)](5.10)

whereλisavectoroftheLagrangianmultiplierofsuitabledimensionandisthesameasthatofequalityconstraint,f(x,u,p).

Thenecessaryconditionsforanoptimalsolutionareasfollows:

Equation(5.13)isthesameasequalityconstraints,

andtheexpressionsfor and arenotveryinvolved.

ConsiderthegeneralloadflowproblemoftheN–Rmethodbyconsideringasetof‘n’non-linearalgebraicequations,

f (x ,x …x )=0fori=1,2…n(5.14)

Letx ,x ,…x betheinitialvalues.

i 1 2 n

1 2 n

T

0 0 0

0 0 0

andlet∆x ,∆x ,…∆x bethecorrections,whichonbeingaddedtotheinitialguess,giveanactualsolution:

∴f (x +Δx ,+x +Δx ,…,x +Δx )=0fori=1,2…n(5.15)

ByexpandingtheseequationsaccordingtoTaylor’sseriesaroundtheinitialguess,weget

where arethederivativesoff with

respecttox ,x …x evaluatedat(x ,x ,…x ).

Neglectinghigherorderterms,wecanwriteEquation(5.16)inamatrixformas

orinavectormatrixformas

f +J Δx ≅0(5.18)

whereJ isknownastheJacobianmatrixandisobtainedbydifferentiatingthefunctionvector‘f’withrespectto‘x’andevaluatingitatx .

1 2 n

i 1 1 2 2 n n

i

1 2 n 1 2 n

0 0 0

0 0 0 0 0 0

0 0 0

0 0 0

0

0

BycomparingEquations(5.11),(5.12),and(5.13)with

Equations(5.17)and(5.18),itisobservedthat isthe

JacobianmatrixandthepartialderivativesofequalityconstraintswithrespecttodependentvariablesareobtainedasJacobianelementsinaloadflowsolution.

Theequalityconstraintsarebasicallythepowerflowequations,i.e.,realandreactive-powerflowequations:

equalityconstraintsforP–Qbus

P istheequalityconstraintforP–Vbus

‘x’isthedependentvariablelike|V |,δ

Then, maybeexpressedaspartialderivativesof

TheEquations(5.11),(5.12),and(5.13)arenon-linearalgebraicequationsandcanbesolvediterativelybyemployingasimpletechniquethatisa‘gradientmethod’andisalsocalledthesteepestdescentmethod.

Thebasictechniqueemployedinthesteepestdescentmethodistoadjustthecontrolparameters‘u’soastomovefromonefeasiblesolutionpointtoanewfeasiblesolutionpointinthedirectionofthesteepestdescent(ornegativegradient).Here,thestartingpointoffeasiblesolutionisonewhereasetofvalues‘x’(i.e.,dependentvariables)satisfiesEquation(5.13)forgiven‘u’and‘p’.Thenewfeasiblesolutionpointreferstoalocationwherethelowerobjectivefunctionisachieved.

i

i i

Thesemovesaretoberepeatedinthedirectionofnegativegradienttillminimumvalueisreached.Hence,thismethodofobtainingasolutiontonon-linearalgebraicisalsocalledthenegativegradientmethod.

5.2.1Algorithmforcomputationalprocedure

Thealgorithmforobtaininganoptimalsolutionbythesteepestdescentmethodisgivenbelow:

Step1:Makeaninitialguessforcontrolvariables(u ).

Step2:FindthefeasibleloadflowsolutionbytheN–Rmethod.TheN–Rmethodisaniterativemethodandthesolutiondoesnotsatisfytheconstraintequation(5.13).Hence,tosatisfyEquation(5.12),‘x’isimprovedasfollows:

x =x +Δx

∆xisobtainedbysolvingthesetoflinearequationsoftheJacobianmatrixofEquation(5.18)asgivenbelow:

f(x +Δx,y)=f(x ,y)+ (x ,y)Δx=0

ThefinalresultsofStep-2provideafeasiblesolutionof‘x’andtheJacobianmatrix.

Step3:SolveEquation(5.11)forλanditisobtainedas

0

r+1 r

r r r

Step4:SubstituteλfromEquation(5.19)intoEquation(5.12)andcalculatethegradient:

Forcomputingthegradient,theJacobianmatrix, ,

isalreadyknownfromStep2.

Step5:Ifthegradient∇ isnearlyzerowithinthespecifiedtolerance,theoptimalsolutionisobtained.Otherwise,

Step6:Findanewsetofcontrolvariablesas

u =u +Δu(5.21)

whereΔu=−α∇ .(5.22)

Here,∆uisastepinthenegativedirectionofthegradient.

Theparameterαisapositivescalar,whichcontrolsthestepi’s(sizeofsteps),andthechoiceofαisveryimportant.

Toosmallavalueofαguaranteestheconvergencebutslowsdownitsrate.Toohighavalueofitcausesoscillationsaroundtheoptimalsolution.Severalmethodsaresuggestedfordeterminingthebestvalueofαforagivenproblemandforanoptimumchoiceofstepsize.

αisaproblem-dependentconstant.Experienceandproperjudgmentarenecessaryinchoosingavalueofit.

new old

Steps1,2,and5arerepeatedforanewvalueof‘u’tillanoptimalsolutionisreached.

5.3OPTIMALPOWER-FLOWPROBLEMWITHINEQUALITYCONSTRAINTS

5.3.1Inequalityconstraintsoncontrolvariables

InSection5.2,theunconstrainedoptimalpowerflowproblemandthecomputationalprocedureforobtainingtheoptimalsolutionarediscussed.Now,inthissection,theinequalityconstraintsareintroducedoncontrolvariables,andthenthemethodofobtainingasolutiontotheoptimalpowerflowproblemisdiscussed.

Thepermissiblevaluesofcontrolvariables,infact,arealwaysconstrained,suchthat

u ≤u≤u (5.23)

Forexample,iftherealpowerorreactive-powergenerationaretakenascontrolvariables,theninequalityconstraintsbecome

P ≤P ≤P

Q ≤Q ≤Q (5.24)

Infindingtheoptimalpowerflowsolution,Step6ofthealgorithmofSection5.2.1givesthechangeincontrolvariableas

Δu=−α∇

where andthenewcontrolvariable,u =u

+∆u.

min max



new old

Thisnewvalueofcontrolvariablemustbecheckedwhetheritviolatestheinequalityconstraintsonthecontrolvariableornot:

i.e.,u ≤u ≤u

Ifthecorrection∆ucausestoexceedoneofthelimits,‘u ’issetequaltothecorrespondinglimit,i.e.,thenewvalueofu isdeterminedas

otherwisesetu =u +∆u

Afteracontrolvariablereachesanyofthelimits,itscomponentinthegradientshouldcontinuetobecomputedinlateriteration,asthevariablemaycomewithinlimitsatsomelaterstages.

TheoptimalityconditionunderinequalityconstraintscanberewrittenasKuhn–Tuckerconditionsgivenbelow:

Therefore,now,inStep5ofthealgorithmofSection5.2.1,thegradientvectorhastosatisfytheoptimalityconditiongivenbyEquation(5.26).

5.3.2Inequalityconstraintsondependentvariables—penaltyfunctionmethod

i(min) i(new) i(max)

i

i

i(new) i(old) i

Inthissection,theoptimalsolutiontoanoptimalpowerflowproblemwillbeobtainedwiththeintroductionofinequalityconstraintsondependentvariablesandpenaltiesfortheirviolation.

Theinequalityconstraintsondependentvariablesspecifiedintermsofupperandlowerlimitsare

x ≤x≤x (5.27)

wherexisavectorofdependentvariables.

Forexample,ifthebusvoltagemagnitude|V |istakenasadependentvariable,theinequalityconstraintbecomes

|V| ≤|V|≤|V| onaP−Qbus(5.28)

Theabove-mentionedinequalityconstraintscanbehandledconvenientlybyamethodknownasthepenaltyfunctionmethod.Inthismethod,theobjectivefunctionisaugmentedbypenaltiesfortheviolationsofinequalityconstraints.Duetothisaugmentedobjectivefunction,thesolutionliessufficientlyclosetotheconstraintlimitswhentheviolationsoftheselimitshavetakenplace.Thepenaltyfunctionmethod,inthiscase,isvalidsincetheseconstraintsareseldomrigidlimitsinthestrictsensebutare,infact,softlimits(e.g.,|V|≤1.0onaP–Qbusreallymeans|V|shouldnotexceed1.0toomuchand|V|=1.01maystillbepermissible).

Wheninequalityconstraintsareviolated,theobjectivefunctioncanbemodifiedbyaugmentingpenaltiesas

whereω isthepenaltyintroducedforeachoftheviolatedinequalityconstraints.

Asuitablepenaltyfunctionisdefinedas

(min) (max)

i

min max

j

whereγ iscalledapenaltyfactorsinceitcontrolsthedegreeofpenaltyandisarealpositivenumber.

Aplotofthepenaltyfunction,whichisproposed,isshowninFig.5.1.Theplotclearlyindicateshowtherigidlimitsarereplacedbysoftlimits.Thenecessaryoptimalityconditions:

FIG.5.1Penaltyfunction

wouldnowbemodifiedasgivenbelow,whiletheconditionofEquation(5.13),i.e.,loadflowequations,remainsunchanged:

j

Intheaboveequations,thevector canbe

calculatedfromthepenaltyfunctionω .

Thevector canbeobtainedfromEquation(5.30)

andwouldcontainonenon-zerotermcorrespondingtodependentvariablex .

Thevector ,sincethepenaltyfunctionson

dependentvariablesareindependentofcontrolvariables.

Ifwechooseahighervalueforγ ,thepenaltyfunctionω canbemadesteepersothatthesolutionliesclosertotherigidlimits,buttheconvergencebecomespoorer.Innormalpractice,itisrequiredtostartwithalowervalueofγ andthenincreaseitduringtheoptimizationprocessifthesolutionviolatesconstraintsaboveacertaintolerancelimit.

ItisconcludedthatthesolutiontooptimalpowerflowproblemcanbeachievedbysuperimposingtheN–Rmethodofloadflowontheoptimalpowerflowproblemwithrespecttorelevantinequalityconstraints.Thesesolutionsareoftenrequiredforsystemplanningandoperation.

KEYNOTES

j

j

j

j

j

Theoptimalpowerflowproblem:

1. referstoloadflow,whichgivesmaximumsystemsecuritybyminimizingtheoverloads,

2. aimsatminimumoperatingcostandminimumlosses,3. shouldbebasedonoperationalconstraints,and4. isastaticoptimizationproblemwithcostfunctionasthescalarobjective

function.

ThesolutiontechniqueforoptimalpowerflowproblemproposedbyDommelandTinneyhasthefollowingthreemajorsteps:

1. itisbasedontheloadflowsolutionbytheN–Rmethod.2. afirst-ordergradientmethodconsistsofanalgorithmthatadjuststhegradient

forminimizingtheobjectivefunction.3. useofpenaltyfunctionstoaccountforinequalityconstraintsondependent

variables.

ForP–Vbuses,Pand|V|arespecifiedasfunctionsofsomevectorsandarerepresentedasf(x,y)=0wherexisavectorofdependentvariablesandyisavectorofindependentvariables.Thevectorofindependentvariablescanberepresentedasthecombinationofvectorofcontrolvariables‘u’andvectoroffixedor

disturbanceoruncontrollablevariables‘p’andisexpressedas

Thecontrolparametersare:

1. voltagemagnitudeatP–Vbuses,2. P atgeneratorbuseswithcontrollablepower,

3. slackbusvoltageandregulatingtransformertapsettingasadditionalcontrolvariables,and

4. inthecaseofbuseswithreactive-powercontrol,Q istakenascontrolvariable.

Optimalpowerflowproblemcanbestatedas

minC=C(x,u)

subjecttoequalityconstraints:

f(x,u,p)=0


1. ThesolutiontechniqueproposedbyDommelandTinneyfortheoptimalpowerflowproblemisbasedonthreemajorsteps.Whatarethey?

1. LoadflowsolutionbyN–Rmethod.2. Afirst-ordergradientmethod.3. Useofpenaltyfunctionstoaccountforinequalityconstraintsondependent

variables.

2. Writetheexpressionsofpowerflowequalityconstraintsintermsofoptimalpowerflowproblem.

Gi

Gi

3. Writethenecessaryconditionsforobtaininganoptimalsolutiontotheoptimalpowerflowproblemwithoutinequalityconstraints.

4. Whatisanoptimalpowerflowproblem?

1. Ageneraloptimizationproblemreferstoloadflow,whichgivesmaximumsystemsecuritybyminimizingtheoverloads.

2. Optimalpowerflowproblemisastaticoptimizationproblemwithcostfunctionasascalarobjectivefunction.

5. WhichparametersareobtainedasJacobianelementsinanoptimalpowerflowproblem?

Partialderivativesofequalityconstraintswithrespectto

dependentvariables.i.e.,

6. Whatisthebasictechniqueemployedinthesteepestdescentmethod?

Toadjustthecontrolvariablessoastomovefromonefeasiblesolutionpointtoanewfeasiblesolutionpointwherethelowerobjectivefunctionisachieved.

7. Whythesteepestdescentmethodiscalledthenegativegradientmethod?

Themovesfromonefeasiblesolutionpointtoanewfeasiblepointaretoberepeatedinthedirectionofnegativegradienttillaminimumvalueisreached.

8. Whatistheeffectoftoosmallavalueofαandtoohighvalueofαontheconvergenceofasolution?

Thetoosmallvalueofαguaranteestheconvergencebutslowsdowntherateofconvergence,whereastoohighavalueofitcausesanoscillatorysolutionaroundtheoptimalsolution.

9. WhenwilltheKuhn–Tuckerconditionsbecomeoptimalityconditions?

Whileintroducinginequalityconstraintsoncontrolvariables.

10. Whenwillthepenaltyfunctionmethodbeadoptedinsolvingoptimalpower-flowproblem?

Whileintroducinginequalityconstraintsondependentvariables.

11. Whatistheeffectofchoosingahighervalueforγ thepenalty

factor?

Thepenaltyfunctionω canbemadesteepersothatthesolution

liesclosertotherigidlimits,butconvergencebecomespoorer.


1. ThesolutiontechniqueforanoptimalpowerflowproblemproposedbyDommelandTinneyhasthestepsbasedon:

1. loadflowsolutionbytheN–Rmethod.2. afirst-ordergradientmethod.3. useofpenaltyfunctionstoaccountforinequalityconstraintsondependent

variables.4. non-linearitiespresentintheoperationmethods

1. (i)and(iii)2. (ii)and(iii)3. Allexcept(iii)4. Allexcept(iv).

2. AccordingtotheDommelandTinneytechnique,________methodisemployedforobtainingtheoptimalsolution.

1. Divergencemethod.2. Kuhn–Tuckermethod.3. First-ordergradientmethod.4. Lagrangianmultipliermethod.

3. Inanoptimalpowerflowsolution,theobjectivefunctionmin

subjecttotheequalityconstraints:

1.

2.

3.

j

j

4.

4. Theequalityconstraintsofanoptimalpowerflowproblemarespecifiedasfunctionf(x,y)=0,wherexis:

1. Vectorofdependentvariables.2. Vectorofindependentvariables.3. Vectorofcontrolvariables.4. Vectorofuncontrolledvariables.

5. Theequalityconstraintsofanoptimalpowerflowproblemarespecifiedasfunctionf(x,y)=0,whereyis:

1. Vectorofdependentvariables.2. Vectorofindependentvariables.3. Vectorofcontrolvariables.4. Vectorofuncontrolledvariables.

6. Theindependentvariablesare:

1. Controlvariables.2. Disturbancevariables.3. Both(a)and(b).4. Noneofthese.

7. Thecontrolparameterinanoptimalpowerflowproblemis:

1. VoltagemagnitudeattheP–Vbus.2. P andQ atthegeneratorbus.

3. Slackbusvoltage.4. Allofthese.

8. Ifxisthevectorofdependentvariables,yisthevectorofindependentvariables,uisthevectorofcontrolvariables,andpisthevectorofdisturbancevariables,thenamongthefollowingwhichiscorrect?

1.

2.

Gi Gi

3.

4. Noneofthese.

9. Inanoptimalpowerflowsolution,theequalityconstraintsarebasically:

1. Voltageequations.2. Powerflowequations.3. Currentflowequations.4. Both(a)and(c).

10. WhichofthefollowingisobtainedasJacobianelementsinaload-flowsolution?

1. Partialderivativesofequalityconstraintswithrespecttodependent

variables,

2. Partialderivativesofequalityconstraintswithrespecttoindependent

variables,

3. Partialderivativesofequalityconstraintswithrespecttocontrolvariables,

4. Partialderivativesofequalityconstraintswithrespecttocontrolled

variables,

11. Inanoptimalpowerflowproblem,thebasictechniqueistoadjustthecontrolvariableusoastomovefromonefeasiblesolutionpointtoanewsolutionpointwithalowervalueofobjectivefunction.Thistechniqueis:

1. Steepestdescentmethod.2. Negativegradientmethod.3. Either(a)or(b).4. Noneofthese.

12. Thenewsetofcontrolvariablesisu =u =+∆u.Thechange

incontrolvariable∆uisexpressedas

1. Δu=−α∇2. Δu=−α3. Δu=−∇α4. Noneofthese.

13. αisaparameterandtoosmallavalueofαresultsinthefollowing:

1. Guaranteestheconvergence.2. Slowsdowntherateofconvergence.3. Increasestherateofconvergence.

new old

1. Only(a).2. (b)Only.3. (a)and(c).4. (a)and(b).

14. ________valueofαcausesanoscillatorysolutionaroundtheoptimalsolution.

1. Toohigh.2. Toolow.3. Inbetweentoohighandtoolow.4. Noneofthese.

15. TheKuhn–Tuckercondition if

1. U ≤U ≤U .

2. U =U .

3. U =U .

4. Noneofthese.

16. if

1. U ≤U ≤U .

2. U =U .

3. U =U .

4. Noneofthese.

17. if

1. U ≤U ≤U .

2. U =U .

3. U =U .

4. Noneofthese.

18. Theinequalityconstraintsondependentvariablesareconvenientlyhandledby________method.

1. Penaltyfunction.2. Kuhn–Tucker.3. Newton–Raphson.4. Noneofthese.

19. Theinequalityconstraintlimitsareusuallynotvery________limit(soft/rigid)butareinfact________limits(soft/rigid).

20. Intheabove,thepenaltyintroduced(ω )foreachviolationof

________constraint.

1. Equality.2. Inequality.3. Either(a)or(b).4. Noneofthese.

21. Fortheoptimalpowerflowproblem,theequalityconstraintsare

i(min) i i(max)

i i(min)

i i(max)

i(min) i i(max)

i i(min)

i i(max)

i(min) i i(max)

i i(min)

i i(max)

j

specifiedasfunction,f(x,y)=0,where:

1. xisavectorofadependentvariable.

yisavectorofanindependentvariable.

2. xisavectorofanindependentvariable.

yisavectorofadependentvariable.

3. xisavectorofadependentandanindependentvariable.

yisavectorofaconstant.

4. xisavectorofacontrolvariable.

yisavectorofanuncontrolledvariable.

22. Toobtaintheoptimalsolutiontoanoptimalpowerflowproblem,asimpletechniquethatcanbeemployedis

1. Apositivegradientmethod.2. Negativegradientmethod.3. Fastdecoupledmethod.4. Priorityordering.

23. Thepenaltyintroducedforeachviolatedinequalityconstraintisω .Forahighervalueofω ,

1. Thepenaltyfunctioncanbemadesteeper.2. Thesolutionliesclosertotherigidlimits.3. Rateofconvergencebecomespoorer.4. Rateofconvergencebecomeshigher.

1. (a)and(b).2. (a)and(c).3. Allexcept(d).4. Allofthese.

24. Inanoptimalpowerflowsolution,theequalityconstraintsarespecifiedasafunctionof:

1. Vectorofdependentvariables.2. Vectorofindependentvariables.3. Vectorofconstants.4. Both(a)and(b).

25. Controlvariable‘u’anddisturbancevariable‘p’comeunder:

1. Dependentvariables.2. Independentvariables.3. Both(a)and(b).4. Noneofthese.

26. Theoptimalpowerflowproblemwithinequalityconstraintsondependentvariablescanbesolvedconvenientlyby

1. Negativegradientmethod.2. Costfunctionmethod.3. Penaltyfunctionmethod.4. Steepestdescentmethod.

27. Penaltyfunctionsondependentvariablesare________ofthecontrolvariables.

1. Dependent.2. Independent.3. Dependentinonecaseandindependentonanothercase.

j j

4. Noneofthese.

28. Theoptimalpowerflowproblem:

1. Referstotheloadflowthatgivesmaximumsystemsecuritybyminimizingtheoverloads.

2. Aimsatminimumoperatingcostandminimumlosses.3. Shouldbebasedonoperationalconstraints.4. Allofthese.

29. Theoptimalpowerflowproblemis:

1. Astaticoptimizationproblemwiththecostfunctionasascalarobjectivefunction.

2. Adynamicoptimizationproblemwiththecostfunctionasascalarobjectivefunction.

3. Fullystaticandpartiallydynamicoptimizationproblemwiththecostfunctionasanobjectivefunction.

4. Noneofthese.

30. Forahighervalueofthepenaltyfactor,

1. Thepenaltyfunctioncanbemadesteeper.2. Thesolutionliesclosertotherigidlimit.3. Convergencebecomespoorer.4. Allofthese.

REVIEWQUESTIONS

1. Discussoptimalpowerflowproblemswithoutinequalityconstraints.

2. Obtainanoptimalpowerflowsolutionwithinequalityconstraintsoncontrolvariables.

3. Explainthepenaltyfunctionmethodofobtaininganoptimalpowerflowsolutionwithinequalityconstraintsondependentvariables.

4. Developanalgorithmforobtainingtheoptimalpowerflowsolutionwithoutinequalityconstraintsbythesteepestdescentmethod.

6

Hydro-ThermalScheduling

OBJECTIVES


knowtheimportanceofhydro-thermalco-ordinationdevelopthemathematicalmodelingoflong-termhydro-thermalco-ordinationstudytheKirchmayer’smethodforshort-termhydro-thermalco-ordinationstudytheadvantagesofhydro-thermalplantscombination

6.1INTRODUCTION

Nostateorcountryisendowedwithplentyofwatersourcesorabundantcoalandnuclearfuel.Forminimumenvironmentalpollution,thermalgenerationshouldbeminimum.Hence,amixofhydroandthermal-powergenerationisnecessary.Thestatesthathavealargehydro-potentialcansupplyexcesshydro-powerduringperiodsofhighwaterrun-offtootherstatesandcanreceivethermalpowerduringperiodsoflowwaterrun-offfromotherstates.Thestates,whichhavealowhydro-potentialandlargecoalreserves,canusethesmallhydro-powerformeetingpeakloadrequirements.Thismakesthethermalstationstooperateathighloadfactorsandtohavereducedinstalledcapacitywiththeresulteconomy.Instates,whichhaveadequatehydroaswellasthermal-powergenerationcapacities,powerco-ordinationtoobtainamosteconomicaloperatingstateisessential.Maximumadvantageofcheaphydro-powershouldbetakensothatthecoalreservescanbeconservedandenvironmentalpollutioncanbeminimized.Thewholeorapartofthebaseloadcanbe

suppliedbytherun-offriverhydro-plants,andthepeakortheremainingloadisthenmetbyapropermixofreservoir-typehydro-plantsandthermalplants.Determinationofthisbyapropermixisthedeterminationofthemosteconomicaloperatingstateofahydro-thermalsystem.Thehydro-thermalco-ordinationisclassifiedintolong-termco-ordinationandshort-termco-ordination.

6.2HYDRO-THERMALCO-ORDINATION

Initially,thereweremostlythermalpowerplantstogenerateelectricalpower.Thereisaneedforthedevelopmentofhydro-powerplantsduetothefollowingreasons.

1. Duetotheincrementofpowerintheloaddemandfromallsidessuchasindustrial,agricultural,commercial,anddomestic.

2. Duetothehighcostoffuel(coal).3. Duetothelimitedrangeoffuel.

Thehydro-plantscanbestartedeasilyandcanbeassignedaloadinveryshorttime.However,inthecaseofthermalplants,itrequiresseveralhourstomaketheboilers,superheater,andturbinesystemreadytotaketheload.Forthisreason,thehydro-plantscanhandlefast-changingloadseffectively.Thethermalplantsincontrastareslowinresponse.Hence,duetothis,thethermalplantsaremoresuitabletooperateasbaseloadplants,leavinghydro-plantstooperateaspeakloadplants.

FIG.6.1Fundamentalhydro-thermalsystem

Themaximumadvantageofcheaphydro-powershouldbetakensothatthecoalreservescanbeconservedandenvironmentalpollutioncanbeminimized.Inahydro-thermalsystem,thewholeorapartofthebaseloadcanbesuppliedbytherun-offriverhydro-plantsandthepeakortheremainingloadisthenmetbyaproperco-ordinationofreservoir-typehydro-plantsandthermalplants.

Theoperatingcostofthermalplantsisveryhighandatthesametimeitscapitalcostislowwhencomparedwithahydro-electricplant.Theoperatingcostofahydro-electricplantislowanditscapitalcostishighsuchthatithasbecomeeconomicalaswellasconvenienttorunboththermalaswellashydro-plantsinthesamegrid.

Inthecaseofthermalplants,theoptimalschedulingproblemcanbecompletelysolvedatanydesiredinstantwithoutreferringtotheoperationatothertimes.Itisastaticoptimizationproblem.

Theoperationofasystemhavingbothhydroandthermalplantsismorecomplexashydro-plantshaveanegligibleoperatingcostbutarerequiredtorunundertheconstraintofavailabilityofwaterforhydro-generationduringagivenperiodoftime.Thisproblemisthe‘dynamicoptimizationproblem’wherethetimefactoristobeconsidered.

Theoptimalschedulingprobleminahydro-thermalsystemcanbestatedastominimizethefuelcostofthermalplantsundertheconstraintofwateravailabilityforhydro-generationoveragivenperiodofoperation.

Considerasimplehydro-thermalsystem,showninFig.6.1,whichconsistsofonehydroandonethermalplantsupplyingpowertoloadconnectedatthecenterinbetweentheplantsandisreferredtoasthefundamentalsystem.

Tosolvetheoptimizationprobleminthissystem,considertherealpowergenerationsoftwoplantsP

andP ascontrolvariables.Thetransmissionpower

lossisexpressedintermsoftheBcoefficientas

6.3SCHEDULINGOFHYDRO-UNITSINAHYDRO-THERMALSYSTEM

1. Incaseofhydro-unitswithoutthermalunitsinthesystem,theproblemissimple.Theeconomicschedulingconsistsofschedulingwaterreleasetosatisfythehydraulicconstraintsandtosatisfytheelectricaldemand.

2. Wherehydro-thermalsystemsarepredominantlyhydro,schedulingmaybedonebyschedulingthesystemtoproduceminimumcostforthethermalsystems.

3. Insystemswherethereisaclosebalancebetweenhydroandthermalgenerationandinsystemswherethehydro-capacityisonlyafractionofthetotalcapacity,itisgenerallydesiredtoschedulegenerationsuchthatthermalgeneratingcostsareminimized.

6.4CO-ORDINATIONOFRUN-OFFRIVERPLANTANDSTEAMPLANT

Arun-offriverhydro-plantoperatesasthewaterisavailableinneededquantities.Theseplantsareprovidedwithasmallpondageorreservoir,whichmakesitpossibletomeetthehourlyvariationofload.

Theratioofrun-offduringtherainyseasontotherun-offduringthedryseasonmaybeaslargeas100.Assuchtherun-offriverplantshaveverylittlefromcapacity.Theusefulnessoftheserun-offriverplantscanbeconsiderablyincreasedifsuchaplantisproperlyco-ordinatedwithathermalplant.Whensuchco-ordinationexists,thehydro-plantmaycarrythebaseloaduptoitsinstalledcapacityduringtheperiodofhighstreamflowsandthethermalplantmaycarrythepeakload.Duringtheperiodofleanflow,thethermalplantsuppliesthebaseloadandthehydro-plantsuppliesthepeakload.Thus,theloadmetbyathermalplantcanbeadjustedto

GThermal

GHydro

conformtotheavailableriverflow.Thistypeofco-ordinationofarun-offriverhydro-plantwithathermalplantresultsinagreaterutilizationfactoroftheriverflowandasavingintheamountoffuelconsumedinthethermalplant.

6.5LONG-TERMCO-ORDINATION

Typicallong-termco-ordinationmaybeextendedfromoneweektooneyearorseveralyears.Theco-ordinationoftheoperationofreservoirhydro-powerplantsandsteamplantsinvolvesthebestutilizationofavailablewaterintermsoftheschedulingofwaterreleased.Inotherwords,sincetheoperatingcostsofhydro-plantsareverylow,hydro-powercanbegeneratedatverylittleincrementalcost.Inacombinedoperationalsystem,thegenerationofthermalpowershouldbedisplacedbyavailablehydro-powersothatmaximumdecrementproductioncostswillberealizedatthesteamplant.Thelong-termschedulingprobleminvolvesthelong-termforecastingofwateravailabilityandtheschedulingofreservoirwaterreleasesforanintervaloftimethatdependsonthereservoircapacitiesandthechronologicalloadcurveofthesystem.Basedonthesefactorsduringdifferenttimesoftheyear,thehydroandsteamplantscanbeoperatedasbaseloadplantsandpeakloadplantsandviceversa.

Forthelong-termdrawdownschedule,abasicbestpolicyselectionmustbemade.Thebestpolicyisthatshouldthewaterbeusedundertheassumptionthatitwillbereplacedataratebasedonthestatisticallyexpectedrateorshouldthewaterbereleasedusingaworst-caseprediction?

Long-termschedulingismadebasedonanoptimizingpolicyinviewofstatisticallytreatedunknownssuchasload,hydraulicinflows,andunitavailability(i.e.,steamandhydro-plants).

Theusefultechniquesemployedforthistypeofschedulingproblemsinclude:

1. thesimulationofanentirelong-termoperationaltimeperiodforagivensetofoperatingconditionsbyusingthedynamicprogrammingmethod,

2. compositehydraulicsimulationmodels,and3. statisticalproductioncostmodels.

Forthelong-termschedulingofahydro-thermalsystem,thereshouldberequiredgenerationtomeettherequirementsofloaddemandandbothhydroandthermalgenerationsshouldbesoscheduledsoastomaintaintheminimumfuelcosts.Thisrequiresthattheavailablewatershouldbeputtoanoptimumuse.

6.6SHORT-TERMCO-ORDINATION

Theeconomicsystemoperationofthermalunitsdependsonlyontheconditionsthatexistfrominstanttoinstant.However,theeconomicschedulingofcombinedhydro-thermalsystemsdependsontheconditionsexistingovertheentireoperatingperiod.

Thistypeofhydro-thermalschedulingisrequiredforonedayoroneweek,whichinvolvesthehour-by-hourschedulingofallavailablegenerationsonasystemtogettheminimumproductioncostforthegiventime.Suchtypesofschedulingproblems,theload,hydraulicinflows,andunitavailabilitiesareassumedtobeknown.

Herealso,theproblemishowtosupplyload,aspertheloadcycleduringtheperiodofoperationsothatgenerationbythermalplantswillbeminimum.Thisconditionwillbesatisfiedwhenthevalueofhydro-powergenerationratherthanitsamountisamaximumoveracertainperiod.Thebasicproblemisthatdeterminingthedegreetowhichtheminimizedeconomyofoperatingthehydro-unitsatotherthanthemaximumefficiencyloadingmaybetoleratedforanincreasedeconomywithanincreasedloadorviceversatoresultinthelowest

totalthermalpowerproductioncostsoverthespecifiedoperatingperiod.

Thefactorsonwhichtheeconomicoperationofacombinedhydro-thermalsystemdependsareasfollows:

Loadcycle.Incrementalfuelcostsofthermalpowerstations.Expectedwaterinflowinhydro-powerstations.Waterheadthatisafunctionofwaterstorageinhydro-powerstations.Hydro-powergeneration.Incrementaltransmissionloss(ITL).

Thefollowingarethefewimportantmethodsforshort-termhydro-thermalco-ordination:

1. Constanthydro-generationmethod.2. Constantthermalgenerationmethod.3. Maximumhydro-efficiencymethod.4. Kirchmayer’smethod.

6.6.1Constanthydro-generationmethod

Inthismethod,ascheduledamountofwaterataconstantheadisusedsuchthatthehydro-powergenerationiskeptconstantthroughouttheoperatingperiod.

6.6.2Constantthermalgenerationmethod

Thermalpowergenerationiskeptconstantthroughouttheoperatingperiodinsuchawaythatthehydro-powerplantsuseaspecifiedandscheduledamountofwaterandoperateonvaryingpowergenerationschedulesduringtheoperatingperiod.

6.6.3Maximumhydro-efficiencymethod

Inthismethod,duringpeakloadperiods,thehydro-powerplantsareoperatedattheirmaximumefficiency;duringoff-peakloadperiodstheyoperateatanefficiencynearertotheirmaximum–efficiencywiththeuseofaspecifiedamountofwaterforhydro-powergeneration.

Kirchmayer’smethodisexplainedinSection6.8.

6.7GENERALMATHEMATICALFORMULATIONOFLONG-TERMHYDRO-THERMALSCHEDULING

Tomathematicallyformulatetheoptimalschedulingprobleminahydro-thermalsystem,thefollowingassumptionsaretobemadeforacertainperiodofoperationT(aday,aweek,orayear):

1. Thestorageofahydro-reservoiratthebeginningandattheendofperiodofoperationTarespecified.

2. Afteraccountingfortheirrigationpurpose,waterinflowtothereservoirandloaddemandonthesystemareknowndeterministicallyasfunctionsoftimewithcertainties.

Theoptimizationproblemhereistodeterminethewaterdischargerateq(t)soastominimizethecostofthermalgeneration.

Objectivefunctionis

Subjecttothefollowingconstraints:

(i)Therealpowerbalanceequation

P (t)+P (t)=P (t)+P (t)+P (t)

i.e.,P (t)+P (t)−P (t)−P (t)=0fort∈(0,T)(6.2)

where

P (t)istherealpowerthermalgenerationattime‘t’,

P (t)therealpowerhydrogenerationattime‘t’,

P (t)realpowerlossattime‘t’,and

GT GH L D D

GT GH L D

GT

GH

L

P (t)therealpowerdemandattime‘t’.

(ii)Wateravailabilityequation:

where

X′(t)isthewaterstorageattime‘t’,

X′(0)thewaterstorageatthebeginningofoperationtime,T,

X′(T)thewaterstorageattheendofoperationtime,T,

J(t)thewaterinflowrate,and

q(t)thewaterdischargerate.

(iii)Realpowerhydro-generation

Therealpowerhydro-generationP (t)isafunctionofwaterstorageX′(t)andwaterdischargerateq(t)

i.e.,P (t)=f(X′(t),q(t))(6.4)

6.7.1Solutionofproblem-discretizationprinciple

Bythediscretizationprinciple,theaboveproblemcanbeconvenientlysolved.TheoptimizationintervalTissub-dividedintoNequalsub-intervalsofΔttimelengthandovereachsub-interval,itisassumedthatallthevariablesremainfixedinvalue.

Thesameproblemcanbereformulatedas

D

GH

GH

subjecttothefollowingconstraints:

(i)Powerbalanceequation

P +P −P −P =0(6.6)

where

P isthethermalgenerationinKthinterval,

P thehydrogenerationinKthinterval,

P thetransmissionpowerlossinKthintervalandisexpressed

as

,and

P istheloaddemandintheKthinterval.

(ii)Wateravailabilityequation:

X′ −X′ −j Δt+q Δt=0(6.7)

whereX′ isthewaterstorageattheendofintervalK,jthewaterinflowrateinintervalK,andq thewaterdischargerateinintervalK.

DividingEquation(6.7)byΔt,itbecomes

X −X −j +q =0forK=1,2…N(6.8)

GT GH L D

GT

GH

L

D

K K K K

K

K

K

K

K (K−1) K K

K K

K

K K−1 K K

where isthewaterstorageindischargeunits.

x andx arespecifiedaswaterstorageratesatthebeginningandattheendoftheoptimizationinterval,respectively.

(iii)Therealpowerhydro-generationinanysub-intervalcanbewrittenas

P =h 1+0.5e(X +X )(q −ρ)(6.9)

where

h =9.81×10 h ′;

h ′isthebasicwaterheadwhichiscorrespondingtodead

storage,

ethewaterheadcorrectionfactortoaccountforthevariationinheadwithstorage,and

ρthenon-effectivedischarge(duetotheneedofwhichahydrogenerationcanrunatno-loadcondition).

Equation(6.9)canbeobtainedasfollows:

P =9.81×10 h (q −ρ)MW

where(q −ρ)istheeffectivedischargeinm /sandhistheaverageheadintheK intervalandisgivenas

whereAistheareaofcross-sectionofthereservoiratthegivenstorage

GH o

o o

o

GH av K

av

0 N

K K K−1 K

−3

K −3 K

K 3 K

th

K K K−1

h =h′ (1+0.5e(X +X ))

where ,whichistabulatedforvariousstorage

values

∴P =h 1+0.5e(X +X )(q

whereh =9.81×10 h′ .

Theoptimizationproblemismathematicallystatedforanysub-interval‘K’bytheobjectivefunctiongivenbyEquation(6.5),whichissubjectedtoequationconstraintsgivenbyEquations(6.6),(6.8),and(6.9).

Intheaboveoptimizationproblem,itisconvenienttochoosewaterdischargesinallsub-intervalsexceptonesub-intervalasindependentvariablesandhydro-generations,thermalgenerations,waterstoragesinallsub-intervalsandexceptwaterdischargeasdependentvariables;i.e.,independentvariablesarerepresentedbyq ,forK=2,3,…,NandforK≠1.DependentvariablesarerepresentedbyP ,P X ,andq ,forK=1,2,…,N.[Sincethewaterdischargeinonesub-intervalisadependentvariable.]

Equation(6.8)canbewrittenforallvaluesofK=1,2,…,N:

i.e.,X –X –j +q =0 forK=1

X –X –j +q =0 forK=2

X –X –j +q =0 fork=N interval

Byaddingtheabovesetofequations,weget

av o

GH o

o o

GT GH

K K K−1

K K K−1 K−ρ

–3

K

K K K 1

1 0 1 1

2 1 2 2

N (N–1) N N th

Equation(6.10)isknownasthewateravailabilityequation.

ForK=2,3,…,N,thereare(N–1)numberofwaterdischarges(q’s),whichcanbespecifiedasindependentvariablesandtheremainingone,i.e.,q ,isspecifiedasadependentvariableanditcanbedeterminedfromEquation(6.10)as

6.7.2Solutiontechnique

Forobtainingasolutiontotheoptimizationprobleminahydro-thermalsystem,anon-linearprogrammingtechniqueinconjunctionwiththefirst-ordergradientmethodisused.

DefinetheLagrangianfunction byaugmentingtheobjectivefunction(costfunction)givenbyEquation(6.5)withequalityconstraintsgivenbyEquations(6.6),(6.8),and(6.9)throughLagrangianmultipliers.

whereλ ,λ ,andλ aretheLagrangianmultipliersthataredualvariables.TheseareobtainedbytakingthepartialderivativesoftheLagrangianfunctionwithrespecttothedependentvariablesandequatingthemtozero.

1 2 3

1

K K K

SubstitutingEquation(6.8)inEquation(6.12)anddifferentiatingtheresultantequationwithrespecttoq ,weget

Fromtheaboveequations,foranysub-interval,theLagrangianmultiplierscanbeobtainedasfollows:

1. λ canbeobtainedfromEquation(6.13),

2. λ canbeobtainedfromEquation(6.14),and

3. λ′ canbeobtainedfromEquation(6.16)andremainingλ can

beobtainedfromEquation(6.15).

ThepartialderivativesoftheLagrangianfunctionwithrespecttoindependentvariablesgivethegradientvector:

Foroptimality,thegradientvectorshouldbezero

,iftherearenoinequalityconstraintsonthe

independentvariables,i.e.,oncontrolvariables(waterdischarges).

1

2

2 2(K≠1)

1

K

K

K

Ifnotwehavetofindoutthenewvaluesofcontrolvariablesthatwilloptimizetheobjectivefunction,thiscanbeachievedbymovinginthenegativedirectionofthegradientvectortoapoint,wherethevalueofobjectivefunctionisnearertotheoptimalvalue.

Itisaniterativeprocessandthisprocessisrepeatedtillallthecomponentsofthegradientvectorareclosertozerowithinaspecifiedtolerance.

6.7.3Algorithm

Step1: Assumeaninitialsetofindependentvariables,q forallsub-intervalsexceptthefirstsub-intervali.e.,q ,q …q

Step2: Obtainthevaluesofdependentvariablesx ,P ,P

andq usingEquations(6.8),(6.9),(6.6),and(6.11),respectively.

Step3: ObtaintheLagrangianmultipliersλ ,λ λ ,andλ

usingEquations(6.13),(6.14),(6.16),and(6.15),respectively.

Step4: Obtainthegradientvector andcheckwhetherall

itselementsareclosetozerowithinaspecifiedtolerance,ifsotheoptimalvalueisreached;ifnot,gotothenextstep.

Step5: Obtainnewvaluesofcontrolvariablesusingthefirst-ordergradientmethod,

GH GT

1 3 2 2

K

2 3

N

K K K

1

K K K

whereαisapositivescalar,whichdefinesthesteplength,andhavingavaluedependsontheproblemonhand,thengotoStep2andrepeattheprocess.

Theinequalityconstraintsoftheproblemondependentandindependentvariablescanbehandledinthecaseofanoptimalpowerflowsolution.InequalityconstraintsonindependentvariableschecktheKuhn–Tuckercondition(giveninoptimalpowerflow,ChapterV).Theinequalityconstraintsondependentvariablescanbehandledbyaugmentingtheobjectivefunctionthroughapenaltyfunction.

Theabove-mentionedsolutionmethodcanbedirectlyextendedtoasystemhavingmultihydroandmultithermalplants.

Drawback:Itrequireslargememorysincetheindependentvariables,dependentvariables,andgradientsneedtobestoredsimultaneously.

Amodifiedtechniqueknownasdecompositionovercomestheabovedrawback.Inthedecompositiontechnique,optimizationiscarriedoutovereachsub-intervalandacompletecycleofiterationisrepeated,ifthewateravailabilityequationdoesnotcheckattheendofthecycle.

Example6.1:Atypicalhydro-thermalsystemisshowninFig.6.2.Foratypicalday,theloadonthesystemvariesinstepsofeighthourseachas9,12,and8MW,respectively.Thereisnowaterinflowintothereservoirofthehydro-plant.Theinitialwaterstorageinthereservoiris120m /sandthefinalwaterstorageshouldbe75m /s,i.e.,thetotalwateravailableforhydro-generationduringthedayis30m /s.

3

3

3

FIG.6.2Fundamentalhydro-thermalsystem

Basicheadis30m.Waterheadcorrectionfactoreisgiventobe0.004.Assumeforsimplicitythatthereservoirisrectangularsothatedoesnotchangewithwaterstorage.Letthenon-effectivewaterdischargebeassumedas3m /s.Thefuelcost-curvecharacteristicsofthethermalplantisC =0.2P 50P 130Rs./hr.Findtheoptimumgenerationschedulebyassumingthetransmissionlossesneglected.

Solution:

Given:

Fuelcostofthethermalplant,C =0.2P +50P +130Rs./hr


Totaltimeofoperation,T =24hr

No.ofsub-intervals,N =3

Durationofeachsub-interval,Δt =8hr

Initialwaterstorageinreservoir,x′(0) =120m /s

Finalwaterstorage,x′(3) =75m /s

T GT GT+

T GT GT

3

2+

2

3

3

Basicwaterhead,h′ =30m

Water-headcorrectionfactor,e =0.04

Non-effectivewaterdischarge,ρ =3m /s

Sincetherearethreesub-intervals,(N−1),thenumberofwaterdischargesofthecorrespondingsub-intervalscanbespecifiedasindependentvariablesandtheremainingoneisspecifiedasadependentvariable,i.e.,thewaterdischargesq andq areconsideredasindependentvariablesanddependentvariableq .

Letusassumetheinitialvaluestobe

q =15m /s

q =15m /s

fortheproblemformulationP ,P ,x,andq aretreatedasindependentvariables.

Thedependentvariableq (waterdischargeinthefirstsub-interval)canbeobtainedbyEquation(6.11).

Wehavethewateravailabilityequation,

x –x –j q =0forK=1,2,…N

Fromtheaboveequation,wehave

x =x +j –q =120–10=110m /s

x =x +j –q =110–15=95m /s

Weknowtherealpowerhydro-generationatanyintervalKbyEquation(6.9):

o

GH GT

3

2 3

1

2 3

3 3

1

1

K k −1 K+ K

1 o 1 1 3

2 1 2 2 3

P = h 1+0.5e(x +x )(q −e)

= 9.81×10 h′ 1+0.5e(x +x )(q −ρ)

P = 9.81×10 ×301+0.5×0.004(x +x )q −ρ

= 9.81×10 ×301+0.5×0.004(110+120)(10−3)

= 3.0077MW

P = 9.81×10 ×301+0.5×0.004(x +x )q −ρ

= 9.81×10 ×301+0.5×0.004(95+110)(15−3)

= 4.9795MW

P = 9.81×10 ×301+0.5×0.004(x +x )q −ρ

= 9.81×10 ×301+0.5×0.004(75+95)(20−3)

= 6.7041MW

Thethermalpowergenerationsduringthesub-intervalsare

P =P −P =9−3.0077=5.9923MW

P =P −P =12−4.9795=7.0205MW

P =P −P =8−6.7041=1.2959MW

λ canbeobtainedfromEquation(6.13):

GH

o

o

GH

GH

GH

GT D GH

GT D GH

GT D GH

1

K K K−1 k

−3 K K−1 K

1 −3 1 o 1

−3

2 −3 2 1 2

−3

3 −3 3 2 2

−3

1 1 1

2 2 2

3 3 3

K

i.e.,

Byneglectingtransmissionlosses,wehave

⇒λ =0.4P +50=0.4×5.9923+50=52.3969Rs./MWh

λ =0.4P +50=0.4×7.0205+50=52.8082Rs./MWh

λ =0.4P +50=0.4×1.2959+50=50.5183Rs./MWh

FromEquation(6.14),

Byneglectingtransmissionlosses,wehave

⇒λ =λ

∴λ =λ =52.3969Rs./MWh

λ =λ =52.8082Rs./MWh

λ =λ =50.5183Rs./MWh


⇒λ =λ h 1+0.5e(2x +j −2q +ρ)

=52.3969×9.81×10 ×301+0.5×0.004(2×120−2×10+3)

(sincej=0)

1 GT

1 GT

1 GT

3 1

3 1

3 1

3 1

2 2 o

1 1

2 2

3 3

K K

1 1

2 2

3 3

1 1 o 1 1

−3

=22.2979Rs./MWh


ForK=1,

∴λ

= λ −λ 0.5h e(q −ρ)−λ 0.5h e(q −ρ)

= 22.2979−52.3969×0.5×9.81×10−3×30×0.004(10−3)

−52.8082×0.5×9.81×10−3×30×0.004(15−3)

= 22.2979−0.5889

= 21.709Rs./MWh

andforK=2

∴λ = λ −λ 0.5h e(q −ρ)−λ 0.5h e(q −ρ)

= 21.709−52.8082×0.5×9.81×10−3×30×0.004(15−3)

−50.5183×0.5×9.81×10−3×30×0.004(20−3)

= 21.709−0.8784

2

2 3 o 3 o

2 2 2 o 3 o

2

2 2 1 2 2

3 2 2 2 3 3

= 20.8305Rs./MWh

i.e.,λ = 22.2979Rs./MWh

λ = 21.709Rs./MWh

λ = 20.8305Rs./MWh

FromEquation(6.17),thegradientvectoris

Ifthetolerancevalueforthegradientvectoris0.1,sincefortheaboveiteration,thegradientvectorisnotzero(≤0.1),i.e.,theoptimalityisnotsatisfiedhere.Then,fortheseconditeration,obtainthenewvaluesofcontrolvariables(q ,forK≠1)byusingthefirst-ordergradientmethodasfollows:

(∵αisapositivescalar)

2

2

2

new

1

2

3

K

Letusconsiderα=0.5,

∴q =(q ) =15−0.5(0.1685)=14.9157m /s

Similarly,q =(q ) =15−0.5(1.4134)=19.2933m /s

andfromEquation(6.11),

⇒q =x –x –(q q )(sincej =0)

q =120–75–(14.9157+19.2933)

=10.791m /s

Toobtaintheoptimalgenerationscheduleinhydro-thermalco-ordination,theprocedureisrepeatedforthenextiterationandcheckedforagradientvector.Ifthegradientvectorbecomeszerowithinaspecifiedtolerance,thenthatwillbetheoptimumgenerationschedule,otherwisetheiterationsaretobecarriedout.

6.8SOLUTIONOFSHORT-TERMHYDRO-THERMALSCHEDULINGPROBLEMS—KIRCHMAYER’SMETHOD

Inthismethod,theco-ordinationequationsarederivedintermsofpenaltyfactorsofbothplantsforobtainingtheoptimumschedulingofahydro-thermalsystemandhenceitisalsoknownasthepenaltyfactormethodofsolutionofshort-termhydro-thermalschedulingproblems.

Let P bethepowergenerationofi thermalplantinMW,

new

new

GTi

2 2 1 3

3 3 1

3

1 o 3 2+ 3 K

1

3

th

th

P bethepowergenerationofj hydro-plantinMW,

betheincrementalfuelcostofi thermalplantin

Rs./MWh,

w bethequantityofwaterusedforpowergenerationatj

hydro-plantinm /s,

betheincrementalwaterrateofj hydro-plantin

m /s/MW,

betheincrementaltransmissionlossofi thermalplant,

betheincrementaltransmissionlossofj hydelplant,

λbetheLagrangianmultiplier,

γ betheconstantwhichconvertstheincrementalwaterrateof

hydelplantjintoanincrementalcost,

nbethetotalnumberofplants,

αbethenumberofthermalplants,

n−αbethenumberofhydro-plants,and

Tbethetimeintervalduringwhichtheplantoperationisconsidered.

GHj

j

j

th

th

th

3

th

3

th

th

Here,theobjectiveistofindthegenerationofindividualplants,boththermalaswellashydelthatthegenerationcost(costoffuelinthermal)isoptimumandatthesametimetotaldemand(P )andlosses(P )arecontinuouslymet.

Asitisashort-rangeproblem,therewillnotbeanyappreciablechangeinthelevelofwaterinthereservoirsduringtheinterval(i.e.,theeffectsofrainfallandevaporationareneglected)andhencetheheadofwaterinthereservoirwillbeassumedtobeconstant.

LetK bethespecifiedquantityofwater,whichmustbeutilizedwithintheintervalTateachhydro-stationj.

Problemformulation

Theobjectivefunctionistominimizethecostofgeneration:

i.e.,

subjecttotheequalityconstraints

and

wherew istheturbinedischargeinthej plantinm /sandK theamountofwaterinm utilizedduringthetimeperiodTinthej hydro-plant.

D L

j

j

j

th 3

3

th

Thecoefficientγmustbeselectedsoastousethespecifiedamountofwaterduringtheoperatingperiod.

Now,theobjectivefunctionbecomes

SubstitutingK fromEquation(6.21)intheaboveequation,weget

ForaparticularloaddemandP ,Equation(6.20)resultsas

Foraparticularhydro-plantx,Equation(6.23)canberewrittenas

Byrearrangingtheaboveequation,weget

FromEquation(6.22),theconditionforminimizationis

j

D

Theaboveequationcanbewrittenas

Forhydro-plantx,

Multiplyingtheaboveequationby ,

Substitutefor fromEquation(6.24)in

Equation(6.27),weget

Rewritingtheaboveequationas

∴ΔP ≠0andΔP ≠0,Equation(6.28)becomes

and

Equations(6.29)and(6.30)canbewrittenintheform:

and

GTi GHj


where(I ) istheincrementalfuelcostofthei thermalplantand(I ) theincrementalwaterrateofthejhydro-plant.

Equations(6.34)and(6.35)maybeexpressedapproximatelyas

C i

W j

th

th

where and aretheapproximate

penaltyfactorsofthei thermalplantandthej hydro-plant,respectively.

Equations(6.34)and(6.35)aretheco-ordinateequations,whichareusedtoobtaintheoptimalschedulingofthehydro-thermalsystemwhenconsideringthetransmissionlosses.

Intheaboveequations,thetransmissionlossP isexpressedas

Thepowergenerationofahydro-plantP isdirectly

proportionaltoitsheadanddischargeratew .

Whenneglectingthetransmissionlosses,theco-ordinationequationsbecome

Example6.2:Atwo-plantsystemhavingasteamplantneartheloadcenterandahydro-plantataremotelocationisshowninFig.6.3.Theloadis500MWfor16hradayand350–MW,for8hraday.

Thecharacteristicsoftheunitsare

C =120+45P +0.075P

w =0.6P +0.00283P m /s

Losscoefficient,B =0.001MW

L

GHj

j

1 GT GT

2 GH 2GH

22

th th

2

3

−1

Findthegenerationschedule,dailywaterusedbythehydro-plant,anddailyoperatingcostofthethermalplantforγ =85.5Rs./m -hr.

Solution:

Given:C =120+45P +0.075P

Co-ordinationequationforthermalunitis

45+0.15P +0.075P

FIG.6.3Atypicaltwo-planthydro-thermalsystem

Forthehydro-unit,theco-ordinationequationis

Sincetheloadisnearertothethermalplant,thetransmissionlossisonlyduetothehydro-plantandthereforeB =B =B =0:

j

1 GT GT

GT GT

TT TH HT

3

2

2

Powerbalanceequation,P +P =P +P andtheconditionforoptimalschedulingis

WhenP =500MW

0.15P +45=85.5(0.6+5.66×10 P )

(0.15P +45)(1−0.002P )=85.5(0.6+5.66×10−3P )

0.15P +45−3×10 P P −0.09P =51.3+0.48393P

0.57393P −0.15P +3×10 P P +6.3=0(6.39)

and

P +P =400+0.001P

P =400+0.001P −P (6.40)

SubstitutingEquation(6.40)inEquation(6.39),weget

0.57393P −0.15(400+0.001P −P )+3×10

P (400+0.001P −P )+6.3=0

Bysolvingtheaboveequation,weget

P =81.876MW

BysubstitutingtheP valueinEquation(6.40),weget

P =424.8MW

P =6.70367MW

WhenP =350MW

GT GH D L

D

GT GH

GT GH

GH

GT GT GH GT

GH

GH GT GT GH

GT GH GT

GT GT GH

GH GH GH

GH GH GH

GH

GH

GT

L

D

−3

−4

−4

2

2

2 −4

2

Equation(6.40)canbemodifiedas

P =350+0.001P −P (6.41)


0.57393P −0.15(350+0.001P −P )+3×10

P (350+0.001P −P )+6.3=0


P =58.5851MW


P =294.847MW

P =3.43221MW

Dailywaterusedbythehydro-plant

w 0.6P +0.00283P m /s

= Dailywaterquantityusedfora500MWloadfor16hr+dailywaterquantityusedfora350MWloadfor8hr

= [0.6×81.876+0.00283×(81.876) ]×14+[0.6×58.586+0.00283×(58.586) ]×8×3600

= 5.21449×10 m

Dailyoperatingcostofthethermalplantis:

C =

(120+45P +0.075P )

= Operatingcostofthethermalplantformeetingthe500MWloadfor16hr+operatingcostofthethermalplantformeetingthe350MWloadfor8hr

GT GH GH

GH GH GH

GH GH GH

GH

GH

GT

L

GH GH

1GT GT

2

2 −4

2

2 3

2

2

6 3

2

= [120+45×424.8+0.075(424.8) ]×16+[120+45×294.85+0.075(424.8) ]×8

= Rs.6,83,589.96perday

Example6.3:Atwo-plantsystemthathasahydro-plantneartheloadcenterandasteamplantataremotelocationisshowninFig.6.4.Theloadis400MWfor14hradayand200MW,for10hraday.

Thecharacteristicsoftheunitsare

C =150+60P +0.1P Rs/hr

w =0.8P +0.000333P m3/s

FIG.6.4Atypicaltwo-planthydro-thermalsystem

Losscoefficient,B =0.001MW

Findthegenerationschedule,dailywaterusedbythehydro-plant,andthedailyoperatingcostofathermalplantforγ =77.5Rs./m hr.

Solution:

Equationsforthermalandhydro-plantsare

1 GT GT

2 GH GH

22

j

2

2

2

2

−1

3

Sincetheloadisnearertothehydro-plant,thetransmissionlossisonlyduetothethermalplantandthereforeB =B =B =0:

WhenP =400MW

Thepowerbalanceequationis

P +P = P +P

= 400+0.001P

P = 400+0.001P −P (6.42)

Theconditionforoptimalschedulingproblemis

0.2P +60=77.5(0.8+6.6×10 P )(1−0.002P )

0.2P +60=62+0.051615P −0.124P −1.032×10 P P

0.2P +0.124P +1.032×10 P P −0.0516P−2=0(6.43)

SubstitutingP fromEquation(6.42)inEquation(6.43),weget

HH TH HT

D

GT GH D L

GT

GH GT GT

GT GH

GT

GT GH GT

GH GT

GT GT GH GT GH

GH

2

2

−4

−4

−4

−4

0.2P +0.124P +1.032×10 P (400+0.001P −P )

−0.0516(400+0.001P −P )−2=0


P =55.4MW


P =347.66MW

P =3.069MW

WhenP =200MW

FromEquation(6.42),thepowerbalanceequationbecomes

P =2000.001P −P (6.44)


0.2P +0.124P +1.032×10 P (200+0.001P −P )

−0.0516(200+0.001P −P )−2=0


P =31.575MW


P =169.421MW

P =0.9969MW

Dailyoperatingcostofthethermalplant

C = 150+60P +0.1P

= Dailyoperatingcostofthethermalplantformeetinga400MWloadfor14hr+dailyoperatingcostofthethermal

GT GT GT

GT GT

GT GT

GT

GT

GH

L

D

GH GT GT

GH

GT GT GT

GH GT

GT GT

GT

GH

GH

L

1GT GT

−4

2

2

2

−4

2

2

2

plantformeetinga200MWloadfor10hr

= [150+60×55.4+0.1×(55.4) ]×14+[150+60×31.575+0.1×(31.575) ]×10

= Rs.74,374.80

Dailyoperatingcostofthehydro-plant

w = 0.8P +0.000333P m /s

= Dailywaterquantityusedforthe400MWloadfor14hr+dailywaterquantityusedforthe200MWloadfor10hr

= [0.8×347.66+0.000333×(347.66) ]×14+[0.8×169.421+0.000333×(169.421) ]×10×3600

= 21.2696×10 m

Example6.4:Atwo-plantsystemthathasathermalstationneartheloadcenterandahydro-powerstationataremotelocationisshowninFig.6.5.

Thecharacteristicsofbothstationsare

C =(26+0.045P )P Rs./hr

w =(7+0.004P )P m /s

andγ =Rs.4×10 /m

Thetransmissionlosscoefficient,B =0.0025MW .

Determinethepowergenerationateachstationandthepowerreceivedbytheload

whenλ=65Rs./MWh.

Solution:

GH GH

1 GT GT

2 GH GH

2

22

2

2

2 3

2

2

6 3

3

−4 3

−1

Here,n=2

Transmissionloss,

Sincetheloadisnearthethermalstation,thepowerflowisfromthehydro-stationonly;therefore,B =B =0:

Forthethermalpowerstation,theco-ordinationequationis

FIG.6.5Two-plantsystem

Forahydro-powerstation,theco-ordinationequationis

12 11


P =199.99MW

Transmissionloss,P =B P =0.0025(199.99) =99.993MW

Therefore,thepowerreceivedbytheload,P =P +P −P =433.33+622.38−193.68=533.327MW.

Example6.5:ForthesystemofExample6.4,iftheloadis750MWfor14hradayand500MWfor10hronthesameday,findthegenerationschedule,dailywaterusedbythehydro-plant,andthedailyoperatingcostofthermalpower.

Solution:

Whenload,P =750MW

Thepowerbalanceequation,P +P =P +P

=750+0.0025P

P =750+0.0025P −P (6.45)

Theconditionforoptimalityis

GH

L 22 GH

D GT

GH L

D

GT GH D L

GH

GT GH GH

2 2

2

2

−3 −4

(26+0.09P )(1−5×10 P )=28×10 +32×10 P (6.46)


[26+0.09(750+0.0025P −P )](1−5×10P )=28×10 +32×10 P

−1.125×10 P +6.75×10 P −0.5574P +25.9922=0


P =200MW


P =650MWandP =100MW

Whenload,P =400MW


P =400+0.0022P −P (6.47)

SubstitutingtheaboveequationinEquation(6.46),weget

[26+0.09(400+0.0025P −P )](1−5×10P )=28×10 +32×10 P

−1.125×10 P +6.75×10 P −0.3999P +61.9972=0


P =200MW


P =300MWandP =100MW

Dailyoperatingcostofthehydro-plant

w = (7+0.004P )P m /s

GT GH

GH

GT

GH GH

GH GH

GH GH GH

GH

GH

GT L

D

GT GH GH

GH GH

GH GH

GH GH GH

GH

GH

GT L

2GH GH

−3 −4

−7

2 −3

−4 −7

−6 3 −4 2

2

2 −3

−4 −7

−6 3 −4 2

3

= Dailywaterquantityusedfora750MWloadfor14hr+dailywaterquantityusedfora400MWloadfor10hr

= [7×200+0.004×(200) ]×14+[7×200+0.004×(200) ]×10×3600

= 134.784×10 m


C = (26+0.045P )P Rs./hr

= Dailyoperatingcostofathermalplantformeetinga750MWloadfor14hr+dailyoperatingcostofthermalplantformeetinga400MWloadfor10hr

= [26×650+0.045×(650) ]×14+[26×300+0.045×(300) ]×10

= Rs.6,21,275

Example6.6:Aloadisfeededbytwoplants,oneisthermalandtheotherisahydro-plant.TheloadislocatednearthethermalpowerplantasshowninFig.6.6.Thecharacteristicsofthetwoplantsareasfollows:

C =0.04P +30P +20Rs./hr

w =0.0012P +7.5P m /s

γ =2.5×10 Rs./m


1GT GT

T GT GT

H GH GH

H

2

2

6 3

2

2

2

2 3

–5 3

Thetransmissionlossco-efficientisB =0.0015MW.Determinethepowergenerationofboththermalandhydro-plants,theloadconnectedwhenλ=45Rs./MWh.

Solution:

Given:

Transmissionloss,

Theloadislocatednearthethermalplants;hence,thepowerflowtotheloadisonlyfromthehydro-plant:

i.e.,B =B =0

∴P =B P =B P =0.0015P

Theincrementaltransmissionlossofthethermalplantis

22

11 12

L 22 G2 22 GH GH

−1

2 2 2

Penaltyfactorofthethermalplant,

Theincrementaltransmissionlossofthehydro-plantis

Penaltyfactorofthehydro-plant,

Theconditionforhydro-thermalco-ordinationis

and

or(0.000216P +0.675) = (1−0.003P )45

0.135216P = 44.325

GH GH

GH

∴P = 327.809MW

Transmissionloss,P =B P =0.0015(327.809) =161.188MW

Theloadconnected,P =P +P −P =187.5+327.809−161.188=354.121MW

Example6.7:ForExample6.6,determinethedailywaterusedbythehydro-plantandthedailyoperatingcostofthethermalplantwiththeloadconnectedfortotally24hr.

Solution:

FromExample6.6,

Theloadconnected,P =354.121MW

Generationofthethermalplant,P

=187.5MW

Generationofthehydro-plant,P

=327.809MW

Thedailywaterusedis

w = 0.0012P +7.5P m /s

= [0.0012P +7.5P ]×3,600m /hr

= [0.0012P +7.5P ]×3,600×24m /day

SubstitutingthevalueofP =327.809MWintheaboveequation,wehave

GH

L 22 GH

D GT GH L

D

GT

GH

HGH GH

GH GH

GH GH

GH

2 2

2 3

2 3

2 3

w = [0.0012(327.809) +7.5×327.809]×3,600×24

= 223.56×10 m


= (0.04P +30P +20)

Rs./h

= Rs.[0.04(187.5) +30(187.5)+20]×24

= Rs.1,69,230

Example6.8:Inatwo-plantoperationsystem,thehydro-plantoperatesfor8hrduringeachdayandthesteamplantoperatesthroughouttheday.Thecharacteristicsofthesteamandhydro-plantsare

C =0.025P +14P +12Rs./hr

w =0.002P +28P m /s

Whenbothplantsarerunning,thepowerflowfromthesteamplanttotheloadis190MWandthetotalquantityofwaterusedforthehydro-plantoperationduring8hris220×10 m .

Determinethegenerationofahydro-plantandcostofwaterused.Neglectthetransmissionlosses.

Solution:

Thecostofthethermalplantis

C =(0.025P +14P +12)Rs./hr

Theincrementalfuelcostofthethermalplantis

H

GT GT

T GT GT

H GH GH

T GT GT

2

6 3

2

2

2

2 3

6 3

2

andforthehydro-plant,w =(0.002P +28P )m /s

Theincrementalwaterflowis

Forhydro-thermalscheduling,theoptimalconditionis

(sincelossesareneglected,L =1)

Powerflowtotheloadfromthethermalplant,P =190MW(given).BysubstitutingthevalueofP =190MWintheaboveequation,weget

λ=0.05(190)+14=23.5Rs./MWh

Thetotalquantityofwaterusedduringaone-houroperationis

w =0.0012P +7.5P m /s

=[0.0012P +7.5P ]×3,600m /hr

Foran8-hroperation,thequantityofwaterusedis

H GH GH

T

GT

GT

HGH GH

GH GH

2

3

2 3

2 3

Letthecostofwaterbeγ Rs./hr/m /s.

FromEquation(6.48)

Example6.9:Atwo-plantsystemwithnotransmissionlossshowninFig.6.7(a)istosupplyaloadshowninFig.6.7(b).

Thedataofthesystemareasfollows:

C =(32+0.03P )P

w =(8+0.004P )P m /s

Themaximumcapacityofthehydro-plantandthesteamplantare450and250MW,respectively.Determinethegeneratingscheduleofthesystemsothat150.3478millionm waterisusedduringthe24-hrperiod.

H

1 GT GT

2 GH GH

3

3

3

FIG.6.7(a)Two-plantsystem;(b)dailyloadcurve

Solution:

(i)Constanthydro-generation

IfP isthehydro-powerinMWgeneratedin24hr,thenwehave

(8+0.004P )P ×24×60×60 = 150.3478×10

8P +0.004P = 1,740.136

0.004P +8P −1,740.136 = 0


GH

GH GH

GH GH

GH GH

6

2

2

P =197.929MW

Duringthepeakloadof600MW

Hydro-generation,P =197.929MW

Thermalgeneration,P =600-P =600-197.929=402.071MW

Duringoff-peakloadof400MW

Hydro-generation,P =197.929MW

Thermalgeneration,P =400-P =400-197.929=402.071MW

Therunningcostofasteamplantfor24hris

C = (32+0.03P )P ×12/at +(32+0.03P )P ×

12/at

= (32+0.03×402.071)402.071×12+(32+0.03×202.071)202.071×12

= Rs.3,04,888.288

(ii)Constantthermalgeneration

IfP isthehydro-powerduringthepeakloadperiod

(P –200)isthehydro-powerduringtheoff-peakloadperiod

Givenw =(8+0.004P )P m /s

(8+0.004P )P +[8+0.004(P −200)](P −200)×12×3,600=150.3478×10

Aftersimplification,weget

8×10 P +14.4P –4,920.273=0

∴P =93.74793MW

GH

GH

GT GH

GH

GT GH

1GT GT 600MW GT GT

400MW

GH

GH

2 GH GH

GH GH GH GH

GH GH

GH

3

6

−3 2

Thegenerationschedulingisgivenasfollows:

Hydro Thermal(P –P )

Peak(600) 293.75MW 306.25MW

Off-peak(400)

93.75MW 306.25MW

Thesteamplantoperatingcostfor24hris

C = (32+0.03P )P

= (32+0.03×306.25)306.25×12+(32+0.03×306.25)306.25×12

= Rs.3,02,728.125

(iii)Equalincrementalplantcosts

LetP′ andP′ bethesteamgenerationandhydro-generationduringpeakloads,P″ andP″ thesteamgenerationandhydro-generationduringoff-peakloads,respectively.

Forpeakloadconditions:

Thevalueofλ′shouldbesochosenastomake

D GH

1GT GT

GT GH

GT GH

P′ +P′ =600(6.51)

Foroff-peakperiods:

Thevalueofλ″shouldbechosensoastomake

P″ +P″ (6.54)

Forthewholeoperatingperiod,γ shouldbechosensoastousethesamevalueofwater,i.e.,150.3478millionm duringthe24-hrperiod.

(8+0.004P′ )P′ +(8+0.004P″ )P″ ×12×3,600=150.3478×10 (6.55)

Alltheaboveequationscanbesolvedbyahit-and-trailoraniterativemethod:

P′ = 276.362MW,P′ =323.638MW

λ′ = 48.58172Rs./MWh

−(8+0.004×323.638)×323.638+3,480.273=(8+0004P‴ )P″

8P′ = +0.004P″ =472.2


GT GH

GT GH

2

GH GH GH GH

GT GH

GH GH

GH GH

3

6

2

P″ +57.38MWP″ = 342.62MW

λ″ = 52.5572Rs./MWh

γ = 6.2131Rs./hr/m3/s

Thethermaloperatingcost

C = (32+0.03P )P

= (32+0.03×276.362)×276.362×12+(32+0.03×342.62)×342.62×12

= Rs.3,07,444.279

(iv)Maximumhydro-efficiencymethod

Letitbeassumedthatthemaximumefficiencyofahydro-unitoccursat275MW.Therefore,thehydro-powerplantsupplyis275MWduringthepeakload.Theamountofwaterusedduringpeakloadhours:

w =(8+0.004P )P m /s

=(8+0.004×275)×275×12×3,600=108.108×10

Wateravailableforoff-peakhydro-generation:

=totalwateravailable–wateravailableatpeakload

=150.34×10 −108.108×10 =42.2398×10 m

Therealpowergenerationofahydro-plantP duringoff-peakhoursisfoundbyusing

GH GT

2

1GT GT

2GH GH

GH

3

6

6 6 6 3

(8+0.004P )P ×12×3600=4,22,39,800

0.004P +8P –977.77=0

P =115.5461MW

Thegenerationschedulingisgivenasfollows:

Hydro Thermal(P –P )

Peak(600) 273MW 325MW

Off-peak(400)

115.546MW 284.4539MW

Thedailyoperatingcostofathermalplant

C = (32+0.03P )P

= (32+0.03×325)325×12+(32+0.03×284.4539)284.4539×12

= Rs.12,43,023.55

Example6.10:Athermalstationandahydro-stationsupplyanareajointly.Thehydro-stationisrun16hrdailyandthethermalstationisrunthrough24hr.Theincrementalfuelcostcharacteristicsofthethermalplantare

C =6+12P +0.04P Rs./hr

Iftheloadonthethermalstation,whenbothplantsareinoperation,is350MW,theincrementalwaterrate

ofahydro-powerplant .The

totalquantityofwaterutilizedduringa16-hroperationofthehydro-plantis450millionm .Findthegeneration

GH GH

GH GH

GH(Off-peakload)

D GH

1GT GT

T GT GT

2

2

3

ofthehydro-plantandcostofwateruse.Assumethatthetotalloadonthehydro-plantisconstantforthe16-hrperiod.

Solution:

Given:C =6+12P +0.04P

P =350MW(given)

∴12+0.08×350=λ

λ=40Rs./MWh

Thetotalquantityofwaterusedduring16hrofoperationofahydro-plantis

(28+0.03P )P ×16×3,600=450×10

0.03P +28P =7,812.5

0.03P +28P –7,812.5=0


P =224.849MW

Ifthecostofwaterusedisγ,thenwehave

γ(28+0.03P )=λ

γ(28+0.03×224.849)=40

∴γ=1.15122Rs./hr/m /s

6.9ADVANTAGESOFOPERATIONOFHYDRO-THERMALCOMBINATIONS

Thefollowingadvantagesareobtainedbyoperationcombinationofhydro-thermalpowerplants.

6.9.1Flexibility

Thepowersystemreliabilityandsecuritycanbeobtainedbythecombinedoperationofhydroand

T GT GT

GT

GH GH

GH GH

GH GH

GH

GH

2

6

2

2

3

thermalunits.Itprovidesthereservecapacitytomeettherandomphenomenaofforcedoutageofunitsandunexpectedloadimpliedonasystem.

Thermalplantsrequireanappreciabletimeforstartingandforbeingputintoservice.Hydro-plantscanbestartedandputintooperationveryquicklywithloweroperatingcosts.Hence,itisrequiredtooperatehydro-plantseconomicallyasbase-loadplantsaswellaspeakloadplants.Hydro-plantsaremostpreferabletooperateaspeakloadplantssuchthattheiroperationimprovestheflexibilityofthesystemoperationandmakesthethermalplantoperationeasier.

6.9.2Greatereconomy

Therun-offriverhydro-plantswouldgenerallymeettheentireorpartofthebaseloads,andthermalplantsshouldbesetuptoincreasethefirmcapacityofthesystem.Theremainingpowerdemandcanbemetbyacombinationofreservoir-typehydro-plants,thermalplants,andnuclearplants.Ineverypowersystem,acertainratioofhydro-powertototalpowerdemandwillresultinaminimumoverallcostofsupply.

6.9.3Securityofsupply

Wateravailabilitymustdependontheseason.Itishighduringtherainyseasonandmaybereducedduetotheoccurrenceofdraughtduringlongerplants.Problemsariseinthethermalpowerplantoperationduetotransportationofcoal,unavailabilityoflabor,etc.Itisfoundthattheforcedoutagesofhydro-plantsarefewcomparedtothoseinthermalplants.

Theabovefactssuggestedtheoperationofhydro-thermalsystemstomaintainthereliabilityandsecurityofsupplytotheconsumers.

6.9.4Betterenergyconservation

Duringheavyrun-offperiods,thegenerationofhydro-powerismore,whichresultsintheconservationoffossilfuels.Duringdraughtperiods,moresteampowerhastobegeneratedsuchthattheavailabilityofwaterneedstheminimumneedslikedrinkingandagriculturalevents.

6.9.5Reservecapacitymaintenance

Fortheoperationofapowersystem,itisnecessarythateverysystemhassomecertainreservecapacitytomeettheforcedoutagesandunexpectedloaddemands.Bythecombinedoperationofhydroandthermalplants,thereservecapacitymaintenanceisreduced.

Example6.11:MATLABprogramonhydro-thermalschedulingwithoutinflowandwithoutloss.Findtheoptimumgenerationforahydro-thermalsystemforatypicalday,whereinloadvariesinthreestepsof8hreachas15,25,and8MW,respectively.Thereisnowaterinflowintothereservoirofthehydro-plant.Theinitialwaterstorageinthereservoiris180m /sandthefinalwaterstorageshouldbe100m /s.Thebasicheadis35mandthewater-headcorrectionfactoreis0.005.Assumeforsimplicitythatthereservoirisrectangularsothatρdoesnotchangewithwaterstorage.Letthenon-effectivewaterdischargebeassumedas4m /s.Theincrementalfuelcost(IFC)ofthethermal

powerplantis .Further

transmissionlossesmaybeneglected.

PROGRAMISUNDERTHEFILENAMEhydrothermal.m

3 3

3

RESULTS:

pgh=12.4706 21.8178 5.4672

pgt=2.5294 3.1822 2.5328

netPG=15 25 8

iter=15

Example6.12:MATLABprogramonhydro-thermalschedulingwithinflowandwithoutlosses.Findtheoptimumgenerationforahydro-thermalsystemforatypicalday,whereinloadvariesinthreestepsof8hreachas15,25,and8MW,respectively.Thereiswaterinflowintothereservoirofthehydro-plantinthreeintervalsof2,4,and3m /s.Theinitialwaterstorageinthereservoiris180m /sandthefinalwaterstorageshouldbe100m /s.Thebasicheadis35–mandthewater-headcorrectionfactoreis0.005.Assumeforsimplicitythatthereservoirisrectangularsothatρdoesnotchangewithwaterstorage.Letthenon-effectivewaterdischargebeassumedas4m /s.TheIFC

ofthethermalpowerplantis

Furthertransmissionlossesmaybeneglected.

3

3

3

3

PROGRAMISUNDERTHEFILENAMEhydrothermalinflow.m

RESULTS:

pgh=14.0553 23.6463 7.3583

pgt=0.9447 1.3537 0.6417

netPG=15 25 8

iter=15

Example6.13:MATLABprogramonhydro-thermalschedulingwithoutinflowandwith

losses.Findtheoptimumgenerationforahydro-thermalsystemforatypicalday,whereinloadvariesinthreestepsof8hreachas15,25,and8MW,respectively.Thereisnowaterinflowintothereservoirofthehydro-plant.Theinitialwaterstorageinthereservoiris180m /sandthefinalwaterstorageshouldbe100m /s.Thebasicheadis35mandthewater-headcorrectionfactoreis0.005.Assumeforsimplicitythatthereservoirisrectangularsothatρdoesnotchangewithwaterstorage.Letthenon-effectivewaterdischargebeassumedas4m /s.TheIFCofthethermalpower

plantis .Furthertransmission

lossesareconsideredandaretakenasfollows:

.

PROGRAMISUNDERTHEFILENAMEhydrothermalloss.m

3

3

3

RESULTS:

pgh=12.4706 21.8178 5.4672

pgt=2.5294 3.1822 2.5328

grad=0 −0.0765 0.0826

netPG=15 25 8

iter=15

Example6.14:MATLABprogramonhydro-thermalschedulingwithinflowandwithlosses.Findtheoptimumgenerationforahydro-thermalsystemforatypicalday,whereinloadvariesinthreestepsof8hreachas15,25and8MW,respectively.Thereiswaterinflowintothereservoirofthehydro-plantinthreeintervalsof2,4,3m /s.Theinitialwaterstorageinthereservoiris180m /sandthefinalwaterstorageshouldbe100m /s.Thebasicheadis35mandthewater-headcorrectionfactoreis0.005.Assumeforsimplicitythatthereservoirisrectangularsothatρdoesnotchangewithwaterstorage.Letthenon-effectivewaterdischargebeassumedas4m /s.TheIFCofthe

thermalpowerplantis .Further

transmissionlossesareconsideredandaretakenas

follows:

PROGRAMISUNDERTHEFILENAMEhydrothermalinflowloss.m

3

3

3

3

RESULTS:

pgh=14.0553 23.6463 7.3583

pgt=0.9447 1.3537 0.6417

netPG=15 25 8

iter=15

KEYNOTES

Theoptimalschedulingprobleminthecaseofthermalplantscanbecompletelysolvedatanydesiredinstantwithoutreferringtotheoperationatothertimes.Itisastaticoptimizationproblem.

Theoptimalschedulingprobleminthehydro-thermalsystemisadynamicoptimizationproblemwherethetimefactoristobeconsidered.Theoptimalschedulingprobleminahydro-thermalsystemcanbestatedasminimizingthefuelcostofthermalplantsundertheconstraintofwateravailabilityforhydro-generationoveragivenperiodofoperation.Themethodsofhydro-thermalco-ordinationare:

Constanthydro-generationmethod.Constantthermalgenerationmethod.Maximumhydro-efficiencymethod.Kirchmayer’smethod.

Constanthydro-generationmethod—Ascheduledamountofwaterataconstantheadisusedsuchthatthehydro-powergenerationiskeptconstantthroughouttheoperatingperiod.Constantthermalgenerationmethod—Thermalpowergenerationiskeptconstantthroughouttheoperatingperiodinsuchawaythatthehydro-powerplantsuseaspecifiedandscheduledamountofwaterandoperateonvaryingpowergenerationscheduleduringtheoperatingperiod.Maximumhydro-efficiencymethod—Duringpeak-loadperiods,thehydro-powerplantsareoperatedattheirmaximumefficiency;duringoff-peakloadperiods,theyoperateatanefficiencynearertotheirmaximumefficiencywiththeuseofaspecifiedamountofwaterforhydro-powergeneration.Kirchmayer’smethod—Theco-ordinationequationsarederivedintermsofpenaltyfactorsofbothplantsforobtainingtheoptimumschedulingofthehydro-thermalsystemandhenceitisalsoknownasthepenaltyfactormethodofsolutionofshort-termhydro-thermalschedulingproblems.Long-termhydro-thermalschedulingproblemscanbesolvedbythediscretizationprinciple.Inthelong-termhydro-thermalschedulingproblem,itisconvenienttochoosewaterdischargesinallsub-intervalsexceptonesub-intervalasindependentvariablesandhydro-generations,thermalgenerations,waterstoragesinallsub-intervals,andexceptedwaterdischargeasdependentvariables,

i.e.,Independentvariablesarerepresentedbyq ,forK=2,3,…,N

≠1

DependentvariablesarerepresentedbyP ,P ,X andq ,forK

=1,2,…,N.[Sincethewaterdischargeinonesub-intervalisadependentvariable.]

Foroptimalityoflong-termhydro-thermalscheduling,thegradient

vectorshouldbezero,i.e., .


GH GT

K

K K K, 1

1. Whyistheoptimalschedulingprobleminthecaseofthermalplantsreferredtoasastaticoptimizationproblem?

Optimalschedulingproblemcanbecompletelysolvedatanydesiredinstantwithoutreferringtotheoperationatothertimes.

2. Theoptimizationprobleminthecaseofahydro-thermalsystemisreferredtoasadynamicproblem.Whyisitso?

Theoperationofthesystemhavinghydroandthermalplantshavenegligibleoperationcostsbutisrequiredundertheconstraintofwateravailabilityforhydro-generationoveragivenperiodoftime.

3. Whatisthestatementofoptimizationproblemofhydro-thermalsystem?

Minimizethefuelcostofthermalplantsundertheconstraintofwateravailabilityforhydro-generationoveragivenperiodoftime.

4. Intheoptimalschedulingproblemofahydro-thermalsystem,whichvariablesareconsideredascontrolvariables?

Thermalandhydro-powergenerations(P andP ).

5. Fast-changingloadscanbeeffectivelymetbywhichtypeofplants?

Hydro-plants.

6. Generally,whichtypeofplantsaremoresuitabletooperateasbase-loadandpeakloadplants?

Thermalplantsaresuitedforbase-loadplantsandhydro-plantsaresuitedforpeakloadplants.

7. Wholeorpartofthebaseloadcanbesuppliedbywhichtypeofhydro-plants?

Run-offrivertype.

8. Thepeakloadorremainingbaseloadismetbywhichtypeofplants?

Aproperco-ordinationofreservoir-typehydro-plantsandthermalplants.

9. Intheoptimalschedulingproblemofahydro-thermalsystem,

GT GH

whatparametersareassumedtobeknownasthefunctionoftimewithcertainty?

Waterinflowtothereservoirandloaddemand.

10. Whatisthemathematicalstatementoftheoptimizationprobleminthehydro-thermalsystem?

Determinethewaterdischargerateq(t)soastominimizethecostofthermalgeneration.

11. Writetheobjectivefunctionexpressionofhydro-thermalschedulingproblem.

12. Writetheconstraintequationsofthehydro-thermalschedulingproblem.

P (t)+P (t)−P (t)−P (t)=0

fort∈(O,T)—Realpowerbalanceequation

—P (t)=(X‴(t),q(t))

13. Bywhichprinciplecantheoptimalschedulingproblemofahydro-thermalsystembesolved?

Discretizationprinciple.

14. Writetheexpressionforrealpowerhydro-generationinanysub-interval‘K’?

P =h 1+0.5e(X +X )(q −ρ)

15. Definethetermsoftheaboverealpowerhydro-generation.

P =h 1+0.5e(X +X )(q −ρ)

whereh =9.81×10 h ,h isthebasicwaterheadthat

correspondstodeadstorage,ethewater-headcorrectionfactortoaccountforthevariationinheadwithwaterstorage,X thewaterstorageatintervalk,q thewaterdischargeatintervalk,andρthenon-effectivedischarge.

GT GH L D

GH

GH o

GH o

0 0 o

K K K−1 K

K K K−1 K

–3 1 1

k

k

16. Intheoptimalschedulingproblemofahydro-thermalsystem,whichvariablesareusedtochooseasindependentvariables?

Waterdischargesinallsub-intervalsexceptonesub-interval:

i.e.,e ,forq ,q ⋯q

wherek=2,3…N(kissub-interval).

17. Whichparametersareusedasdependentvariables?

Thermal,hydro-generations,waterstoragesatallsub-intervals,andwaterdischargeatexceptedsub-intervalsareusedasdependentvariables,

i.e.,P ,P X ,andq

18. Insolvingtheoptimalschedulingproblemofahydro-thermalsystem,for‘N’sub-intervals(i.e.,k=1,2,…,N),N−1numberofwaterdischargesq’scanbespecifiedasindependentvariablesexceptonesub-interval.Writetheexpressionforwaterdischargeintheexceptedsub-interval,whichistakenasadependantvariable.

19. Whichtechniqueisusedtoobtainthesolutiontotheoptimizationproblemofthehydro-thermalsystem?

Anon-linearprogrammingtechniqueinconjunctionwithafirst-ordernegativegradientmethodisusedtoobtainthesolutiontotheoptimizationproblem.

20. WritetheexpressionforaLagrangianfunctionobtainedbyaugmentingtheobjectivefunctionwithconstraintequationsinthecaseofahydro-thermalschedulingproblem.

21. Whatisthegradientvector?

ThepartialderivativesoftheLagrangianfunctionwithrespecttoindependentvariablesare

k≠1

GT GH

k 2 3 N

k k k 1

22. Whatistheconditionforoptimalityinthecaseofahydro-thermalschedulingproblem?

Thegradientvectorshouldbezero:

23. Theconditionforoptimalityinahydro-thermalschedulingproblemisthatthegradientvectorshouldbezero.Ifthisconditionisviolated,howwillweobtaintheoptimalsolution?

Findthenewvaluesofcontrolvariables,whichwilloptimizetheobjectivefunction.Thiscanbeachievedbymovinginthenegativedirectionofthegradientvectortoapointwherethevalueoftheobjectivefunctionisnearertoanoptimalvalue.

24. Forasystemwithamultihydroandamultithermalplant,thenon-linearprogrammingtechniqueinconjunctionwiththefirst-ordergradientmethodisalsodirectlyapplied.However,whatisthedrawback?

Itrequireslargememorysincetheindependentanddependantvariables,andgradientsneedtobestoredsimultaneously.

25. Bywhichmethodcanthedrawbackofthenon-linearprogrammingtechniquebeovercomewhenappliedtoamultihydroandmultithermalplantsystemandwhatisitsprocedure?

Bythemethodofdecompositiontechnique.Inthistechnique,theoptimizationiscarriedoutovereachsub-intervalandacompletecycleofiterationisreplaced,ifthewateravailabilityequationdoesnotcheckattheendofthecycle.

26. Forshort-rangeschedulingofahydro-thermalplant,whichmethodisuseful?

Kirchmayer’smethodorthepenaltyfactormethodisusefulforshort-rangescheduling.

27. WhatisKirchmayer’smethodofobtainingtheoptimumschedulingofahydro-thermalsystem?

InKirchmayer’smethodorthepenaltyfactormethod,theco-ordinationequationsarederivedintermsofpenaltyfactorsofbothhydroandthermalplants.

28. Whatistheconditionforoptimalityinahydro-thermalschedulingproblemwhenconsideringtransmissionlosses?

whereirepresentsthethermalplantandjrepresentsthehydro-plant.

29. Whatisthemeaningoftheterms and ?

istheincrementalcostoftheiththermalplantand

istheincrementalwaterrateofthejthhydro-plant.

30. Whatisshort-termhydro-thermalco-ordination?

Short-termhydro-thermalco-ordinationisdoneforafixedquantityofwatertobeusedinacertainperiod(i.e.,24hr).

31. Whataretheschedulingmethodsforshort-termhydro-thermalco-ordination?

1. Constanthydro-generationmethod.2. Constantsteamgenerationmethod.3. Maximumhydro-efficiencymethod.4. Equalincrementalproductioncostsandsolutionofco-ordination

equations(Kirchmayer’smethod).

32. Whatisthesignificanceoftheco-efficientγ ?

γ representstheincrementalwaterratesintoincrementalcosts

whichmustbesoselectedastousethedesiredamountofwaterduringtheoperatingperiod.

33. Writetheconditionforoptimalityintheproblemofashortrangehydro-thermalsystemaccordingtoKirchmayer’smethodwhen

j

j

neglectingtransmissionlosses.

34. Whatisthesignificanceofterms and ?

isthepenaltyfactorofthei thermalplantand

isthepenaltyfactorofthej hydro-plant

Thesetermsareverymuchusefulingettingtheoptimalityinahydro-thermalschedulingproblem,whichissolvedbyKirchmayer’smethod.

35. Writetheconditionforoptimalityinanoptimalschedulingproblemofashortrangehydro-thermalsystemwithapproximatepenaltyfactors.


1. Whencomparedtoahydro-electricplant,theoperatingcostofthethermalplantisvery_____anditscapitalcostis_____.

1. High,low.2. High,high.

th

th

3. Low,low.4. Low,high.

2. Whencomparedtoathermalplant,theoperatingcostandcapitalcostofahydro-electricplantare:

1. Highandlow.2. Lowandhigh.3. Bothhigh.4. Bothlow.

3. Theoptimalschedulingprobleminthecaseofthermalplantsis:

1. Staticoptimizationproblem.2. Dynamicoptimizationproblem.3. Staticaswellasdynamicoptimizationproblem.4. Eitherstaticordynamicoptimizationproblem.

4. Theoperationofthesystemhavinghydroandthermalplantsismorecomplex.Inthiscase,theoptimalschedulingproblemis:

1. Staticoptimizationproblem.2. Dynamicoptimizationproblem.3. Staticaswellasdynamicoptimizationproblem.4. Eitherstaticordynamicoptimizationproblem.

5. Theoptimalschedulingprobleminthecaseofathermalplantcanbecompletelysolvedatanydesiredinstant:

1. Withreferencetotheoperationatothertimes.2. Withoutreferencetotheoperationatothertimes.3. Case(a)orcase(b)thatdependsonthesizeoftheplant.4. Noneofthese.

6. Thetimefactorisconsideredinsolvingtheoptimizationproblemof_____.

1. Hydroplants.2. Thermalplants.3. Hydro-thermalplants.4. Noneofthese.

7. Theobjectivefunctiontotheoptimizationprobleminahydro-thermalsystembecomes:

1. Minimizethefuelcostofthermalplants.2. Minimizethetimeofoperation.3. Maximizethewateravailabilityforhydro-generation.4. Allofthese.

8. Theoptimalschedulingproblemofahydro-thermalsystemissolvedundertheconstraintof:

1. Fuelcostofthermalplantsforthermalgeneration.2. Timeofoperationoftheentiresystem.3. Wateravailabilityforhydro-generationoveragivenperiod.4. Availabilityofcoalforthermalgenerationoveragivenperiod.

9. Tosolvetheoptimizationprobleminahydro-thermalsystem,whichofthefollowingvariablesareconsideredascontrolvariables?

1. P andP .Dthermal Ghydro

2. Q andQ .

3. P andP .

4. P andP .

10. Inwhichsystemisthegenerationscheduledgenerallysuchthattheoperatingcostsofthermalgenerationareminimized?

1. Systemswherethereisaclosebalancebetweenhydroandthermalgeneration.

2. Systemswherethehydro-capacityisonlyafractionofthetotalcapacity.3. Both(a)and(b).4. Noneofthese.

11. Thermalplantsaremoresuitabletooperateas_____plantsleavinghydro-plantstooperateas_____plants.

1. Baseload,baseload.2. Peakload,peakload.3. Peakload,baseload.4. Baseload,peakload.

12. Inhydro-thermalsystems,thewholearepartofthebaseloadthatcanbesuppliedby:

1. Run-offriver-typehydro-plants.2. Reservoir-typehydro-plants.3. Thermalplants.4. Reservoir-typehydro-plantsandthermalplantswithproperco-ordination.

13. Inahydro-thermalsystem,thepeakloadcanbemetby:

1. Run-offriver-typehydro-plants.2. Reservoir-typehydro-plants.3. Thermalplants.4. Reservoir-typehydro-plantsandthermalplantswithproperco-ordination.

14. Foranoptimalschedulingproblem,itisassumed,whichparameterisknowndeterministicallyasafunctionoftime?

1. Waterinflowtothereservoir.2. Powergeneration.3. Loaddemand.4. Both(a)and(c).

15. Inahydro-thermalsystem,theoptimizationproblemisstatedasdetermining_____soastominimizethecostofthermalgeneration.

1. Loaddemand(P ).

2. Waterstorage(X).3. Waterdischargerate(q(t)).4. Waterinflowrate(J(t)).

16. Whichofthefollowingequationsisconsideredasaconstrainttotheoptimizationproblemofahydro-thermalsystem?

1. Realpowerbalanceequation.2. Wateravailabilityequation.3. Realpowerhydro-generationasafunctionofwaterstorage.4. Allofthese.

17. Thewateravailabilityequationis:

Dthermal Dhydro

Gthermal Dhydro

Gthermal Ghydro

D

1.

2. P (t)+P (t)−P (t)−P (t)=0,t∈(0,T).3. P (t)=f(X′(t),q(t)).

4. Noneofthese.

18. Intheoptimizationproblemofahydro-thermalsystem,theconstraintrealpowerhydro-generationisafunctionof:

1. Waterinflowrate(J(t)).2. Waterstorage(X).3. Waterdischargerate(q(t))

1. (i)and(ii).2. (ii)and(iii).3. (i)and(iii).4. Noneofthese.

19. Theoptimizationschedulingproblemofahydro-thermalsystemcanbeconvenientlysolvedby_____principle.

1. Dependence.2. Discretization.3. Dividing.4. Noneofthese.

20. Inthediscretizationprinciple,therealpowerhydro-generationatanysub-interval‘k’canbeexpressedas:

1. P =h 1+0.5e(X +X )(q −ρ)

2. P =h 1−0.5e(X +X )(q −ρ)

3. P =h 1+0.5e(X +X )(q −ρ)

4. P =h 1+0.5e(X +X )(q −ρ)

21. Intheoptimizationproblemofahydro-thermalsystem,whichofthefollowingareclosedasindependentvariables?

1. Waterstoragesinallsub-intervalsexceptonesub-interval.2. Waterinflowsinallsub-intervalsexceptonesub-interval.3. Waterdischargesinallsub-intervalsexceptonesub-interval.4. Hydroandthermalgenerations,waterstoragesatallsub-intervals,and

waterdischargeatonesub-interval.

22. Intheoptimalschedulingproblemofahydro-thermalsystem,whichofthefollowingareclosedasdependentvariables?

1. Waterstoragesinallsub-intervalsexceptonesub-interval.2. Waterinflowsinallsub-intervalsexceptonesub-interval.3. Waterdischargesinallsub-intervalsexceptonesub-interval.4. Hydroandthermalgenerations,waterstoragesatallsub-intervals,and

waterdischargeatonesub-interval.

23. Toobtainthesolutiontotheoptimizationproblemofahydro-thermalsystem,whichofthefollowingtechniqueisused?

1. Non-linearprogrammingtechniqueinconjunctionwiththefirst-ordergradientmethod.

2. Linearprogrammingtechniqueinconjunctionwiththefirst-ordergradientmethod.

3. Non-linearprogrammingtechniqueinconjunctionwiththemultiple-order

GH GH L D

GH

GH o

GH o

GH o

GH o

K k−1 k k

K k−1 k k

K k−1 k k

K k−1 k−1 k

gradientmethod.4. Linearprogrammingtechniqueinconjunctionwiththemultiple-order

gradientmethod.

24. Inahydro-thermalsystemforoptimality,theconditionis:

1. Gradientvectorshouldbezero.2. Gradientvectorshouldbepositive.3. Gradientvectorshouldbenegative.4. Noneofthese.

25. Formultihydroandmultithermalplants,theoptimizationproblemcanbesolvedbyamodifiedtechnique,whichisknownas:

1. Discretizationtechnique.2. Decompositiontechnique.3. Decoupledtechnique.4. Noneofthese.

26. InKirchmayer’smethodofsolutionofoptimizationprobleminahydro-thermalsystem,theco-ordinationequationsarederivedintermsof_____ofbothplants.

1. Powergenerations.2. Powerdemands.3. Penaltyfactors.4. Allofthese.

27. γ isusedasaconstant,inanoptimizationproblemofahydro-

thermalsystem,whichconverts:

1. Fuelcostofathermalplantintoanincrementalfuelcost.2. Incrementalwaterrateofahydro-plantintoanincrementalcost.3. Incrementalwaterinflowrateintoanincrementaldischargerate.4. Noneofthese.

28. Thepowergenerationofahydro-plantP isdirectly

proportionalto:

1. Planthead.2. Waterdischargeω .

3. Both(a)and(b).4. Noneofthese.

29. Themainadvantagesoftheoperationofahydro-thermalsystemare:

1. Greatereconomy.2. Securityofsupplyandflexibility.3. Betterenergyconservation.4. Reductioninreservecapacitymaintenance.

Regardingtheabovestatement,whichiscorrect?

1. (i)and(ii).2. (ii)and(iii).3. allexcept(iii).4. Allofthese.

30. Theco-ordinationequationsusedtoobtaintheoptimalschedulingofahydro-thermalsystemwhenconsideringtransmissionlossesare:

j

GH

j

1.

2.

3.

4. Noneofthese.

31. Asfaraspossible,hydro-plantsareusedforbase-loadoperationsince:

1. Theircapitalcostishigh.2. Theiroperationiseasy.3. Theircapitalcostislow.4. Theirefficiencyislow.

32. Athermalplantgivesminimumcostperunitofenergygeneratedwhenusedasa_____plant.

1. Peakload.2. Base-loadplant.3. Simultaneouslyasbase-loadandpeakloadplant.4. Noneofthese.

33. Inthecombinedoperationofsteamplantandrun-offriverplants,thesitesofhydroandsteamplantscanbefoundwiththehelpof_____.

1. Demandcurve.2. Input–outputcurve.3. Loadcurve.4. Chronologicalloadcurve.

34. Long-termhydro-thermalco-ordinationcanbedoneby:

1. Plottingthebasicrulecurve.2. Plottingnospill-rulecurve.3. Plottingthefullreservoirstoragecurve.4. Allofthese.

35. _____hydro-thermalco-ordinationisdonefortheavailablewaterandistobeusedinagivenperiod(24hr).

1. Long-term.2. Short-term.3. Both(a)and(b).4. Noneofthese.

36. Hydro-thermalco-ordinationisnecessaryonlyincountrieswith:

1. Amplecoalresources.2. Amplewaterresources.3. Both(a)and(b).4. Noneofthese.

37. Inshort-termhydro-thermalco-ordination,

1. Nospill-rulecurveisused.2. Spill-rulecurveisused.3. Herenorulecurveisusedduetoconstraints.4. Noneofthese.

38. Theunitsofincrementalwaterrateare:

1. m /s-MW.

2. m -s/MW.3. m-s/MW.

4. m -s/MW.

39. Hydro-generationisafunctionof:

1. Waterhead.2. Waterdischarge.3. Waterinflow.4. Both(a)and(b).

40. Inthelong-termhydro-thermalco-ordination,

1. Basicrulecurveisplotted.2. Nospillcurve.3. Nofullreservoirstoragecurve.4. Allofthese.

41. Inthecombinedoperationofasteamandarun-offriverplant,thesizesofhydroandsteamplantscanbeobtainedwiththehelpof:

1. Loadcurve.2. Demandcurve.3. Chronologicalloadcurve.4. Noneofthese.

REVIEWQUESTIONS

1. Explainthehydro-thermalco-ordinationanditsimportance.2. Describethetypesofhydro-thermalco-ordination.3. Whatarethefactorsonwhicheconomicoperationofacombined

hydro-thermalsystemdepends?4. Whataretheimportantmethodsofhydro-thermalco-

ordination?Explaintheminbrief.5. Explainthemathematicalformulationoflong-termhydro-

thermalscheduling.6. Explainthesolutionmethodoflong-termhydro-thermal

schedulingbydiscretizationprinciple.

3

3

2

7. Developanalgorithmforthesolutionoflong-termhydro-thermalschedulingproblem.

8. Derivetheconditionforoptimalityofshort-termhydro-thermalschedulingproblem.

9. Whataretheadvantagesofhydro-thermalplantscombinations?

PROBLEMS

1. ThesystemshowninFig.6.8(a)istosupplyaloadshowninFig.6.8(b).Thedataofthesystemareasfollows:

C =(16+0.01P )P Rs./hr

w =(4+0.0035P )P m /s

Themaximumcapacityofahydro-plantandasteamplantare400and270MW,respectively.Determinethegeneratingscheduleofthesystemsothat130.426millionm waterisusedduringthe24-hrperiod.

2. Athermalstationandahydro-stationsupplyanareajointly.Thehydro-stationisrun12hrdailyandthethermalstationisrunthroughout24hr.Theincrementalfuelcostcharacteristicofthethermalplantis

C =3+5P +0.02P Rs/hr

Iftheloadonthethermalstation,whenbothplantsareinoperation,is250MW,theincrementalwaterrateofthehydro-powerplantis

Thetotalquantityofwaterutilizedduringthe12-hroperationofahydro-plantis450millionm .Findthegenerationofthehydro-plantandthecostofwaterused.Assumethatthetotalloadonthehydro-plantisconstantforthe12-hrperiod.

3. Atwo-plantsystemthathasathermalstationneartheloadcenterandahydro-powerstationataremotelocationisshowninFig.6.9.

Thecharacteristicsofbothstationsare:

C =(20+0.03P )P Rs./hr

w =(8+0.002P )P m /s

T GT GT

2 GH GH

T GT GT

T GT GT

2 GH GH

3

3

3

3

3

andγ =Rs.5×10-4/m

Thetransmissionlossco-efficient,B =0.0005.

1. iftheloadis700MWfor15-hradayand500MWfor9hronthesameday,findthegenerationschedule,dailywaterusedbyhydro-plant,andthedailyoperatingcostofthethermalpower.

2. Determinethepowergenerationateachstationandthepowerreceivedbytheloadwhenλ=50Rs./MWh.

4. Atwo-plantsystemthathasahydro-stationneartheloadcenterandathermalpowerstationataremotelocationisshowninFig.6.10.

Thecharacteristicsofbothstationsare

C =(20+0.03P )P Rs./hr

w =(8+0.002P )P m /s

andγ =Rs.5.5/m

Thetransmissionlossco-efficient,B =0.0005.

1. Iftheloadis700MWfor15hradayand500MWfor9hronthesameday,findthegenerationschedule,dailywaterusedbythehydro-plant,andthedailyoperatingcostofthermalpower.

2. Determinethepowergenerationateachstationandthepowerreceivedbytheloadwhenλ=50Rs./MWh.

2

22

T GT GT

2 GH GH

2

22

3

3

3

FIG.6.8IllustrationforProblem1;(a)two-plantsystem;(b)dailyloadcurve



7

LoadFrequencyControl-I

OBJECTIVES

Afterreadingthischapter,youshouldbeabletobeableto:

studythegoverningcharacteristicsofageneratorstudytheloadfrequencycontrol(LFC)developthemathematicalmodelsfordifferentcomponentsofapowersystemobservethesteadystateanddynamicanalysisofasingle-areapowersystemwithandwithoutintegralcontrol

7.1INTRODUCTION

Inapowersystem,bothactiveandreactivepowerdemandsareneversteadyandtheycontinuallychangewiththerisingorfallingtrend.Steaminputtoturbo-generatorsorwaterinputtohydro-generatorsmust,therefore,becontinuouslyregulatedtomatchtheactivepowerdemand,failingwhichthemachinespeedwillvarywithconsequentchangeinfrequencyanditmaybehighlyundesirable.Themaximumpermissiblechangeinfrequencyis±2%.Also,theexcitationofthegeneratorsmustbecontinuouslyregulatedtomatchthereactivepowerdemandwithreactivepowergeneration;otherwise,thevoltagesatvarioussystembusesmaygobeyondtheprescribedlimits.Inmodernlargeinterconnectedsystems,manualregulationisnotfeasibleandthereforeautomaticgenerationandvoltageregulationequipmentisinstalledoneachgenerator.Thecontrollersaresetforaparticularoperatingconditionandtheytakecareofsmallchangesinloaddemandwithoutexceedingthelimitsoffrequencyandvoltage.As

thechangeinloaddemandbecomeslarge,thecontrollersmustbereseteithermanuallyorautomatically.

7.2NECESSITYOFMAINTAININGFREQUENCYCONSTANT

Constantfrequencyistobemaintainedforthefollowingfunctions:

AlltheACmotorsshouldrequireconstantfrequencysupplysoastomaintainspeedconstant.Incontinuousprocessindustry,itaffectstheoperationoftheprocessitself.Forsynchronousoperationofvariousunitsinthepowersystemnetwork,itisnecessarytomaintainfrequencyconstant.Frequencyaffectstheamountofpowertransmittedthroughinterconnectinglines.Electricalclockswillloseorgaintimeiftheyaredrivenbysynchronousmotors,andtheaccuracyoftheclocksdependsonfrequencyandalsotheintegralofthisfrequencyerrorislossorgainoftimebyelectricclocks.

7.3LOADFREQUENCYCONTROL

Loadfrequencycontrol(LFC)isthebasiccontrolmechanisminthepowersystemoperation.Wheneverthereisavariationinloaddemandonageneratingunit,thereismomentarilyanoccurrenceofunbalancebetweenreal-powerinputandoutput.Thisdifferenceisbeingsuppliedbythestoredenergyoftherotatingpartsoftheunit.

Thekineticenergyofanyunitisgivenby

whereIisthemomentofinertiaoftherotatingpartandωtheangularspeedoftherotatingpart.

IfKEreduces,ωdecreases;thenthespeedfalls,hencethefrequencyreduces.ThechangeinfrequencyΔfissensedandthroughaspeed-governorsystem,itisfedbacktocontrolthepositionoftheinletvalveoftheprime

mover,whichisconnectedtothegeneratingunit.Itchangestheinputtotheprimemoversuitablyandtriestobringbackthebalancebetweenthereal-powerinputandoutput.Hence,itcanbestatedthatthefrequencyvariationisdependentonthereal-powerbalanceofthesystem.

TheLFCalsocontrolsthereal-powertransferthroughtheinterconnectingtransmissionlinesbysensingthechangeinpowerflowthroughthetielines.

7.4GOVERNORCHARACTERISTICSOFASINGLEGENERATOR

Primemoversdrivingthegeneratorsarefittedwithgovernors,whichareregardedasprimarycontrolelementsintheLFCsystem.Governorssensethechangeinaspeedcontrolmechanismtoadjusttheopeningofsteamvalvesinthecaseofsteamturbinesandtheopeningofwatergatesinthecaseofwaterturbines.ThecharacteristicsofatypicalgovernorofasteamturbineareshowninFig.7.1,whichislinearizedbydottedlinesforstudyingthesystembehavior.

FIG.7.1Characteristicsofatypicalgovernorofasteamturbine

Theamountofspeeddropastheloadontheturbineisincreasedfromnoloadtoitsfull-loadvalueis(N –N),

o

whereN isthespeedatnoloadandNisthespeedatratedload.

Thesteady-statespeedregulationinperunitisgivenby

ThevalueofRvariesfrom2%to6%foranygeneratingunit.Sincethefrequencyandspeedaredirectlyrelated,thespeedregulationcanalsobeexpressedastheratioofthechangeinfrequencyfromnoloadtoitsfullloadtotheratedfrequencyoftheunit:

i.e.,

Ifthereisa4%speedregulationofaunit,thenforaratedfrequencyof50Hz,therewillbeadropof2Hzinfrequency.

IfthegenerationisincreasedbyΔP duetoastaticfrequencydropofΔf,thenthespeedregulationcanbedefinedastheratioofthechangeinfrequencytothecorrespondingchangeinreal-powergeneration:

i.e.,

TheunitofRistakenasHzperMW.Inpractice,powerismeasuredinperunitandhenceRisinHz/p.u.MW.

InFig.7.2,theturbineisoperatingwith99%ofno-loadspeedat25%offull-loadpowerandiftheloadisincreasedto50%,thespeeddropsto98%.Let‘A’betheinitialoperatingpointoftheturbineat50%loadandiftheloadisdroppedto25%,thespeedincreasesto99%.Inordertokeepthespeedat25%oftheloadsameasat

o

G

‘A’,thegovernorsettinghastobechangedbychangingthespringtensioninthefly-balltypeofgovernor.Thiswillresultinspeedcharacteristicsindicatedbythedottedlineparalleltothefirstoneandbelowit,passingthroughthepointA′,whichisthepointofintersectionofthenewspeedlineand25%loadline.Hence,theturbinecanbeadjustedtocarryanygivenloadatanydesiredspeed.

FIG.7.2Speed-governorsetting

Thistypeofshiftingthespeedorfrequencycharacteristicparalleltoitselfisknownassupplementarycontrol.Itisadoptedinon-linecontroltoensureproperloaddivisionamongtherunningunitsandtomaintainthesystemfrequency.Thereisanothermethodofchangingtheslopeofthegovernorcharacteristics.ThisisachievedbychangingtheratiooftheleverL(referthespeedcontrolmechanism)ofthegovernorandtherebyadjustingtheparameterRtoensureproperco-ordinationwiththeotherunitsofthesystem.Thisadjustmentcanbemadeduringtheoff-lineconditiononly.

7.5ADJUSTMENTOFGOVERNORCHARACTERISTICOFPARALLELOPERATINGUNITS

Whentwogeneratorsarerunninginparallel,thegovernorcharacteristicofthefirstunit(Line1)isshowntowardstheright,whilethatofthesecondunit(Line2)isshowntowardstheleftofthefrequencyaxisasshowninFig.7.3.

Thecharacteristicsareobviouslydifferentandhencecorrespondingtotheratedfrequencyf ,thetwounitscarryloadsP andP sothatthesystemloadP =P +P .IfthesystemloadisnowincreasedtoP′ ,thesystemfrequencywilldropdowntof′,sincetheunitscanonlyincreasetheiroutputbydecreasingthespeed.

Torestorethesystemfrequency,thecharacteristicofoneoftheunitssayofUnit1needstobeshiftedupwardsasindicatedbythedottedcharacteristic,sothatitcancarrytheincreasedload.TheshareofUnit1willbeP ′andthatofUnit2willbeP sothattheincreasedtotalload,P ′=P ′+P .

FIG.7.3Sharingofloadbytwounits(parallel)withaspeed-governorcharacteristicssetting

r

1 2 D 1 2

D

1

2

D 1 2

7.6LFC:(P–fCONTROL)

TheLFC,alsoknownasgenerationcontrolorP–fcontrol,dealswiththecontrolofloadingofthegeneratingunitsforthesystematnormalfrequency.Theloadinapowersystemisneverconstantandthesystemfrequencyremainsatitsnominalvalueonlywhenthereisamatchbetweentheactivepowergenerationandtheactivepowerdemand.Duringtheperiodofloadchange,thedeviationfromthenominalfrequency,whichmaybecalledfrequencyerror(Δf),isanindexofmismatchandcanbeusedtosendtheappropriatecommandtochangethegenerationbyadjustingtheLFCsystem.Itisbasicallycontrollingtheopeningoftheinletvalvesoftheprimemoversaccordingtotheloadingconditionofthesystem.Inthecaseofamulti-areasystem,theLFCsystemalsomaintainsthespecifiedpowerinterchangesbetweentheparticipatingareas.Inasmallersystem,thiscontrolisdonemanually,butinlargesystemsautomaticcontroldevicesareusedintheloopoftheLFCsystem.

TheLFCsystem,however,doesnotconsiderthereactivepowerflowinthesystemeventhoughthereactivepowerflowisalsoaffectedtosomeextentduringthefluctuatingloadcondition.Butsincethereisnocounterpartofthereactivepowerinthemechanicalsideofthesystem,itdoesnotcomewithintheloopoftheLFCsystem.

7.7Q–VCONTROL

Inthiscontrol,theterminalvoltageofthegeneratorissensedandconvertedintoproportionateDCsignalandthencomparedtoDCreferencevoltage.TheerrorinbetweenaDCsignalandaDCreferencevoltage,i.e.,Δ|V |istakenasaninputtotheQ–Vcontroller.AcontroloutputΔQ isappliedtotheexciter.

7.8GENERATORCONTROLLERS(P–fANDQ–VCONTROLLERS)

i

ci

TheactivepowerPismainlydependentontheinternalangleδandisindependentofthebusvoltagemagnitude|V|.ThebusvoltageisdependentonmachineexcitationandhenceonreactivepowerQandisindependentofthemachineangleδ.Changeinthemachineangleδiscausedbyamomentarychangeinthegeneratorspeedandhencethefrequency.Therefore,theloadfrequencyandexcitationvoltagecontrolsarenon-interactiveforsmallchangesandcanbemodeledandanalyzedindependently.

Figure7.4givestheschematicdiagramofloadfrequency(P–f)andexcitationvoltage(Q–V)regulatorsofaturbo-generator.TheobjectiveoftheMWfrequencyortheP–fcontrolmechanismistoexertcontroloffrequencyandsimultaneouslyexchangeofthereal-powerflowsviainterconnectinglines.Inthiscontrol,afrequencysensorsensesthechangeinfrequencyandgivesthesignalΔf .TheP–fcontrollersensesthechangeinfrequencysignal(Δf )andtheincrementsintie-linerealpowers(ΔP ),whichwillindirectlyprovideinformationabouttheincrementalstateerror(Δδ ).Thesesensorsignals(Δf andΔP )areamplified,mixed,andtransformedintoareal-powercontrolsignalΔP .ThevalvecontrolmechanismtakesΔP astheinputsignalandprovidestheoutputsignal,whichwillchangethepositionoftheinletvalveoftheprimemover.Asaresult,therewillbeachangeintheprimemoveroutputandhenceachangeinreal-powergenerationΔP .ThisentireP–fcontrolcanbeyieldedbyautomaticloadfrequencycontrol(ALFC)loop.

TheobjectiveoftheMVAr-voltageorQ–Vcontrolmechanismistoexertcontrolofthevoltagestate|V |.AvoltagesensorsensestheterminalvoltageandconvertsitintoanequivalentproportionateDCvoltage.ThisproportionateDCvoltageiscomparedwithareferencevoltageV bymeansofacomparator.TheoutputobtainedfromthecomparatoriserrorsignalΔ|V |andis

i

i

tie

i

i tie

ci

ci

Gi

i

iref

i

givenasinputtoQ–Vcontroller,whichtransformsittoareactivepowersignalcommandΔQ andisfedtoacontrollableexcitationsource.Thisresultsinachangeintherotorfieldcurrent,whichinturnmodifiesthegeneratorterminalvoltage.ThisentireQ–Vcontrolcanbeyieldedbyanautomaticvoltageregulator(AVR)loop.

FIG.7.4SchematicdiagramofP–fcontrollerandQ–Vcontroller

Inadditiontovoltageregulatorsatgeneratorbuses,equipmentisusedtocontrolvoltagemagnitudeatotherselectedbuses.Tap-changingtransformers,switchedcapacitorbanks,andstaticVArsystemscanbeautomaticallyregulatedforrapidvoltagecontrol.

7.9P–fCONTROLVERSUSQ–VCONTROL

AnystaticchangeintherealbuspowerΔP willaffectonlythebusvoltagephaseangles(δ )(sinceP∝δ),butwillleavethebusvoltagemagnitudesalmostunaffected.

StaticchangeinthereactivebuspowerΔQ affectsessentiallyonlythebusvoltagemagnitudes(sinceQ∝V),butleavethebusvoltagephaseanglesalmostunchanged.

ci

i

i

i

2

Staticchangeinreactivebuspowerataparticularbusaffectsthemagnitudeofthatbusvoltagemoststrongly,butinlessdegreethemagnitudesofthebusvoltagesatremotebuses.

7.10DYNAMICINTERACTIONBETWEENP–fANDQ–VLOOPS

Inastaticsense,forsmalldeviations,thereisalittleinteractionbetweenP–fandQ–Vloops.Duringdynamicperturbations,weencounterconsiderablecouplingbetweentwocontrolloopsfortwofollowingreasons:

Asthevoltagemagnitudefluctuatesatabus,therealloadofthatbus

willlikewisechangeasaresultofthevoltageloadcharacteristic .

Asthevoltagemagnitudefluctuatesatabus,thepowertransmittedoverthelinesconnectedtothatbuswillchange.Inotherwords,thechangeinQ–Vloopwillaffectthegeneratedemf,whichalsoaffectsthemagnitudeofrealpower.

AdynamicperturbationintheQ–Vloopwillthusaffectthereal-powerbalanceinthesystem.Ingeneral,theQ–VloopismuchfasterthantheP–floopduetothemechanicalinertialconstantsintheP–floop.IfitcanbeassumedthatthetransientsintheQ–VloopareessentiallyoverbeforetheP–floopreacts,thenthecouplingbetweenthetwoloopscanbeignored.

7.11SPEED-GOVERNINGSYSTEM

ThespeedgovernoristhemainprimarytoolfortheLFC,whetherthemachineisusedalonetofeedasmallersystemorwhetheritisapartofthemostelaboratearrangement.Aschematicarrangementofthemainfeaturesofaspeed-governingsystemofthekindusedonsteamturbinestocontroltheoutputofthegeneratortomaintainconstantfrequencyisasshowninFig.7.5.

FIG.7.5Speed-governorsystem

Itsmainpartsorcomponentsareasfollows:

Fly-ballspeedgovernor:Itisapurelymechanical,speed-sensitivedevicecoupleddirectlytoandbuildsdirectlyontheprimemoverstoadjustthecontrolvalveopeningvialinkagemechanism.Itsensesaspeeddeviationorapowerchangecommandandconvertsitintoappropriatevalveaction.Hence,thisistreatedastheheartofthesystem,whichcontrolsthechangeinspeed(frequency).Asthespeedincreases,theflyballsmoveoutwardsandthepointBonlinkagemechanism

movesupwards.Thereversewillhappenifthespeeddecreases.

Thehorizontalrotatingshaftonthelowerleftmaybeviewedasanextensionoftheshaftofaturbine-generatorsetandhasafixedaxisasshowninFig.7.5.Theverticalshaft,abovethefly-ballmechanism,alsorotatesbetweenfixedbearings.Althoughitsaxisisfixed,itcanmoveupanddown,transferringitsverticalmotiontothepilotpointB.

Hydraulicamplifier:Itisnothingbutasingle-statehydraulicservomotorinterposedbetweenthegovernorandvalve.Itconsistsofapilotvalveandthemainpiston.Withthisarrangement,hydraulicamplificationisobtainedbyconvertingthemovementoflow-powerpilotvalveintomovementofhigh-powerlevelmainpiston.

Inhydraulicamplification,alargemechanicalforceisnecessarysothatthesteamvalvecouldbeopenedorclosedagainsthigh-pressureinletsteam.

Speedchanger:Itprovidesasteady-statepoweroutputsettingfortheturbines.Itsupwardmovementopenstheupperpilotvalvesothatmoresteamisadmittedtotheturbineundersteadyconditions.Thisgivesrisetohighersteady-statepoweroutput.Thereversewillhappenifthespeedchangermovesdownward.

Linkagemechanism:Thesearelinkedfortransformingthefly-ballsmomenttotheturbinevalve(steamvalve)throughahydraulicamplifier.

ABCisarigidlinkpivotedatpointBandCDEisanotherrigidpivotedlinkatpointD.LinkDEprovidesafeedbackfromthesteamvalvemoment.

Thespeed-governingsystemisbasicallycalledtheprimarycontrolloopintheLFC.Ifthecontrolvalvepositionisindicatedbyx ,asmallupwardmovementofpointEdecreasesthesteamflowbyaconsiderable

E

amount.ItismeasuredintermsofvalvepowerΔP .ThisflowdecrementgetstranslatedintodecrementinturbinepoweroutputΔP .

Withthehelpoflinkagemechanism,thepositionofthepilotvalvecanbechangedinthefollowingthreedifferentways:

1. Directlybythespeedchanger:AsmallupwardmomentoflinkagepointAcorrespondstoadecreaseinthesteady-statepoweroutputorreferencepowerΔP .

2. Indirectlythroughthefeedbackduetothepositionchangesinthemainsystem.

3. IndirectlythroughfeedbackduetothepositionchangesinlinkagepointEresultingfromachangeinspeed.

7.11.1Speed-governingsystemmodel

Inthissection,wedevelopthemathematicalmodelbasedonsmalldeviationsaroundanominalsteadystate.LetusassumethatthesteamisoperatingundersteadystateandisdeliveringpowerP fromthegeneratoratnominalspeedorfrequencyf .

Underthiscondition,theprimemovervalvehasaconstantsettingχ ,thepilotvalveisclosed,andthelinkagemechanismisstationary.Now,wewillincreasetheturbinepowerbyΔP withthehelpofthespeedchanger.Forthis,themovementoflinkagepointAmovesdownwardbyasmalldistanceΔx andisgivenby

Δx =KΔP (7.1)

WiththemovementΔx ,thelinkpointCmoveupwardsbyanamountΔx andsodoesthelinkpointDbyanamountΔx upwards.DuetothemovementoflinkpointD,thepilotvalvemovesupwards,thenthehigh-pressureoilisadmittedintothecylinderofthehydraulicamplifierandflowsontothetopofthemainpiston.Duetothis,thepistonmovesdownwardbyanamountΔxandresultsintheopeningofthesteamvalve.Duetothe

v

T

ref

G

E

C

A

A C

A

C

D

E

0

o

0

openingofthesteamvalve,theflowofsteamfromtheboilerincreasesandtheturbinepoweroutputincreases,whichleadstoanincreaseinpowergenerationbyΔP .TheincreasedpoweroutputcausesanacceleratingpowerinthesystemandthereisaslightincreaseinfrequencysaybyΔfifthesystemisconnectedtoafinitesize(i.e.,notconnectedtoinfinitebus).

Nowwiththeincreasedspeed,theflyballsofthegovernormovedownwards,thuscausingthelinkpointBtomoveslightlydownwardsbyasmalldistanceΔxproportionaltoΔf.DuetothedownwardmovementoflinkpointB,thelinkpointCalsomovesdownwardsbyanamountΔx ,whichisalsoproportionaltoΔf.

ItshouldbenotedthatallthedownwardmovementsareassumedtobepositiveindirectionsasindicatedinFig.7.5.Nowmodeltheaboveeventsmathematically.

ThenetmovementoflinkpointCcontributestwofactorsasfollows:

1. Δx contribution:TheloweringofspeedchangerbyanamountΔ

x resultsintheupwardmomentoflinkpointCproportionaltoΔx :

i.e.,Δx′ =Δx l =−Δx l

or

SubstitutingΔx fromEquation(7.1)intheaboveequation,weget

where

2. Δfcontribution:IncreaseinfrequencyΔfcausesanoutwardmomentofflyballsandinturncausesthedownwardmovementofpointBbyanamountΔx ,whichisproportionaltoK ′Δf,i.e.,

G

B

C

A

A A

C A AB C BC

A

B 2

movementofpoint‘C’withpoint‘A’remainingfixedatΔx is

∴Δx″ =K Δf(7.3)

Therefore,thenetmovementoflinkpointCcanbeexpressedas

Δx =Δx′ +Δx″C(7.4)

SubstitutingthevaluesofΔx′ andΔx″ fromEquations(7.2)and(7.3)inEquation(7.4),weget

Δx =−K ΔP +K Δf(7.5)

TheconstantsK andK dependuponthelengthoflinkagearmsABandBCandalsodependupontheproportionalconstantsofthespeedchangerandthespeedgovernor.

ThemovementoflinkpointD,Δx istheamountbywhichthepilotvalveopensanditiscontributedbythemovementofpointC,Δx ,andmovementofpointE,Δx .

Therefore,thenetmovementofpointDcanbeexpressedas

Δx =Δx′ +Δx″ (7.6)

whereΔx′ (l +l )=Δx (l )

andΔx″ (l +l )=Δx (l )

A

C 2

C C

C C

C 1 C 2

1 2

D

C

E

D D D

D CD DE C DE

D CD DE E CD

SubstitutingthevaluesofΔx′ andΔx″ fromEquations(7.7)and(7.8)inEquation(7.6),weget

Δx =K Δx +K Δx (7.9)

ThemovementΔx ,resultsintheopeningofthepilotvalve,whichleadstotheadmissionofhigh-pressureoilintothehydraulicamplifiercylinder;thenthedownwardmovementofthemainpistontakesplaceandthusthesteamvalveopensbyanamountΔx .

TwoassumptionsaremadetorepresentthemathematicalmodelofthemovementofpointE:

1. Themainpistonandsteamvalvehavesomeinertialforces,whicharenegligiblewhencomparedtotheexternalforcesexertedonthepistonduetohigh-pressureoil.

2. Becauseofthefirstassumption,theamountofoiladmittedintothecylinderisproportionaltotheportopeningΔx ,i.e.,thevolumeofoil

admittedintothecylinderisproportionaltothetimeintegralofΔx .

ThemovementΔx isobtainedas

whereAistheareaofcross-sectionofthepiston:

where

D D

D 3 C 4 E

D

E

D

D

E

TheconstantK dependsuponthefluidpressureandthegeometryoftheorificeandcylinderofthehydraulicamplifier.

InEquation(7.10),thenegativesignrepresentsthemovementsoflinkpointsDandEintheoppositedirections.Forexample,thesmalldownwardmovementofΔx causesthemovementΔx inthepositivedirection(i.e.,upwards).

TakingtheLaplacetransformofEquations(7.5),(7.9),and(7.10),weget

Δx (s)=−K Δ (s)+K ΔF(s) (7.11)

Δx (s)=−K Δx (s)+K Δx (s) (7.12)

(7.13)

EliminatingΔx (s)andΔx (s)intheaboveequationsandsubstitutingΔx (s)fromEquation(7.12)inEquation(7.13),weget

SubstitutingΔx (s)fromEquation(7.11)intheaboveequation,weget

5

D E

C 1 PC 2

D 3 C 4 E

C D

D

C

FIG.7.6Blockdiagrammodelofaspeed-governorsystem


whereR isthespeedregulationofthegovernor

itisalsotermedasregulationconstantorsetting,

thegainofthespeedgovernor,and

thetimeconstantofthespeedgovernor.Normally,τ ≤100ms.

sg

Equation(7.15)canberepresentedinablockdiagrammodelasshowninFig.7.6,whichisthelinearizedmodelofthespeed-governormechanism.

Fromtheblockdiagram, isthenet

inputtothespeed-governorsystemandΔx (s)istheoutputofthespeedgovernor.

7.12TURBINEMODEL

WeareinterestednotintheturbinevalvepositionbutinthegeneratorpowerincrementΔP .ThechangeinvalvepositionΔx causesanincrementalincreaseinturbinepowerΔP andduetoelectromechanicalinteractionswithinthegenerator,itwillresultinanincreasedgeneratorpowerΔP ,i.e.,ΔP =ΔP ,sincethegeneratorincrementallossisneglected.Thisoverallmechanismisrelativelycomplicatedparticularlyifthegeneratorvoltagesimultaneouslyundergoeswildswingduetomajornetworkdisturbances.

Atpresent,wecanassumethatthevoltagelevelisconstantandthetorquevariationsaresmall.ThenanincrementalanalysiswillgivearelativelysimpledynamicrelationshipbetweenΔx andΔP .Suchananalysisrevealsconsiderabledifferences,notonlybetweensteamturbinesandhydro-turbines,butalsobetweenvarioustypes(reheatandnon-reheat)ofsteamturbines.Therefore,thetransferfunction,relatesthechangeinthegeneratedpoweroutputwithrespecttothechangeinthevalveposition,varieswiththetypeoftheprimemover.

7.12.1Non-reheat-typesteamturbines

Figure7.7(a)showsasingle-stagenon-reheattypesteamturbine.

Inthismodel,theturbinecanbecharacterizedbyasinglegainconstantK andasingletimeconstantτ as

E

G

E

T

G T G

E G

t t

FIG.7.7(a)Single-stagenon-reheat-typesteamturbine;(b)blockdiagramrepresentationofanon-reheat-typesteamturbine;(c)transferfunctionrepresentationofspeedcontrolmechanismofageneratorwithanon-

reheat-typesteamturbine

Typically,thetimeconstantτ liesintherangeof0.2to2.

Onopeningthesteamvalve,thesteamflowwillnotreachtheturbinecylinderinstantaneously.Thetimedelayexperiencedinthisisintheorderof2sinthesteampipe.


WecanrepresentEquation(7.17)byablockdiagramasshowninFig.7.7(b).

t t

t

Figure7.7(c)showsthelinearizedmodelofanon-reheat-typeturbinecontrollerincludingthespeed-governormechanism.

FromFig.7.7(c),thecombinedtransferfunctionoftheturbineandthespeed-governormechanismwillbe

Therefore,

Ingeneral,itisobtainedthattheturbineresponseislowwiththeresponsetimeofseveralseconds.

FIG.7.8Blockdiagramofasimplifiedturbinegovernor

7.12.2Incrementalorsmallsignalforaturbine-governorsystem

LetthecommandincrementalsignalbeΔP .Theninthesteadystate,wegetΔP =K K ΔP .LetK K =1;theblockdiagramofFig.7.7(c)isreducedtothatshowninFig.7.8.

Thisblockdiagramgivesthederivationofanincrementalorsmallsignalmodel.Themodelisadoptedforlargesignalusebyaddingasaturation-typenon-linearelement,whichintroducestheobviousfactthatthesteamvalvemustoperatebetweencertainlimits.The

C

G sg t C sg t

valvecanneitherbemoreopenthanfullyopennormoreclosedthanfullyclosed.

ThismodelofFig.7.8mayalsobemodifiedtoaccountforreheatcyclesintheturbineandmoreaccuraterepresentationoffluiddynamicsinthesteaminletpipesorinthehydraulicturbinesinthepenstock.

7.12.3Reheattypeofsteamturbines

Moderngeneratingunitshavereheat-typesteamturbinesasprimemoversforhigherthermalefficiency.

Figure7.9showsatwo-stagereheat-typesteamturbine.

Insuchturbines,steamathighpressureandlowtemperatureiswithdrawnfromtheturbineatanintermediatestage.Itisreturnedtotheboilerforresuperheatingandthenreintroducedintotheturbineatlowpressureandhightemperature.Thisincreasestheoverallthermalefficiency.Mostly,twofactorsinfluencethedynamicresponseofareheat-typesteamturbine:

1. Entrainedsteambetweentheinletsteamvalveandthefirststateofturbine.

2. Thestorageactioninthereheater,whichcausestheoutputofthelow-pressurestagetolagbehindthatofthehigh-pressurestage.

FIG.7.9Atwo-stagereheattypeofasteamturbine

Thus,inthiscase,theturbinetransferfunctionischaracterizedbytwotimeconstants.Itinvolvesanadditionaltimelagτ associatedwiththereheaterinadditiontotheturbinetimeconstantτ .Hence,theturbinetransferfunctionwillbeofasecondorderandisgivenby

Thetimeconstantτ hasavalueintherangeof10sandapproximatesthetimedelayforchargingthereheatsectionoftheboiler.K isareheatcoefficientandisequaltotheproportionoftorquedevelopedinthehigh-pressuresectionoftheturbine:

K =(1–fractionofthesteamreheated)

WhenthereisnoreheatK =1andthetransferfunctionreducestoasingletimeconstantgiveninEquation(7.16).

ThetransferfunctionsasgivenbyEquations(7.16)and(7.18)givegoodrepresentationwithinthefirst20sfollowingtheincrementaldisturbance.Theydonotaccountfortheslowerboilerdynamics.Togetaneasyanalyzation,itcanbeassumedthattheprimemoverorturbineismodeledbyasingleequivalenttimeconstantτasgiveninEquation(7.16).

7.13GENERATOR–LOADMODEL

Thegenerator–loadmodelgivestherelationbetweenthechangeinfrequency(Δf)asaresultofthechangeingeneration(ΔP )whentheloadchangesbyasmallamount(ΔP ).

Whenneglectingthechangeingeneratorloss,ΔP =ΔP (changeinturbinepoweroutput),net-surpluspower

r

t

r

r

r

r

t

G

D

G

T

atthebusbar=(ΔP –ΔP ).Thissurpluspowercanbeabsorbedbythesystemintwodifferentways:

(i)Byincreasingthestoredkineticenergyofthe

generatorrotoratarate

LetW bethestoredKEbeforethedisturbanceatnormalspeedandfrequencyf ,andW betheKEwhenthefrequencyis(f +Δf).

SincethestoredKEisproportionaltothesquareofspeedandthefrequencyisproportionaltothespeed,

Neglectinghigher-orderterms,since issmall:

Differentiatingtheaboveexpressionwithrespectto‘t’,weget

G D

KE

KE

0

0

0

LetHbetheinertiaconstantofagenerator(MW-s/MVA)andP theratingoftheturbo-generator(MVA):

W =H×P (MW-sorM-J)(7.22)

Hence,Equation(7.21)becomes

(ii)Theloadonthemotorsincreaseswithincreaseinspeed.Theloadonthesystembeingmostlymotorload,hencesomeportionofthesurpluspowerisobservedbythemotorloads.Therateofchangeofloadwithrespecttofrequencycanberegardedasnearlyconstantforsmallchangesinfrequency.

i.e.,

wheretheconstantBistheareaparameterinMW/Hzandcanbedeterminedempirically.Bispositiveforapredominantlymotorload.

Now,thesurpluspowercanbeexpressedas

FromEquations(7.23)and(7.24),theaboveequationcanbemodifiedas

DividingthroughoutbyP ofEquation(7.25),weget

r

KE r

r

0

TakingLaplacetransformonbothsides,weget

where isthepowersystemtimeconstant

(normally20s)and thepowersystemgain.

Equation(7.26)canberepresentedinablockdiagrammodelasgiveninFig.7.10.

Theoverallblockdiagramofanisolatedpowersystemisobtainedbycombiningindividualblockdiagramsofaspeed-governorsystem,aturbinesystem,andagenerator–loadmodelandisasshowninFig.7.11.

Thisrepresentationbeingathird-ordersystem,thecharacteristicequationforthesystemwillbeofthethirdorder.

FIG.7.10Blockdiagramrepresentationofagenerator–loadmodel

FIG.7.11Completeblockdiagramrepresentationofanisolatedpowersystem

Example7.1:Twogeneratingstations1and2havefull-loadcapacitiesof200and100MW,respectively,atafrequencyof50Hz.Thetwostationsareinterconnectedbyaninductionmotorandsynchronousgeneratorsetwithafull-loadcapacityof25MWasshowninFig.7.12.ThespeedregulationofStation-1,Station-2,andinductionmotorandsynchronousgeneratorsetare4%,3.5%,and2.5%,respectively.Theloadsonrespectivebusbarsare750and50MW,respectively.Findtheloadtakenbythemotor-generatorset.

Solution:

LetapowerofAMWflowfromStation-1toStation-2:

∴TotalloadonStation-1=(75+A)MW

TotalloadonStation-2=(50−A)

%dropinspeedatStation-1=

%dropinspeedatStation-2=

ThereductioninfrequencywillresultduetothepowerflowfromStation-1throughtheinterconnectorofM-Gset.

∴%dropinspeedatM-Gset=

(reductioninfrequencyatStation-1+reductioninfrequencyatM-Gset)

=(reductioninfrequencyatStation-2)

0.02(75+A)+0.1A = 0.035(50−A)

1.5+0.02A+0.1A = 1.75−0.03A

0.02A+0.1A+0.03A = 175−1.5=0.25

0.15A = 0.25

A = 1.666MW

i.e.,apowerofA = 1.666MWflowsfromStation-1toStation-2.

∴TotalloadatStation-1=75+A = 75+1.666=76.666MW

TotalloadatStation-2=50−A = 50−1.666=48.334MW


Example7.2:A125MVAturbo-alternatoroperatoronfullloadoperatesat50Hz.Aloadof50MWissuddenlyreducedonthemachine.Thesteamvalvestotheturbinecommencetocloseafter0.5sduetothetimelaginthegovernorsystem.Assumingtheinertiatobeconstant,H=6kW-sperkVAofgeneratorcapacity,calculatethechangeinfrequencythatoccursinthistime.

Solution:

Bydefinition,

∴Energystoredatnoload=6×125×1,000=750MJ

Excessiveenergyinputtorotatingpartsin0.5s=50×0.5×1,000=25MJ

Asaresultofthis,thereisanincreaseinthespeedofthemotorandhenceanincreaseinfrequency:

7.14CONTROLAREACONCEPT

Inrealpractice,thesystemofasinglegeneratorthatfeedsalargeandcomplexareahasrarelyoccurred.Severalgeneratorsconnectedinparallel,locatedalsoatdifferentlocations,willmeettheloaddemandofsuchageographicallylargearea.Allthegeneratorsmayhavethesameresponsecharacteristicstothechangesinloaddemand.

Itispossibletodivideaverylargepowersystemintosub-areasinwhichallthegeneratorsaretightlycoupledsuchthattheyswinginunisonwithchangeinloadorduetoaspeed-changersetting.Suchanarea,whereallthegeneratorsarerunningcoherentlyistermedasacontrolarea.Inthisarea,frequencymaybesameinsteadystateanddynamicconditions.Fordevelopingasuitablecontrolstrategy,acontrolareacanbereducedtoasinglegenerator,aspeedgovernor,andaloadsystem.

7.15INCREMENTALPOWERBALANCEOFCONTROLAREA

Inthissection,weshalldevelopadynamicmodelintermsofincrementalpowerandfrequencydynamicsofacontrolarea‘i’connectedviatielinesasshowninFig.7.13.

Nowassumethatcontrolarea‘i’experiencesarealloadchangeΔP (MW).Duetotheactionsoftheturbinecontrollers,itsoutputincreasesbyΔP (MW).Thenet-surpluspowerinthearea(ΔP –ΔP )willbeabsorbedbythesysteminthreeways:

ByincreasingtheareakineticenergyW attherate

Byanincreasedloadconsumption.Alltypicalloads(becauseofthedominanceofmotorloads)experienceanincrease,

withspeedorfrequency.

ByincreasingtheflowofpowerviatielineswiththetotalamountΔP MW,whichisdefinedpositiveforoutflowfromthearea.

Di

Gi

Gi Di

KE, i

tie,i

FIG.7.13Interconnectedcontrolarea

Hence,thenet-surpluspowercanbeexpressedas

ΔP isthedifferencebetweenscheduledrealpowerandactualrealpowerthroughinterconnectedlinesanditistakenastheinputtotheLFCsystem.

7.16SINGLEAREAIDENTIFICATION

Thefirsttwotermsontheright-handsideofEquation(7.27)representagenerator–loadmodel(withthesubscript‘i’absent).Ifthethirdtermisabsent,itmeansthatthereisnointerchangeofpowerbetweenarea‘i’andanyotherarea.Thus,itbecomesasingle-areacase.Asingleareaisacoherentareainwhichallthegeneratorsswinginunisontothechangesinloadorspeed-changersettingsandinwhichthefrequencyisassumedtobeconstantthroughoutbothinstaticanddynamicconditions.Thissinglecontrolareacanberepresentedbyanisolatedpowersystemconsistingofaturbine,itsspeedgovernor,generator,andload.

7.16.1Blockdiagramrepresentationofasinglearea

Theblockdiagramofanisolatedpowersystem,whichinessenceisasingle-areasystem,isthesameastheblockdiagramgiveninFig.7.11.

tie

7.17SINGLEAREA—STEADY-STATEANALYSIS

TheblockdiagramofanLFCofanisolatedpowersystemofathird-ordermodelisrepresentedinFig.7.11.

Therearetwoincrementalinputstothesystemandtheyare:

1. Thechangeinthespeed-changerposition,ΔP (referencepower

input).2. Thechangeintheloaddemand,ΔP .

Inthissection,wewillanalyzetheresponseofasingle-areasystemtosteady-statechangesbythreeways:

1. Constantspeed-changerpositionwithvariableloaddemand(uncontrolledcase).

2. Constantloaddemandwithvariablespeed-changerposition(controlledcase).

3. Variablespeed-changerpositionaswellasloaddemand.

7.17.1Speed-changerpositionisconstant(uncontrolledcase)

WiththemodelgiveninFig.7.11andwithΔP =0,theresponseofanuncontrolledsingleareaLFCcanbeobtainedasfollows.

Letusconsiderasimplecasewhereinthespeedchangerhasafixedsetting,whichmeansΔP =0andtheloaddemandalonechanges.Suchanoperationisknownasfreegovernoroperationoruncontrolledcasesincethespeedchangerisnotmanipulated(orcontrolledtoachievebetterfrequencyconstancy).

Forasuddenstepchangeofloaddemand,

Forsuchanoperation,thesteady-statechangeoffrequencyΔfistobeestimatedfromtheblockdiagramofFig.7.14as

C

D

C

C

Applyingthefinalvaluetheorem,wehave

FIG.7.14BlockdiagramrepresentationofanisolatedpowersystemsettingΔP =0

ThegainK isfixedfortheturbineandK isfixedforthepowersystem.ThegainK ofthespeedgovernoriseasilyadjustablebychangingthelengthsofvariouslinksofthelinkagemechanism.K issoadjustedsuchthatK K ≈1.

ThereforeEquation(7.29)canbesimplifiedas:

C

t ps

sg

sg

sg t

Alsoweknowfromthedynamicsofthegenerator–

loadmodel,

where

inp.u.MW/unitchangeinfrequency

wherethefactor andisknownasthearea

frequencyresponsecharacteristic(AFRC)orareafrequencyregulationcharacteristic.

Equation(7.30)givesthesteady-stateresponseoffrequencytothechangesinloaddemand.Thespeedregulationisusuallysoadjustedthatchangesinfrequencyaresmall(oftheorderof5%)fromnoloadtofullload.Figure7.15givesthelinearrelationshipbetweenfrequencyandloadforafreegovernor

operation,withspeedchangessettogiveascheduledfrequencyof100%atfullload.

Thedrooportheslopeoftherelationshipis

PowersystemparameterBisgenerallymuchlessthan

sothatBcanbeneglectedinEquation(7.30),

whichresultsin

Δf=−R(ΔP )(7.31a)

Thedroopofthefrequencycurveisthusmainlydeterminedbythespeed-governorregulation(R).

FIG.7.15Steady-stateloadfrequencycharacteristicsofaspeed-governingsystem

Theincreaseinloaddemand(ΔP )ismetundersteady-stateconditionspartlybytheincreasedgeneration(ΔP )duetotheopeningofthesteamvalve

D

D

G

andpartlybythedecreasedloaddemandduetodroopinfrequency.

Theincreaseingenerationisexpressedas

SubstitutingΔffromEquation(7.30),weget

Andadecreaseinthesystemloadisexpressedas

FromEquations(7.31(b))and(7.31(c)),itisobservedthatcontributionofthedecreaseinthesystemloadismuchlessthantheincreaseingeneration.

7.17.2Loaddemandisconstant(controlledcase)

Considerastepchangeinaspeed-changerpositionwiththeloaddemandremainingfixed:

i.e.,

Thesteady-statechangeinfrequencycanbeobtainedfromtheblockdiagramofFig.7.16:

G

Thesteady-statevalueisobtainedbyapplyingthefinal-valuetheorem:

FIG.7.16BlockdiagramrepresentationofanisolatedpowersystemsettingΔP =0

D

7.17.3Speedchangerandloaddemandarevariables

Bysuperposition,ifthespeed-changersettingischangedbyΔP whiletheloaddemandalsochangesbyΔP ,thesteady-statechangeinfrequencyisobtainedfromEquations(7.30)and(7.32)as

Fromtheaboveequation,wecanobservethatthechangeinloaddemandcausesthechangesinfrequency,whichcanbecompensatedbychangingthepositionofthespeedchanger.

IfΔP =ΔP ,thenΔfwillbecomezero.

7.18STATICLOADFREQUENCYCURVES

Theblockdiagramrepresentationofaturbine-speed-governormodelisshowninFig.7.17(a)andtheirstaticloadfrequencycurvesareshowninFig.7.17(b).

ThecurverelatespowergenerationP andfrequencyfwithcontrolparameterP .

FromtheblockdiagramshowninFig.7.17(a),wegetthestaticalgebraicrelationfromwhichthelocalshapeofthespeed-powercurvesmaybeinferred.

c D

C D

G

C

FIG.7.17(a)Blockdiagramofaturbine-speed-governormodel;(b)staticloadfrequencycurvesfortheturbinegovernor

Figure7.17(b)givesthetwostaticload-frequencycurves.Adjustpowergeneration,P ,byusingaspeedermeter(speedchanger)uptoP =P ,whereP isthedesiredcommandpoweratsynchronousspeedω (f ).Withfreegovernoroperation(i.e.,ΔP =0),thefixedspeed-changerpositionP predictsthestraight-linerelationship.Thisstraightline(1)hasaslopeof–R.

Togetmoregenerationatthesamesynchronousspeedofω (f ),adjustP toP withaspeedermeter.Thisresultsintheloadfrequencycurve(2).Thespeed

G

G C1 C1

C

C1

C1 C2

0 0

0 0

regulationRreferstothevariationinfrequencywithpowergeneration.Bettertheregulationresults,lessthedroopsspeed-power(load)characteristicsofLFC.

7.19DYNAMICANALYSIS

Themeaningofdynamicresponseishowthefrequencychangesasafunctionoftimeimmediatelyafterdisturbancebeforeitreachesthenewsteady-statecondition.Theanalyzationofdynamicresponserequiresthesolutionofdynamicequationofthesystemforagivendisturbance.Thesolutioninvolvesthesolutionofdifferentequationsrepresentingthedynamicbehaviorofthesystem.

TheinverseLaplacetransformofΔF(s)givesthevariationoffrequencywithrespecttotimeforagivenstepchangeinloaddemand.Comparingtherelativevaluesoftimeconstants,wecanreducethethirdorderedmodeltoafirstorderedsystem.

ForapracticalLFCsystem,

τ <τ <<τ

Typicalvaluesare:

τ =0.4s

τ =0.5s

τ =20s

Ifτ andτ areconsiderednegligiblecomparedtoτandbyadjustingK K =1,theblockdiagramofLFCofthepowersystemofanisolatedsystemisreducedtoafirst-ordersystemasshowninFig.7.18withΔP =0foranuncontrolledcase.

FromFig.7.18,thechangeinfrequencyisgivenby

sg t Ps

sg

t

Ps

sg t Ps

sg t

c

FIG.7.18First-orderapproximateblockdiagramofLFCofanisolatedarea

FIG.7.19Dynamicresponseoffrequencychange(Δf)forastep-loadchange

Theplotofchangeinfrequencyversustimeforafirst-orderapproximationandexactresponseareshowninFig.7.19:

ΔP =0.01p.u,K =100,R=3,τ =0.4s,τ =0.5s,andτ =20s

Example7.3:Anisolatedcontrolareaconsistsofa200-MWgeneratorwithaninertiaconstantofH=5kW-s/kVAhavingthefollowingparameters(Fig.7.20(a)):

D ps sg t

Ps

Powersystemgainconstant,K =100

Powersystemtimeconstant,τ =20s

Speedregulation,R=3

Normalfrequency,f =50Hz

Obtainthefrequencyerrorandplotthegraphofdeviationoffrequencywhenastep-loaddisturbanceof(i)0.5%,(ii)1%,and(iii)2%isapplied(Fig.7.20(b)).

FromFig.7.20(b),thesteady-statechangeinfrequencyΔf =–0.0145Hz.

Similarly(ii)forastep-loadchangeof1%,thesteady-statechangeinfrequencyΔf =–0.029Hz;(iii)forastep-loadchangeof2%,steady-statechangeinfrequencyΔf =–0.0583Hz

Example7.4:ForExample7.3,showtheeffectofgovernoractionandturbinedynamics(Fig.7.21(a)),iftheyarenottobeneglectedandgiventhatτ =0.4sandτ =0.5sforastep-loadchangeof(i)0.5%and(ii)1%(Fig.7.21(b)).

FromFig.7.21(b),thesteady-statechangeinfrequencyΔf =–0.0235Hz.

Similarly(ii)forastep-loadchangeof1%,thesteady-statechangeinfrequencyΔf =–0.047Hz.

Example7.5:ObtaintheresultantfrequencyplotwhencombiningExamples7.3and7.4forastep-loaddisturbanceof0.5%(Figs.7.22(a)and(b)).

ps

ps

ss

ss

ss

sg

t

ss

ss

0

FIG.7.20(a)Simulationblockdiagramofsingleareawithoutaspeed-governorsystem;(b)responseofthechangeinfrequencyforFig.7.20(a)for

astep-loadchangeof0.5%

FIG.7.21(a)Simulationblockdiagramofsingleareawithaspeed-governorsystem;(b)responseofthechangeinfrequencyforFig.7.21(a)for

astep-loadchangeof0.5%

FIG.7.22(a)Simulationblockdiagramofasingleareawithoutandwithaspeed-governorsystem;(b)responseofthechangeinfrequencyforFig.

7.22(a)forastep-loadchangeof0.5%

Example7.6:Anisolatedcontrolareaconsistsofa200-MWgeneratorwithaninertiaconstantofH=5kw-s/kVAhavingthefollowingparameters(Fig.7.23(a)):



Speedregulation,R=3


Governortimeconstant,τ =0.4s

Turbinetimeconstant,τ =0.5s

Obtainthefrequencyerrorandplotthegraphofdeviationoffrequencywhenastepchangeof1%inthespeed-changerpositionisapplied(Fig.7.23(b)).

ps

ps

sg

t

0

FIG.7.23(a)Simulationblockdiagramofasimulatedsystemwithastepchangeinthespeed-changerposition;(b)frequencyresponseforExample

7.6




Speedregulation,R=3

ps

ps

0




Obtainthefrequencyerrorandplotthegraphofdeviationoffrequencywhenastepchangeof1%inboththespeed-changerpositionandtheloadisapplied(Fig.7.24(b)).

FIG.7.24(a)Simulationblockdiagramofasingle-areasystemwithPCandPD;(b)frequencyresponseofExample7.5

Example7.8:Findthestaticfrequencydropiftheloadissuddenlyincreasedby25MWonasystemhavingthe

sg

t

0

followingdata:

RatedcapacityP =500MW

OperatingLoadP =250MW

InertiaconstantH=5s

GovernorregulationR=2Hzp.u.MW

Frequencyf=50Hz

Alsofindtheadditionalgeneration.

Solution:

Assumingthefrequencycharacteristictobelinear,wehave

∂P /∂fexpressedinp.u.,

Areafrequencyresponsecharacteristic(AFRC)

Thestaticfrequencydrop

Hence,thesystemfrequencydropsto(50–0.098)=49.902Hz.

Theamountofadditionalgeneration

r

D

D

Whilethesuddenincreaseinloadis25MW,theincreaseingenerationis24.5MWand0.5MWisthelossofloadduetothedropinfrequency.

7.20REQUIREMENTSOFTHECONTROLSTRATEGY

Thefollowingarethebasicrequirementsneededforthecontrolstrategy:

Thesystemfrequencycontrolisobtainedthroughaclosedloop.Sincestabilityisthemajorproblemassociatedwithaclosed-loopcontrol,maintenanceofthestabilitywillbethemainobjective.Thefrequencydeviationduetoastep-loadchangeshouldreturntozero.Thecontrolthatoffersaboveiscalled‘isochronouscontrol’.Inaddition,thecontrolshouldkeepthemagnitudeofthetransientfrequencydeviationtoaminimum.Theintegralofthefrequencyerrorshouldnotexceedacertainmaximumvalue.

Isochronouscontrolensuresthatthesteady-statefrequencyerrorfollowingastep-loadchangewillbezero.However,nocontrolcaneliminatetransientfrequencyerror.Thetimeerrorofsynchronousclocksisproportionaltotheintegralofthistransientfrequencyerror.Therefore,itisnecessarytoputalimitonthevalueofthisintegral.

Thetotalloadshouldbedividedamongtheindividualgeneratorsofthecontrolareaforoptimumeconomy.

Thefirstthreerequirementsaresatisfiedwhentheadditionoftheintegral-controltothesystemtakesplace.

7.20.1Integralcontrol

Theintegralcontroliscomposedofafrequencysensorandanintegrator.ThefrequencysensormeasuresthefrequencyerrorΔfandthiserrorsignalisfedintothe

integrator.Theinputtotheintegratoriscalledthe‘AreaControlError’(i.e.,ACE=Δf).

TheACEisthechangeinareafrequency,whichwhenusedinanintegral-controlloop,forcesthesteady-statefrequencyerrortozero.

Theintegratorproducesareal-powercommandsignalΔP andisgivenby

ΔP =−K ∫Δfdt (7.33)

=−K ∫(ACE)dt

ThesignalΔP isfedtothespeed-changercausingittomove.Here,K iscalledtheintegralgainconstant,whichcontrolstherateofintegration.ThefrequencysensorandtheintegratorareconnectedinthesystemasaclosedcontrolloopasshownintheblockdiagraminFig.7.25.

Figure7.25consistsofFig.7.11augmentedbyadditionalloopsshowingthegenerationofACEanditsuseinchangingtheareacommandpowers;Risthespeed-regulationfeedbackparameter.ΔP (s),ΔP (s),andΔF(s)aretheincrementalchangesinthegeneration,systemload,andfrequency,respectively.TheblockdiagramofFig.7.25isthesingle-areapowersystem(isolatedpowersystem)withintegralcontrolforsmallincrementalchanges.

Thenegativesignintheintegralcontrollerisforproducinganegativeordecreasecommandforapositivefrequencyerror.ThegainconstantK ispositiveandcontrolstherateofintegration,andthusthespeedoftheresponseofthecontrolloop.Theintegratorisanelectronicintegratorofthesametypeasusedinanalogcomputers.

C

C I

I

C

I

G D

I

Inviewofhardware,wecanunderstandthepresenceoftheintegratorbyconsideringtheACEvoltagesdistributedtothespeedchangers(speedermotors)ofindividualgeneratorunitsthatparticipateinsupplementarycontrolwithinagivenarea.ThesemotorsturnatarateofθproportionaltotheACEvoltageandcontinuetoturnuntiltheyaredriventozero.

FIG.7.25ProportionalplusintegralcontrolofLFCofasingle-areasystem

Theintegralcontrolwillgiverisetozerosteady-statefrequencyerror(Δf =0)duetoastep-loadchange.Aslongastheerrorremains,theintegratoroutputwillincrease,causingthespeedchangertomove.Theintegratoroutputandthusthespeed-changerpositionattainaconstantvalueonlywhenthefrequencyerrorhasbeenreducedtozero.Thisisprovedthroughasimplifiedmathematicalanalysisasfollows.

7.21ANALYSISOFTHEINTEGRALCONTROL

Thefollowingassumptionsaremadeinordertoobtainasimpleanalysis.Theseassumptionsdonotdistorttheessentialfeatures.Also,theerrorsintroducedonaccountoftheseassumptionsaffectonlythetransientandnotthesteady-stateresponse.

Assumptions

steadystate

Thetimeconstantofthespeed-governormechanismτ andthatof

theturbineτ arebothneglected,i.e.,τ =τ =0.

Thespeedchangerisanelectromechanicaldeviceandhenceitsresponseisnotinstantaneous.However,itisassumedtobeinstantaneousinthepresentanalysis.Allnon-linearitiesintheequipment,suchasdeadzone,etc.,areneglected.ThegeneratorcanchangeitsgenerationΔP asfastasitis

commandedbythespeedchanger.TheACEisacontinuoussignal.

TheLaplacetransformationofEquation(7.33)gives

and,forastepchangeofloaddemandΔP ,

FromtheblockdiagramofFig.7.25,wehave

and

SubstitutingforΔP (s)fromEquation(7.36)intheaboveequation,weget

sg

t sg t

G

D

G

Equation(7.38)becomesafourth-ordersystem.

Thesteady-statevalueofΔf(t)canbeobtainedbyapplyingthefinal-valuetheorem,viz.,

Hence,thestatic-orsteady-statefrequencyerrorwillbezerowithintegralcontrol.

ThenatureoftransientvariationofΔf(t)canbefoundbytakingtheinverseLaplacetransformofEquation(7.39).Accordingtoassumption(i),Equation(7.39)simplifiesto

ThenatureofΔf(t)dependsontherootsofthecharacteristicequationofEquation(7.39)

Theaboveequationcanberewrittenas

where isapositiverealnumber

and

ThenatureoftherootsofEquation(7.41)dependsonwhetherω =0,ω >0,orω <0.

Case(i):ω =0

Thecharacteristicequationhasarepeatedroot(viz.,αrepeatedtwice).Hence,theexpressionforΔf(t)containstermsofthetype

e andte

2 2 2

2

–αt –αt

Consequently,theresponse[viz.,Δf(t)]isacriticallydampedone.Forthiscriticalcase,

SolvingtheaboveforK ,weget

Case(ii):ω >0

Now,(s+α) =–ω ,whereω isapositiverealnumber.

(s+α)=+jω

(or)s=(–α±jω)

ThetimeresponseΔf(t)willthereforeconsistofdampedoscillatorytermsofthetype

e sinωtande cosωt.

Thiscaseiscalledasupercriticalcase.Inthiscase,K>K .

Case(iii):ω >0

Then,ω isanegativerealnumber.

So,(s+α) =–ω isapositiverealnumber

=γ (say)

∴(s+α)=+γ[sinceγ<α]

ors=(−α+γ)or(−α−γ)

=β1or−β2(say)

Accordingly,inthiscase,thetimeresponseΔf(t)willcomprisetermsofthetype

I

I

Icrit

2

2 2 2

–αt –αt

2

2

2 2

2

−β t1 −β t2

e ande

Hence,theresponsewillbedampedandnon-oscillatory.Thecontrol,inthiscase,iscalledthesubcriticalintegralcontrol.Inthiscase,K <K .

Inallthethreecasesdescribedabove,Δf(t)willapproachzero.Thiswasprovedearlierusingthefinal-valuetheorem.Itcanbeobservedthatthetransientfrequencyerrordoesremainfinite.Thisisaproofthatthecontrolisbothstableandisochronous.Thus,thefirsttwocontrolrequirementsstatedearlier(Section7.20)arefulfilledwiththisintegralcontrol.Thiscontrolisalsocalled‘proportionalplusintegralcontrol’.Theproportionalcontrolisprovidedbytheclosedloopofgainconstantof1/R.

Theactualsimulatedtimeresponsesofasingle-areacontrolsystemwithandwithoutintegralcontrolareasshowninFig.7.26.

FIG.7.26DynamicresponseofLFCofasingle-areasystemwithandwithoutintegralcontrolaction

7.22ROLEOFINTEGRALCONTROLLERGAIN(K )SETTING

I Icrit

I

−β t1 −β t2

Theroleplayedbythegainsettingofanintegralcontrollerinthecontroloffrequencyerrorisdescribedbelow.

Withsubcriticalingainsettings(i.e.,K <K ),asluggish,non-oscillatoryresponseisobtained.TheslownessoftheresponsemakestheintegralofΔf(t),andhencethetimeerror,relativelylarge.However,withthissetting,thegeneratorneednot‘chase’rapidloadfluctuations,whichultimatelycauseequipmentwear.

7.23CONTROLOFGENERATORUNITPOWEROUTPUT

ThecollectiveperformanceofallgeneratorsinthesystemisstudiedbyassumingtheequivalentgeneratorhavinganinertiaconstantofH tobeequaltothesumoftheinertiaconstantsofallthegeneratingunits.Similarly,theeffectsofthesystemloadsarelumpedintoasingledampingconstantB.

Forasystemhaving‘n’generatorsandacompositeload-dampingconstantofBthesteady-statefrequencydeviationfollowingaloadchangeΔP is

Thecompositefrequencyresponsecharacteristicofthesystemis

ItisnormallyexpressedinMW/Hz.Sometimes,itisreferredtoasthestiffnessofthesystem.


I Icrit

eq

D



Speedregulation,R=3




Obtainthefrequencyerrorandplotthegraphofdeviationoffrequencywhenastep-loadchangeof0.48p.u.withanintegralcontrolleractionofk =0.1isapplied(Fig.7.27(b)).




Speedregulation,R=2.5

ps

ps

sg

t

i

ps

ps

0

FIG.7.27(a)Simulationblockdiagramforasingle-areasystemwithintegralcontrolaction;(b)frequencyresponsecharacteristicsofExample

7.9




Integratorgainconstant,k =0.15

Obtainthefrequencyerrorandplotthegraphofdeviationoffrequencywhenastep-loaddisturbanceof2%withandwithouttheintegralcontrolactionisapplied(Fig.7.28(b)).

sg

t

i

0

FIG.7.28(a)Simulationdiagramofasingle-areasystemwithoutandwithintegralcontrolaction;(b)frequencyresponsecharacteristicsofExample

7.10

Example7.11:GivenasingleareawiththreegeneratingunitsasshowninFig.7.29:

Unit Rating(MVA) SpeeddroopR(perunitonunitbase)

1 100 0.010

2 500 0.015

3 500 0.015

TheunitsareloadedasP =80MW;P =300MW;P=400MW.AssumeB=0;whatisthenewgenerationoneachunitfora50-MWloadincrease?RepeatwithB=1.0p.u.(i.e.,1.0p.u.onloadbase).

Solution:

1 2 3

1.

withB=0;atacommonbaseof1,000MVA

FIG.7.29Asingleareawiththreegeneratingunits

f=f +Δf

=50−652.17×10 (50)=49.96Hz

Changesinunitgeneration:

0

−6

Newgeneration:

P ′=P +ΔP =80+6.52=86.52MW

P ′=P +ΔP =300+21.74=321.74MW

P ′=P +ΔP =400+21.74=421.74MW

2. withB=1p.u.(onloadbase)

Changesinunitgeneration:

Newgeneration:

P ′=P +ΔP =80+6.44=86.44MW

P ′=P +ΔP =300+21.459=321.459MW

P ′=P +ΔP =400+21.459=421.459MW

1 1 1

2 2 2

3 3 3

1 1 1

2 2 2

3 3 3

FIG.7.30Singleareawithtwogeneratingunits

Example7.12:GivenasingleareawithtwogeneratingunitsasshowninFig.7.30:

Unit Rating(MVA) SpeeddroopR(perunitonunitbase)

1 400 0.04

2 800 0.05

TheunitssharealoadofP =200MW;P =500MW.Theunitsareoperatinginparallel,sharing700MWat1.0(50Hz)frequency.Theloadisincreasedby130MW.

WithB=0,findthesteady-statefrequencydeviationandthenewgenerationoneachunit.

WithB=0.804,findthesteady-statefrequencydeviationandthenewgenerationoneachunit.

Solution:

1.

Atacommonbaseof1,000MVA:

1 2

Changeinunitgeneration:

Newgeneration:

P ′=P +ΔP =200+50=250MW

P ′=P +ΔP =500+80=580MW

2. WithB=0.804(onloadbase)

Changeinunitgeneration:

Newgeneration:

P ′=P +ΔP =200+48.5=248.5MW

P ′=P +ΔP =500+77.6=577.6MW

1 1 1

2 2 2

1 1 1

2 2 2

Example7.13:A500-MWgeneratorhasaspeedregulationof4%.Ifthefrequencydropsby0.12Hzwithanunchangedreference,determinetheincreaseinturbinepower.Andalsofindbyhowmuchthereferencepowersettingshouldbechangediftheturbinepowerremainsunchanged.

Solution:

Case1:

Speedregulation,

Givenadropinfrequency,Δf=–0.12Hz

Increaseinturbinepower,

∴Turbinepowerincrease,ΔP=30MW

Case2:

Iftheturbinepowerremainsunchanged,thereferencepowersettingatthepointoftheblockdiagrammustbechangedsuchthatthesignaltotheincreaseingenerationisblocked:

Example7.14:Twogeneratingunitshavingthecapacities600and900MWandareoperatingata50Hzsupply.Thesystemloadincreasesby150MWwhenboththegeneratingunitsareoperatingatabouthalfoftheircapacity,whichresultsinthefrequencyfallingby0.5Hz.Ifthegeneratingunitsaretosharetheincreasedloadinproportiontotheirratings,whatshouldbetheindividualspeedregulations?Whatshouldtheregulationsbeifexpressedinp.u.Hz/p.u.MW?

Solution:

RatedcapacityofUnit-1=600MW

RatedcapacityofUnit-2=900MW

Systemfrequency,f=50Hz

Systemloadincrement,ΔP=150MW

Fallinginfrequency,Δf=0.5Hz

Weknowthat

Iftheloadissharedinproportionaltotheirratings,

∴FromEquation(7.43),

Itisobservedthatthespeedregulationsinp.u.Hz/p.u.MWareattainingthesamevalue,evenwhentheyarebasedontheirindividualratingsandtheyhavedifferentregulations.

Example7.15:Asingle-areasystemhasthefollowingdata:

Speedregulation,R=4Hz/p.u.MW

Dampingcoefficient,B=0.1p.u.MW/Hz

Powersystemtimeconstant,T =10s

Powersystemgain,K =75Hz/p.u.MW

Whena2%loadchangeoccurs,determinetheAFRCandthestaticfrequencyerror.Whatisthevalueofthesteady-statefrequencyerrorifthegovernorisblocked?

Solution:

P

P

Staticfrequencyerror

Ifthegovernorisblocked,thefeedbackloopwillnotbepresent;therefore,Rwillbecomeinfinite:

Staticfrequencyerror

i.e.,frequencyfallsby0.0571Hz.

∴Newfrequency,f′=50−0.0571

=49.94Hz

Observation:

Withspeed-governoraction:

Frequencyfallsby0.0571Hz

∴Newfrequency,f′=50–0.0571

=49.94Hz

Withoutspeed-governoraction:

Frequencyfallsby0.2Hz

∴Newfrequency,f′=50–0.2=49.8Hz

Fromtheaboveresults,itisnotedthatthespeed-governoractionisnecessaryforobtainingareductioninthesteady-statefrequencyerror.

Example7.16:A200-MVAsynchronousgeneratorisoperatedat3,000rpm,50Hz.Aloadof40MWissuddenlyappliedtothemachineandthestationvalvetotheturbineopensonlyafter0.4sduetothetimelaginthegeneratoraction.Calculatethefrequencytowhichthegeneratedvoltagedropsbeforethesteamflowcommencestoincreasesoastomeetthenewload.GiventhatthevalveofHofthegeneratoris5.5kW-sperkVAofthegeneratorenergy.

Solution:

Given:

Ratingofthegenerator=200MVA

Loadappliedonthem/c=40MW

Timetakenbythevalvetoopen=0.4s

H=5.5kW-s/kVA

=11×10 s

Energystoredatno-load=5.5×200×1,000=1,100MW-s=1,100MJ

Beforethesteamvalveopens,theenergylostbytherotor=40×0.4=16MJ.

Theenergylostbytherotorresultsinareductioninthespeedoftherotorandhencethereductioninfrequency.

5

Weknow

∴Frequencyattheendof0.4s=

Example7.17:Twogeneratorsofrating100and200MWareoperatedwithadroopcharacteristicof6%fromnoloadtofullload.Determinetheloadsharedbyeachgenerator,ifaloadof270MWisconnectedacrosstheparallelcombinationofthosegenerators.

Solution:

Thetwogeneratorsareoperatingwithparallelconnection;the%dropinfrequencyfromtwogeneratorsduetodifferentloadsmustbesame.

Letpowersuppliedby(100MW)Generator-1=x

Percentagedropinfrequency=6%

∴PercentagedropinthespeedofGenerator-1

Totalloadacrosstheparallelconnection=270MW

Powersuppliedby(200MW)Generator-2=(270–x)

∴PercentagedropinthespeedofGenerator-2

Percentdropinfrequency(orspeed)ofbothmachinesmustbethesame:


x=90MW

∴LoadsharedbyGenerator-1(100MWunit)=90MW

LoadsharedbyGenerator-2(200MWunit)=270–x

=270–90=180MW

KEYNOTES

Necessityofmaintainingfrequencyconstant

1. AlltheACmotorsshouldbegivenaconstantfrequencysupplysoastomaintainthespeedconstant.

2. Incontinuousprocessindustry,itaffectstheoperationoftheprocessitself.3. Forsynchronousoperationofvariousunitsinthepowersystemnetwork,itis

necessarytomaintainthefrequencyconstant.4. Frequencyaffectstheamountofpowertransmittedthroughinterconnecting

lines.

Loadfrequencycontrol(LFC)isthebasiccontrolmechanisminthepowersystemoperationwheneverthereisavariationinloaddemandonageneratingunitmomentarilyifthereisanoccurrenceofunbalancebetweenreal-powerinputandoutput.Thisdifferenceisbeingsuppliedbythestoredenergyoftherotatingpartsoftheunit.Primemoversdrivingthegeneratorsarefittedwithgovernors,whichareregardedasprimarycontrolelementsintheLFCsystem.Governorssensethechangeinaspeedcontrolmechanismtoadjusttheopeningofsteamvalvesinthecaseofsteamturbinesandtheopeningofwatergatesinthecaseofwaterturbines.Thesteady-statespeedregulationinperunitisgivenby

ThevalueofRvariesfrom2%to6%foranygeneratingunit.

ThespeedgovernoristhemainprimarytoolfortheLFC,whetherthemachineisusedalonetofeedasmallersystemorwhetheritisapartofthemostelaboratearrangement.

Itsmainpartsarefly-ballspeedgovernor,hydraulicamplifier,speedchanger,andlinkagemechanism.

Controlareaispossibletodivideaverylargepowersystemintosub-areasinwhichallthegeneratorsaretightlycoupledsuchthattheyswinginunisonwithchangeinloadorduetoaspeed-changer

setting.Suchanarea,whereallthegeneratorsarerunningcoherentlyistermedasacontrolarea.Asingleareaisacoherentareainwhichallthegeneratorsswinginunisontothechangesinloadorspeed-changersettingsandinwhichthefrequencyisassumedtobeconstantthroughoutbothinstaticanddynamicconditions.Dynamicresponseishowthefrequencychangesasafunctionoftimeimmediatelyafterdisturbancebeforeitreachesthenewsteady-statecondition.Thecanalizationofdynamicresponserequiresthesolutionofadynamicequationofthesystemforagivendisturbance.Integralcontrolconsistsofafrequencysensorandanintegrator.ThefrequencysensormeasuresthefrequencyerrorΔfandthiserrorsignalisfedintotheintegrator.Theinputtotheintegratoriscalledthe‘areacontrolerror(ACE)’.TheACEisthechangeinareafrequency,whichwhenusedinanintegral-controlloopforcesthesteady-statefrequencyerrortozero.


1. Whatistheeffectofspeedofageneratoronitsfrequency?

Theeffectofspeedofageneratoronitsfrequencyis

wherepisthenumberofpolesandNthespeedinrpm.

2. Whyshouldthesystemfrequencybemaintainedconstant?

Constantfrequencyistobemaintainedforthefollowingfunctions:

1. AlltheACmotorsshouldbegivenconstantfrequencysupplysoastomaintainthespeedconstant.

2. Incontinuousprocessindustry,itaffectstheoperationoftheprocessitself.3. Forsynchronousoperationofvariousunitsinthepowersystemnetwork,it

isnecessarytomaintainthefrequencyconstant.

3. Whatisthenatureofthegenerator–loadfrequencycharacteristic?

Thenatureofthegeneratorisdroopingstraight-linecharacteristics.

4. Howdoloadfrequencycharacteristicschangeduringon-linecontrol?

Byshiftingtheloadfrequencycharacteristicsasawholeupordownvaryingtheinletvalveopeningoftheprimemover.

5. Howdoloadfrequencycharacteristicschangeduringoff-linecontrol?

Bychangingtheslopeoftheloadcharacteristicsbyvaryingtheleverratioofthespeedgovernor.

6. StatewhyP–fandQ–Vcontrolloopscanbetreatedasnon-interactive?

TheactivepowerPismainlydependentontheinternalangleδandisindependentofbusvoltagemagnitude|V|.ThebusvoltageisdependentonmachineexcitationandhenceonreactivepowerQandisindependentofthemachineangleδ.Thechangeinthemachineangleδiscausedbyamomentarychangeinthegeneratorspeedandhencethefrequency.Therefore,theloadfrequencyandexcitationvoltagecontrolsarenon-interactiveforsmallchangesandcanbemodeledandanalyzedindependently.

7. Whatwillbetheorderofthesystemfornon-reheatsteamturbineandreheatturbine?

Theorderofthesystemfornon-reheatandreheatsteamturbinearefirstorderandsecondorder,respectively.

8. Whatarethetransferfunctionsofnon-reheatsteamturbineandreheatturbine?Whatwillbethevalueoftheirtimeconstants?

Thetransferfunctionofnon-reheattypeofsteamturbineis

Thetransferfunctionofreheattypeofsteamturbineis

Thetimeconstantτ hasavalueintherangeof10s.

9. Underwhatconditionwillthemodeldevelopedforaturbinebevalid?

Theconditionfortheturbineisthefirst20sfollowingtheincrementaldisturbance.

10. Explainthecontrolareaconcept.

Itispossibletodivideaverylargepowersystemintosub-areasinwhichallthegeneratorsaretightlycoupledsuchthattheyswinginunisonwithchangeinloadorduetoaspeed-changersetting.Suchanarea,whereallthegeneratorsarerunningcoherently,istermedthecontrolarea.Inthisarea,frequencymaybesameinsteady-stateanddynamicconditions.Fordevelopingasuitablecontrolstrategy,acontrolareacanbereducedtoasinglegenerator,aspeedgovernor,andaloadsystem.

11. Whatismeantbysingle-areapowersystem?

Asingleareaisacoherentareainwhichallthegeneratorsswinginunisontothechangesinloadorspeed-changersettingsandinwhichthefrequencyisassumedtobeconstantthroughoutboth

r

instaticanddynamicconditions.Thissinglecontrolareacanberepresentedbyanisolatedpowersystemconsistingofaturbine,itsspeedgovernor,generator,andload.

12. WhatismeantbydynamicresponseinLFC?

Themeaningofdynamicresponseishowthefrequencychangesasafunctionoftimeimmediatelyafterdisturbancebeforeitreachesthenewsteady-statecondition.

13. Whatismeantbyuncontrolledcase?

Foruncontrolledcase,ΔP =0;i.e.,constantspeed-changer

positionwithvariableload.

14. Whatistheneedofafly-ballspeedgovernor?

Thisistheheartofthesystem,whichcontrolsthechangeinspeed(frequency).

15. Whatistheneedofaspeedchanger?

Itprovidesasteady-statepoweroutputsettingfortheturbines.Itsupwardmovementopenstheupperpilotvalvesothatmoresteamisadmittedtotheturbineundersteadyconditions.Thisgivesrisetohighersteady-statepoweroutput.Thereversehappensfordownwardmovementofthespeedchanger.

16. Whatismeantbyareacontrolerror?

Theareacontrolerror(ACE)isthechangeinareafrequency,whichwhenusedinanintegral-controlloopforcesthesteady-statefrequencyerrortozero.

17. Whatisthenatureofthesteady-stateresponseoftheuncontrolledLFCofasinglearea?

Thenatureofthesteady-stateresponseofasingleareaisthelinearrelationshipbetweenfrequencyandloadforfreegovernoroperation.

18. HowandwhydoyouapproximatethesystemforthedynamicresponseoftheuncontrolledLFCofasinglearea?

ThecharacteristicequationoftheLFCofanisolatedpowersystemisthirdorder,dynamicresponsethatcanbeobtainedonlyforaspecificnumericalcase.

However,thecharacteristicequationcanbeapproximatedasfirstorderbyexaminingtherelativemagnitudesofthedifferenttimeconstantsinvolved.

19. Whatarethebasicrequirementsofaclosed-loopcontrolsystememployedforobtainingthefrequencyconstant?

Thebasicrequirementsareasfollows:

1. Goodstability;2. Frequencyerror,accompanyingastep-loadchange,returnstozero;3. Themagnitudeofthetransientfrequencydeviationshouldbeminimum;4. Theintegralofthefrequencyerrorshouldnotexceedacertainmaximum

value.

C

20. Whatarethebasiccomponentsofanintegralcontroller

Itconsistsofafrequencysensorandanintegrator.

21. Whyshouldtheintegratorofthefrequencyerrornotexceedacertainmaximumvalue?

Thefrequencyerrorshouldnotexceedamaximumvaluesoastolimittheerrorofsynchronousclocks.

22. Whataretheassumptionsmadeinthesimplifiedanalysisoftheintegralcontrol?

1. Thetimeconstantofthespeed-governingmechanismτ andthatofthe

turbinearebothneglected,i.e.,itisassumedthatτ =τ =0.

2. Thespeedchangerisanelectromechanicaldeviceandhenceitsresponseisnotinstantaneous.However,itisassumedtobeinstantaneousinthepresentanalysis.

3. Allnon-linearitiesintheequipment,suchasdeadzone,etc.,areneglected.4. ThegeneratorcanchangeitsgenerationΔP asfastasitiscommandedby

thespeed-changer.5. TheACEisacontinuoussignal.

23. Statebrieflyhowthetimeresponseofthefrequencyerrordependsuponthegainsettingoftheintegralcontrol.

IfK islessthanitscriticalvalue,thentheresponsewillbe

dampednon-oscillatory.Δf(t)reducestozeroinalongertime.Hence,theresponseissluggish.Thisisanoverdampedcase.Thisisthesubcriticalcaseofintegralcontrol.

IfK isgreaterthanitscriticalvalue,thetimeresponsewouldbe

dampedoscillatory.Δf(t)approacheszerofaster.Thisisanunderdampedcase.Thisisthesupercriticalcaseofthecontrol.

IfK equalsitscriticalvalue,nooscillationswouldbepresentin

thetimeresponseandΔf(t)approacheszeroinlesstimethaninthesubcriticalcase.Theintegralofthefrequencyerrorwouldbetheleastinthiscase.


1. Iftheloadonanisolatedgeneratorisincreasedwithoutincreasingthepowerinputtotheprimemover:

1. Thegeneratorwillslowdown.2. Thegeneratorwillspeedup.3. Thegeneratorvoltagewillincrease.4. Thegeneratorfield.

2. Governorsofcontrollingthespeedofelectric-generatingunitsnormallyprovide:

1. Aflat-speedloadcharacteristic.2. Anincreaseinspeedwithanincreasingload.3. Adecreaseinspeedwithanincreasingload.4. None

3. WhentwoidenticalAC-generatingunitsareoperatedinparallel

sg

sg t

G

I

I

I

ongovernorcontrol,andonemachinehasa5%governordroopandtheothera10%droop,themachinewiththegreatergovernordroopwill:

1. Tendtotakethegreaterportionoftheloadchanges.2. Sharetheloadequallywiththeothermachine.3. Tendtotakethelesserportionoftheloadchanges.4. None.

4. OnLFCinstallations,errorsignalsaredevelopedproportionaltothefrequencyerror.Ifthefrequencydeclines,theerrorsignalwillactto:

1. Increasetheprimemoverinputtothegenerators.2. Reducetheprimemoverinputtothegenerators.3. Increasegeneratorvoltages.4. None.

5. IfKEreduces

1.wdecreases.2. Speedfalls.3. Frequencyreduces.4. All.

6. ThechangingofslopeofaspeedgovernercharacteristicisacheviedbychangingtheratioofleverLofgovernerandcanbemadeduring

1. On-lineconditiononly.2. Off-lineconditiononly.3. Both(a)and(b).4. Either(a)or(b).

7. UnitofRis________.

1. Hz/MVAr.2. Hz/MVA.3. Hz/MW.4. Hz-s.

8. UnitofBis________.

1. MVAr/Hz.2. MVA/Hz.3. MW/Hz.4. MW-s.

9. UnitofHofasynchronousmachineis:

1. MJ/MW.2. MJ/MVA.3. MJ/s.4. MW-s.

10. KEandfrequencyofasynchronousmachinearerelatedas:

1. KE=f.2. KE=1/f.

3. KE=f .4. Noneofthese.

11. InputsignalstoanALFCloopis________.

2

1. ΔP

2. ΔP


12. Twomaincontrolloopsingeneratingstationsare:

1. ALFC.2. AVR.3. Both(a)and(b).4. Noneofthese.

13. Thespeedregulationcanbeexpressedas

1. Ratioofchangeinfrequencyfromno-loadtofullloadtotheratedfrequencyoftheunit.

2. Ratioofchangeinfrequencytothecorrespondingchangeinreal-powergeneration.

3. (a)and(b).4. Noneofthese.

14. InanALFCloop,Δfcanbereducedusing________controller.

1. Differential.2. Integral.3. Proportional.4. Allofthese.

15. Timeconstantofapowersystewhencomparedtoaspeedgovernoris:

1. Less.2. More.3. Same.4. Noneofthese.

16. Δfisoftheorderof________Hz.

1. 0to0.05.2. –0.05to0.3. Both(a)and(b).4. Noneofthese.

17. Inapowersystem________arecontinuouslychanging.

1. Activeandreactivepowergeneration.2. Activeandreactivepowerdemands.3. Voltageanditsangle.4. Allofthese.

18. Inanormalstate,thefrequencyandvoltagearekeptatspecifiedvaluesthatcarefullymaintainabalancebetween:

1. Real-powerdemandandreal-powergeneration.2. Reactivepowerdemandandreactivepowergeneration.3. Both.4. Noneofthese.

19. Real-powerbalancewillcontrolthevariationsin________.

1. Voltage.2. Frequency.3. Both.4. Noneofthese.

ref

D

20. Theexcitationsofthegeneratorsmustbecontinuouslyregulated:

1. Tomatchthereactivepowergenerationswithreactivepowerdemand.2. Tocontrolthevariationsinvoltage.3. Both.4. Noneofthese.

21. ________isthebasiccontrolmechanisminthepowersystem.

1. LFC.2. Voltage.3. Both.4. Noneofthese

22. Settingofspeed-loadcharacteristicparalleltoitselfisknownas________anditsadaptedason-linecontrol.

1. Primarycontrol.2. Supplementarycontrol.3. Basic.4. Allofthese.

23. ThebasicfunctionofLFCis:

1. Tomaintainfrequencyforvariationsinreal-powerdemand.2. Tomaintainvoltageforvariationsinreactivepowerdemand.3. Tomaintainbothvoltageandfrequencyforvariationsinreal-power

demand.4. Tomaintainbothvoltageandfrequencyforvariationsinreal-power

demand.

24. Thedegreeofunbalancebetweenreal-powergenerationandreal-powerdemandisindicatedbytheindex:

1. SpeedregulationR.2. Changeinvoltage,3. Frequencyerror.4. None.

25. TheLFCsystem________inthesystem.

1. Doesconsiderthereactivepowerflow.2. Doesnotconsiderthereactivepowerflow.3. Doesnotconsiderthereal-powerflow.

26. ________controlstheexcitationvoltageandmodifiestheexcitation.

1. Changeinreal-power,ΔP .

2. ChangeinfrequencyΔ.3. Changeintie-linepower,ΔP .

4. ChangeinreactivepowerΔQ .

27. Thep–fcontrollerisemployedto:

1. Controlthefrequency.2. Monitortheactivepowerflowsininterconnection.3. Controlthevoltage.

1. Only(a).2. Only(b).3. (b)and(c).4. (a)and(b).

a

tie

ci

28. Whichofthefollowingiscorrectregardingp–fcontroller?

1. Itsensesthefrequencyerror.2. Itchangesthetie-linepowers.3. ProvidestheinformationaboutincrementalerrorinpowerangleΔδ.

1. (a)and(b).2. (b)and(c).3. (a)and(c).4. Allofthese.

29. Thecontrolsignalthatwillchangethepositionoftheinletvalveoftheprimemoveris:

1. ΔP .

2. ΔP .

3. ΔP .

4. Noneofthese.

30. TheobjectiveofQ–Vcontrolleristotransformthe:

1. Terminalvoltageerrorsignalintoareactivepowercontrolsignal,ΔQ .

2. Terminalvoltageerrorsignalintoareal-powercontrolsignal,ΔP .

3. Frequencyerrorsignalintoareal-powercontrolsignal,ΔP .

4. Noneofthese.

31. TheactivepowerPis:

1. Mainlydependentontheinternaltorqueangle,δ.2. Almostindependentofthevoltagemagnitude.3. totallydependentonboththetorqueangleandthevoltage.4. Mainlydependentonvoltageandindependentoftorqueangle,δ.

1. (a)and(d).2. (b)and(c).3. (a)and(b).4. Only(d).

32. ThebusvoltageVis:

1. Dependentontheinternaltorqueangle,δ.2. Almostindependentofactivepower,P.3. Dependentonmachineexcitationandhenceonreactivepower.4. Almostindependentofinternaltorqueangle,δ.

1. and(d).2. (b)and(c).3. (a)and(b).4. (c)and(d).

33. Usuallyp–fcontrollerandQ–Vcontrollerfor________change,canbeconsideredas________type.

1. Dynamic,non-interacting.2. Static,interacting.3. Static,non-interacting.4. Noneofthese.

34. AVRloopis________controlmechanism.

1. Slow.2. Faster.3. Slowinsomecasesandfasterinsomeothercases.4. Noneofthese.

ci

gi

di

ci

ci

ci

35. ALFCloopis________controlmechanism.

1. Slow.2. Faster.3. Slowaswellasfast.4. Noneofthese.

36. Whichofthefollowingindicatesthelarge-signalanalysisofpowersystemdynamics?

1. Largeandsuddenvariationsinthesystemvariablesduetosuddendisturbances.

2. Mathematicalmodelisasetofnon-lineardifferentialequations.3. Mathematicalmodelisasetoflineardifferentialequations.4. Smallandgradualvariationsofsystemvariables.

1. (a)and(b).2. (b)and(c).3. (c)and(d).4. Noneofthese.

37. Laplacetransformmethodsareemployedtodeterminetheresponseofthesystemin________analysis.

1. Largesignal.2. Smallsignal.3. Both.4. Noneofthese.

38. Asignalareasystemisoneinwhich:

1. Itisnotconnectedtoanyothersystem.2. Totaldemandonthesystemshouldbefullymetbyitsownlocal

generation.3. Allgeneratorsswingtogether.4. Allofthese.

39. Inasignalareasystem,allgeneratorsworkingremaininsynchronismmaintainingtheirrelativepowerangles;suchagroupofgeneratorsiscalled________.

1. Swinggroup.2. Synchrogroup.3. Coherentgroup.4. Noneofthese.

40. Theheartofthespeedgovernorsystem,whichcontrolsthechangeinspeedis:

1. Linkagemechanism.2. Fly-ballspeedgovernor.3. Speedchanger.4. Hydraulicamplifier.

41. Inahydraulicamplifier:

1. High-power-levelpilotvalvemomentisconvertedintolow-power-levelmainpistonmovement.

2. Low-powerpilotvalvemomentisconvertedintolow-power-levelpistonmovement.

3. Low-power-levelpilotvalvemomentisconvertedintohigh-power-levelpistonmovement.

4. Low-power-levelpilotvalvemomentisconvertedintohigh-power-levelpilotvalvemoment.

42. Linkagemechanismprovides:

1. Themomentofcontrolvalveinpropositionaltotheinletsteam.2. Thefeedbackfromthecontrolvalvemoment.3. Both(a)and(b).4. Noneofthese.

43. Theprimarycontrolloopingeneratorcontrolis:

1. Linkagemechanism.2. Fly-ballspeedgovernor.3. Speedchanger.4. Hydraulicamplifier.

44. Thepositionofthepilotvalvecanbeaffectedthroughlinkagemechanismin________way.

1. Directlybythespeedchanger.2. Indirectlythroughfeedbackduetopositionchangesofthemainsystem.3. IndirectlythroughfeedbackduetopositionchangesofthelinkagepointE

resultingfromachangeinspeed.4. Allofthese.

45. Fornon-reheattypeofsteamturbine,themathematicalmodelis:

1.

2.

3.

4. Noneofthese.

46. Inreheattypeofsteamturbine,

1. Steamathighpressurewithlowtemperatureistransformedintosteamatlowpressurewithhighertemperature.

2. Steamatlowpressurewithhighertemperatureistransformedintosteamathighpressurewithlowtemperature.

3. Steamatlowpressurewithlowtemperatureistransformedintosteamathighpressurewithhighertemperature.

4. Noneofthese.

47. Transferfunctionofreheattypeofsteamturbineisof________order.

1. First.2. Second.3. Third.4. Noneofthese.

48. Transferfunctionofnon-reheattypeofsteamturbineisof________order.

1. First.2. Second.3. Third.4. Noneofthese.

49. Thesurpluspower(ΔP –ΔP )canbeabsorbedbyasystem:

1. ByincreasingthestoredKЄofthesystemattherate

2. Bymotorloads.3. Thereisnoabsorptionofsurpluspowerbythesystem.4. Both(a)and(b).

50. TheblockdiagramoftheLFCofanisolatedpowersystemisof________model.

1. First.2. Second.3. Third.4. Fourth.

REVIEWQUESTIONS

1. DeveloptheblockdiagramoftheLFCofasingle-areasystem.2. Comparethesteadystateanddynamicoperationsofanisolated

system.3. Drawtheschematicdiagramofaspeed-governingsystemand

explainitscomponentsonthedynamicresponseofanuncontrolledsystemwithnecessaryequations.Hence,obtainthetransferfunctionofaspeed-governingsystem.

4. Howdothegovernorcharacteristicsoftheprimemoveraffectthecontrolofsystemfrequencyandsystemload?

5. Explainwhyitisnecessarytomaintainthefrequencyofthesystemconstant.

6. WhatdoyoumeanbyLFC?7. Drawaneatsketchofatypicalturbinespeed-governingsystemandderiveitsblockdiagramrepresentation.

8. Forasingle-areasystem,showthatthestaticerrorinfrequencycanbereducedtozerousingfrequencycontrolandcommentonthedynamicresponseofanuncontrolledsystemwithnecessaryequations.

9. Explainthep–fandQ–Vcontrolloopsofpowersystem.10. WhatismeantbycontrolareaandACE?11. ExplainclearlyaboutproportionalplusintegralLFCwithablock

diagram.12. Discusstheadverseeffectsofchangeinthevoltageandthe

frequencyofapowersystem.Mentiontheacceptablerangesofthesechanges.

PROBLEMS

1. A250-MVAsynchronousgeneratorisoperatingat1,500rpm,50Hz.Aloadof50MWissuddenlyappliedtothemachineandthestationvalvetotheturbineopensonlyafter0.35sduetothetimelaginthegeneratoraction.Calculatethefrequencyatwhichthe

G D

generatedvoltagedropsbeforethesteamflowcommencestoincreasetomeetthenewload.GiventhatthevalveofHofthegeneratoris3.5kW-sperkVAofthegeneratorenergy.

2. TwogeneratingstationsAandBhavefull-loadcapacitiesof250and100MW,respectively.Theinterconnectorconnectingthetwostationshasaninductionmotor/synchronousgenerator(PlantC)offull-loadcapacity30MW;percentagechangesofspeedsofA,B,andCare4,3,and2,respectively.TheloadsonbusbarsAandBareMWand50MW,respectively.DeterminetheloadtakenbyPlantCandindicatethedirectionofthepowerflow.

3. A750-MWgeneratorhasaspeedregulationof3.5%.Ifthefrequencydropsby0.1Hzwithanunchangedreference,determinetheincreaseinturbinepower.Andalsofindbyhowmuchthereferencepowersettingshouldbechangediftheturbinepowerremainsunchanged.

8

LoadFrequencyControl-II

OBJECTIVES


developtheblockdiagrammodelsforatwo-areapowersystemobservethesteadystateanddynamicanalysisofatwo-areapowersystemwithandwithoutintegralcontroldevelopthedynamic-statevariablemodelforsingle-area,two-area,andthree-areapowersystemnetworks

8.1INTRODUCTION

Anextendedpowersystemcanbedividedintoanumberofloadfrequencycontrol(LFC)areas,whichareinterconnectedbytielines.Suchanoperationiscalledapooloperation.Apowerpoolisaninterconnectionofthepowersystemsofindividualutilities.Eachpowersystemoperatesindependentlywithinitsownjurisdiction,buttherearecontractualagreementsregardinginternalsystemexchangesofpowerthroughthetielinesandotheragreementsdealingwithoperatingprocedurestomaintainsystemfrequency.Therearealsoagreementsrelatingtooperationalprocedurestobefollowedintheeventofmajorfaultsoremergencies.Thebasicprincipleofapooloperationinthenormalsteadystateprovides:

1. Maintainingofscheduledinterchangesoftie-linepower:Theinterconnectedareassharetheirreservepowertohandleanticipatedloadpeaksandunanticipatedgeneratoroutages.

2. Absorptionofownloadchangebyeacharea:Theinterconnectedareascantoleratelargerloadchangeswithsmallerfrequencydeviationsthantheisolatedpowersystemareas.

ForanalyzingthedynamicsoftheLFCofann-areapowersystem,primarilyconsidertwo-areasystems.

Twocontrolareas1and2areconnectedbyasingletielineasshowninFig.8.1.

FIG.8.1Twocontrolareasinterconnectedthroughasingletieline

Here,thecontrolobjectiveistoregulatethefrequencyofeachareaandtosimultaneouslyregulatethepowerflowthroughthetielineaccordingtoaninterareapoweragreement.

Inthecaseofanisolatedcontrolarea,thezerosteady-stateerrorinfrequency(i.e.,Δf =0)canbeobtainedbyusingaproportionalplusintegralcontroller,whereasintwo-controlareacase,proportionalplusintegralcontrollerwillbeinstalledtogivezerosteady-stateerrorinatie-linepowerflow(i.e.,ΔP =0)inadditiontozerosteady-stateerrorinfrequency.

Forthesakeofconvenience,eachcontrolareacanberepresentedbyanequivalentturbine,generator,andgovernorsystem.

Inthecaseofasinglecontrolarea,theincrementalpower(ΔP −ΔP )wasconsideredbytherateofincreaseofstoredKEandincreaseinarealoadcausedbytheincreaseinfrequency.

Inatwo-areacase,thetie-linepowermustbeaccountedfortheincrementalpowerbalanceequationofeacharea,sincethereispowerflowinoroutoftheareathroughthetieline.

PowerflowoutofControlarea-1canbeexpressedas

steadystate

TL

G D

where∣E ∣and∣E ∣arevoltagemagnitudesofArea-1andArea-2,respectively,δ andδ arethepoweranglesofequivalentmachinesoftheirrespectiveareas,andX isthetie-linereactance.

Ifthereischangeinloaddemandsoftwoareas,therewillbeincrementalchangesinpowerangles(Δδ andΔδ ).Then,thechangeinthetie-linepoweris

Therefore,changeinincrementaltie-linepowercanbeexpressedas

where

1 2

1 2

TL

1

2

T isknownasthesynchronizingcoefficientorthestiffnesscoefficientofthetie-line.

Equation(8.3)canbewrittenas

where Statictransmissioncapacityofthe

tieline.

Considerthechangeinfrequencyas

Inotherwords,

Hence,thechangesinpoweranglesforAreas-1and2are

and

12

Sincetheincrementalpoweranglesarerelatedintermsofintegralsofincrementalfrequencies,Equation(8.2)canbemodifiedas

Δf andΔf aretheincrementalfrequencychangesofAreas-1and2,respectively.Similarly,theincrementaltie-linepoweroutofArea-2is

where

DividingEquation(8.6)byEquation(8.3),weget

Therefore,T =a T

andhence∆P =a ∆P (8.7)

FromEquation(7.25)(LFC-1),surpluspowerinp.u.is

Foratwo-areacase,thesurpluspowercanbeexpressedinp.u.as

1 2

21 12 12

TL2 12 TL1

TakingLaplacetransformonbothsidesofEquation(8.8),weget

Rearrangingtheaboveequationasfollows,weget

where

BycomparingEquation(8.9)withsingle-areaEquation(7.26),theonlyadditionaltermistheappearanceofsignal∆P (S)

Equation(8.9),canberepresentedinablockdiagrammodelasshowninFig.8.2.TakingLaplacetransformationonbothsidesofEquation(8.4),weget

TL1

FIG.8.2BlockdiagramrepresentationofEquation(8.9)(forControlarea-1)

FIG.8.3BlockdiagramrepresentationofEquations(8.10)and(8.11)

ForControlarea-2,wehave

TheblockdiagramrepresentationofEquations(8.10)and(8.11)isshowninFig.8.3.

8.2COMPOSITEBLOCKDIAGRAMOFATWO-AREACASE

BythecombinationofbasicblockdiagramsofControlarea-1andControlarea-2andwiththeuseofFigs.8.2and8.3,thecompositeblockdiagramofatwo-areasystemcanbemodeledasshowninFig.8.4.

8.3RESPONSEOFATWO-AREASYSTEM—UNCONTROLLEDCASE

Foranuncontrolledcase,∆P =∆P =0,i.e.,thespeed-

changerpositionsarefixed.

8.3.1Staticresponse

Inthissection,thechangesordeviations,whichresultinthefrequencyandtie-linepowerundersteady-stateconditionsfollowingsuddenstepchangesintheloadsinthetwoareas,aredetermined.

FIG.8.4Blockdiagramrepresentationofatwo-areasystemwithanLFC

Let∆P ,∆P besudden(incremental)stepchanges

intheloadsofControlarea-1andControlarea-2,

c1 c2

D1 D2

simultaneously.

∆P ,∆P aretheincrementalchangesinthe

generationinArea-1andArea-2asaresultoftheloadchanges.

Δfisthestaticchangeinfrequency.Thiswillbethesameforboththeareasand∆P isthestaticchangein

thetie-linepowertransmittedfromArea-1toArea-2.Sinceonlythestaticchangesarebeingdetermined,theincrementalchangesingenerationcanbedeterminedbythestaticloopgains.So,wehave

and forstaticchanges(8.13)

Forthetwoareas,thedynamicsaredescribedby:

and

Understeady-stateconditions,wehave

G1 G2

TL1

AftersubstitutingEquations(8.12),(8.13),and(8.16)inEquations(8.14)and(8.15),weget

and

Since∆P =−a ∆P and∆f =∆f =∆f,fromEquation

(8.17),wehave

Substituting∆P fromEquation(8.18(a))inEquation

(8.18),weget

SubstitutingΔffromEquation(8.18(b))inEquation(8.18(a)),weget

TL2 12 TL1 1 2

TL1

Equations(8.18(b))and(8.18(c))aremodifiedas

Tie-linefrequency,

Tie-linepower,

where

Equations(8.19)and(8.20)givethevaluesofthestaticchangesinfrequencyandtie-linepower,respectively,asaresultofsuddenstep-loadchangesinthetwoareas.Itcanbeobservedthatthefrequencyandtie-linepowerdeviationsdonotreducetozeroinanuncontrolledcase.

Considertwoidenticalareas,

B =B =B,β =β =β,R =R =Randa =+1

Hence,fromEquations(8.19)and(8.20),wehave

and

IfasuddenloadchangeoccursonlyinArea-2(i.e.,∆P

=0),thenwehave

and

Equations(8.23)and(8.24)illustratetheadvantagesofpooloperation(i.e.,gridoperation)asfollows:

Equations(8.19)representsthechangeinfrequencyaccordingtothechangeinloadineitherofatwo-areasysteminterconnectedbyatieline.Whenconsideringthatthosetwoareasareidentical,Equation(8.19)becomesEquation(8.21).Hence,itisconcludedthatifaloaddisturbanceoccursinonlyoneoftheareas(i.e.,∆P =0or∆P =

0),thechangeinfrequency(Δf)isonlyhalfofthesteady-stateerror,whichwouldhaveoccurredwithnointerconnection(i.e.,anisolatedcase).Thus,withseveralsystemsinterconnected,thesteady-statefrequencyerrorwouldbereduced.Halfoftheaddedload(inArea-2)issuppliedbyArea-1throughthetieline.

Theabovetwoadvantagesrepresentthenecessityofinterconnectingthesystems.

1 2 1 2 1 2 12

D1

D1 D2

8.3.2Dynamicresponse

Todescribethedynamicresponseofthetwo-areasystemasshowninFig.8.4,asystemofseventh-orderdifferentialequationsisrequired.Thesolutionoftheseequationswouldbetedious.However,someimportantcharacteristicscanbebroughtoutbyananalysisrenderedsimplebythefollowingassumptions.Apowersystemoftwoidenticalcontrolareasisconsideredfortheanalysis:

1. τ =τ =0forboththeareas.

2. Thedampingconstantsoftwoareasareneglected,

i.e.,B =B =0

Byvirtueofthesecondassumption,Equations(8.14)and(8.15)become

TakingLaplacetransformationonbothsidesofEquations(8.25)and(8.26)andbyrearrangement,weget

FromtheblockdiagramofFig.8.4,thefollowingequationscanbeobtained:

gt t

1 2

( ,sincetwocontrolareasareidentical)

BysolvingEquations(8.27)–(8.30),weget

Fromtheaboveequation,thefollowingobservationscanbemade:

(i)Thedenominatorisoftheform:

S +2αs+ω =(s+α) +(ω −α )(8.32)

where

andαandω arebothrealandpositive.Hence,itcanbeconcludedfromtherootsofcharacteristicequationthatthetimeresponseisstableanddamped.

Thethreeconditionsare:

Ifα=ω ,systemiscriticallydamped

2 2

n

2 2 2

2

α>ω ,systembecomesoverdamped

whereα=dampingfactorordecrementofattenuation

ω =dampedangularfrequency

SinceparameterαalsodependsonB,but in

practice,therefore,theeffectofcoefficientBisneglectedondamping.

(ii)Afteradisturbance,thechangeintie-linepoweroscillatesatthedampedangularfrequency.

(iii)Thedampingofthetie-linepowervariationisstronglydependentupontheparameterα,whichisequal

to .Sincef andHareessentiallyconstant,the

dampingisafunctionoftheRparameters.IftheRvalueislow,dampingbecomesstrongandviceversa.

Thetransientchangeinthetie-linepowerwillbeofundampedoscillationsoffrequency,ω =ω.

IfR=∞,i.e.,ifthespeedgovernorisnotpresent(α=0),thevariationinfrequencydeviationandthetie-linepowerwouldbeasshowninFig.8.5.

Itcanbeseenthatthesteady-statefrequencydeviationisthesameforboththeareasanddoesnotvanish.Thetie-linepowerdeviationalsodoesnotbecomezero.

n

n

0

o

0

Althoughtheaboveapproximateanalysishasconfirmedstability,ithasbeenfoundthroughmoreaccurateanalysesthatwithcertainparametercombinations,thesystembecomesunstable.

FIG.8.5Frequencydeviationandtie-linepowerchangefollowingastep-loadchangeinArea-2(twoareasareidentical)

Example8.1:Atwo-identicalareapowersystemhasthefollowingparameters(Fig.8.6(a)):




ps

ps

0




Integrationtimeconstant,k =0.15

Biasparameter,b=0.326

2πT =0.08

Plotthechangeinthetie-linepowerandchangeinfrequencyofcontrol-area1ifthereexistsastep-loadchangeof2%inArea-1(Fig.8.6(b)).

sg

t

i

12

0

FIG.8.6(a)Simulationblockdiagramforatwo-identicalareasystemofExample8.1;(b)frequencyandtie-lineresponseforExample8.1

Example8.2:Atwo-areapowersystemhasthefollowingparameters(Fig.8.7(a)):

ForArea-1:


Powersystemtimeconstant,Ƭ =20s



Governortimeconstant,Ƭ =0.2s

Turbinetimeconstant,Ƭ =0.4s



ForArea-2:


ps

ps

sg

t

i

ps

0

Powersystemtimeconstant,Ƭ =22s

Speedregulation,R=3


Governortimeconstant,Ƭ =0.3s

Turbinetimeconstant,Ƭ =0.5s



2πT =0.08

Plotthechangeinthetie-linepowerandchangeinfrequencyofControl-area1ifthereexistsastep-loadchangeof2%inArea-1(Fig.8.7(b)).

ps

sg

t

i

12

0

FIG.8.7(a)SimulationblockdiagramofExample8.2;(b)Frequencyandtie-linepowerresponseofExample8.2

Example8.3:Determinethefrequencyofoscillationsofthetie-linepowerdeviationforatwo-identical-areasystemgiventhefollowingdata:

R=3.0Hz/p.u.;H=5s;f =60Hz

Thetie-linehasacapacityof0.1p.u.andisoperatingatapowerangleof45°.

Solution:

Thesynchronizing-powercoefficientofthelineisgivenby

T =P cosδ =0.1×cos45°=0.0707p.u.

Hence,thefrequencyofoscillationsisgivenby

m 12

0

012

8.4AREACONTROLERROR—TWO-AREACASE

Inasingle-areacase,ACEisthechangeinfrequency.Thesteady-stateerrorinfrequencywillbecomezero(i.e.,Δf =0)whenACEisusedintheintegral-controlloop.

Inatwo-areacase,ACEisthelinearcombinationofthechangeinfrequencyandchangeintie-linepower.Inthiscasetomakethesteady-statetie-linepowerzero(i.e.,ΔP =0),anotherintegral-controlloopforeachareamustbeintroducedinadditiontotheintegralfrequencylooptointegratetheincrementaltie-linepowersignalandfeeditbacktothespeed-changer.

Thus,forControlarea-1,wehave

ACE =∆P +b ∆f (8.33)

whereb =constant=areafrequencybias.TakingLaplacetransformonbothsidesofEquation(8.33),weget

ACE (s)=∆P (s)+b ∆F (s)(8.34)

Similarly,forControlarea-2,wehave

ACE (s)=∆P (s)+b ∆F (s)(8.35)

8.5COMPOSITEBLOCKDIAGRAMOFATWO-AREASYSTEM

ss

TL

1 TL1 1 1

1

1 TL1 1 1

2 TL2 2 2

(CONTROLLEDCASE)

BythecombinationofbasicblockdiagramsofControlarea-1andControlarea-2andwiththeuseofFigs.8.2and8.3,thecompositeblockdiagramofatwo-areasystemcanbemodeledasshowninFig.8.4.Figure8.8canbeobtainedbytheadditionofintegralsofACE andACE totheblockdiagramshowninFig.8.4.Itrepresentsthecompositeblockdiagramofatwo-areasystemwithintegral-controlloops.Here,thecontrolsignals∆P (s)and∆P (s)aregeneratedbytheintegrals

ofACE andACE .Thesecontrolerrorsareobtainedthroughthesignalsrepresentingthechangesinthetie-linepowerandlocalfrequencybias.

8.5.1Tie-linebiascontrol

Thespeed-changercommandsignalswillbeobtainedfromtheblockdiagramshowninFig.8.6as

and

TheconstantsK andK arethegainsofthe

integrators.Thefirsttermsontheright-handsideofEquations(8.36)and(8.37)constituteandareknownastie-linebiascontrols.Itisobservedthatfordecreasesinbothfrequencyandtie-linepower,thespeed-changerpositiondecreasesandhencethepowergenerationshoulddecrease,i.e.,iftheACEisnegative,thentheareashouldincreaseitsgeneration.

So,theright-handsidetermsofEquations(8.36)and(8.37)areassignedanegativesign.

1

2

c1 c2

1 2

I1 I2

8.5.2Steady-stateresponse

Thatthecontrolstrategy,describedintheprevioussection,eliminatesthesteady-statefrequencyandtie-linepowerdeviationsthatfollowastep-loadchange,canbeprovedasfollows:

FIG.8.8Two-areasystemwithintegralcontrol

Letthestepchangesinloads∆P and∆P

simultaneouslyoccurinControlarea-1andControlarea-2,respectively,orineitherarea.Anewstaticequilibrium

D1 D2

state,i.e.,steady-stateconditionisreachedsuchthattheoutputsignalofallintegratingblockswillbecomeconstant.Inthiscase,thespeed-changercommandsignals∆P and∆P havereachedconstantvalues.This

obviouslyrequiresthatboththeintegrands(inputsignals)inEquations(8.36)and(8.37)bezero.

Inputofintegratingblock is

∆P +b ∆f =0(8.38)

Inputofintegratingblock is

∆P +b ∆f =0(8.39)

andinputofintegratingblock is

∆f −∆f =0(8.40)

Equations(8.38)and(8.39)aresimultaneouslysatisfiedonlyfor∆P =∆P =0and∆f =∆f

=0.

Thus,underasteady-statecondition,changeintie-linepowerandchangeinfrequencyofeachareawillbecomezero.Toachievethis,ACEsinthefeedbackloopsofeachareaareintegrated.

Therequirementsforintegralcontrolactionare:

1. ACEmustbeequaltozeroatleastonetimeinall10-minuteperiods.2. AveragedeviationofACEfromzeromustbewithinspecifiedlimits

basedonapercentageofsystemgenerationforall10-minuteperiods.

c1 c2

TL (ss)1 1 1(ss)

TL (ss)2 2 2(ss)

1 2

TL (ss)1 TL (ss)2 1(ss) 2(ss)

Theperformancecriteriaalsoapplytodisturbanceconditions,anditisrequiredthat:

1. ACEmustreturntozerowithin10-minuteperiods.2. Correctivecontrolactionmustbeforthcomingwithin1minuteofa

disturbance.

8.5.3Dynamicresponse

Thedeterminationofthedynamicresponseofthetwo-areamodelshowninFig.8.6ismoredifficult.Thisisduetothefactthatthesystemofequationstobesolvedisoftheorderofnine.Therefore,actualsolutionisnotattempted.Buttheresultsobtainedfromanapproximateanalysisofatwo-identical-areapowersystemforthreedifferentvaluesoftheparameter‘b’,arepresentedinFigs.8.9(a),(b),and(c).

ThegraphsofFig.8.9(a)correspondtothecaseofb=0.Itcanbeseenthatthetie-linepowerdeviationreducestozerowhilethefrequencydoesnot.

ThegraphsofFig.8.9(b)correspondtotheotherextremecaseofb=∞.Now,thefrequencyerrorvanishes.But,thetie-linepowerdoesnotvanish.

ThegraphsofFig.8.9(c)showanintermediatecasewhereinboththefrequencyandthetie-linepowererrorsdecreasetozero.Thisisthedesiredcase.

Therefore,itcanbeconcludedthatthestabilityisnotalwaysguaranteed.Hence,thereisaneedforproperparameterselectionandadjustmentoftheirvalues.

8.6OPTIMUMPARAMETERADJUSTMENT

ThegraphsgiveninFig.8.9(c)stresstheneedforproperparametersettings.ThechoiceofbandK constantsaffectsthetransientresponsetoloadchanges.Thefrequencybiasbshouldbehighenoughsuchthateachareaadequatelycontributestofrequencycontrol.Itisprovedthatchoosingb=βgivessatisfactoryperformanceoftheinterconnectedsystem.

I

TheintegratorgainK shouldnotbetoohigh,otherwise,instabilitymayresult.AlsothetimeintervalatwhichLFCsignalsaredispatched,twoormoreseconds,shouldbelowenoughsothatLFCdoesnotattempttofollowrandomorspuriousloadchanges.

FIG.8.9Approximatedynamicresponseoftwo-identical-areapowersystemswiththreedifferentvaluesofbparameters

First,asetofparameters,whichensurestabilityofthecontrol,isselected.Forexample,b andb cannotbothbezero,i.e.,oneofthemshouldbechosenforthecontrolstrategy.Later,thevaluesoftheseparametersare

I

1 2

adjustedsothatabestoranoptimumresponseisobtained.Inotherwords,thevaluesofparameters,whichgiverisetoanoptimumresponse,aretobedetermined.

Theprocedureisasfollows:

Thepopularerrorcriterion,knownastheintegralofthesquarederrors(ISE),ischosenforthecontrolparameters∆f ,∆f ,and∆P .Foratwo-areasystem,the

ISEcriterionfunctionCwouldbe

whereα ,α ,andα aretheweightfactors,whichprovideappropriateimportance,i.e.,weightagetotheerrors∆P ,Δf ,andΔf .Thereisnoneedtochoose∆P ,

since∆P =a ∆P .

SinceΔf andΔf behaveinasimilarmanner,weneedtoconsideronlyoneofthem.So,letusconsiderΔf only.Thisisaparameterselection.Then,α =0.Also,letα =α.Since,weareinterestedonlyintherelativemagnitudesofCforparametersetting,wecansetα =1.

Withthese,Equation(7.41)reducesto

Foratwo-areasystem,∆P andΔf wouldbe

functionsoftheintegratorgainconstantsK andK as

wellasthefrequencybiasparametersb andb .

Theprocedureforobtainingtheoptimumparametervalueswouldbeasfollows:

First,aconvenientandsuitablevalueischosenfortheweightfactor‘α’.Then,fordifferentassumedvaluesof

1 2 TL1

1 2 3

TL1 1 2 TL2

TL2 12 TL1

1 2

1

3 2

1

TL2 1

I1 I2

1 2

K ,K ,b ,andb ,thevaluesof∆P andΔf are

determinedatdifferentinstantsoftime.Withthesevaluesandα,thevalueofCiscomputedusingEquation(8.42).ThesetofvaluesofK ,K ,b ,andb forwhichC

isaminimumistheoptimumone.

Ifweconsiderthetwoidenticalareas,thenthenumberofparametersreducestotwo,viz.,K =K =K andb =

b =b.Inthiscase,valuesofCfordifferentvaluesofKandbcanbeplottedasshowninFig.8.10.Ascanbeseen,thevariationofCwithK fordifferentfixedvaluesofbisplottedtogetafamilyofcurvescalledconstant-bcontours.Forillustration,onlythreecurvesareshowninFig.8.10.Inpractice,anumberofcurveshavetobedeterminedanddrawn.ItcanbeseenthatCisminimumforb=0.2andK =1.0.

Inthiscase,theoptimumcontrolstrategywould,therefore,be

and

Inpractice,thefrequencyandtie-linepowerdeviationsaremeasuredatfixedintervalsoftimeinasample-datafashion.Thesamplingrate(therateatwhichthefrequencydeviationandtie-linepowerdeviationsamplesaremeasured)shouldbesufficientlyhightoavoiderrorsduetosampling.

Note:LFCprovidesenoughcontrolduringnormalchangesinloadandfrequency,i.e.,changesthatarenottoolarge.Duringemergencies,whenlargeimbalancesbetweengenerationandloadoccur,LFCisbypassedand

I1 I2 1 2 TL1 1

I1 I2 1 2

I1 I2 1 1

2 I

I

I

otheremergencycontrolsareapplied,whichisbeyondthescopeofthisbook.

FIG.8.10Constantb-contoursoftheISEcriterionfunctionC

8.7LOADFREQUENCYANDECONOMICDISPATCHCONTROLS

EconomicloaddispatchandLFCplayavitalroleinmodernpowersystem.InLFC,zerosteady-statefrequencyerrorandafast,dynamicresponsewereachievedbyintegralcontrolleraction.Butthiscontrolisindependentofeconomicdispatch,i.e.,thereisnocontrolovertheeconomicloadingsofvariousgeneratingunitsofthecontrolarea.

Somecontroloverloadingofindividualunitscanbeexercisedbyadjustingthegainfactors(K )oftheintegralsignaloftheACEasfedtotheindividualunits.Butthisisnotasatisfactorysolution.

Asuitableandsatisfactorysolutionisobtainedbyusingindependentcontrolsofloadfrequencyandeconomicdispatch.Theloadfrequencycontrollerprovidesafast-actingcontrolandregulatesthesystemaroundanoperatingpoint,whereastheeconomic

I

dispatchcontrollerprovidesaslow-actingcontrol,whichadjuststhespeed-changersettingseveryminuteinaccordancewithacommandsignalgeneratedbythecentraleconomicdispatchcomputer.

EDC—economicdispatchcontroller

CEDC—centraleconomicdispatchcomputer

Thespeed-changersettingischangedinaccordancewiththeeconomicdispatcherrorsignal,(i.e.,P −P )convenientlymodifiedbythesignal∫ACEdtatthatinstantoftime.Thecentraleconomicdispatchcomputer(CEDC)providesthesignalP ,andthissignalistransmittedtothelocaleconomicdispatchcontroller(EDC).Thesystemtheyoperatewitheconomicdispatcherrorisonlyforveryshortperiodsoftimebeforeitisreadilyused(Fig.8.11).

ThistertiarycontrolcanbeimplementedbyusingEDCandEDCworksonthecostcharacteristicsofvariousgeneratingunitsinthearea.Thespeed-changersettingsareonceagainoperatedinaccordancewithaneconomicdispatchcomputerprogram.

G(desired)

G(actual)

G(desired)

FIG.8.11Loadfrequencyandeconomicdispatchcontrolofthecontrolareaofapowersystem

TheCEDCsareprovidedatacentralcontrolcenter.Thevariablepartoftheloadiscarriedbyunitsthatarecontrolledfromthecentralcontrolcenter.Medium-sizedfossilfuelunitsandhydro-unitsareusedforcontrol.Duringpeakloadhours,lesserefficientunits,suchasgas-turbineunitsordieselunits,areemployedinaddition;generatorsoperatingatpartialoutput(withspinningreserve)andstandbygeneratorsprovideareservemargin.

Thecentralcontrolcentermonitorsinformationincludingareafrequency,outputsofgeneratingunits,

andtie-linepowerflowstointerconnectedareas.ThisinformationisusedbyALFCinordertomaintainareafrequencyatitsscheduledvalueandnettie-linepowerflowoutoftheareaatitssheddingvalue.Raiseandlowerreferencepowersignalsaredispatchedtotheturbinegovernorsofcontrolledunits.

Economicdispatchisco-ordinatedwithLFCsuchthatthereferencepowersignalsdispatchedtocontrolledunitsmovetheunitstowardtheireconomicloadingandsatisfyLFCobjectives.

8.8DESIGNOFAUTOMATICGENERATIONCONTROLUSINGTHEKALMANMETHOD

Amoderngigawattgeneratorwithitsmultistagereheatturbine,includingitsautomaticloadfrequencycontrol(ALFC)andautomaticvoltageregulator(AVR)controllers,ischaracterizedbyanimpressivecomplexity.Whenallitsnon-negligibilitydynamicsaretakenintoaccount,includingcross-couplingbetweencontrolchannels,theoveralldynamicmodelmaybeofthetwentiethorder.

Thedimensionalitybarriercanbeovercomebymeansofcomputer-aidedoptimalcontroldesignmethodsoriginatedbyKalman.Acomputer-orientedtechniquecalledoptimumlinearregulator(OLR)designhasproventobeparticularlyusefulinthisregard.

TheOLRdesignresultsinacontrollerthatminimizesbothtransientvariableexcursionsandcontrolefforts.Intermsofpowersystem,thismeansoptimallydampedoscillationwithminimumwearandtearofcontrolvalves.

OLRcanbedesignedusingthefollowingsteps:

1. Castingthesystemdynamicmodelinstate-variableformandintroducingappropriatecontrolforces.

2. Choosinganintegral-squared-errorcontrolindex,theminimizationofwhichisthecontrolgoal.

3. Findingthestructureoftheoptimalcontrollerthatwillminimizethe

chosencontrolindex.

8.9DYNAMIC-STATE-VARIABLEMODEL

TheLFCmethodsdiscussedsofararenotentirelysatisfactory.Inordertohavemoresatisfactorycontrolmethods,optimalcontroltheoryhastobeused.Forthispurpose,thepowersystemmodelmustbeinastate-variablemodel.

8.9.1Modelofsingle-areadynamicsysteminastate-variableform

Fromtheblockdiagramofanuncontrolledsingle-areasystemshowninFig.8.12,wegetthefollowing‘s-domain’equations:

Intimedomain,theaboveequationscanbeexpressedas

FIG.8.12State-spacemodelofasingle-areasystem

Letuschoosethestatevariables ,input,u=

ΔP ,anddisturbance,d=ΔP

Theaboveequationsarewritteninastate-variableform:

Thestateequationwouldthenbewritteninamoregeneralformas

Ẋ=Ax+Bu+Jp

C D

wherexisthen-dimensionalstatevector,uthem-dimensionalcontrol-forcevector=[u]=[ΔP ],andpthedisturbanceforcevector=[p]=[ΔP ].

8.9.2Optimumcontrolindex(I)

Theoptimumlinearregulatordesignisbasedontheintegral-squared-errorindexoftheform.

whereq’sandr’sarepositivepenaltyfactors.LetusconsiderindexIforasingle-areasystemas

Here,theideaofoptimumcontrolistominimizetheindex(I).Considerthetermq (Δf) ,squaringoffrequencyerrorwillcontributeto‘I’independentofitssign.IfΔfisdoubled,itscontributionto‘I’willquadruple.TheintegralcausesΔftoaddto‘I’duringitsentireduration.Thepenaltyfactorsq distributethepenaltyweightamongthestate-variableerrors.Iftheerrorinaparticularvariableisoflittlesignificance,wesimplysetitspenaltyfactortozero.

Similarly,inthecaseofcontrol-forceincrements,thepenaltyfactorsr distributethepenaltiesamongthemcontrolforce.(Here,noneofr’saresettozero.)Iftheyaresettozero,thenthecontrolforcewillassumeaninfinitemagnitudewithoutaffecting‘I’.Aninfinitecontrolforcecoulddoitscorrectingjobinzerotime.Thiswouldobviouslybeaveryunrealisticregulator.

Ifallq’sandr’sconstitutethediagonalelementsofthetwopenaltymatrices,thenwehave

c

D

3

i

i

2

Index-Iinacompactformis

8.9.3Optimumcontrolproblemandstrategy

ThetaskthatanOLRoranoptimumcontrollermustperformisthefulfillmentoftheoptimumcontrolproblem,whichcanbestatedasfollows:considerasystemthatisinitiallyundersteady-statecondition.Ifitisdisturbedbyasetofsteptypeofdisturbances,itgoesthroughatransientstatefirst.Itisrequiredthat,aftertheexpiryofthetransientperiod,itshouldreturntotheoriginalornewprescribedsteady-statecondition.Theproblemistodeterminethesetofcontrolforceswhichwillnotonlytakethesystemtotheoriginalornewprescribedsteadystate,butwillalsodosobysimultaneouslyminimizingthechosencontrolcriterionfunctionoroptimumcontrolindex(I).

Thevalueof whichfulfillstheaboveoptimumcontrolrequirement,isthedesiredoptimumcontrolstrategy.AnoptimumcontrollerorOLRistheonethatcarriesouttheabovestrategy.

8.9.4Dynamicequationsofatwo-areasystem

Fromtheblockdiagramofuncontrolledtwo-areasystemsshowninFig.8.4,getthefollowing‘s-domain’equations:

whereX (s)andX (s)aretheLaplacetransformsofthe

movementsofthemainpositionsinthespeed-governingmechanismsofthetwoareas.

BytakinginverseLaplacetransformfortheaboveequations,wegetasetofsevendifferentialequations.Thesearethetime-domainequations,whichdescribethesmall-disturbancedynamicbehaviorofthepowersystem.

Considerthefirstequation,

TakingtheinverseLaplacetransformoftheaboveequation,weget

E1 E2

Inasimilarway,theremainingequationscanberearrangedandaninverseLaplacetransformisfound.Then,theentiresetofdifferentialequationsis

8.9.4.1Statevariablesandstate-variablemodel

Thestatevariablesareaminimumnumberofthosevariables,whichcontainsufficientinformationaboutthepasthistorywithwhichallfuturestatesofthesystemcanbedeterminedforknowncontrolinputs.Forthetwo-areasystemunderconsideration,thestatevariableswouldbeΔf ,Δf ,∆X ,∆X ,∆P ,∆P and∆P ;seven

innumber.Denotingtheabovevariablesbyx ,x ,x ,x ,x ,x ,andx andarrangingtheminacolumnvectoras

1 2 E1 E2 sg1 sg2 TL1

1 2 3 4

5 6 7

where iscalledastatevector.

Thecontrolvariables∆P and∆P aredenotedbythe

symbolsu andu ,respectively,as

whereūiscalledthecontrolvectororthecontrol-forcevector.

Thedisturbancevariables∆P and∆P ,sincethey

createperturbationsinthesystem,aredenotedbyp andp ,respectively,as

where iscalledthedisturbancevector.

Theabovestateequationscanbewritteninamatrixformas

c1 c2

1 2

D1 D2

1

2

where ;i=1,2,3,…7.

Theabovematrixequationcanbewritteninthevectorformas

where[A]iscalledthesystemmatrix,[B]theinputdistributionmatrix,and[J]thedisturbancedistributionmatrix.

Inthepresentcase,theirdimensionsare(7×7),(7×2),and(7×2),respectively.Equation(8.45)isashorthandformofEquation(8.44),andEquation(8.44)

constitutesthedynamic‘state-variablemodel’oftheconsideredtwo-areasystem.

Thedifferentialequationscanbeputintheaboveformonlyiftheyarelinear.Ifthedifferentialequationsarenon-linear,thentheycanbeexpressedinthemoregeneralformas

8.9.5State-variablemodelforathree-areapowersystem

TheblockdiagramrepresentationofthismodelisshowninFig.8.13.

FIG.8.13Three-areamodel

Fromtheblockdiagram,thefollowingequationsarewrittenas

TakingtheinverseLaplacetransformfortheaboveequations,whichwegetinasimilarway,theremainingequationscanberearrangedandtheinverseLaplacetransformcanbefound.Then,theentiresetofdifferentialequationsis

and

and

and

Theaboveequationsarewritteninavectorformasshownbelow:

Ẋ=[A]X+[B]ū+[J]

where

whereXiscalledastatevector.

Thecontrolvariables∆P ,∆P ,and∆P aredenoted

bythesymbolsu ,u ,andu ,respectively,as

whereuiscalledthecontrolvectororthecontrol-forcevector.

Thedisturbancevariables∆P ,∆P ,and∆P ,since

theycreateperturbationsinthesystem,aredenotedbyp ,p ,andp ,respectively,as

c1 c2 c1

1 2 3

D1 D2 D1

1 2 3

where iscalledthedisturbancevector.

where

m = 2π(T +T )

n = 2π(T +T )

o = 2π(T +T )

p = −2πT

q = −2πT

r = −2πT

12 13

21 23

31 32

12

13

21

0 0

0 0

0 0

0

0

0

s = −2πT

t = −2πT

ū = −2πT

23

31

32

0

0

0

8.9.6Advantagesofstate-variablemodel

Thestate-variablemodelingofapowersystemoffersthefollowingadvantages:

1. Moderncontroltheoryisbaseduponthisstandardform.2. Byarrangingsystemparametersintomatrices[A],[B],and[J],avery

organizedmethodologyofsolvingsystemequations,eitheranalyticallyorbycomputer,isdeveloped.Thisisimportantforlargesystemswherealackoforganizationeasilyresultsinerrors.

Example8.4:TwointerconnectedArea-1andArea-2havethecapacityof2,000and500MW,respectively.Theincrementalregulationanddampingtorquecoefficientforeachareaonitsownbaseare0.2p.u.and0.8p.u.,respectively.Findthesteady-statechangeinsystemfrequencyfromanominalfrequencyof50Hzandthechangeinsteady-statetie-linepowerfollowinga750MWchangeintheloadofArea-1.

Solution:

RatedcapacityofArea-1=P =2,000MW

RatedcapacityofArea-2=P =500MW

Speedregulation,R=0.2p.u.

Nominalfrequency,f=50Hz

ChangeinloadpowerofArea-1,ΔP =75MW

Speedregulation,R=0.2=0.2p.u.×50=10Hz/p.u.MW

Dampingtorquecoefficient,B=0.8p.u.MW/p.u.Hz

ChangeinloadofArea-1,∆P =75MW

p.u.changeinloadofArea-1

1(rated)

2(rated)

1

D1


Steady-statechangeinsystemfrequency,

where

Steady-statechangeintie-linepowerfollowingloadchangeinArea-1:

Example8.5:SolveExample8.4,withoutgovernorcontrolaction.

Solution:

Withoutthegovernorcontrolaction,R=0

Steady-statechangeintie-linepowerfollowingloadchangeinArea-1:

Itisobservedfromtheresultthatthepowerflowthroughthetielineisthesameinboththecasesofwithgovernoractionandwithoutgovernoraction,sinceitdoesnotdependonspeedregulationR.

Example8.6:Findthenatureofdynamicresponseifthetwoareasoftheaboveproblemareofuncontrolledtype,followingadisturbanceineitherareaintheformofastepchangeinelectricload.TheinertiaconstantofthesystemisgivenasH=3sandassumethatthetieline

hasacapacityof0.09p.u.andisoperatingatapowerangleof30 beforethestepchangeinload.

Solution:

Given:

Speedregulation,R=0.2p.u.=0.2×50=10Hz/p.u.MW

Dampingcoefficient,B=0.8p.u.MW/p.u.Hz

Inertiaconstant,H=3s

Nominalfrequency,f =50Hz

Tie-linecapacity,

Fromthetheoryofdynamicresponse,weknowthat

Itisobservedthatthedampedoscillationtypeofdynamicresponsehasresultedsinceα<ω :

∴Dampedangularfrequencyn

o

0

∴Dampedfrequency=f

Example8.7:Twocontrolareashavethefollowingcharacteristics:

Area-1: Speedregulation=0.02p.u.

Dampingcoefficient=0.8p.u.

RatedMVA=1,500

Area-2: Speedregulation=0.025p.u.

Dampingcoefficient=0.9p.u.

RatedMVA=500

Determinethesteady-statefrequencychangeandthechangedfrequencyfollowingaloadchangeof120MW,whichoccursinArea-1.Alsofindthetie-linepowerflowchange.

Solution:

GivenR =0.1p.u.;R =0.098p.u.

B =0.8p.u.;B =0.9p.u.

P =1,500MVA;P =1,500MVA

ChangeinloadofArea-1,

d

1 2

1 2

1rated 2rated

∆P =120MW,∆P =0


∴Steady-statefrequencychange,

i.e.,Steady-statechangeinfrequency,∆f = 0.0012415×50

= 0.062Hz

∴Newvalueoffrequency,f=f −∆f = 50−0.062

= 49.937Hz

Steady-statechangeintie-linepower

D1 D2

ss

ss

0

Example8.8:InExample8.6,ifthedisturbancealsooccursinArea-2,whichresultsinachangeinloadby75MW,determinethefrequencyandtie-linepowerchanges.

Solution:





Steady-statefrequencychange,

∴Steady-statefrequencychange=0.002×50=0.1Hz

∴Newvalueoffrequency=f −Δf =50−0.1=49.899Hz

Steady-statechangeintie-linepower,

D1

D2

ss

0

Example8.9:Twoareasofapowersystemnetworkareinterconnectedbyatieline,whosecapacityis250MW,operatingatapowerangleof45 .Ifeachareahasacapacityof2,000MWandtheequalspeed-regulationcoefficiencyof3Hz/p.u.MW,determinethefrequencyofoscillationofthepowerforastepchangeinload.AssumethatbothareashavethesameinertiaconstantsofH=4s.Ifastep-loadchangeof100MWoccursinoneoftheareas,determinethechangeintie-linepower.

Solution:

Given:

Tie-linecapacity,P =250MW

Powerangleoftwoareas,(δ −δ )=457°

Capacityofeacharea,P =2,000MW

Speed-regulationcoefficient=R =R =R=3Hz/p.u.MW


tie(max)

1 2

rated

1 2

o

0 0

Since,α<ω ,thedynamicresponsewillbeofadampedoscillationtype.

Dampedangularfrequency,

∴Frequencyofoscillation,

Ifastep-loadchangeof100MWoccursinanyoneoftheareas,thetotalloadchangewillbesharedequallybybothareassincethetwoareasareequal,i.e.,apowerof

willflowfromtheotherareaintothearea

wherealoadchangeoccurs.

Example8.10:TwopowerstationsAandBofcapacities75and200MW,respectively,areoperatinginparallelandareinterconnectedbyashorttransmissionline.ThegeneratorsofstationsAandBhavespeedregulationsof4%and2%,respectively.Calculatetheoutputofeachstationandtheloadontheinterconnectionif

1. theloadoneachstationis100MW,

n

2. theloadsonrespectivebusbarsare50and150MW,and3. theloadis130MWatStationAbusbaronly.

Solution:

Given:

CapacityofStation-A=75MW

CapacityofStation-B=200MW

SpeedregulationofStation-Agenerator,R =4%

SpeedregulationofStation-Bgenerator,R =2%

(a)Iftheloadoneachstation=100MW

i.e.,P +P =100+100=200MW(8.47)

Speedregulation

∴(1−f)=0.0001P (8.49)


0.000533P =0.0001P

5.33P =P (8.50)

P +P =200


A

B

1 2

2

1 2

1 2

1 2

Thepowergenerationsandtie-linepowerareindicatedinFig.8.14(a).

(b)Iftheloadonrespectivebusbarsare50and150MW,thenwehave

i.e.,P +P =50+150=200MW

5.33P =P

P +5.33P =200

⇒6.33P =200

P =31.6MW

∴P =200−31.60=168.4MW

Thepowergenerationsandtie-linepowerareindicatedinFig.8.14(b).

1 2

1 2

1 1

1

1

2

FIG.8.14(a)IllustrationforExample8.10;(b)illustrationforExample8.10;(c)illustrationforExample8.10

(c)Iftheloadis130MWatAonly,thenwehave

P +P =130

5.33P =P

1 2

1 2

∴P =5.33P =130

6.33P =130

⇒P =20.537MW

∴P =130−P =109.462MW

Thepowergenerationsandtie-linepowerareindicatedinFig.8.14(c).

Example8.11:Thetwocontrolareasofcapacity2,000and8,000MWareinterconnectedthroughatieline.TheparametersofeachareabasedonitsowncapacitybaseareR=1Hz/p.u.MWandB=0.02p.u.MW/Hz.Ifthecontrolarea-2experiencesanincrementinloadof180MW,determinethestaticfrequencydropandthetie-linepower.

Solution:

CapacityofArea-1=2,000MW


Taking8,000MWasbase,

∴SpeedregulationofArea-1,

DampingcoefficientofArea-1,

SpeedregulationofArea-2,R =1Hz/p.u.MW

DampingcoefficientofArea-2,B =0.02p.u.MW/Hz

1 1

1

1

2 1

2

2

GivenanincrementofArea-2inload,

∆P =0

∴Staticchangeinfrequency,

Staticchangeintie-linepower,

Note:Here,a valuedeterminationisnotrequiredsincevaluesofR ,B ,andβ areobtainedaccordingtothebasevalues.

Alternatemethod:

Find Then,obtainthe∆f and

∆P values.

Here,thereisnoneedtoobtain,R ,B ,R ,andBseparately.

Example8.12:TwogeneratingstationsAandBhavingcapacities500and800MW,respectively,areinterconnectedbyashortline.Thepercentagespeedregulationsfromno-loadtofullloadofthetwostationsare2and3,respectively.Findthepowergenerationat

D1

12

1 1 1

(ss)

tie(ss)

1 1 2 2

eachstationandpowertransferthroughthelineiftheloadonthebusofeachstationis200MW.

Solution:

Givendata:

CapacityofStation-A=500MW

CapacityofStation-B=800MW

PercentagespeedregulationofStation-A=2%=0.02

PercentagespeedregulationofStation-B=3%=0.03

Loadonbusofeachstation=P =P =200MW

Totalload,P =400MW

SpeedregulationofStation-A:

SpeedregulationofStation-B:

LetP bethepowergenerationofStation-AandPthepowergenerationofStation-B:

P =Totalload−P =(400−P )

⇒0.002P =0.001875(400−P )

=0.75=193.55MW

(0.002+0.001875)P =0.75

⇒ P = 193.55MW

P = 206.45MW

DA DB

D

GA GB

GB GA GA

GA GA

GA

GA

GB

P = 193.55MW

∴ P = 206.45MW

ThepowertransferthroughthelinefromStation-Btostation-A

=P −(loadatbusbarofB)

=206.45−200

=6.45MW

Example8.13:Twocontrolareasof1,000and2,000MWcapacitiesareinterconnectedbyatieline.Thespeedregulationsofthetwoareas,respectively,are4Hz/p.u.MWand2.5Hz/p.u.MW.Considera2%changeinloadoccursfor2%changeinfrequencyineacharea.Findsteady-statechangeinfrequencyandtie-linepowerof10MWchangeinloadoccursinbothareas.

Solution:



SpeedregulationofArea-1,R =4Hz/p.u.MW(on1,000-MWbase)


Letuschoose2,000MWasbase,2%changeinloadfor2%changeinfrequency

DampingcoefficientofArea-1,

Similarly,dampingcoefficientofArea-2on2,000-MWbase

GA

GB

GB

1

2

SpeedregulationofArea-1on2,000-MWbase=R


Ifa10-MWchangeinloadoccursinArea-1,thenwehave

Steady-statechangeinfrequency,

Steady-statechangeinfrequency,

orΔf =−0.007633×50=0.38Hz

Steady-statechangeintie-linepower:

1

2

(ss)

i.e.,thepowertransferof7.938MWisfromArea-2toArea-1.

Ifa10-MWchangeinloadoccursinArea-2,thenwehave

∴Steady-statechangeinfrequency,

Steady-statechangeintie-linepower:

i.e.,Apowerof2.061MWistransferredfromArea-1toArea-2.

Example8.14:Twosimilarareasofequalcapacityof5,000MW,speedregulationR=3Hz/p.u.MW,andH=5sareconnectedbyatielinewithacapacityof500MW,andareoperatingatapowerangleof45 .Fortheabovesystem,thefrequencyis50Hz;find:

1. Thefrequencyofoscillationofthesystem.

o

2. Thesteady-statechangeinthetie-linepowerifastepchangeof100MWloadoccursinArea-2.

3. Thefrequencyofoscillationofthesysteminthespeed-governorloopisopen.

Solution:

Given:

Capacityofeachcontrolarea=P P =500MW

Speedregulation,R=2Hz/p.u.MW


Powerangle=45

Supplyfrequency,f =50Hz

(a)Stiffnesscoefficient,

Sinceα<ω ,dampedoscillationswillbepresent.

∴Dampedangularfrequency,

1(rated) 2(rated)

n

o

0

(b)Sincethetwoareasaresimilar,eachareawillsupplyhalfoftheincreasedload:

∴β =β

ΔP =50MWfromArea-1toArea-2.

Ifthespeed-governorloopisopen,then

Dampedangularfrequency,

KEYNOTES

AnextendedpowersystemcanbedividedintoanumberofLFCareas,whichareinterconnectedbytielines.Suchanoperationiscalledapooloperation.

Thebasicprincipleofapooloperationinthenormalsteadystateprovides:

1. Maintainingofscheduledinterchangesoftie-linepower.2. Absorptionofownloadchangebyeacharea.

Theadvantagesofapooloperationareasfollows:

1. Halfoftheaddedload(inArea-2)issuppliedbyArea-1throughthetieline.2. Thefrequencydropwouldbeonlyhalfofthatwhichwouldoccuriftheareas

wereoperatingwithoutinterconnection.

Thespeed-changercommandsignalswillbe:

and

1 2

tie

TheconstantsK andK arethegainsoftheintegrators.Thefirst

termsontheright-handsideoftheaboveequationsconstitutewhatisknownasatie-linebiascontrol.

Theloadfrequencycontrollerprovidesafast-actingcontrolandregulatesthesystemaroundanoperatingpoint,whereastheEDCprovidesaslow-actingcontrol,whichadjuststhespeed-changersettingseveryminuteinaccordancewithacommandsignalgeneratedbytheCEDC.


1. Whataretheadvantagesofapooloperation?

Theadvantagesofapooloperation(i.e.,gridoperation)are:

1. Halfoftheaddedload(inArea-2)issuppliedbyArea-1throughthetieline.

2. Thefrequencydropwouldbeonlyhalfofthatwhichwouldoccuriftheareaswereoperatingwithoutinterconnection.

2. Withoutspeed-changerpositioncontrol,canthestaticfrequencydeviationbezero?

No,thestaticfrequencydeviationcannotbezero.

3. Statetheadditionalrequirementofthecontrolstrategyascomparedtothesingle-areacontrol.

Thetie-linepowerdeviationduetoastep-loadchangeshoulddecreasetozero.

4. WritedowntheexpressionsfortheACEs.

TheACEofAreas-1and2are:

ACE (S)=∆P (S)+b ∆F (S).

ACE (S)=∆P (S)+b ∆F (S).

5. Whatisthecriterionusedforobtainingoptimumvaluesforthecontrolparameters?

Integralofthesumofthesquarederrorcriterionistherequiredcriterion.

6. Givetheerrorcriterionfunctionforthetwo-areasystem.

12 12

1 TL1 1 1

2 TL2 2 2

7. Whatistheorderofdifferentialequationtodescribethedynamicresponseofatwo-areasysteminanuncontrolledcase?

Itisrequiredforasystemofseventh-orderdifferentialequationstodescribethedynamicresponseofatwo-areasystem.Thesolutionoftheseequationswouldbetedious.

8. WhatisthedifferenceofACEinsingle-areaandtwo-areapowersystems?

Inasingle-areacase,ACEisthechangeinfrequency.Thesteady-stateerrorinfrequencywillbecomezero(i.e.,Δf =0)whenACE

isusedinanintegral-controlloop.

Inatwo-areacase,ACEisthelinearcombinationofthechangeinfrequencyandchangeintie-linepower.Inthiscasetomakethesteady-statetie-linepowerzero(i.e.,ΔP =0),anotherintegral-

controlloopforeachareamustbeintroducedinadditiontotheintegralfrequencylooptointegratetheincrementaltie-linepowersignalandfeeditbacktothespeed-changer.

9. Whatisthemaindifferenceofloadfrequencyandeconomicdispatchcontrols?

Theloadfrequencycontrollerprovidesafast-actingcontrolandregulatesthesystemaroundanoperatingpoint,whereastheEDCprovidesaslow-actingcontrol,whichadjuststhespeed-changersettingseveryminuteinaccordancewithacommandsignalgeneratedbytheCEDC.

10. Whatarethestepsrequiredfordesigninganoptimumlinearregulator?

Anoptimumlinearregulatorcanbedesignedusingthefollowingsteps:

1. Castingthesystemdynamicmodelinastate-variableformandintroducingappropriatecontrolforces.

2. Choosinganintegral-squared-errorcontrolindex,theminimizationofwhichisthecontrolgoal.

3. Findingthestructureoftheoptimalcontrollerthatwillminimizethechosencontrolindex.


1. ChangesinloaddivisionbetweenACgeneratorsoperationinparallelareaccomplishedby:

ss

TL

1. Adjustingthegeneratorvoltageregulators.2. Changingenergyinputtotheprimemoversofthegenerators.3. Loweringthesystemfrequency.4. Increasingthesystemfrequency.

2. WhentheenergyinputtotheprimemoverofasynchronousACgeneratoroperatinginparallelwithotherACgeneratorsisincreased,therotorofthegeneratorwill:

1. Increaseinaveragespeed.2. Retardwithrespecttothestator-revolvingfield.3. Advancewithrespecttothestator-revolvingfield.4. Noneofthese.

3. Whentwoormoresystemsoperateonaninterconnectedbasis,eachsystem:

1. Candependontheothersystemforitsreserverequirements.2. Shouldprovideforitsownreservecapacityrequirements.3. Shouldoperateina‘flatfrequency’mode.

4. Whenaninterconnectedpowersystemoperateswithatie-linebias,theywillrespondto:

1. Frequencychangesonly.2. Bothfrequencyandtie-lineloadchanges.3. Tie-lineloadchangesonly.

5. Inatwo-areacase,ACEis:

1. Changeinfrequency.2. Changeintie-linepower.3. Linearcombinationofboth(a)and(b).4. Noneoftheabove.

6. AnextendedpowersystemcanbedividedintoanumberofLFCareas,whichareinterconnectedbytielines.Suchanoperatoriscalled

1. Pooloperation.2. Bankoperation.3. (a)and(b).4. None.

7. Forthestaticresponseofatwo-areasystem,

1. ∆P =∆ .

2. ∆P =0.

3. ∆P =0.

4. Both(b)and(c).

8. Areaoffrequencyresponsecharacteristic‘β’is:

1. 1/R.2. B.3. B+1/R.4. B-1/R.

9. Thetie-linepowerequationisΔP =_____

1. T(Δδ +Δδ ).

2. T/(Δδ +Δδ ).

ref1 ref2ref1ref2

12

1 2

1 2

3. T/(Δδ -Δδ ).

4. T(Δδ1-Δδ2).

10. Theunitofsynchronizingcoefficients‘T’is:

1. MW-s.2. MW/s.3. MW-rad.4. MW/rad.

11. Foratwo-areasystem,ΔfisrelatedtoincreasedsteploadM and

M withareafrequencyresponsecharacteristicsβ andβ is:

1.M +M /β +β .

2. (M +M )(β +β ).

3. -(M +M )/(β +β ).

4. Noneofthese.

12. Tie-linepowerflowfortheabovequestion(11)isΔP =_____

1. (β M +β M )/β +β .

2. (β M -β M )/β +β .

3. (β M -β M )/β +β .

4. Noneofthese.

13. Advantageofapooloperationis:

1. Addedloadcanbesharedbytwoareas.2. Frequencydropreduces.3. Both(A)and(B).4. Noneofthese.

14. Dampingoffrequencyoscillationsforatwo-areasystemismorewith:

1. Low-R.2. High-R.3. R=α.4. Noneofthese.

15. ACEequationforageneralpowersystemwithtie-linebiascontrolis:

1. ΔP +B Δf .

2. ΔP -B Δf .

3. ΔP /B Δf .

4. Noneofthese.

16. Foratwo-areasystemΔf,ΔP ,R ,R ,andDarerelatedasΔf=

_____

1. ΔP /R +R .

2. -ΔP /(1/R +R +B).

3. -ΔP /(B+R +1/R ).

4. Noneofthese.

17. Ifthetwoareasareidentical,thenwehave:

1. Δf =1/Δf .

2. Δf Δf =2.

3. Δf =Δf .

4. Noneofthese.

1 2

1

2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

12

1 2 2 1 1 2

1 2 2 1 1 2

1 1 2 2 1 2

ij i i

ij i i

ij i i

L 1 2

L 1 2

L 1 2

L 1 2

1 2

1 2

1 2

18. Tie-linebetweentwoareasusuallywillbea_____line.

1. HVDC.2. HVAC.3. NormalAC.4. Noneofthese.

19. Dynamicresponseofatwo-areasystemcanberepresentedbya_____ordertransferfunction.

1. Third.2. Second.3. First.4. Zero.

20. ControlofALFCloopofamulti-areasystemisachievedbyusing_____mathematicaltechnique.

1. Rootlocus.2. Bodeplots.3. Statevariable.4. Nyquistplots.

REVIEWQUESTIONS

1. Obtainthemathematicalmodelingofthelinepowerinaninterconnectedsystemanditsblockdiagram.

2. Obtaintheblockdiagramofatwo-areasystem.3. Explainhowthecontrolschemeresultsinzerotie-linepower

deviationsandzero-frequencydeviationsundersteady-stateconditions,followingastep-loadchangeinoneoftheareasofatwo-areasystem.

4. Deducetheexpressionforstatic-errorfrequencyandtie-linepowerinanidenticaltwo-areasystem.

5. Explainabouttheoptimaltwo-areaLFC.6. Whatismeantbytie-linebiascontrol?7. Derivetheexpressionforincrementaltie-linepowerofanareainanuncontrolledtwo-areasystemunderdynamicstateforastep-loadchangeineitherarea.

8. Drawtheblockdiagramforatwo-areaLFCwithintegralcontrollerblocksandexplaineachblock.

9. Whatarethedifferencesbetweenuncontrolled,controlled,andtie-linebiasLFCofatwo-areasystem.

10. Explainthemethodinvolvedinoptimumparameteradjustmentforatwo-areasystem.

11. ExplainthecombinedoperationofanLFCandanELDCsystem.

PROBLEMS

1. Twointerconnectedareas1and2havethecapacityof250and600MW,respectively.Theincrementalregulationanddampingtorquecoefficientforeachareaonitsownbaseare0.3and0.07p.u.respectively.Findthesteady-statechangeinsystem

frequencyfromanominalfrequencyof50Hzandthechangeinsteady-statetie-linepowerfollowinga850MWchangeintheloadofArea-1.

2. Twocontrolareasof1,500and2,500MWcapacitiesareinterconnectedbyatieline.Thespeedregulationsofthetwoareas,respectively,are3and1.5Hz/p.u.MW.Considerthata2%changeinloadoccursfora2%changeinfrequencyineacharea.Findthesteady-statechangeinthefrequencyandthetie-linepowerof20MWchangeinloadoccurringinbothareas.

3. Findthenatureofdynamicresponseifthetwoareasoftheaboveproblemareofuncontrolledtype,followingadisturbanceineitherareaintheformofastepchangeinanelectricload.TheinertiaconstantofthesystemisgivenasH=2sandassumethatthetielinehasacapacityof0.08p.u.andisoperatingatapowerangleof35 beforethestepchangeinload.

o

9

ReactivePowerCompensation

OBJECTIVES


knowtheneedofreactivepowercompensationdiscusstheobjectivesofloadcompensationdiscusstheoperationofuncompensatedandcompensatedtransmissionlinesdiscusstheconceptofsub-synchronousresonance(SSR)studythevoltage-stabilityanalysis

9.1INTRODUCTION

InanidealAC-powersystem,thevoltageandthefrequencyateverysupplypointwouldremainconstant,freefromharmonicsandthepowerfactor(p.f.)wouldremainunity.Fortheoptimumperformanceataparticularsupplyvoltage,eachloadcouldbedesignedsuchthatthereisnointerferencebetweendifferentloadsasaresultofvariationsinthecurrenttakenbyeachone.

Mostelectricalpowersystemsintheworldareinterconnectedtoachievereducedoperatingcostandimprovedreliabilitywithlesserpollution.Inapowersystem,thepowergenerationandloadmustbalanceatalltimes.Tosomeextent,itisself-regulating.Ifanunbalancebetweenpowergenerationandloadoccurs,thenitresultsinavariationinthevoltageandthefrequency.Ifvoltageisproppedupwithreactivepowersupport,thentheloadincreaseswithaconsequentdropinfrequency,whichmayresultinsystemcollapse.Alternatively,ifthereisaninadequatereactivepower,thesystem’svoltagemaycollapse.

Here,thequalityofsupplymeansmaintainingconstant-voltagemagnitudeandfrequencyunderallloadingconditions.Itisalsodesirabletomaintainthethree-phasecurrentsandvoltagesasbalancedaspossiblesothatunderheatingofvariousrotatingmachinesduetounbalancingcouldbeavoided.

Inathree-phasesystem,thedegreetowhichthephasecurrentsandvoltagesarebalancedmustalsobetakenintoconsiderationtomaintainthequalityofsupply.

Toachievetheabove-mentionedrequirementsfromthesupplypointofviewaswellastheloads,whichcandeterioratethequalityofsupply,weneedloadcompensation.

LoadcompensationisthecontrolofreactivepowertoimprovethequalityofsupplyinanAC-powersystembyinstallingthecompensatingequipmentneartheload.

9.2OBJECTIVESOFLOADCOMPENSATION

Theobjectivesofloadcompensationare:

1. p.f.Correction.2. Voltageregulationimprovement.3. Balancingofload.

9.2.1p.f.Correction

Generally,loadcompensationisalocalproblem.Mostoftheindustrialloadsabsorbthereactivepowersincetheyhavelaggingp.f.’s.Theloadcurrenttendstobelargerthanitisrequiredtosupplytherealpoweralone.So,p.f.correctionofloadisachievedbygeneratingreactivepowerascloseaspossibletotheload,whichrequiresittogenerateitatadistanceandtransmitittotheload,asthisresultsnotonlyinalargeconductorsizebutalsoinincreasedlosses.Itisdesirabletooperatethesystemnearunityp.f.economically.

9.2.2Voltageregulationimprovement

Allloadsvarytheirdemandforreactivepower,althoughtheydifferwidelyintheirrangeandrateofvariation.Thevoltagevariationisduetotheimbalanceinthegenerationandconsumptionofreactivepowerinthesystem.Ifthegeneratedreactivepowerismorethanthatbeingconsumed,voltagelevelsgoupandviceversa.However,ifbothareequal,thevoltageprofilebecomesflat.Thevariationindemandforreactivepowercausesvariation(orregulation)inthevoltageatthesupplypoint,whichcaninterferewithanefficientoperationofallplantsconnectedtothatpoint.So,differentconsumersconnectedtothatpointgetaffected.Toavoidthis,thesupplyutilityplacesboundstomaintainsupplyvoltageswithindefinedlimits.Theselimitsmayvaryfromtypically±6%averagedoveraperiodofafewminutesorhours.

Toimprovevoltageregulation,weshouldstrengthenthepowersystembyincreasingthesizeandnumberofgeneratingunitsaswellasbymakingthenetworkmoredenselyinterconnected.Thisapproachwouldbeuneconomicandwouldintroduceproblemssuchashighfaultlevels,etc.Inpractice,itismuchmoreeconomictodesignthepowersystemaccordingtothemaximumdemandforactivepowerandtomanagethereactivepowerbymeansofcompensatorslocally.

9.2.3Loadbalancing

Mostpowersystemsarethree-phasedandaredesignedforbalancedoperationsincetheirunbalancedoperationgivesrisetowrongphase-sequencecomponentsofcurrents(negativeandzero-sequencecomponents).Suchcomponentsproduceundesirableresultssuchasadditionallossesinmotors,generators,oscillatingtorqueinACmachines,increasedripplesinrectifiers,saturationoftransformers,excessivenaturalcurrent,andsoon.Theseundesirableeffectsarecausedmainlyduetotheharmonicsproducedunderanunbalanced

operation.Tosuppresstheseharmonics,certaintypesofequipmentincludingcompensatorsareprovided,whichyieldthebalancedoperationofthepowersystem.

9.3IDEALCOMPENSATOR

Anidealcompensatorisadevicethatcanbeconnectedatornearasupplypointandinparallelwiththeload.Themainfunctionsofanidealcompensatorareinstantaneousp.f.correctiontounity,eliminationorreductionofthevoltageregulation,andphasebalanceoftheloadcurrentsandvoltages.Inperformingtheseinterdependentfunctions,itwillconsumezeropower.

Thecharacteristicsofanidealcompensatorareto:

provideacontrollableandvariableamountofreactivepowerwithoutanydelayaccordingtotherequirementsoftheload,maintainaconstant-voltagecharacteristicatitsterminals,andshouldoperateindependentlyinthethreephases.

9.4SPECIFICATIONSOFLOADCOMPENSATION

Thespecificationsofloadcompensationare:

Maximumandcontinuousreactivepowerrequirementintermsofabsorbingaswellasgeneration.Overloadratingandduration.Ratedvoltageandlimitsofvoltagebetweenwhichthereactivepowerratingmustnotbeexceeded.Frequencyanditsvariation.Accuracyofvoltageregulationrequirement.Specialcontrolrequirement.Maximumharmonicdistortionwithcompensationinseries.Emergencyprocedureandprecautions.Responsetimeofthecompensatorforaspecifieddisturbance.Reliabilityandredundancyofcomponents.

9.5THEORYOFLOADCOMPENSATION

Inthissection,relationshipsbetweenthesupplysystem,theload,andthecompensatorweretobedeveloped.Thesupplysystem,theload,andthecompensatorcanbemodeledindifferentways.Here,thesupplysystemismodeledasaThevenin’sequivalentcircuitwithreactivepowerrequirements.Thecompensatorismodeledasa

variableimpedance/asavariablesource(orsink)ofreactivecurrent/power.Accordingtorequirements,theselectionofmodelusedforeachcomponentcanbevaried.

Theassumptionmadeindevelopingtherelationshipsbetweensupplysystem,theload,andthecompensatoristhattheloadandsystemcharacteristicsarestatic/constant(or)changingslowlysothatphasorrepresentationcanbeused.

9.5.1p.f.Correction

Considerasingle-phaseloadwithadmittanceY =G +jB withasourcevoltageasshowninFig.9.1(a).

TheloadcurrentI isgivenby

I =V (G +jB )=V G +jV B =I +jI

whereI istheactivecomponentoftheloadcurrent

I thereactivecomponentoftheloadcurrent.

FIG.9.1Representationofsingle-phaseload;(a)withoutcompensation;(b)withcompensation

Apparentpoweroftheload,S =V I *

=V G −jV B

L L

L

L

L s L L s L s L a r

a

r

L s L

S L S L

2 2

=P +jQ

whereP istheactivepoweroftheload

Q thereactivepoweroftheload.

Forinductiveloads,B isnegativeandQ ispositivebyconvention.

Thecurrentsuppliedtotheloadislargerthanwhenitisnecessarytosupplytheactivepoweralonebythefactor

Theobjectiveofthep.f.correctionistocompensateforthereactivepower,i.e.,locallyprovidingacompensatorhavingapurelyreactiveadmittancejB inparallelwiththeloadasshowninFig.9.1(b).Thecurrentsuppliedfromthesourcewiththecompensatoris

I =I +I

=V (G +jB )–V (jB )

=V G =I (∵B =B )

whichmakesthep.f.tounity,sinceI isinphasewiththesourcevoltageV .

Thecurrentofthecompensator,I =V Y =–jV B

Theapparentpowerofthecompensator,S =V I *

S =jV B =−jQ (∵S =P −jQ ,forpurecompensationP =0)

L L

L

L

L L

C

s L C

s L L s C

s L a L C

a

s

c s c s c

c s c

c s C C C C C

C

2

Weknowthat

Q =P tanϕ

Forafullycompensatedsystem,i.e.,Q =Q

Thedegreeofcompensationisdecidedbyaneconomictrade-offbetweenthecapitalcostofthecompensatorandthesavingsobtainedbythereactivepowercompensationofthesupplysystemoveraperiodoftime.

9.5.2Voltageregulation

Itisdefinedastheproportionalchangeinsupplyvoltagemagnitudeassociatedwithadefinedchangeinloadcurrent,i.e.,fromno-loadtofullload.Itiscausedbythevoltagedropinthesupplyimpedancecarryingtheloadcurrent.

Whenthesupplysystemisbalanced,itcanberepresentedassingle-phasemodelasshowninFig.

9.2(a).Thevoltageregulationisgivenby ,where

|V |istheloadvoltage.

9.5.2.1Withoutcompensator

Fromthephasordiagramofanuncompensatedsystem,showninFig.9.2(b),thechangeinvoltagesisgivenby

ΔV=V −V =Z I (9.1)

whereZ =R +jX andtheloadcurrent,

L L L

L C

L

s L s L

s s s

SubstitutingZ andI inEquation(9.1),weget

FIG.9.2(a)Circuitmodelofanuncompensatedloadandsupplysystem;(b)phasordiagramforanuncompensatedsystem

FromEquation(9.3),itisobservedthatthechangeinvoltagedependsonbothrealandreactivepowersoftheloadconsideringthelineparameterstobeconstant.

9.5.2.2Withcompensator

Inthiscase,apurelyreactivecompensatorisconnectedacrosstheloadasshowninFig.9.3(a)tomakethevoltageregulationzero,i.e.,thesupplyvoltage(|V |)equalstheloadvoltage(|V |).ThecorrespondingphasordiagramisshowninFig.9.3(b).

s L

s

L

FIG.9.3(a)Circuitmodelofacompensatedloadandsupplysystem;(b)phasordiagramforacompensatedsystem

Thesupplyreactivepowerwithacompensatoris

Q =Q +Q

Q isadjustedinsuchawaythat∆V=0

i.e.,|V |=|V |

FromEquations(9.1)and(9.3),weget

s C L

C

s L

Simplifyingandrearrangingequation(9.4),

Theaboveequationcanberepresentedinacompactformas

aQ +bQ +c=0

where

ThevalueofQ isfoundusingtheaboveequationbyusingthecompensatorreactivepowerbalanceequation|V |=|V |andQ =Q –Q .

Here,thecompensatorcanperformasanidealvoltageregulator,i.e.,themagnitudeofthevoltageisbeingcontrolled,itsphasevariescontinuouslywiththeloadcurrent,whereasthecompensatoractingasap.f.correctorreducesthereactivepowersuppliedbythesystemtozeroi.e.,Q =0=Q +Q .

Equation(9.3)canbereducedto

s s

C

s L C s L

s L C

2

So,∆Visindependentoftheloadreactivepower.Fromthis,weconcludethatapurereactivecompensatorcannotmaintainbothconstantvoltageandunityp.f.simultaneously.

9.6LOADBALANCINGANDP.F.IMPROVEMENTOFUNSYMMETRICALTHREE-PHASELOADS

Thethirdobjectiveofloadcompensationisthebalancingofunbalancedthree-phaseloads.Wefirstmodeltheloadasadelta-connectedadmittancenetworkforageneralunbalancedthree-phaseloadasshowninFig.9.4inwhichtheadmittancesY ,Y andY arecomplexandunequal.

Inthiscase,supplyvoltagesareassumedtobebalanced.AnyungroundedY-connectedloadcanberepresentedbyadelta-connectedloadbymeansoftheY-∆transformation.

Acompensatorcanbeanypassivethree-phaseadmittancenetwork,whichwhencombinedinparallelwiththeloadwillpresentarealandbalancedloadwithrespecttothesupply.

9.6.1p.f.Correction

Eachloadadmittancecanbemadepurelyrealbyconnecting,inparallel,acompensatingsusceptanceequaltothenegativeoftheloadsusceptanceinthatbranchofthedelta-connectedloadasshowninFig.9.5(a).

Ifloadadmittance,Y =G +jB ,thenthecompensatingsusceptanceB =−B isconnectedacrossY :

L L L

L L L

C L

L

ab bc cd

ab ab ab

ab ab

ab

Aninductivesusceptancebetweenphases‘c’and‘a’asshowninFig.9.5(a)is

Now,thelinecurrentswillbebalancedandareinphasewiththeirrespectivephasevoltages.Thecompensatedsingle-phaseloadwithapositivesequenceequivalentcircuitisshowninFig.9.5(a).

FIG.9.4Unbalancedthree-phaseload

FIG.9.5(a)Connectionofp.f.correctingsusceptance;(b)resultantunbalancedrealloadwithunityp.f.

Similarly,thecompensatingsusceptance,B =−BandB =−B areconnectedacrossY andY ,respectively,asshowninFig.9.5(a).

Figure9.5(b)showstheresultantunbalancedrealloadwithunityp.f.

9.6.2Loadbalancing

Now,wemakethisrealunbalancedloadtoabalancedone.Todothis,letusconsiderasingle-phaseload(G )(asshowninFig.9.6(a))ofthe∆-connectedloadasshowninFig.9.5(b).Thethree-phasepositivesequencelinecurrentscanbebalancedbyconnectingcapacitivesusceptancebetweenphases‘b’and‘c’andtogetherwiththeinductivesusceptancebetweenphases‘c’and‘a’asshowninFig.9.6(b).

c L

c L L L

L

bc bc

cd ca bc ca

ab

FIG.9.6(a)Single-phaseunityp.f.loadbeforepositivesequencebalancing;(b)positivesequencebalancingofasingle-phaseu.p.f.load

FIG.9.7(a)Idealthree-phasecompensatingnetworkwithcompensatoradmittances;(b)equivalentcircuitofrealandbalancedcompensatedload

admittance

Similarly,therealadmittancesintheremainingphases‘bc’and‘ca’canbebalanced.

Theresultantcompensatoradmittance(susceptance)representedbyanequivalentcircuitisshowninFig.9.7(a).

B =−B (p.f.correction)+(G +G )/ (loadbalancing)

c L L L

ab ab ca bc

bc bc ab ca

B =−B +(G +G )/

B =−B +(G +G )/

Theresultingcompensatedloadadmittanceispurelyrealandbalanced,asshownintheequivalentcircuitofFig.9.7(b).

9.7UNCOMPENSATEDTRANSMISSIONLINES

Anelectrictransmissionlinehasfourparameters,whichaffectitsabilitytofulfillitsfunctionaspartofapowersystemandtheseareaseriescombinationofresistance,inductance,shuntcombinationofcapacitance,andconductance.TheseparametersaresymbolizedasR,L,C,andG,respectively.Theseparametersaredistributedalongthewholelengthatanyline.Eachsmalllengthatanysectionofthelinewillhaveitsownvaluesandconcentrationofallsuchparametersforthecompletelengthoflineintoasingleoneisnotpossible.Theseareusuallyexpressedasresistance,inductance,andcapacitanceperkilometer.

Shuntconductancethatismostlyduetothebreakageovertheinsulatorsisalmostalwaysneglectedinapowertransmissionline.Theleakagelossinacableisuniformlydistributedoverthelengthofthecable,whereasitisdifferentinthecaseofoverheadlines.Itislimitedonlytotheinsulatorsandisverysmallundernormaloperatingconditions.So,itisneglectedforanoverheadtransmissionline.

9.7.1Fundamentaltransmissionlineequation

Consideraverysmallelementoflength∆xatadistanceofxfromthereceivingendoftheline.Letzbetheseriesimpedanceperunitlength,ytheshuntadmittanceperunitlength,andlthelengthoftheline.

c L L L

c L L L

bc bc ab ca

ca ca bc ab

FIG.9.8Representationofatransmissionlineonasingle-phasebasis

Then,

Z=zl=totalseriesimpedanceoftheline

Y=yl=totalshuntadmittanceoftheline

ThevoltageandcurrentatadistancexfromthereceivingendareVandI,andatdistancex+∆xareV+∆VandI+∆I,respectively(Fig.9.8).

So,thechangeofvoltage,∆V=Iz∆x,wherez∆xistheimpedanceoftheelementconsidered:

Similarly,thechangeofcurrent,∆I=Vy∆x,wherey∆xistheadmittanceoftheelementconsidered:

DifferentiatingEquation(9.6)withrespecttox,weget

Substitutingthevalueof fromEquation(9.7)in

Equation(9.8),weget

Equation(9.9)isasecond-orderdifferentiatialequationanditssolutionis

DifferentiatingEquation(9.10)withrespecttox,weget



V(x)=Ae +Be (9.13)


γx −γx

where isknownascharacteristicimpedanceor

surgeimpedanceand isknownasthepropagation

constant.

TheconstantsAandBcanbeevaluatedbyusingtheconditionsatthereceivingendoftheline.

Theconditionsare

atx=0,V=V andI=I

SubstitutingtheaboveconditionsinEquations(9.11)and(9.12),weget

∴V =A+B(9.15)

and


Now,substitutingthevaluesofAandBinEquations(9.13)and(9.14),thenweget

r r

r

whereV andI arethevoltagesandcurrentsatanydistancexfromthereceivingend.

Foralosslesslineγ=jβandthehyperbolicfunctions,i.e.,coshγx=coshjβx=cosβxandsinhγx=sinhjβx=jsinβx.

Therefore,Equations(9.17)and(9.18)canbemodifiedas

V =V cosβx+jI Z sinβx(9.19)

and

whereβistheelectricallengthoftheline(radiansorwavelength)

9.7.2Characteristicimpedance

Thequantity isacomplexnumberasyandzarein

complex.

(x) (x)

(x) r r C

ItisdenotedbyZ orZ .Ithasthedimensionofimpedance,since

Thisquantitydependsuponthecharacteristicofthelineperunitlength.Itis,therefore,calledcharacteristicimpedanceoftheline.Italsodependsuponthelengthoftheline,radius,andspacingbetweentheconductors.Foralosslessline,r=g=0,thecharacteristicimpedancebecomes

Thecharacteristicimpedanceisalsocalledthesurgeornaturalimpedanceoftheline.

Theapproximatevalueofthesurgeimpedanceforoverheadlinesis400Ωandthatforundergroundcablesis40Ω,andthetransformershaveseveralthousandohmsastheirsurgeimpedance.Surgeimpedanceistheimpedanceofferedtothepropagationofavoltageorcurrentwaveduringitstravelalongtheline.

9.7.3Surgeimpedanceornaturalloading

Thesurgeimpedanceloading(SIL)ofatransmissionlineistheMWloadingofatransmissionlineatwhichanaturalreactivepowerbalanceoccurs(zeroresistance).

Transmissionlinesproducereactivepower(MVAr)duetotheirnaturalcapacitance.TheamountofMVArproducedisdependentonthetransmissionline’scapacitivereactance(X )andthevoltage(kV)atwhichthelineisenergized.

C 0

C

NowtheMVArproducedis

Transmissionlinesalsoutilizereactivepowertosupporttheirmagneticfields.Themagneticfieldstrengthisdependentonthemagnitudeofthecurrentflowinthelineandthenaturalinductivereactance(X )oftheline.TheamountofMVArusedbyatransmissionlineisafunctionofthecurrentflowandinductivereactance.

TheMVArusedbyatransmissionline=I X (9.22)

TransmissionlineSILissimplytheMWloading(ataunityp.f.)atwhichthelineMVArusageisequaltothelineMVArproduction.Fromtheabovestatement,theSILoccurswhen

AndtheEquation(9.23)canberewrittenas

Theterm isthe‘surgeimpedance’.

Thetheoreticalsignificanceofthesurgeimpedanceisthatifapurelyresistiveloadthatisequaltothesurge

L

L

2

impedancewereconnectedtotheendofatransmissionlinewithnoresistance,avoltagesurgeintroducedtothesendingendofthelinewouldbeabsorbedcompletelyatthereceivingend.Thevoltageatthereceivingendwouldhavethesamemagnitudeasthesending-endvoltageandwouldhaveavoltagephaseanglethatislaggingwithrespecttothesendingendbyanamountequaltothetimerequiredtotravelacrossthelinefromthesendingtothereceivingend.

Theconceptofsurgeimpedanceismorereadilyappliedtotelecommunicationsystemsratherthantopowersystems.However,wecanextendtheconcepttothepowertransferredacrossatransmissionline.Thesurgeimpedanceloading(powertransmittedatthiscondition)orSIL(inMW)isequaltothevoltagesquared(inkV)dividedbythesurgeimpedance(inohms):

Note:Inthisformula,theSILisdependentonlyonthevoltage(kV)ofthelineisenergizedandthelinesurgeimpedance.ThelinelengthisnotafactorintheSILorsurgeimpedancecalculations.Therefore,theSILisnotameasureofatransmissionlinepowertransfercapabilityasitneithertakesintoaccountthelinelengthnorconsidersthestrengthofthelocalpowersystem.

ThevalueoftheSILtoasystemoperatorisrealizedwhenalineisloadedaboveitsSIL,itactslikeashuntreactorabsorbingMVArfromthesystemandwhenalineisloadedbelowitsSIL,itactslikeashuntcapacitorsupplyingMVArtothesystem.

9.8UNCOMPENSATEDLINEWITHOPENCIRCUIT

Inthissection,weshalldiscussthecases:(a)voltageandcurrentprofiles,(b)symmetricallineatno-load,and(c)underexcitedoperationofgenerators.

9.8.1Voltageandcurrentprofiles

Alosslesslineisenergizedatthesendingendandisopen-circuitedatthereceivingend.

FromEquations(9.19)and(9.20)withI =0

∴V(x)=V cosβx(9.25)

Voltageandcurrentatthesendingendaregivenbyequationswithx=las

V =V ,I =I

Equations(9.25)and(9.26)aremodifiedas

V =V cosθ(9.27)

whereθ=βl

FromEquations(9.25)and(9.26),thevoltageprofileequationis

Andthecurrentprofileequation,

9.8.2Thesymmetricallineatno-load

r

r

(x) s (x) (s)

s r

Thisissimilartoanopen-circuitedlineenergizedatoneend.Thisisalineidenticalatbothends,butwithnopowertransfer.Supposetheterminalvoltagesaremaintainedassamevalues,

i.e.,V =V

FromEquations(9.19)and(9.20)withx=l,wehave

V =V cosθ+jZ I sinθ(9.31)

Theelectricalconditionsarethesame(V =V );therewouldnotbeanypowertransfer.Therefore,bysymmetryI =I .

SubstitutingtheaboveconditioninEquations(9.32),weget


FromEquation.(9.34),wehave

s r

s r c r

s r

s r

AcomparisonofEquations(9.35)and(9.36)withEquation(9.28)showsthatthelineisequivalenttotwoequalhalvesconnectedback-to-back.Halftheline-chargingcurrentissuppliedfromeachend.Bysymmetry,themidpointcurrentiszero,whereasthemidpointvoltageisequaltotheopen-circuitvoltageofthelinehavinghalfthetotallength.

FromEquation(9.31)themidpointvoltageis

9.8.3Underexcitedoperationofgeneratorsduetoline-charging

Noloadatthereceivingends,i.e.,I =0.Thechargingreactivepoweratthesendingendis

Q =Im(V I *)


Line-chargingcurrentatthesendingend,

∴Line-chargingpoweratthesendingend,Q =–Ptanθ

Fora300-kmline,Q isnearly43%ofthenaturalloadexpressedinMVA.At400kV,thegeneratorswouldhavetoabsorb172MVAr.

Thereactivepowerabsorptioncapabilityofasynchronousmachineislimitedfortworeasons:

r

s s s

s 0

s

Theheatingoftheendsofthestatorcoreincreasesduringunderexcitedoperation.Thereducedfieldcurrentresultsinreducedinternale.m.fofthemachineandthisweakensthestability.

Usingacompensator,thisproblemcanbereducedbytwoways:

Ifthelineismadeupoftwo(or)moreparallelcircuits,one(or)moreofthecircuitscanbeswitchedoffunderlightload(or)open-circuitconditions.Ifthegeneratorabsorptionislimitedbystabilityandnotbycore-endheating,theabsorptionlimitcanbeincreasedbyusingarapidresponseexcitationsystem,whichrestoresthestabilitymarginswhenthesteady-statefieldcurrentislow.Theunderexcitedoperationofgeneratorscansetamorestringentlimittothemaximumlengthofanuncompensatedlinethantheopen-circuitvoltagerise.

9.9THEUNCOMPENSATEDLINEUNDERLOAD

Inthissection,theeffectsoflinelength,loadpower,andp.f.onvoltageaswellasreactivepowerarediscussed.

9.9.1Radiallinewithfixedsending-endvoltage

Aload(P +jQ )atthereceivingendofaradiallinedrawsthecurrent.

i.e.,

FromEquation(9.19),withx=l,foralosslessline,thesending-andreceiving-endvoltagesarerelatedas

IfV isfixed,thisquadraticequationcanbesolvedforV .ThesolutiongiveshowV varieswiththeloadanditsp.f.aswellaswiththelinelength.

SeveralfundamentalimportantpropertiesofACtransmissionareevidentfromFig.9.9

r r

s

r r

Foreachloadp.f.,thereisamaximumtransmissiblepower.Theloadp.f.hasastronginfluenceonthereceiving-endvoltage.Uncompensatedlinesbetweenabout150-kmand300-kmlongcanbeoperatedatnormalvoltageprovidedthattheloadp.f.ishigh.Longerlines,withlargevoltagevariations,areimpracticalatallp.f.’sunlesssomemeansofvoltagecompensationisprovided.

Themidpointvoltagevariationonasymmetrical300-kmlineisthesameasthereceiving-endvoltagevariationsona150-kmlinewithaunityp.f.load.

9.9.2Reactivepowerrequirements

Fromthelinevoltageandthelevelofpowertransmission,thereactivepowerrequirementscanbedetermined.Itisveryimportanttoknowthereactivepowerrequirementsbecausetheydeterminethereactivepowerratingsofthesynchronousmachinesaswellasthoseofanycompensatingequipment.Ifanyinductiveloadisconnectedatthesendingendoftheline,itwillsupportthesynchronousgeneratorstoabsorbtheline-chargingreactivepower.Withtheabsenceofthecompensatingequipment,thesynchronousmachinesmustabsorborgeneratethedifferencebetweenthelineandthelocalloadreactivepowers.

FIG.9.9Magnitudeofreceiving-endvoltageasafunctionofloadandloadp.f.

Theequationsofvoltageandcurrentforthesending-endhalfofthesymmetricallineis

Thepoweratmidpointis

P +jQ =V I *=P=transmittedpower

SinceQ =0,becausenoreactivepowerflowsatthemidpoint.

Thepoweratthesendingendis

P +jQ =V I *

m m m m

m

s s s s

SubstitutingV andI fromEquations(9.38)and(9.39)intheaboveequation,weget

Ifthelineisassumedtobelossless,thenP =P =P:

Theaboveexpressiongivestherelationbetweenthemidpointvoltageandthereactivepowerrequirementsofthesymmetricalline.

Iftheterminalvoltagesarecontinuouslyadjustedsothatthemidpointvoltage,

V =V =1.0p.u.atalllevelsofpowertransmission

9.9.3Theuncompensatedlineunderloadwithconsiderationofmaximumpowerandstability

ConsiderEquation(9.37)as

IfV istakenasreferencephasor,then:

V =V e =V (cosδ+jsinδ)(9.41)

s s

s r

m 0

r

s s s

jδ

whereδisthephaseanglebetweenV andV andiscalledtheloadangle(or)thetransmissionangle.

EquatingrealandimaginarypartsofEquations(9.40)and(9.41),weget

Foranelectricallyshortline,sinθ=θ=βl:

Then, theseriesreactanceofthe

line:

where

9.10COMPENSATEDTRANSMISSIONLINES

Thechangeintheelectricalcharacteristicsofatransmissionlineinordertoincreaseitspowertransmissioncapabilityisknownaslinecompensation.Whilesatisfyingtherequirementsforatransmissionsystem(i.e.,synchronism,voltagesmustbekeptnear

s r

theirratedvalues,etc.),acompensationsystemideallyperformsthefollowingfunctions:

Itprovidestheflatvoltageprofileatalllevelsofpowertransmission.Itimprovesthestabilitybyincreasingthemaximumtransmissioncapacity.Itmeetsthereactivepowerrequirementsofthetransmissionsystemeconomically.

Thefollowingtypesofcompensationsaregenerallyusedfortransmissionlines:

1. Virtual-Z .

2. Virtual-θ.3. Compensationbysectioning.

Theeffectivenessofacompensatedsystemisgaugedbytheproductofthelinelengthandmaximumtransmissionpowercapacity.Compensatedlinesenablethetransmissionofthenaturalloadoverlargerdistances,andshortercompensatedlinescancarryloadsmorethanthenaturalload.

TheflatvoltageprofilecanbeachievediftheeffectivesurgeimpedanceofthelineismodifiedastoavirtualvalueZ ′,forwhichthecorrespondingvirtualnatural

load isequaltotheactualload.Thesurge

impedanceoftheuncompensatedlineis ,which

canbewrittenas ,iftheseriesand/ortheshuntreactanceX and/orX aremodified,respectively.Then,thelinecanbemadetohavevirtualsurgeimpedanceZ ′andavirtualnaturalloadP’forwhich

0

0

L c

0

Compensationofline,bywhichtheuncompensatedsurgeimpedanceZ ismodifiedtovirtualsurgeimpedanceZ ′,iscalledvirtualsurgeimpedancecompensationorvirtualZ compensation.

OncealineiscomputedforZ ,theonlywaytoimprovestabilityistoreducetheeffectivevalueofθ.Twoalternativecompensationstrategieshavebeendevelopedtoachievethis.

ApplyseriescapacitorstoreduceX andtherebyreduceθ,sinceθ=βl

= atfundamentalfrequency.Thismethodiscalled

line-lengthcompensation(or)θcompensation.Dividethelineintoshortersectionsthataremore(or)lessindependentofoneanother.Thismethodiscalledcompensationbysectioning.Itisachievedbyconnectingconstantvoltagecompensationsatintervalsalongtheline.

9.11SUB-SYNCHRONOUSRESONANCE

Inthissection,variouseffectsduetosub-synchronousresonancearediscussedindetail.

9.11.1Effectsofseriesandshuntcompensationoflines

Theobjectiveofseriescompensationistocancelpartoftheseriesinductivereactanceofthelineusingseriescapacitors,whichresultsinthefollowingfactors.

1. Increaseinmaximumtransferablepowercapacity.2. Decreaseintransmissionangleforconsiderableamountofpower

transfer.3. Increaseinvirtualsurgeimpedanceloading.

Fromapracticalpointofview,itisdesirablenottoexceedseriescompensationbeyond80%.Ifthelineiscompensatedat100%,thelinebehavesasapurelyresistiveelementandwouldresultinseriesresonanceevenatfundamentalfrequencysincethecapacitivereactanceequalstheinductivereactance,anditwouldbedifficulttocontrolvoltagesandcurrentsduring

0

0

0

0

L

disturbances.Evenasmalldisturbanceintherotoranglesoftheterminalsynchronousmachinewouldresultinflowoflargecurrents.

Thelocationofseriescapacitorsisdecidedpartlybyeconomicalfactorsandpartlybytheselectivityoffaultcurrentsastheywoulddependuponthelocationoftheseriescapacitor.Thevoltageratingofthecapacitorwilldependuponthemaximumfaultcurrentthatlikelyflowsthroughthecapacitor.

Thenetinductivereactanceofthelinebecomes

X =X −X

Theconnectionofthetransmissionlineandtheseriescapacitorbehaveslikeaseriesresonancecircuitwithinductivereactanceoflineinserieswiththecapacitanceoftheseriescapacitor.

Theeffectsofseriesandshuntcompensationofoverheadtransmissionlinesareasfollows:

Forafixeddegreeofseriescompensation,thecapacitiveshuntcompensationdecreasesthevirtualsurgeimpedanceloadingoftheline.However,theinductiveshuntcompensationincreasesthevirtualsurgeimpedanceanddecreasesthevirtualsurgeimpedanceloadingoftheline.Iftheinductiveshuntcompensationis100%,thenthevirtualsurgeimpedancebecomesinfiniteandtheloadingiszero,whichimpliesthataflatvoltageprofileexistsatzeroloadsandtheFerrantieffectcanbeeliminatedbytheuseofshuntreactors.Underaheavy-loadcondition,theflatvoltageprofilecanbeobtainedbyusingshuntcapacitors.Aflatvoltageprofilecanbeobtainedbyseriescompensationforheavyloadingcondition.Voltagecontrolusingseriescapacitorsisnotrecommendedduetothelumpednatureofseriescapacitors,butnormallytheyarepreferredforimprovingthestabilityofthesystem.Seriescompensationhasnoeffectontheload-reactivepowerrequirementsofthegeneratorand,therefore,theseries-compensatedlinegeneratesasmuchline-chargingreactivepoweratnoloadascompletelyuncompensatedlineofthesamelength.Ifthelengthofthelineislargeandneedsseriescompensationfromthestabilitypointofview,thegeneratoratthetwoendswillhavetoabsorbanexcessivereactivepowerand,therefore,itisimportantthattheshuntcompensation(inductive)mustbeassociatedwithseriescompensation.

lnet l sc

9.11.2ConceptofSSRinlines

Consideratransmissionlinecompensatedbyaseriescapacitorsconnection.

LetL ,L ,andL betheinductanceoftheline,generator,andtransformer,respectively.LetC bethecapacitanceoftheseriescapacitor,X thetotalinductivereactance(X +X +X ),andX thereactanceoftheseriescapacitor.

Thenaturalfrequencyofoscillationoftheabove-saidseriesresonancecircuitisgivenbytherelation

TheinductivereactanceofthesystemisX =2πfL

Capacitivereactanceofthecapacitoris

Therefore,thenaturalfrequencyofoscillationintermsofX andX isexpressedas

wherefistheratedfrequency.

Theterm representsthedegreeofseries

compensationanditvariesbetween25%and65%;therefore,thenaturalfrequencyofoscillationbecomeslesswhencomparedtothenaturalfrequency(f <f),i.e.,

L g t

sc

L

L g t sc

l

l c

0

theseriesresonancewilloccuratsub-synchronousfrequency.

Therearethreetypesofsub-synchronousoscillationsthathavebeenidentifiedduetoSSRconditions.

9.11.2.1SSRduetoinductiongeneratoreffect

Thetransientcurrentsatthesub-harmonicfrequencyresultedinaseries-compensatednetworkduetoaswitchingoperationorafault.Thesesub-harmonicfrequencycurrentsassumedangerouslyhighvaluesandevenbecomeunstable.Theunstableoperationisexhibitedintheformofanegativeresistanceintheequivalentcircuitofsynchronousandinductionmotors.

Consideringaround-rotorsynchronousmachine,thesub-harmonicfrequencyoperationcanbestudiedwiththehelpofitsequivalentcircuitasshowninFig.9.10.

FIG.9.10Equivalentcircuitofasynchronousmachineforsub-harmonicoperation

Thesub-harmoniccurrentsareexcitedinthestatorwindingofthesynchronousmachineduetosomedisturbancesandthesesub-harmoniccurrentswouldgenerallybeunbalanced.

Thepositivesequencecomponentoftheseunbalancedsub-harmoniccurrentswillproduceamagneticfield,whichrotatesinthesamedirectionofrotationoftherotorbutwithaspeedN<N .Themachinebehavesasaninductiongeneratorasfarassub-harmoniccurrentsareconsidered.Duetothisspeeddeviationbetweenthe

s

rotorandthemagneticfield,theslip willbe

present.Sincef <f,theslipSbecomesnegative.

Therefore,theequivalentresistanceofthedamperwindingandthesolidrotorresistancewhenreferredto

thestatorside,i.e., becomesnegativeandtherefore

providesnegativedamping.

Iftheseriescompensationisveryhigh,theslipSwouldturnouttobeverysmallandhencetheequivalentresistancebecomesveryhighandmaybecomelargeenoughtohavetotalresistanceofthesystem,whichisnegative.Therefore,itprovidesnegativedampingofthesub-harmoniccurrents,andvoltagemaybuilduptodangerouslyhighvalues.SeveralmeasuresaretobetakentopreventSSRinthesystem.

9.11.2.2SSRduetotorsionalinteractionbetweenelectricalandmechanicalsystems

Thesub-harmoniccurrentsproducefieldrotationsinthedirectionwithrespecttotherotorandmainfieldandwhichinturnproducesanalternatingtorqueontherotoratthefrequency(f–f ).Ifthisfrequencydifferencecoincideswithoneofthenaturaltensionalresonancesofthemachineshaftsystem,tensionaloscillationsmaybeexcitedandthisoperationisentirelyknownasSSR.Itmeansthatwheneverthenaturalfrequencyofthemechanicaloscillationoftherotorequals(f–f ),mechanicalresonancewouldtakeplace.Hence,SSRistreatedasacombinedelectrical–mechanicalresonancephenomenon.

Thecurrentsofhighfrequencymayproducetorqueinsomeoftheshafts,whichmayhavethesamenaturalfrequencyasthetorquefrequency(calledswing

0

0

0

frequency,whichisthelowestfrequencyofnaturaloscillationofanequivalentsystemofaturbinecylinderandagenerator)suchthattheseshaftsmaybreakdownduetothetwistingaction.Hence,resonantfrequenciesmaybeextendeduptoseveralhundredsofHz.Thelargemultiple-stagesteamturbinesthathavemorethanonetensionalmodesinthefrequencyrangeof0.5HzaremoresusceptibletoSSR.

IntheSSRphenomenon,iftheresonancefrequencycoincideswiththeswingfrequency,thewholeturbine-generatorassemblymaycomeoutfromitsfoundation,and/orifthefrequencyofthetorquedevelopedcoincideswiththenaturalfrequencyofoscillationofsomeshaftsandifoscillationsbuildupsufficiently,itresultsinthebreakingoftheshaft.

9.11.2.3SSRduetolargedisturbances

Duetothelargedisturbanceslikeanyswitchingoperationoranyfaultcondition,theconditionofSSRoccursinthesystemevenifoscillationsaredampedout.ThisSSRconditionresultsina‘lowcyclefatigue’conditioninamechanicalsystemorslowdeteriorationofthemechanicalsystemduetothereductioninlifeoftheshafts.

ThecorrectivemeasuresforSSRare:

1. Bypassingaseriesofcapacitors.2. Useofverysensitiverelaystodetectevensmalllevelsofsub-

harmoniccurrents.3. Modulationofgeneratorfieldcurrenttoprovideanincreasedpositive

dampingsub-harmonicfrequency.

9.12SHUNTCOMPENSATOR

Ashunt-connectedstaticVArcompensator,composedofthyristor-switchedcapacitors(TSCs)andthyristor-controlledreactors(TCRs),isshownFig.9.11.Withproperco-ordinationofthecapacitorswitchingandreactorcontrol,theVAroutputcanbevariedcontinuouslybetweenthecapacitiveandinductiverating

oftheequipment.Thecompensatorisnormallyoperatedtoregulatethevoltageofthetransmissionsystemataselectedterminal,oftenwithanappropriatemodulationoptiontoprovidedampingifpoweroscillationisdetected.

FIG.9.11StaticVArcompensatoremployingTSCsandTCR

9.12.1Thyristor-controlledreactor

Ashunt-connectedthyristor-controlledinductorhasaneffectivereactance,whichisvariedinacontinuousmannerbypartial-conductioncontrolofthethyristorvalve.

Withtheincreaseinthesizeandcomplexityofapowersystem,fastreactivepowercompensationhasbecomenecessaryinordertomaintainthestabilityofthesystem.Thethyristor-controlledshuntreactorshavemadeitpossibletoreducetheresponsetimetoafewmilliseconds.Thus,thereactivepowercompensator

utilizingthethyristor-controlledshuntreactorsbecomepopular.Anelementarysingle-phaseTCRisshownFig.9.12.

ItconsistsofafixedreactorofinductanceLandabidirectionalthyristorvalve.Thethyristorvalvecanbebroughtintoconductionbytheapplicationofagatepulsetothethyristor,andthevalvewillbeautomaticallyblockedimmediatelyaftertheACcurrentcrosseszero.Thecurrentinthereactorcanbecontrolledfrommaximumtozerobythemethodoffiringanglecontrol.Partialconductionisobtainedwithahighervalueoffiringangledelay.Theeffectofincreasingthegatingangleistoreducethefundamentalcomponentofcurrent.Thisisequivalenttoanincreaseintheinductanceofthereactor,reducingitscurrent.Asfarasthefundamentalcomponentofcurrentisconcerned,theTCRisacontrollablesusceptance,andcan,therefore,beusedasastaticcompensator.

Thecurrentinthiscircuitisessentiallyreactive,laggingthevoltageby90°andthisiscontinuouslycontrolledbythephasecontrolofthethyristors.Theconductionanglecontrolresultsinanon-sinusoidalcurrentwaveforminthereactor.Inotherwords,theTCRgeneratesharmonics.Foridenticalpositiveandnegativecurrenthalf-cycletime,onlyoddharmonicsaregeneratedasshowninFig.9.13.Byusingfilters,wecanreducethemagnitudeofharmonics.

TCR’scharacteristicsare:

Continuouscontrol.Notransients.Generationofharmonics.

FIG.9.12TCR

FIG.9.13TCRwaveform

9.12.2Thyristor-switchedcapacitor

Ashunt-connectedTSCshowsthataneffectivereactanceisvariedinastep-wisemannerbyfull-orzero-conductionoperationofthethyristorvalve.

TheTSCisalsoasub-setofSVCinwhichthyristor-basedACswitchesareusedtoswitchinandswitchoutshuntcapacitorsunitsinordertoachievetherequiredstepchangeinthereactivepowersuppliedtothesystem.Unlikeshuntreactors,shuntcapacitorscannotbeswitchedcontinuouslywithavariablefiringanglecontrol.

DependingonthetotalVArrequirement,anumberofcapacitorsareused,whichcanbeswitchedintooroutof

thesystemindividually.ThecontrolisdonecontinuouslybysensingtheloadVArs.Asingle-phaseTSCisshowninFig.9.14.

Itconsistsofacapacitor,abidirectionalthyristorvalve,andrelativelysmallsurgecurrentinthethyristorvalveunderabnormaloperatingconditions(e.g.,controlmalfunctioncausingcapacitorswitchingata‘wrongtime’);itmayalsobeusedtoavoidresonancewithsystemimpendenceatparticularfrequencies.Theproblemofachievingtransient-freeswitchingofthecapacitorsisovercomebykeepingthecapacitorschargedtothepositiveornegativepeakvalueofthefundamentalfrequencynetworkvoltageatalltimeswhentheyareinthestand-bystate.Theswitching-on-transientisthenselectedatthetimewhenthesamepolarityexistsinthecapacitorvoltage.Thisensuresthatswitchingontakesplaceatthenaturalzeropassageofthecapacitorcurrent.Theswitchingthustakesplacewithpracticallynotransients.Thisiscalledzero-currentswitching.

FIG.9.14TSC

FIG.9.15TSCwaveforms

Switchingoffacapacitorisaccomplishedbysuppression-offeringpulsestotheanti-parallelthyristorssothatthethyristorswillswitchoffassoonasthecurrentbecomeszero.Inprinciple,thecapacitorwillthenremainchargedtothepositiveornegativepeakvoltageandbepreparedforanewtransient-freeswitching-onasshowninFig.9.15.

TSC’scharacteristicsare:

Steepedcontrol.Notransients.Noharmonics.Lowlosses.Redundancyandflexibility.

9.13SERIESCOMPENSATOR

IntheTSCscheme,increasingthenumberofcapacitorbanksinseries,controlsthedegreeofseriescompensation.Toaccomplishthis,eachcapacitorbankiscontrolledbyathyristorbypassswitchorvalve.Theoperationofthethyristorswitchesisco-ordinatedwithvoltageandcurrentzero-crossing;thethyristorswitchcanbeturnedontobypassthecapacitorbankwhentheappliedACvoltagecrosseszero,anditsturn-offhastobeinitiatedpriortoacurrentzeroatwhichitcanrecoveritsvoltage-blockingcapabilitytoactivatethecapacitor

bank.Initially,capacitorischargedtosomevoltage,whileswitchingtheSCR’s,theymaygetdamagedbecauseofthisinitialvoltage.InordertoprotecttheSCR’sfromthiskindofdamage,resistorisconnectedinserieswithcapacitorasshowninFig.9.16.

Inafixedcapacitor,theTCRschemeasshowninFigs.9.17and9.18,thedegreeofseriescompensationinthecapacitiveoperatingregionisincreased(ordecreased)byincreasing(ordecreasing)thecurrentintheTCR.MinimumseriescompensationisreachedwhentheTCRisswitchedoff.TheTCRmaybedesignedforasubstantiallyhighermaximumadmittanceatfullthyristorconductionthanthatofthefixedshunt-connectedcapacitor.Inthiscase,theTCR,timewithanappropriatesurge-currentratingcanbeusedessentiallyasabypassswitchtolimitthevoltageacrossthecapacitorduringfaultsandthesystemcontingenciesofsimilareffects.

FIG.9.16Seriescompensator

FIG.9.17Thyristor-controlledcapacitor

FIG.9.18TCR

Controllableseriescompensationcanbehighlyeffectiveindampingpoweroscillationandpreventingloopflowsofpower.

Theexpressionforpowertransferredisgivenby

whereV isthesending-endvoltage,V thereceiving-endvoltage,δtheanglebetweenV andV ,andX=X –X .

s r

s r L C

Ininterconnectedpowersystems,theactualtransferofpowerfromoneregiontoanothermighttakeunintendedroutesdependingonimpedancesoftransmissionlinesconnectingtheareas.Controlledseriescompensationisausefulmeansforoptimizingpowerflowbetweenregionsforvaryingloadingandnetworkconfigurations.Itbecomespossibletocontrolpowerflowsinordertoachieveanumberofgoalsthatarelistedbelow:

Minimizingsystemlosses.Reductionofloopflows.Eliminationoflineoverloads.Optimizingloadsharingbetweenparallelcircuits.Directingpowerflowsalongcontractualpaths.

9.14UNIFIEDPOWER-FLOWCONTROLLER

IntheUPFC,anACvoltagegeneratedbyathyristor-basedinverterisinjectedinserieswiththephasevoltage.InFig.9.19,Converter-2performsthemainfunctionoftheUPFCbyinjecting,via.,aseriestransformer,anACvoltagewithcontrollablemagnitudeandaphaseangleinserieswiththetransmissionline.ThebasicfunctionofConverter-1istosupplyorabsorbtherealpowerdemandedbyConverter-2atthecommonDClink.Itcanalsogenerateorabsorbcontrollablereactivepowerandprovideanindependentshunt-reactivecompensationfortheline.Converter-2eithersuppliesorabsorbstherequiredreactivepowerlocallyandexchangestheactivepowerasaresultoftheseriesinjectionvoltage.

FIG.9.19TheUPFC

FIG.9.20ImplementationoftheUPFCusingtwovoltagesourceinverterswithadirectvoltagelinksimultaneously

Generally,theimpedancecontrolwouldcostlessandbemoreeffectivethanthephaseanglecontrol,exceptwherethephaseangleisverysmallorverylargeorvarieswidely.

Ingeneral,ithasthreecontrolvariablesandcanbeoperatedindifferentmodes.Theshunt-connected

converterregulatesthevoltagebus‘i’inFig.9.20andtheseries-connectedconverterregulatestheactiveandreactivepoweroractivepowerandthevoltageattheseries-connectednode.Inprinciple,aUPFCisabletoperformthefunctionsoftheotherFACTSdevices,whichhavebeendescribed,namelyvoltagesupport,power-flowcontrol,andanimprovedstability.

9.15BASICRELATIONSHIPFORPOWER-FLOWCONTROL

Thebasicconceptofcontrollingpowertransmissioninrealtimeassumestheavailablemeansforrapidlychangingthoseparametersofthepowersystem,whichdeterminethepowerflow.Toconsiderthepossibilitiesforapower-flowcontrol,powerrelationshipsforthesimpletwo-machinemodelareshowninFigs.9.21and9.22.

Figure9.22showsthesending-andreceiving-endgeneratorswithvoltagephasorsV andV ,inductive

transmissionlineimpedance(X )intwosections ,and

generalizedpower-flowcontrolleroperated(forconvenience)atthemiddleoftheline.Thepower-flowcontrollerconsistsoftwocontrollableelements,i.e.,avoltagesource(V )andacurrentsource(I )areconnectedinseriesandshunt,respectively,withthelineatthemidpoint.BoththemagnitudeandtheangleofthevoltageV arefreelyvariables,whereasonlythemagnitudeofcurrentI isvariable;itsphaseangleisfixedat90°withrespecttothereferencephasorofmidpointvoltageV .Thebasicpower-flowrelationisshowninFig.9.22byusingFACTScontrollerinanormaltransmissionsystem.

S r

L

xy x

xy

x

m

FIG.9.21Simpletwo-machinepowersystemwithageneralizedpower-flowcontroller

FIG.9.22Power-flowrelation

Thefourclassicalcasesofpowertransmissionareasfollows:

1. Withoutlinecompensation.2. Withseriescapacitivecompensation.3. Withshuntcompensation.4. Withphaseanglecontrol,

TheycanbeobtainedbyappropriatelyspecifyingV andI inthegeneralizedpower-flowcontroller.

9.15.1Withoutlinecompensation

Considerthatthepower-flowcontrollerisoff,i.e.,bothV andI arezero.Then,thepowertransmittedbetweenthesending-andreceiving-endgeneratorscanbeexpressedbythewell-knownformula:

xy

x

xy x

whereδistheanglebetweenthesending-andreceiving-endvoltagephasors.PowerP isplottedagainstangleδinFig.9.23.

FIG.9.23Thebasicpowertransmissiononcharacteristicsforfourdifferentcases

9.15.2Withseriescapacitivecompensation

Assumeparallelcurrentsource,I =0,andseriesvoltagesource,V =–jnX I,i.e.,thevoltageinsertedinserieswiththelinelagsthelinecurrentby90°withanamplitudethatisproportionaltothemagnitudeofthelinecurrentandthatofthelineimpedance.Inotherwords,thevoltagesourceactsatthefundamental

(1)

x

xy L

frequencypreciselyasaseries-compensatingcapacitor.Thedegreeofseriescompensatingisdefinedbycoefficientn(i.e.,0≤n≤1).Withthis,Pagainstδrelationshipbecomes

9.15.3Withshuntcompensation

ConsiderthatseriesvoltagesourceV =0andparallel

currentsource i.e.,thecurrentsource

I drawsjustenoughcapacitivecurrenttomakethemagnitudeofthemidpointvoltageV equaltoV.Inotherwords,thereactivecurrentsourceactslikeanidealshuntcompensator,whichsegmentsthetransmissionlineintotwoindependentparts,eachwithanimpedance

of bygeneratingthereactivepowernecessarytokeep

themidpointvoltageconstant,independentlyofangleδ.Forthiscaseofidealmidpointcompensation,thePagainstδrelationcanbewrittenas

9.15.4Withphaseanglecontrol

AssumethatI =0andV =±jV tanα,i.e.,avoltage(V )withtheamplitude±jV tanα,isaddedinquadraturetothemidpointvoltage(V )toproducethedesiredαphaseshift.Thebasicideabehindthephaseshifteristokeepthetransmittedpoweratadesiredlevelindependentlyofangleδinapre-determinedoperating

xy

x

m

x xy m

xy m

m

range.Thus,forexample,thepowercanbekeptatitspeakvalueafterangleδisπ/2bycontrollingtheamplitudeofthequadraturevoltageV sothattheeffectivephaseangle(δ–α)betweenthesending-andreceiving-endvoltagesstaysatπ/2.Inthisway,theactualtransmittedpowermaybeincreasedsignificantlyeventhoughthephaseshifterdoesnotincreasethesteady-statepowertransmissionlimit.Considering(δ–α)astheeffectivephaseanglebetweenthesending-endandthereceiving-endvoltage,thetransmittedpowercanbeexpressedas

fromFig.9.23,itcanbeseenthatthepowerinwithoutcompensatingislessshownintheP curve.PowerisincreasedbyusingtheseriescapacitorcompensationshownintheP curve.ThepoweranglecurvewiththeshuntcompensatorisshownintheP curve,inthiscase,powerisincreasedanditseemsthatvoltageisincreased.TheconceptofthephaseanglecontrolisshownintheP curvebyshiftingthecurvehigher,andpowercanbeobtained.

9.16COMPARISONOFDIFFERENTTYPESOFCOMPENSATINGEQUIPMENTFORTRANSMISSIONSYSTEMS

Thecomparisonamongdifferenttypesofcompensatingequipmentfortransmissionsystemsistabulatedbelow(Table9.1).

TABLE9.1Comparisonofdifferenttypesofcompensatingequipmentfortransmissionsystems

Compensatingequipment

Advantages Disadvantages

Switchedshuntreactor Simpleinprincipleandconstruction

Fixedinvalue

xy

(1)

(2)

(3)

(4)

Switchedshuntcapacitor

Simpleinprincipleandconstruction

Fixedinvalue-switchingtransients.Requiredovervoltageprotectionandsub-harmonicfilters.Limitedoverloadcapacity

Seriescapacitor Simpleinprinciple.Performancerelativelysensitivetolocation.Hasusefuloverloadcapability

High-maintenancerequirements.Slowcontrolresponse

Compensatingequipment

Advantages Disadvantages

Synchronouscondenser Fullycontrollable.Lowharmonics

Performancesensitivetolocation.Requiresstrongfoundations

Polyphase-saturatedreactor

Veryruggedconstruction.Largeoverloadcapability.Noeffectonfaultlevel.Lowharmonics

Essentiallyfixedinvalue.Performancesensitivetolocationandnoisy

TCR Fastresponse.Fullycontrollable.Noeffectonfaultlevel.Canberapidlyrepairedafterfailures

Generatorharmonicsperformancesensitivetolocation

TSC Canberapidlyrepairedafterfailures.Noharmonics

Noinherentabsorbingcapabilitytolimitovervoltages.Complexbusworkandcontrolslow-frequencyresonancewithsystem.Performancesensitivetolocation

9.17VOLTAGESTABILITY—WHATISIT?

Voltageinstabilitydoesnotmeantheproblemoflowvoltageinsteady-statecondition.Asamatteroffact,itispossiblethatthevoltagecollapsemaybeprecipitatedeveniftheinitialoperatingvoltagesmaybeatacceptablelevels.

Voltagecollapsemaybeeitherfastorslow.FastvoltagecollapseisduetoinductionmotorloadsorHVDCconverterstationsandslowvoltagecollapseisduetoon-loadtapchangerandgeneratorexcitationlimiters.

Voltagestabilityisalsosometimestermedloadstability.Thetermsvoltageinstabilityandvoltagecollapseareoftenusedinterchangeably.

Itistobeunderstoodthatthevoltagestabilityisasub-setofoverallpowersystemstabilityandisadynamicproblem.Thevoltageinstabilitygenerallyresultsinmonotonically(oraperiodically)decreasingvoltages.Sometimes,thevoltageinstabilitymaymanifestasundamped(ornegativelydamped)voltageoscillationspriortovoltagecollapse.

9.17.1Voltagestability

Definition:Apowersystematagivenoperatingstateandsubjectedtoagivendisturbanceisvoltagestableifvoltagesneartheloadsapproachpost-disturbanceequilibriumvalues.Thedisturbedstateiswithintheregionsofattractionsofstablepost-disturbanceequilibrium.

Theconceptofvoltagestabilityisrelatedtothetransientstabilityofapowersystem.

9.17.2Voltagecollapse

Followingvoltageinstability,apowersystemundergoesvoltagecollapseifthepost-disturbanceequilibriumvoltagesneartheloadarebelowacceptablelimits.Thevoltagecollapsemaybeeithertotalorpartial.

Theabsenceofvoltagestabilityleadstovoltageinstabilityandresultsinprogressivedecreaseofvoltages.Whendestabilizingcontrols(suchasOLTC)reachlimitsorduetoothercontrolactions(undervoltageloadshedding),thevoltagesarestabilized(atacceptableorunacceptablelevels).Thus,abnormalvoltagelevelsinthesteadystatemaybetheresultofvoltageinstability,whichisadynamicphenomenon.

9.18VOLTAGE-STABILITYANALYSIS

Thevoltage-stabilityanalysisiscarriedoutbyloadflowmethods,whicharebasicallypost-disturbancepower-flowmethods.Besidesthesemethods,P–VcurvesandQ–Vcurvesaretheotherpower-flow-basedmethodsgenerallyusedforvoltage-stabilityanalysis.Bythesetwomethods,thesteady-stateloadabilitylimitsaredetermined,whicharerelatedtovoltagestability.

9.18.1P–Vcurves

TheconceptualanalysisofvoltagestabilityisusefulcarriedoutbyusingP–Vcurves.Theseareusefulforthestudyofanalysisofradialsystems.

ThismethodisalsoapplicableforalargeinterconnectednetworktowhichthetotalloadconnectedisPandthevoltageofthecriticalbusisV.ThetotalloadPmaybethepowertransferredoveratransmissionline.Thevoltageatseveralbusescanbeplotted.

9.18.1.1InterpretationofP–Vcurves

ConsideraradiallinewithanasynchronousloadisasshowninFig.9.24(a).

TheloadisP +jQ atthereceivingendkeepingthesending-endvoltageV constant.

Eventheradiallineisconnectedwiththeasynchronousload,themaximumpowercanbe

L L

s

transmittedovertheline.

Letusconsiderthattheloadisu.p.floadandthesending-endvoltagesourceandlineformavoltagesourcewithanopen-circuitvoltageV andimpedance(R+jX)andatthereceivingendavariableresistiveloadRisconnectedsuchthatthep.f.isunity(Fig.9.24(b)).

FIG.9.24(a)AradiallineterminatedthroughanasynchronousloadP +

jQ ;(b)equivalentcircuitofFig.9.24withu.p.f.load

Theshort-circuitcurrent,

Short-circuitp.f.,

Thetotalloadcurrent,

Thepowerdeliveredtotheload,

oc

L

L

L

Conditionformaximumpowerdeliveredis

∴R =Z,istheconditionformaximumpowerdelivered

SubstitutingthisconditioninEquation(9.42),wegetthemaximumpoweras

NowV istheopen-circuitvoltage,i.e.,V whenI =0.

Letxbethedistancefromthesendingendandlbethelengthoftheline

L

oc r r

Foralosslessline,r=0andg=0,thenthevoltageatdistancexfromthesendingendbecomes

V =V cosβ(l−x)+jZ I sinβ(l−x)

whereβisthephaseconstant

Supposethelineisopencircuitedatthereceivingend,i.e.,I =0,

Similarly,theshort-circuitcurrentI isthevalueofIwhenV =0

Assumingthelinetobelossless,cosϕ =0

Equation(9.43)representslociofmaximumpowerfordifferentlinelengthsatunityp.f.

Thereceiving-endcurrent,

Thesending-endvoltageoftheline,ifassumingthelinetobelossless,nowbecomes

(x) r c r

r

sc r

r

sc

Forafixedsending-endvoltageV andthefixedlinelength,Equation(9.44)isquadraticinV andthuswillhavetworoots.Figure9.25showsagraphicalrelation

between asafunctionofnormalizedloading .

FromFig.9.25,itisobservedthatthemaximumpowercanbetransmittedforeachloadp.f.andforanyloading,therearetwodifferentvaluesofV .

Thenormaloperationofthepowersystemisalongtheupperpartofthecurvewherethereceiving-endvoltageisnearly1.0p.u.Theloadisincreasedbydecreasingtheeffectiveresistanceoftheloaduptothemaximumpower;theproductofloadvoltageandcurrentincreasesasthesystemisstable.Asthepointofmaximumpowerisreached,afurtherreductionineffectiveloadresistancereducesthevoltagemorethantheincreaseincurrentandtherefore,thereisaneffectivereductioninpowertransmission.Thevoltagefinallycollapsestozeroandthesystematthereceivingendiseffectivelyshort-circuitedandtherefore,thepowertransmittediszero(pointatorgininFig.9.25).

ItisobservedfromFig.9.25thatthepowertransmittediszerobothatPointKandPoint0.PointKcorrespondingtotheopencircuitandPoint0correspondingtotheshortcircuitandineithercasethepowertransmittediszero.

s

r

r

Figure9.25showstherelationbetween asafunction

ofnormalizedloading ,whereP isthesurge

impedanceload.Theserelationcurvesbetween and

areknownasnormalizedP–Vcurves.TheseP–Vcurvesaredifferentfordifferentp.f.’s.Atmoreleadingp.f.’s,themaximumpowerishigherandforthattheshuntcompensationisprovided.ThenosevoltageoftheP–Vcurvehasthecriticalvoltageatthereceivingendformaximumpowertransfer.Withleadingp.f.,thecriticalvoltageishigher,whichisaveryimportantaspectofvoltagestability.

FIG.9.25 asafunctionofnormalizedloading

c

Themaindisadvantageoftheload-flowsolutionforP–Vcurvesisthatitislikelytodivergenearthemaximumpower-transferpointorthenosepointoftheP–Vcurve.Aload-flowsolutionisatvariousP–Vbusesorgeneratorsbusesforparticulargenerations.However,whentheloadchanges,theschedulingofgenerationatvariousgeneratorbusesalsochanges.Thisisyetanotherdisadvantageoftheload-flowsolutionmethod.

9.18.2Conceptofvoltagecollapseproximateindicator

ConsiderthatapurelyinductiveloadisconnectedtoasourcethroughalosslesslineasshowninFig.9.26.

Here,P=0andδ=0sincetheloadispurelyinductive;therefore,thereactivepoweratthereceiving

endisexpressedas .

Theconditionformaximumreactivepowertransferis

.

FIG.9.26Radiallineconnectedwithpurelyinductiveloadtosource

whereV isthereceiving-endvoltageformaximumreactivepowertransfer.

Therefore,themaximumreactivepowerisexpressedas

Ashalfofthedropwillbeacrossthelineandanother

halfacrosstheload,X =X;hence, i.e.,the

maximumreactivepoweristransferredwhentheloadreactanceisequaltothelinereactanceorthesourcereactance.

Theshort-circuitreactivepowerofthelineis ;

hence,thenormalizedmaximumreactivepoweris

Also

NowQ =Q +I X

critical

load

s r

2

Since

Hence,

MultiplyingbothsidesofEquation(9.45)by ,weget

Now

Thedifferentiationofthesending-endreactivepowerQ withrespecttothereceiving-endreactivepowerQ ,s r

i.e., ,isknownasthevoltagecollapseproximate

indicator(VCPI)ofaradialline.

Thereceiving-endvoltagevariesfromV atnoloadto

atmaximumloadQ .However,VCPIisunityatno

loadsinceatnoloadQ =0and andit

isinfinityatmaximumloadsinceatthisloadQ =Q

andhence

Itisclearthatnearthemaximumload,anextremelylargeamountofreactivepowerisrequiredatthesendingendtosupplyanincrementalincreaseinload.Thus,VCPIisaverysensitiveindicatorofimpedingvoltagecollapse.There,latedquantitiesreactivereserveactivationandreactivelossesarealsosensitiveindicators.

9.18.3Voltage-stabilityanalysis:Q–Vcurves

Q–VcurvescanbeobtainedfromthenormalizedP–Vcurves.

Let

Forconstantvaluesofp,wenotethetwopairsofvaluesofvandqforeachp.f.andreportthesevalues.TheresultoftheseplotsisshowninFig.9.27.

s

max

r

r rmax

FIG.9.27NormalizedQ–Vcurvesforfixedsourceandreactivenetworkloadsareatconstantpower

Thecriticalvoltageishighforloadings(i.e.,visabove1p.u.forp=1p.u.).Theright-handsideofthecurvesindicatesthenormaloperatingconditionswheretheapplicationofshuntcapacitorsraisesthevoltage.Thesteep-slopedlinearportionsoftherightsideofthecurveareequivalenttothefigurebelow(rotateFig.9.28clock-wiseby90°).

FIG.9.28Systemapproximatevoltage-reactivepowercharacteristic

TheQ–Vcurvesforlargesystemsareobtainedbyaseriesofpower-flowsimulations.Q–Vcurvesresultfromtheplotofvoltageatatestorcriticalbusversusreactivepoweronthesamebus.Generally,theP–Vbusorgeneratorbushasreactivepowerconstraintsforaload-flowsolution.Afictitioussynchronouscondenserisrepresentedatthetestbusandallowsthebustohaveanyreactivepowerforafixedpandv.ThevalueofvatthebuschangesforobtaininganotherpointontheQ–Vcurve,andobtainsreactivepowerflowfordifferentscheduledvoltagesatthebus.Scheduledvoltageatthebusisanindependentvariableandformsanabscissavariable.Thecapacitivereactivepowerrequiredtomaintainthescheduledvoltageatthebusisadependentvariableandisplottedinthepositiveverticaldirection.Withouttheapplicationofshuntreactivecompensationatthetestbus,theoperatingpointisatthezeroreactivepointcorrespondingtotheremovalofthefictitioussynchronouscondenser.

9.19DERIVATIONFORVOLTAGE-STABILITYINDEX

Consideratypicalbranchconsistingofsending-andreceiving-endbusesasshowninFig.9.29.

Currentflowingthroughthebranch,

Therealtermoftheaboveequationis

V V cos(δ −δ )=V +(RP+XQ)

andtheimaginarypartis

V V sin(δ −δ )=XP−RQ

Squaringandaddingtheabovetwoterms,weget

FIG.9.29Single-linemodeloftypicalbranch

TheaboveequationisaquadraticequationofV .ThesystemistobestableifV ≥0.

Itispossiblewhen

b −4ac≥0

i.e.,[2(RP+XQ)−V )] −4(P +Q )(R +X )≥0

or4R p +4X Q +4R×PQ+V −4V (RP+XQ)

S R 1 2 R

S R 1 2

R

R

S

S S

2

2

2

2

2 2 2 2 2 2

2 2 2 2 4 2

−4R p −4X p −4R Q −4X Q ≥0

Simplifyingtheaboveequation,weget

V −4V (RP−XQ)−4(PX−RQ) ≥0

or4(PX−RQ) +4V (RP+XQ)≤V

DividingbothsidesoftheaboveequationbyV ,weget

whereL=stabilityindex

Forstablesystems,L≤1.

Example9.1:A440V,3-ϕdistributionfeederhasaloadof100kWatlaggingp.f.withtheloadcurrentof200A.Ifthep.f.istobeimproved,determinethefollowing:

1. uncorrectedp.f.andreactiveloadand2. newcorrectedp.f.afterinstallingashuntcapacitorof75kVAr.

Solution:

1. Uncorrectedp.f.

2. Correctedp.f.

2 2 2 2 2 2 2 2

S S

S S

S

4 2 2

2 2 4

4

Example9.2:Asynchronousmotorhavingapowerconsumptionof50kWisconnectedinparallelwithaloadof200kWhavingalaggingp.f.of0.8.Ifthecombinedloadhasap.f.of0.9,whatisthevalueofleadingreactivekVAsuppliedbythemotorandatwhatp.f.isitworking?

Solution:

Let:

p.f.angleofmotor=ϕ

p.f.angleofload=ϕ =cos (0.8)=36.87°

Combinedp.f.angle(bothmotorandload),ϕ=cos(0.9)=25.84°

tanϕ =tan36°87′=0.75;tanϕ=tan25°84′=0.4842

Combinedpower,P=200+50=250kW

TotalkVArofacombinedsystem=Ptanϕ =250×0.4842=121.05

LoadkVAr=200×tanϕ =200×0.75=150

∴LeadingkVArsuppliedbysynchronousmotor=150–121.05=28.95

p.f.angleatwhichthemotorisworking,ϕ =tan28.95/50=30.07°

p.f.atwhichthemotorisworking=cosϕ =0.865(lead)

Example9.3:A3-ϕ,5-kWinductionmotorhasap.f.of0.85lagging.Abankofcapacitorisconnectedindeltaacrossthesupplyterminalandp.f.raisedto0.95lagging.DeterminethekVArratingofthecapacitorineachphase.

Solution:

Theactivepoweroftheinductionmotor,P=5kW

1

2

2

1

2

1

1

–1

–

1

–1

Whenthep.f.ischangedfrom0.85lagto0.95lagbyconnectingacondenserbank,theleadingkVArtakenbythecondenserbank=P(tanϕ –tanϕ )

=5(0.6197–0.3287)=1.455

∴Ratingofcapacitorconnectedineachphase=1.455/3=0.485kVAr

Example9.4:A400V,50Hz,3-ϕsupplydelivers200kWat0.7p.f.lagging.Itisdesiredtobringthelinep.f.to0.9byinstallingshuntcapacitors(Fig.9.30).Calculatethecapacitanceiftheyare:(a)starconnectedand(b)deltaconnected.

Solution:

1. Forstarconnection:

Phasevoltage,V =400/ =230.94V/ph

Loadcurrent,

Theactivecomponentofcurrent,I =Icosϕ =412.39×0.7=288.68A

Reactivecomponentofcurrent,

FIG.9.30Phasordiagram

Forafixedload,letustake:

2 1

ph

a 1

Thereactivecomponentofloadcurrentwithoutcapacitor=I tanϕ

Thereactivecomponentofloadcurrentwithcapacitor=I tanϕ

Currenttakenbythecapacitorinstalledforimprovingp.f.,I =I (tan

ϕ –tanϕ )

=288.68(tan(cos 0.7)−tan(cos 0.9))

=154.7A

Thevalueofcapacitortobeconnected,

2. Fordeltaconnection:

Phasevoltage,V =400V

Loadcurrent,

Phasecurrent,

Theactivecomponentofphasecurrent,I =Icosϕ =238.09×0.7=166.663A

Currenttakenbythecapacitorinstalledforimprovingp.f.,I =I

(tanϕ –tanϕ )

=166.663(tan(cos 0.7)−tan(cos 0.9))

=89.312A

Thevalueofcapacitortobeconnected,

Example9.5:A3-ϕ500HP,50Hz,11kVstar-connectedinductionmotorhasafullloadefficiencyof85%atlaggingp.f.of0.75andisconnectedtoafeeder.Ifthep.f.ofloadisdesiredtobecorrectedto0.9lagging,determinethefollowing:

a 1

a 2

C a

1 2

ph

a 1

C a

1 2

−1 −1

−1 −1

1. sizeofthecapacitorbankinkVArand2. capacitanceofeachunitifthecapacitorsareconnectedinΔaswellas

inY.

Solution:

Inductionmotoroutput=500HP

Efficiencyη=85%,

η=output/input

Inputoftheinductionmotor,P = output/η=500/0.85=588.235HP

= 588.235×746=438.82kW

Initialp.f.(cosϕ ) = 0.75⇒tanϕ =0.88

Correctedp.f.(cosϕ ) = 0.9⇒tanϕ =0.48

LeadingkVArtakenbythecapacitorbank,Q = P(tanϕ −tanϕ )

= 438.82(0.88−0.48)=175.53kVAr

Case1:Deltaconnection:

Chargingcurrentperphase,

Reactanceofcapacitorbankperphase,

1 1

2 2

c 1

2

Capacitanceofcapacitorbank,

Case2:Starconnection:

I =I =9.213A

Reactanceofcapacitorbankperphase,

Capacitanceofcapacitorbank,

Example9.6:Astar-connected400HP(metric),2,000V,and50Hzmotorworksatap.f.of0.7lag.Abankofmesh-connectedcondensersisusedtoraisethep.f.to0.9lag.Calculatethecapacitanceofeachunitandtotalnumberofunitsrequired,ifeachunitisrated400V,50Hz.Themotorefficiencyis90%(Fig.9.31).

Solution:

Motoroutput =400HP

Supplyvoltage =2,000V

i.e.,V =2,000V

L c

p.f.withoutcondenser =0.7lag

FIG.9.31Circuitdiagram

Efficiencyofmotor,η=0.9;

Forfixedloads,theactivecomponentofcurrentisthesameforimprovedp.f.,whereasthereactivecomponentwillbechanged.

∴Theactivecomponentofcurrentat0.7p.f.lag.,I =Icosϕ

=136.73×0.7=95.71A

Linecurrenttakenbythecapacitorinstalledforimprovingp.f.,I =I (tanϕ −tanϕ )

=95.71(tan(cos 0.7)−tan(cos 0.9))

=51.28A

Thebankofcondensersusedtoimprovethep.f.isconnectedindelta.Thevoltageacrosseachphaseis2,000V,buteachunitofcondenserbankisof400V.So,eachphaseofthebankwillhavefivecondensersconnectedinseriesasshowninFig.9.31.

a1

1

C a 1 2−1 −1

Thecurrentineachphaseofthebank

LetX bethereactanceofeachcondenser.

Thenthechargecurrent,

Capacitanceofeachphaseofthebank

Example9.7:A3-ϕ,50Hz,2,500Vmotordevelops600HP,thep.f.being0.8laggingandtheefficiency0.9.Acapacitorbankisconnectedindeltaacrossthesupplyterminalsandthep.f.israisedtounity.Eachofthecapacitanceunitsisbuiltoffivesimilar500Vcapacitors(Fig.9.32).Determinethecapacitanceofeachcapacitor.


Solution:

Motorinput

c

LeadingkVArsuppliedbythecapacitorbank=P(tanϕ−tanϕ )

=497.33(0.75−0)

=373kVAr

LeadingkVArsuppliedbyeachofthreesets

Currentperphaseofcapacitorbank,

kVArrequired/phase

ButleadingkVArsuppliedbyeachphase=124.33kVAr

1

2

Sinceitisthecombinedcapacitanceoffiveequalcapacitorsjoinedinseries,

Thecapacitanceofeachunit =5×63.32µF

=3,116.6µF

Example9.8:A3-ϕ,50Hz,30-kmtransmissionlinesuppliesaloadof5MWatp.f.0.7laggingtothereceivingendwherethevoltageismaintainedconstantat11kV.Thelineresistanceandinductanceare0.02Ωand0.84mHperphaseperkm,respectively.Acapacitorisconnectedacrosstheloadtoraisethep.f.to0.9lagging(Fig.9.33).Calculate:(a)thevalueofthecapacitanceperphaseand(b)thevoltageregulation.

Solution:

Lengthoftheline =30km

Frequency =50Hz

Load =5MWat0.7laggingp.f.

Receiving-endvoltage,V =11kV

Lineresistanceperphase=0.02Ωperkm =0.02×30=0.6Ω

Reactanceof30-kmlengthperphase,X =2×π×f×L×30

=2×π×50×0.84×10 ×30=7.92Ω

r

–3


Forfixedloads,theactivecomponentofpoweristhesameforimprovedp.f.,whereasthereactivecomponentofpowerwillbechanged.

∴Theactivecomponentofcurrentat0.7p.f.lag,P=5MW

Reactivepower(MVAr)suppliedbythecapacitorbank=P(tanϕ −tanϕ )

=5(1.02−0.484)

=2.679MVAr

Reactivepower(MVAr)suppliedbythecapacitorbank

=

=0.893MVAr=893kVAr

LetCbethecapacitancetobeconnectedperphaseacrosstheload.

kVArrequired/phase

1 2

ButleadingkVArsuppliedbyeachphase=893kVAr

Sending-endvoltagewithimprovedp.f.=V +I(Rcosϕ +jXsinϕ )

Receiving-endvoltage,

Currentwithimprovedp.f.,

∴Sending-endvoltage,V =6,350.85+291.59(0.6×0.9+j7.92×0.436)

=6,508.31+j1006.9V/ph

=6,585.74∠8.79V/ph

=11,406.84V(L-L)

Example9.9:Asynchronousmotorimprovesthep.f.ofaloadof200kWfrom0.7laggingto0.9laggingandatthesametimecarriesanadditionalloadof100kW(Fig.9.34).Find:(i)TheleadingkVArsuppliedbythemotor,(ii)kVAratingofmotor,and(iii)p.f.atwhichthemotoroperates.

r

2 2

s

Solution:

Load,P =200kW

Motorload,P =100kW

p.f.oftheload200kW =0.7lag

p.f.ofthecombinedload(200+100)kW =0.9lag

Combinedload =P +P =200+100=

300kW

∆OABisapowertrianglewithoutadditionalload,∆ODCthepowertriangleforcombinedload,and∆BECforthemotorload.

FromFig.9.34,wehave

(i)LeadingkVArtakenbythemotor = CE

= DE–DC=AB–DC

= 200tan(cos 0.7)–300tan(cos 0.9)

= 200×1.02–300×0.4843

= 58.71KVAr

(ii)kVAratingofmotor =BE

1

2

1 2

–1

–1

=115.96kVA


(iii)p.f.ofmotoratwhichitoperates,cosϕ

=0.862leading

m

Example9.10:A37.3kWinductionmotorhasp.f.0.9andefficiency0.9atfull-load,p.f.0.6,andefficiency0.7athalf-load.Atno-load,thecurrentis25%ofthefull-loadcurrentandp.f.0.1.Capacitorsaresuppliedtomakethelinep.f.0.8athalf-load.Withthesecapacitorsincircuit,findthelinep.f.at(i)full-loadand(ii)no-load.

Solution:

Full-loadcurrent,I =37.3×10 /( V ×0.9×0.9)=26,586/V

Atfullload:

Motorinput,P =37.3/0.9=41.44kW

LaggingkVArdrawnbythemotor,kVAr =P tanϕ =41.44tan(cos 0.9)=20.07

Athalfload:

Motorinput,P =(0.5×37.3)/0.7=26.64kW

LaggingkVArdrawnbythemotor,kVAr =P tanϕ =26.64tan(cos 0.6)=35.52

Atnoload:

No-loadcurrent,I =0.25(full-loadcurrent)=0.25×26,586/V =6,646.5/V

Motorinputatno-load,P = V I cosϕ

= ×6,646.5×V ×0.1/V =1.151kW

LaggingkVArdrawnbythemotor,kVAr =1.151tan(cos 0.1)

=11.452

LaggingkVArdrawnfromthemainsathalf-loadwithcapacitors,

kVAr =26.64tan(cos 0.8)=19.98

1 L

L

1

1 1 1

2

2 2 2

0

L L

o L 0 0

L L

0

2c

3

–1

–1

–1

–1

kVArsuppliedbycapacitors,kVAr =kVAr –kVAr =35.52–19.98=15.54

kVArdrawnfromthemainatfullloadwithcapacitorskVAr =kVAr –kVAr

=20.07–15.54=4.53

1. Linep.f.atfullload

2. kVArdrawnfrommainsatno-loadwithcapacitors=11.452–15.54=–4.088

Linep.f.atno-load=cos(tan –4.088/1.151)=cos(–74.27°)=0.271leading.

Example9.11:Asingle-phasesystemsuppliesthefollowingloads:

1. Lightingloadof50kWatunityp.f.2. Inductionmotorloadof125kWatp.f.0.707lagging.3. Synchronousmotorloadof75kWatp.f.0.9leading.4. Othermiscellaneousloadsof25kWatp.f.0.8lagging.

DeterminethetotalkWandkVAdeliveredbythesystemandthep.f.atwhichitworks.

Solution:

TotalkWoftheload = 50+125+75+25=275kW

kVAroflightingload = 50×0=0

kVArofinductionmotor = −125tan(cos 0.707)=−125.04

kVArofsynchronousmotor = 75tan(cos 0.9)=36.32

c 2 2c

1c 1 c

–1

−1

−1

−1

kVArofmiscellaneousloads = 25tan(cos 0.8)=−18.75

∴TotalkVAroftheload =

0−125.04+36.32−18.75=−107.47

KEYNOTES

Forqualitypower,i.e.,voltageandfrequencyateverysupplypointwouldremainconstant,freefromharmonics,andthepowerfactorwouldremainunity,compensationisrequired.Theobjectivesofloadcompensationare:

1. Power-factorcorrection.2. Voltageregulationimprovement.3. Balancingofload.

Characteristicsoftheidealcompensatorareto:

1. Provideacontrollableandvariableamountofreactivepowerwithoutanydelayaccordingtotherequirementsoftheload.

2. Maintainconstant-voltagecharacteristicsatitsterminals.3. Operateindependentlyinthethreephases.

Voltageregulationisdefinedastheproportionalchangeinsupply-voltagemagnitudeassociatedwithadefinedchangeinloadcurrent,i.e.,fromno-loadtofullload.TheSILofatransmissionlineistheMWloadingofatransmissionlineatwhichanaturalreactivepowerbalanceoccurs(zeroresistance).Voltagestability:Apowersystematagivenoperatingstateandsubjectedtoagivendisturbanceisvoltagestableifvoltagesnearloadsapproachpost-disturbanceequilibriumvalues.Thedisturbedstateiswithintheregionsofattractionsofstablepost-disturbanceequilibrium.Voltagecollapse:Followingvoltageinstability,apowersystemundergoesvoltagecollapseifthepost-disturbanceequilibriumvoltagesneartheloadarebelowtheacceptablelimits.Thevoltagecollapsemaybeeithertotalorpartial.


1. Definetheneedofcompensation.

Formaintainingthequalitypower,i.e.,voltageandfrequencyateverysupplypointwouldremainconstant,freefromharmonics

−1

andthep.f.wouldremainunityandcompensationisneeded.

2. Whataretheobjectivesofloadcompensation?

Theobjectivesofloadcompensationare:

1. p.f.correction.2. Voltageregulationimprovement.3. Balancingofload.

3. Whatarethecharacteristicsofanidealcompensator?

Thecharacteristicsoftheidealcompensatorare:

1. Toprovideacontrollableandvariableamountofreactivepowerwithoutanydelayaccordingtotherequirementsoftheload.

2. Tomaintainaconstant-voltagecharacteristicatitsterminalsand3. Shouldoperateindependentlyinthethreephases.

4. Definethevoltageregulation.

Itisdefinedastheproportionalchangeinsupplyvoltagemagnitudeassociatedwithadefinedchangeinloadcurrent,i.e.,fromnoloadtofullload.

5. Definethesurgeimpedanceloading(SIL)ofatransmissionline.

ItistheMWloadingofatransmissionlineatwhichanaturalreactivepowerbalanceoccurs(zeroresistance).

6. Whatismeantbyvoltagestability?

Apowersystematagivenoperatingstateandsubjectedtoagivendisturbanceisvoltagestableifvoltagesnearloadsapproachpost-disturbanceequilibriumvalues.Thedisturbedstateiswithintheregionsofattractionsofstablepost-disturbanceequilibrium.

7. Whatismeantbyvoltagecollapse?

Followingvoltageinstability,apowersystemundergoesvoltagecollapseifthepost-disturbanceequilibriumvoltagesneartheloadarebelowacceptablelimits.Thevoltagecollapsemaybeeithertotalorpartial.


1. Themajorreasonforlowlaggingp.f.ofsupplysystemisduetotheuseof__________motors.

1. Induction.2. Synchronous.3. DC.4. Noneofthese.

2. Themaximumvalueofp.f.canbe__________.

1. 1.2. 0.9.3. 0.8.4. 0.7.

3. Byimprovingthep.f.ofthesystem,thekilowattsdeliveredbygeneratingstationsare__________.

1. Decreased.2. Increased.3. Notchanged.4. Noneofthese.

4. Powerfactorcanbeimprovedbyinstallingsuchadeviceinparallelwithload,whichtakes:

1. Laggingreactivepower.2. Leadingreactivepower.3. Apparentpower.4. Noneofthese.

5. Themainreasonforlowp.f.ofsupplysystemisduetotheuseof__________.

1. Resistiveload.2. Inductiveload.3. Synchronousmotor.4. Allofthese.

6. Theonlymotorthatcanalsobeworkedatleadingp.f.andcansupplymechanicalpower__________.

1. Synchronousinductiongenerator.2. Synchronousmotor.3. Alternator.4. Noneofthese.

7. Anover-excitedsynchronousmotoronno-loadisknownas__________.

1. Synchronousinductiongenerator.2. Synchronouscondenser.3. Alternator.4. Noneofthese.

8. Forsynchronouscondensers,thep.f.improvementapparatusshouldbelocatedat__________.

1. Sendingend.2. Receivingend.3. Both(a)and(b).4. Noneofthese.

9. Adisadvantageofsynchronouscondenseris:

1. Continuouslossesinmotor.2. Highmaintenancecost.3. Noisy.4. Alloftheabove.

10. Thesmallerthelaggingreactivepowerdrawnbyacircuit,itsp.f.willbe__________.

1. Better.2. Poorer.3. Unity.4. Noneofthese.

11. kVARisequalto__________.

1. kWtanϕ.2. kWsinϕ.3. kVAcosϕ.4. Noneofthese.

12. Foraparticularpower,thecurrentdrawnbythecircuitisminimumwhenthevalueofp.f.is__________.

1. 0.8lagging.2. 0.8leading.3. Unity.4. Noneofthese.

13. Synchronouscapacitorsarenormally____________cooled.

1. Air.2. Oil.3. Water.4. Noneofthese.

14. Toimprovethep.f.of3-ϕcircuits,thesizeofeachcapacitorwhenconnectedindeltawithrespecttowhenconnectedinstaris__________.

1. 1/6th.2. 1/4th.3. 3times.4. 1/3rd.

15. Thep.f.improvementequipmentisalwaysplaced__________.

1. Atthegeneratingstation.2. Nearthetransformer.3. Neartheapparatusresponsibleforlowp.f.4. Nearthebusbar.

16. Asynchronousmachinehashighercapacityfor:

1. Leadingp.f.2. Laggingp.f.3. Itdoesnotdependuponthep.f.ofthemachine.4. Noneofthese.

17. Ifasynchronousmachineisunderexcited,ittakeslaggingVARsfromthesystemwhenitisoperatedasa__________.

1. Synchronousmotor.2. Synchronousgenerator.3. Synchronousmotoraswellasgenerator.4. Noneofthese.

18. Asynchronousphasemodifierascomparedtosynchronousmotorusedformechanicalloadshas__________.

1. Largershaftandhigherspeed.2. Smallershaftandhigherspeed.3. Largershaftandsmallerspeed.4. Smallershaftandsmallerspeed.

19. Thephaseadvancerismountedonthemainmotorshaftandisconnectedinthe__________motor.

1. Rotor.2. Stator.

3. Core.4. Noneofthese.

20. Industrialheatingfurnacessuchasarcandinductionfurnacesoperateon__________.

1. Verylowlaggingp.f.2. Verylowleadingp.f.3. Veryhighleadingp.f.4. Noneofthese.

21. Ifasynchronousmachineisoverexcited,ittakeslaggingVARsfromthesystemwhenitisoperatedas:

1. Synchronousmotor.2. Synchronousgenerator.3. Synchronousmotoraswellasgenerator.4. Noneofthese.

22. Amachinedesignedtooperateatfullloadisphysicallyheavierandiscostlieriftheoperatingp.f.is:

1. Lagging.2. Leading.3. Thesizeandcostdonotdependonp.f.4. Noneofthese.

23. Unitofreactivepoweris:

1. MW.2. MVAr.3. MVA.4. KVA.

24. Reactivepoweris____________power.

1. Wattfull.2. Wattless.3. Loss.4. Noneofthese.

25. Transmissionlineparametersare:

1. R.2. L.3. C.4. Allofthese.

26. OnfundamentalTγ.lineexpressionV(x)=Ae +Be ,γrepresents:

1. Distance(or)length.2. Velocityoflight.3. Propagationconstant.4. Noneofthese.

27. Characteristicimpedanceis_____________.

1.

2.

3.

γx -γx

4.

28. βis____________.

1.

2.

3.4. Allofthese.

29. Advantageofoperatingatnaturalloadis:

1. Insulationisuniformlystressed.2. Reactivepowerbalanceisachieved.3. Both(a)and(b).4. Noneofthese.

30. Anuncompensatedlineonopen-circuitleadsto__________.

1. Ferrantieffect.2. line-chargingcurrentflowingintogeneratorsismore.3. Both(a)and(b).4. Noneofthese.

31. Asymmetricallineatno-loadmeans__________.

1. Nopowertransmission.2. V =V .


32. Duringtheunderexcitedoperationofasynchronousgenerator:

1. Heatingoftheendsofthestatorcoreincreases.2. Reducesfieldcurrent,resultsintheinternalemf,whichcausesweak

stability.3. Both(a)and(b).4. Noneofthese.

33. ForasymmetricallinewithV =V ,themaximumvoltageoccurs

at:

1. Sendingend.2. Receivingend.3. Midpoint.4. Noneofthese.

34. Unitofp.f.is:

1. s.2. m.3. Nounits.4. Noneofthese.

35. Unitoftimeconstantis:

1. m.2. kg.3. s.4. miles.

s r

s 0

36. Powertransmissionthroughalineisimprovedby:

1. Increasingthelinevoltage.2. Decreasingthelinereactance.3. Both(a)and(b).4. Noneofthese.

37. Alineardevicemustsatisfy:

1. Homogeneity.2. Additivity.3. Both(a)and(b).4. Noneofthese.

38. FundamentalrequirementsofAC-powertransmissionis:

1. Synchronousmachinesmustremainstablyinsynchronizer.2. Voltagesmustbekeptneartotheirratedvalues.3. Both(a)and(b).4. Noneofthese.

39. Loadcompensationis:

1. Thecontrolofreactivepowertoimprovequalityofsupply.2. Thecontrolofrealpowertoimprovequalityofsupply.3. Thecontrolofvoltageanditsangletoimprovethequalityofsupply.4. Both(a)and(b).

40. Powerfactorundernaturalloadis:

1. Lagging.2. Leading.3. Unity.4. Noneofthese.

41. Steady-statestabilityofunitoccurswhenδ=__________.

1. 30°.2. 20°.3. 90°.4. 0°.

42. ‘θ’infundamentaltransmissionlineequationis:

1. β.2. ax.3. βl.4. β/a.

43. Ratingofacompensatoris:

1. MVAr.2. Timeofresponse.3. Both(a)and(b).4. Noneofthese.

44. Loadcompensationincludes:

1. p.f.correction.2. Voltageregulation.3. Loadbalancing.4. Allofthese.

45. Forasymmetricalline,thevoltageismoreat:

1. Sendingend.

2. Receivingend.3. Midpoint.4. Allofthese.

46. Loadcompensationcanbeachievedby:

1. Installingthecompensatingequipmentnearthesource.2. Installingthecompensatingequipmentneartheload.3. Either(a)or(b).4. Both(a)and(b).

47. pfcorrectionofloadisachievedby:

1. Generatingreactivepowerascloseaspossibletothesource.2. Generatingreactivepowerascloseaspossibletotheload.3. Generatingrealpowerascloseaspossibletotheload.4. Generatingrealpowerascloseaspossibletothesource.

48. Themainfunctionofanidealcompensatoris:

1. Instantaneouspfcorrectiontounity.2. Elimination(or)reductionofvoltageregulation.3. Phasebalanceoftheloadcurrentsandvoltages.4. All.

49. Theimportantcharacteristicofanidealcompensatoris:

1. Toprovideacontrollableandvariableamountofreactivepowerwithoutanydelay.

2. Tomaintainaconstantvoltagecharacteristicatitsterminals.3. Shouldoperateindependentlyinthethreephases.4. Alltheabove.

50. Characteristicimpedanceofthelinedependsupon:

1. Thecharacteristicofthelineperunitlength.2. Lengthoftheline.3. Radiusandspacingbetweenconductors.4. All.

51. Thesurgeimpedanceloading(SIL)isexpressedas:

1.

2.

3. SIL=(V ) ×surgeimpedance.

4. None.

52. WhenalineisloadedaboveitsSIL,itactslike:

1. ShuntreactorabsorbingMUARfromthesystem.2. ShuntcapacitorsupplyingMUARtothesystem.3. SeriescapacitorsupplyingMUARtothesystem.4. SeriesreactorabsorbingMUARfromthesystem.

53. WhenalineisloadedbelowitsSIL,itactslike:

1. ShuntreactorabsorbingMUARfromthesystem.2. ShuntcapacitorsupplyingMUARtothesystem.3. ShuntcapacitorsupplyingMUARtothesystem.

L-L

2

4. ShuntreactorabsorbingMUARfromthesystem.

54. Ifanyinductiveloadisconnectedatthesendingendoftheline,itwillsupportthesynchronousgenerators:

1. Toabsorbtheline-chargingreactivepower.2. Toabsorbtheload-chargingreactivepower.3. Tosupplytheline-chargingreactivepower.4. Tosupplytheload-chargingreactivepower.

55. Thechangeinelectricalpropertiesofatransmissionlineinordertoincreaseitspowertransmissioncapabilityisknownas:

1. Loadcompensation.2. Linecompensation.3. Loadsynchronism.4. Linesynchronism.

56. ApplyseriescapacitorstoreduceX andtherebyreduceθatthe

fundamentalfrequency.Thismethodiscalled:

1. Line-lengthcompensation(or)θ-compensation.2. Compensationbysectioning.3. Loadbalancing.4. Alltheabove.

57. Seriescompensationresultsin:

1. Increaseinmaximumtransferablepowercapacity.2. Decreaseintransmissionangleforconsiderableamountofpowertransfer.3. Increaseinvirtualsurgeimpedanceloading.4. Alltheabove.

58. Foraheavyloadingcondition,aflatvoltageprofilecanbeobtainedby:

1. Seriescompensation.2. Shuntcompensation.3. (a)or(c).4. None.

59. Inductiveshuntcompensation__________thevirtualsurgeimpedanceand__________thevirtualSILoftheline:

1. Decreases,decreases.2. Decreases,increases.3. Increases,decreases.4. Decreases,increases.

60. Iftheinductiveshuntcompensationis100%then:

1. Flatvoltageprofileexistsatzeroloads.2. Ferrantieffectcanbeeliminated.3. Both(a)and(b).4. None.

61. Sub-synchronousresonance(SSR)istreatedas__________typeofphenomenon.

1. Electrical.2. Mechanical.3. Combinedelectrical–mechanical.4. Dampedfrequencyresonance.

L

62. UPFCisabletoperform:

1. Voltagesupport.2. Powerflowcontrol.3. Improvedstability.4. All.

63. Thevoltagestabilityanalysisiscarriedoutbywhichpowerflow-basedmethod?

1. P–Vcurves.2. Q–Vcurves.3. Both(a)and(b).4. None.

64. Voltagecollapseproximateindicator(VCPI)foraradiallineisdefinedas:

1.

2.

3.

4.

REVIEWQUESTIONS

1. Explaintheobjectivesofloadcompensation.2. Explainthevoltageregulationwithandwithoutcompensators.3. Whatarethespecificationsofloadcompensation?4. Explaintheeffectsonuncompensatedlineunderno-loadand

loadconditions.5. Explaintheeffectsoncompensatedline.6. Explaintheconceptofsub-synchronousresonance.7. Comparethedifferenttypesofcompensatingequipmentfortransmissionsystems.

8. Explaintheconceptsofvoltagestabilityandvoltagecollapse.9. Derivethevoltagestabilityindexofatypicalbranchofapower

system.

PROBLEMS

1. A3-ϕ,5kWinductionmotorhasap.f.of0.8lag.Abankofcapacitorsisconnectedindeltaacrossthesupplyterminalsandp.f.israisedto0.95lag.DeterminethekVArratingofthe

capacitorsconnectedineachphase.2. A3-ϕ,50Hz,400Vmotordevelops100HP,thep.f.being0.7lag

andefficiency93%.Abankofcapacitorsisconnectedindeltaacrossthesupplyterminalsandp.f.israisedto0.95lag.Eachofthecapacitanceunitsisbuiltoffoursimilar100Vcapacitors.Determinethecapacitanceofeachcapacitor.

3. Astar-connected400HP,2,000V,50Hzmotorworksatap.f.of0.75lagging.Abankofmesh-connectedcondensersisusedtoraisethep.f.to0.98lagging.Determinethecapacitanceofeachunitandtotalnumberofunitsrequired;ifeachisrated500V,50Hz.Themotorefficiencyis85%.

4. A3-ϕ,50Hz,3,000Vmotordevelops600HP,thep.f.being0.75laggingandtheefficiency0.95.Abankofcapacitorsisconnectedindeltaacrossthesupplyterminalsandthep.f.raisedto0.98lagging.Eachofthecapacitanceunitsisbuiltoffivesimilar600Vcapacitors.Determinethecapacitanceofeachcapacitor.

10

VoltageControl

OBJECTIVES


obtainanoverviewofvoltagecontrol

discusstheparametersorequipmentscausingreactivepower

understandthemethodsofvoltagecontroland

calculatetheratingofsynchronousphasemodifier

10.1INTRODUCTION

Apowersystemmustbedesignedinsuchawaysoastomaintainthevoltagevariationsattheconsumerterminalswithinspecifiedlimits.Inpractice,alltheequipmentsonthepowersystemaredesignedtooperatesatisfactorilyattheratedvoltagesorwithinspecifiedlimits,atmost±6%attheconsumerterminals.Themainreasonforthevariationinvoltageattheconsumerterminalsisthevariationinloadonthesupplypowersystem.Incaseloadonthesupplysystemincreases,thevoltageattheconsumerterminalsdecreasesduetoanincreaseinvoltagedropinpowersystemcomponentsandviceversawhenloadisdecreased.Mostoftheelectronicequipmentsaresensitivetovoltagevariations;hence,thevoltagemustbemaintainedconstant.Itcanbemaintainedwithinthelimitsbyprovidingvoltage-controlequipment.

10.2NECESSITYOFVOLTAGECONTROL

Thevoltageattheconsumerterminalschangeswiththevariationinloadonthesupplysystem,whichis

undesirableduetothefollowingreasons:

1. Incaseoflightingload,forexample,incandescentlampisacutelysensitivetovoltagechanges.Fluctuationsinvoltagebeyondacertainlevelmayevendecreasethelifeofthelamp.

2. Incaseofpowerloadconsistingofinductionmotors,thevoltagevariationsmaycauseavariationinthetorqueofaninductionmotor,asthetorqueisproportionaltothesquareoftheterminalvoltage.Ifthesupplyvoltageislow,thenthestartingtorqueofthemotorwillbetoolow.

3. Ifthevoltagevariationismorethanaspecifiedvalue,thentheperformanceoftheequipmentssuffersandthelifeoftheequipmentisreduced.

4. Thepictureonatelevisionsetstartsrollingifthevoltageisbelowacertainlevelbecausethefluorescenttuberefusestoglowatlowvoltages.Hence,voltagevariationsmustberegulatedandkepttoaminimumlevel.

Beforediscussingthevariousmethodsofvoltagecontrol,itisveryimportanttoknowaboutthevarioussourcesandsinksofreactivepowerinapowersystem.

TestYourself

Whyisvoltagetolerancemorethanfrequencytolerance?

10.3GENERATIONANDABSORPTIONOFREACTIVEPOWER

1. Synchronousmachine:Thesecanbeusedeithertogenerateorabsorbreactivepower.Theabilitytosupplyreactivepoweris

determinedbytheshort-circuitratio An

overexcitedsynchronousmachinegenerateskVArandactsasashuntcapacitor,whileaunderexcitedsynchronousmachineabsorbsitandactsasashuntreactor.ThemachineisthemainsourceofsupplytothesystemofbothpositiveandnegativeVArs.

2. Overheadlines:Whenfullyloaded,linesabsorbreactivepowerwithacurrentIamperesforalineofreactanceperphaseXohms,theVArsabsorbedareI Xperphase.Onlightloads,theshuntcapacitancesoflongerlinesmaybecomepredominantandthelinesthenbecomeVArgenerators.

3. Transformers:Transformersabsorbreactivepower.ThemathematicalexpressionforthereactivepowerabsorbedbyatransformerisQ =3∣I∣ X VAr.whereX isthetransformer

reactanceperphaseinohmsand∣I∣isthecurrentflowingthroughinamperes.

4. Cables:CablesactasVArgeneratorsbecausetheyhaveaverysmall

T T T

2

2

inductanceandrelativelyverylargecapacitanceduetothenearnessoftheconductors.

5. Loads:Aloadat0.8p.f.impliesareactivepowerdemandof0.75kVArperkWofpower,whichismoresignificantthanthesimplequotingofthep.f.Inplanningpowersystems,itisrequiredtoconsiderreactivepowerrequirementstoascertainwhetherthegeneratorisabletooperateanyrangeofp.f.

10.4LOCATIONOFVOLTAGE-CONTROLEQUIPMENT

Theconsumerapparatusshouldoperatesatisfactorily.Thisisachievedbyinstallingvoltage-controlequipmentatsuitableplaces.

Thevoltage-controlequipmentisplacedintwoormorethantwoplacesinapowersystembecauseofthefollowingreasons:

1. Thepowersystemisacombinationofwide-rangingnetworksandthereisavoltagedropindifferentsectionsofthedistributionandtransmissionsystems.

2. Thevariouscircuitsofapowersystemhavedifferentloadcharacteristics.

Thevoltage-controlequipmentisplacedat:

1. Generatingstations.2. Transformerstations.3. Thefeeders.

Whenpowerissuppliedtoaloadthroughatransmissionlinekeepingthesending-endvoltageconstant,thereceiving-endvoltagevarieswithmagnitudeofloadandp.f.oftheload.Thehighertheloadwithsmallerp.f.,thegreateristhevoltagevariation.

10.5METHODSOFVOLTAGECONTROL

Thedifferentvoltage-controlmethodsare:

1. Excitationcontrol.2. Shuntcapacitors.3. Seriescapacitors.4. Tap-changingtransformers.5. Boosters.6. Synchronouscondensers.

10.5.1Excitationcontrol

Thismethodisusedonlyatthegeneratingstation.Duetothevoltagedropinthesynchronousreactanceofarmature,thealternatorterminalvoltagechangesandhencetheloadonthesupplysystemalsoundergoesachange.Thiscanbemaintainedconstantbychangingthefieldcurrentofthealternator.Thisprocessiscalledexcitationcontrol.Byusinganautomaticorahand-operatedregulator,theexcitationofthealternatorcanbecontrolled.

Inmodernsystems,automaticregulatorispreferred.Thetwomaintypesofautomaticvoltageregulatorsare:

1. Tirrilregulator.2. Brown-Boveriregulator

(a)Tirrilautomaticregulator:Tirrilregulatorisafast-actingelectromagneticalregulatoranditgives±0.5%regulatingdeviationbetweenno-loadandfullloadofanalternator.

Construction:Tirrilvoltageregulatorisavibrating-typevoltageregulatorinwhicharesistanceRisconnectedintheexcitercircuittogettherequiredvalueofvoltagebyadjustingthepropervalueofresistance.Figure10.1showsthemainpartsoftheTirrilvoltageregulator.

FIG.10.1Tirrilautomaticvoltageregulator

Differentialrelay:Itisa‘U’-shaped(horseshoe)relaymagnet.IthastwoidenticalwindingsonbothlimbsasshowninFig.10.1,whichareconnectedacrossthearmatureoftheexciteronlywhenthemaincontactsareclosed.Acapacitorisconnectedinparalleltotherelayforreducingthesparkwhentherelaycontactsareopened.

Excitationsystem:Itconsistsofasolenoidenergizedbythevoltageequaltotheexciterterminalvoltage.Thecounter-balanceforceofanexcitationsolenoidisprovidedbythreesprings,whichareactinginsequenceandareshowninFig.10.1.

Maincontrolunit:ItisasolenoidexcitedfromanACsupply.Thelowerpartofthissolenoidisconnectedwithadashpot,whichprovidesdampingtothemeasuringunit.

Maincontacts:TheseareattachedtotheleversthatareoperatedbymeasuringandexcitationsolenoidsasshowninFig.10.1.Theleverontheleftsideiscontrolledbytheexcitercontrolmagnetandtheleverontherightsideiscontrolledbythemaincontrolmagnet.

Principleofoperation:Undernormaloperatingconditions,i.e.,thesystemisoperatingatpre-setloadandvoltageconditions,themaincontactsareopen.Thefieldrheostatisinthecircuit.Iftheloadonthealternatorincreases,theterminalvoltagedecreases.Whenthepre-setexcitationsettingsofthedeviceislow,them.m.fdevelopedbythemeasuringsystemorthesolenoidislow,causingadisturbanceintheequilibriumand,therefore,maincontactsareclosed.Theseresultsinde-energizationofdifferentialrelayandrelaycontactsareclosed.So,theresistance‘R’inthefieldisshort-circuited.Whenthisisoutofcircuit,totalfieldcurrentflowsthroughtheexciter,andtheexciterterminalvoltageincreases.Thus,thevoltageacrossthealternatorterminalsincreasesduetotheincreaseinalternatorfieldcurrent.

Duetothisincreasedvoltage,thepullofthesolenoidexceedsthespringforceandsothemaincontactsareopenedagainandtheresistanceisinsertedintheexciterfield.Asimilarprocessisrepeatediftheterminalvoltageisreduced.

(b)Brown-Boveriregulator:ThisdiffersfromtheTirrilregulator.Inthis,theresistanceofregulatoriseithergraduallyvariedorvariedinsmallsteps.

Construction:Brown-Boveriregulatorisnotavibratingtype;hence,wearandtearislesswhencomparedtothatofatirrilregulator.ItconsistsoffourmainpartsanditsschematicdiagramisshowninFig.10.2(a).

Controlsystem:Itcontainstwowindings‘P’and‘Q’woundonanannularcoreoflaminatedsteelsheetasshowninFig.10.2(a).Thewindingsareexcitedfromthethree-phasealternatorsupplythroughtheresistancesRandR andresistanceR isinsertedinwinding‘P’.Theratioofresistancetoreactanceisadjustedinsuchawaysoastogetaphaseangledifferencebetweenthecurrentsintwowindings.Thisresultsintheformationofa

c

f se

rotatingmagneticfieldandhencedevelopsanelectromagnetictorqueonthealuminiumdrumD.ThistorquedependsontheterminalvoltageofthealternatorandontheresistancesR andR .ThetorquedecreaseswithincreasedvaluesofR .

Operatingsystem:Itconsistsoftworesistancesectorsmadeupofcontactblocksontheinnersurfaceofroll-contactsegmentsasshowninFig.10.2(b).Contactsegmentsandresistancesectorsaremadetocontactbyusingsprings.ThetworesistancesectorsRandRareconnectedinseries,andthiscombinationisconnectedinserieswithexciterfieldcircuits.Ifthealternatorvoltagechangesfromitspre-determinedvalue,thecontactsegmentsrollontheinsideofresistancesectors,rotatesclock-wiseoranti-clock-wiseundertheactionofthetwowindingsPandQ.

c f

f

FIG.10.2(a)SchematicdiagramofBrown-Boveriregulator;(b)detaileddiagramofBrown-Boveriregulator

Mechanicalcontroltorque:Mechanicaltorqueisproducedbysprings(mainandauxiliary)andisindependentofthepositionofthecontrolsystem.Inasteadydeflectionstate,themechanicaltorqueisequaltotheelectricaltorque,whichisproducedbythecurrentinthesplit-phasewinding.

Dampingtorque:Itconsistsofanaluminiumdisc,whichisrotatedinbetweentwomagnetsMandM,andaspringSisattachedtoit.Whenthereisachangeinthealternatorvoltage,eddycurrentsareproducedinthediscandtorqueisdeveloped;therebycontrollingtheresponseofthemovingsystem.

Principleofoperation:SupposethevoltageofthealternatorterminalsissettothenormalvaluebyadjustingR andR andisinPosition-3onthescale.In

c f

thisposition,themechanicaltorqueisequaltotheelectromagnetictorqueandthemovingsystemisunderequilibrium.

Letusassumethattheterminalvoltageofthealternatorisreducedduetotheriseinload,andthentheelectromagnetictorqueisreduced.Atthisinstant,themechanicaltorqueisgreaterthantheelectromagnetictorqueandthediscstartstorotate(assumeinanti-clock-wisedirection).Duetothis,thepointermovestoPosition-1.Theresistanceintheexciterfieldwillbereduced,whichcausesanincreaseintheexciterfieldcurrent.So,theterminalvoltageofanalternatorincreases.

10.5.2Shuntcapacitorsandreactors

Shuntcapacitorsareusedforlaggingp.f.circuits;whereasreactorsareusedforleadingp.f.circuitssuchasthosecreatedbylightlyloadedcables.Inbothcases,theeffectistosupplytherequiredreactivepowertomaintainthevaluesofthevoltage.Apartfromsynchronousmachines,staticshuntcapacitorsofferthecheapestmeansofreactivepowersupplybutthesearenotasflexibleassynchronouscondensers.

Capacitorsareconnectedtoabusbarortothetertiarywindingofamaintransformer.Inthismethod,asthevoltagefalls,theVArsproducedbyashuntcapacitororreactoralsofalls.Thus,theireffectivenessfallswhenneeded.Alsoforlightloads,whenthevoltageishigh,thecapacitoroutputislargeandthevoltagetendstobecomeexcessive.theviewofathree-phasecapacitorbankona11-kVdistributionlineisshowninFig.10.3.

11.5.3Seriescapacitors

Capacitorsareinstalledinserieswithtransmissionlines(showninFig.10.4)inordertoreducevoltagedrop.The

seriescapacitorscompensatethereactancevoltagedropinthelinebyreducingnetreactance.Acapacitorinserieswithatransmissionlineservingalaggingp.f.loadwillcauseariseinvoltageastheloadincreases.Thep.f.oftheloadthroughtheseriescapacitorandlinemustbelaggingifthevoltagedropistodecreaseappreciably.Thevoltageontheloadsideoftheseriescapacitorisraisedabovethesourceside,actingtoimprovethevoltageregulationofthefeeder.Sincethevoltageriseordropoccursinstantaneouslywithvariationsintheload,theseriescapacitorresponseasavoltageregulatorisfasterandsmootherthantheregulators.

Themaindrawbackofthiscapacitoristhehighvoltageproducedacrossthecapacitorterminalsundershort-circuitconditions.ThedropacrossthecapacitorisI X ,whereI isthefaultcurrentwhichismanytimesthefull-loadcurrentundercertaincircuitconditions.Itisessential,therefore,thatthecapacitoristakenoutofserviceasquicklyaspossible.Asparkgapwithahigh-speedcontactorcanbeusedtoprotectthecapacitorundertheseconditions.

f c

f

FIG.10.3Viewofathree-phasecapacitorbankona11-kVdistributionline

FIG.10.4Circuitdiagramwithoutandwithseriescompensation

Figures10.4and10.5showthelineanditsvoltagephasordiagramswithoutandwithseriescompensation.Thevoltagedropofthelinewithoutaseriescapacitorisapproximatelygivenby

V =I Rcosϕ+I X sinϕd r r L

FIG.10.5PhasordiagramsofFig.10.4

andthevoltagedropwithaseriescapacitor,

V =I Rcosϕ+I (X −X )sinϕ

whereX isthecapacitivereactanceoftheseriescapacitor.Acomparisonbetweenshuntandseriescapacitorsistabulatedbelow(Table10.1).

10.5.4Tap-changingtransformers

Atap-changingtransformerisastaticdevicehavinganumberoftapsettingsonitssecondarysideforobtainingdifferentsecondaryvoltages.Thebasicfunctionofthisdeviceistochangethetransformationratio,wherebythevoltageinthesecondarycircuitisvariedmakingpossiblevoltagecontrolatallvoltagelevelsatanyload.Thesupplymaynotbeinterruptedwhentapchangingisdonewithandwithoutload.

Typesoftap-changingtransformersare:

1. Off-loadtap-changingtransformer.2. On-loadtap-changingtransformer.

TABLE10.1Comparisonofshuntandseriescapacitors

Shuntcapacitor Seriescapacitor

1.Suppliesfixedamountofreactivepowertothesystematthepointwheretheyareinstalled.Itseffectisfeltinthecircuitfromthelocationtowardssupplysourceonly

1.Quantumofcompensationisindependentofloadcurrentandinstantaneouschangesoccur.Itseffectisfromitslocationtowardstheloadend

d r r L C

c

2.Itreducesthereactivepowerflowinginthelineandcauses:

1. Improvementofp.f.ofasystem

2. Voltageprofileimprovement

3. DecreaseskVAloadingonsource,i.e.,generators,transformers,andlineuptolocationandthusprovidesanadditionalcapacity

2.Itiseffective:

1. Ontielines,thepowertransferisgreater

2. Specifically,suitableforsituationswhenflickersduetorespectiveloadfunctionsoccur

3.Thelocationhastobeasneartotheloadpointaspossible.Inpractice,duetothehighcompensationrequired,itisfoundtobeeconomicaltoprovidegroupcompensationonlinesandsub-stations

3.Asathumbrule,thebestlocationis1/3rdofelectricalimpedancefromthesourcebus

4.AsfixedkVArissupplied,thismaysometimesresultinovercompensationinthelight-loadperiod.SwitchedkVArbanksarecomparativelycostlierthanfixedkVArandbecomenecessary

4.Asfull-loadcurrentistopassthrough,thecapacity(currentrating)shouldbemorethantheloadcurrent

5.Asthep.f.approachesunity,largercompensationisrequiredfortheimprovementofp.f.

5.Asseriescapacitorscarryfaultcurrent,specialprotectionisrequiredtoprotectfromfaultcurrent

6.Wherelinesareheavilyloaded,compensationrequiredwillbemore

6.Causessuddenrisesinvoltageatthelocation

7.Costofcompensationislowerthanthatofthecostrequiredforseriescapacitor

7.Costofaseriescapacitorishigherthanthatofashuntcapacitor

10.5.4.1Off-loadtap-changingtransformers

Thesimpletap-changingarrangementofatransformerisshowninFig.10.6.ThevoltagecanbevariedbyvaryinganumberoftappingsonthesecondarysideofthetransformerasshowninFig.10.6.

Figure10.6referstotheoff-loadtap-changingtransformer,whichrequiresthedisconnectionofthetransformerfromtheloadwhenthetapsettingistobechanged.

Theoutputofthesecondarysideofthetransformerchangeswiththechangeinthetappositionofthesecondarywinding.Thesecondaryvoltageisminimumwhenthemovablearmmakescontactwithstud1,whereasitismaximumwhenitisinpositionN.Whentheloadonthetransformerincreases,thevoltageacrosssecondaryterminalsdecreases.Thiscanbeincreasedtothedesiredvaluebyaddingthenumberofturnsonthesecondaryofthetransformerbychangingtaps.

FIG.10.6Off-loadtap-changingtransformerarrangement

Thus,inthecaseoftap-changingtransformers,themaindrawbackisthatthetapsarechangedonlyaftertheremovaloftheload.Thiscanbeovercomebyusinganon-loadtap-changingtransformerwithreactors.

10.5.4.2On-loadtap-changingtransformer

Tosupplyuninterruptedpowertotheload(consumer),tapchanginghastobeperformedwhenthesystemisonload.Thesecondarywindinginatap-changingtransformerconsistsoftwoidenticalparallelwindingswithsimilartappings.Forexample,1,2,…,Nand1′,2′,…,N′arethetappingsonboththeparallelwindingsofsuchatransformer.Thesetwoparallelwindingsare

controlledbyswitchesS andS asshowninFig.10.7(a).Inthenormaloperatingconditions,switchesS ,S ,andtappings1and1′areclosed,i.e.,boththesecondarywindingsofthetransformerareconnectedinparallel,andeachwindingcarrieshalfofthetotalloadcurrentbyanequalsharing.Thesecondarysideofthetransformerisataratedvoltageundernoload,whentheswitchesSandS areclosedandmovablearmsmakecontactwithstud1and1′,whereasitismaximum(abovetheratedvalue)undernoload,whenthemovablearmsareinpositionNandN′.Thevoltageatthesecondaryterminaldecreaseswithanincreaseintheload.Tocompensateforthedecreasedvoltages,itisrequiredtochangeswitchesfrompositions1and1′topositions2and2′(numberofturnsonthesecondaryisincreased).Forthis,openanyoneoftheswitchesS andS ,assumingthatS isopened.Atthisinstant,thesecondarywindingcontrolledbyswitchS carriesfull-loadcurrentthroughonewinding.Then,thetappingischangedtoposition2onthewindingofthedisconnectedtransformerandclosetheswitchS .Afterthis,switchS isopenedfordisconnectingitswinding,andchangethetappingpositionfrom1′to2′andthenswitchS isclosed.Similarly,tappingpositionscanbechangedwithoutinterruptingthepowersupplytotheconsumers.Theonlinetap-changingtransformerisshowninFig.10.7(b).

a b

a b

a

b

a b

a

b

a b

b

FIG.10.7(a)On-loadtap-changingtransformerarrangement;(b)on-linetap-changingtransfomer

Thismethodhasthefollowingdisadvantages:

1. Itrequirestwowindingswithratedcurrent-carryingcapacityinsteadofonewinding.

2. Itrequirestwooperationsforthechangeofasinglestep.3. Complicationsareintroducedinthedesigninordertoobtainahigh

reactancebetweentheparallelwindings.

10.5.5Boostertransformers

Theboostertransformerperformsthefunctionofboostingthevoltage.Itcanbeinstalledatasub-stationoratanyintermediatepointofline.

InthecircuitshowninFig.10.8(a),PandQarethetworelays.Thesecondaryoftheboostertransformerisconnectedinserieswiththelinewhosevoltageistobe

controlled,andtheprimaryoftheboostertransformerissuppliedfromaregulatingtransformerwithon-loadtap-changinggear.TheboostercanbebroughtintothecircuitbytheclosureofrelayQandtheopeningofrelayP,andviceversaasshowninFig.10.8(a).Thesecondaryoftheboostertransformerinjectsavoltageinphasewiththelinevoltages.Bychangingthetappingontheregulatingtransformer,themagnitudeofV canbechangedandthusthefeedervoltageV canberegulated.Theviewofboosteranddistributiontransformerconnection(lefttoright)isshowninFig.10.8(b).

Advantages

1. Itcanbeinstalledatanyintermediatepointinthesystem.2. Theratingoftheboostertransformerisabout10%thatofthemain

transformer(productofcurrentandinjectedvoltage).

Disadvantages

Whenusedinconjunctionwithmaintransformer:

1. Moreexpensivethanatransformerwithon-loadtapchanges.2. Lessefficientduetolossesinbooster.3. Requiresmorespace.

Q

F

FIG.10.8(a)Boostertransformer;(b)viewofboosteranddistributiontransfomerconnection(lefttoright)

10.5.6Synchronouscondensers

Asynchronouscondenser(synchronousphasemodifier)isasynchronousmotorrunningwithoutmechanicalload.Itisconnectedinparallelwiththeloadatthereceivingendoftheline.Dependinguponitsexcitation,iteithergeneratesorabsorbsthereactivepower.Ittakesleadingcurrentwhenitsfieldisoverexcited,i.e.,abovenormalspeedandtakeslaggingcurrentwhenitisunderexcited.Thus,thecurrentdrawnbyasynchronousphasemodifiercanbevariedfromlaggingtoleadingbyvaryingitsexcitation.Itisaveryconvenientdevicetokeepthereceiving-endvoltage

constantunderanyconditionofload.Italsoimprovesthep.f.andtheoutputcanvarysmoothly.

TABLE10.2Comparisonofsynchronouscondenserandstaticcapacitors

Synchronouscondenser Staticcapacitors

1.Harmonicsinthevoltagedoesnotexist

1.Largeharmonicsareproducedinthesystem

2.Powerfactorvariationisstepless(uniform)

2.Powerfactorvariesinsteps

3.Itallowsoverloadingforashortperiod

3.Itdoesnotallowanyoverloading

4.Powerlossismore 4.Powerlossisless

5.ItismoreeconomicalinthecaseoflargekVAr

5.ItismoreeconomicalforsmallkVArrequirement

6.Failurerateislessand,therefore,thisismorereliable

6.Failurerateismoreand,therefore,itislessreliable

Asynchronousphasemodifierhasasmallershaftandbearingandhigherspeedascomparedtoasynchronousmotorusedformechanicalloads.Asynchronousphasemodifierhashigheroverallefficiencyascomparedwithasynchronousmotor.

Advantages

1. Flexibilityforuseinallloadconditionsbecausewhenthemachineisunderexcited,itconsumesreactivepower.

2. ThereisasmoothvariationofreactiveVArsbysynchronouscapacitors.

3. Itcanbeoverloadedforshortperiods.

Disadvantages

1. Possibilityoffallingoutofcontrolincaseofsuddenchangesinvoltage.

2. Thesemachinesaddtoshort-circuitcapacityofthesystemduringfaultcondition.

AcomparisonbetweensynchronouscondenserandstaticcapacitorsispresentedinTable10.2.

10.6RATINGOFSYNCHRONOUSPHASEMODIFIER

Anexpressionofsending-endvoltageintermsoftransmissionlineconstantsis

where

= V ∠δ=sending-endvoltage

= receiving-endvoltage(referencephasor)

Ī = I ∠−φ =receiving-endcurrent

Ā = A∠α

= B∠βarethelineconstants

Equation(10.1)canbewritteninaphasorformas

V

∠δ= AV ∠α+BI ∠(β−φ )

= AV cosα+jAV sinα+BI cos(β−φ )+jBI

sin(β−φ )

(10.2)

TherealpartofEquation(10.2)is

s

s

r

r r r

s r r r

r r r r r

r

V cosδ=AV cosα+BI cos(β−φ )(10.3)

andtheimaginarypartis

V sinδ=AV sinα+BI sin(β−φ )(10.4)

SquaringandaddingEquations(10.3)and(10.4),weget

V = A V +B I +2AB I cosαcos(β−φ )+2ABV I sin

αsin(β−φ )

= A V +B I +2ABV I cos(α−β+φ )

= A V +B I +2ABV I [cos(α−β)cosφ −sin(α−β)

sinφ ]

(10.5)

Realpoweratreceivingend,P =V I cosϕ

Reactivepoweratreceivingend,Q =V I sinϕ

Receiving-endcurrentcanbewrittenas

I =I cosφ −jI sinφ (∵laggingp.f.)

=I −jI

∴I =I +I

where

SubstitutingtheabovequantitiesinEquation(10.5),wehave

V =A V +B I +2ABP cos(α−β)−2ABQ sin(α−β)(10.6)

InEquation(10.6),I isreplacedbyI andIexpressions,

s r r r

s r r r

s

r r r r r r r

r

r r r r r

r r r r r

r

r r r r

r r r r

r r r r r

p q

r p q

s r r r r

r p q

2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2

2 2 2 2 2

2 2 2

Equation(10.7)isusefulforcalculatingthesending-endvoltagebyknowingthevaluesofA,B,α,β,P ,Q ,andV(or)sometimesthesending-endandreceiving-endvoltagesarefixedandA,B,α,β,P ,andQ aregiven.Itisrequiredtofindouttheratingofthephasemodifier.Inthiscase,therequiredquantityisQ ,whereQ isthenetreactivepoweratthereceivingendandnotthereactivepowerfortheload.So,ifthenetreactivepowerrequiredtomaintaincertainvoltagesatthetwoendsisknown,theratingofthephasemodifiercanbefound.

Example10.1:A3-ϕoverheadlinehasresistanceandreactanceperphaseof25and90Ω,respectively.Thesupplyvoltageis145kVwhiletheload-endvoltageismaintainedat132kVforallloadsbyanautomaticallycontrolledsynchronousphasemodifier.IfthekVArratingofthemodifierhasthesamevalueforzeroloadsasforaloadof50MW,findtheratingofthesynchronousphasemodifier.

Solution:

WehaveV =AV +BI (10.9)

Fromgivendata:

Sending-endvoltage,

Receiving-endvoltage,

r r r

r r

r r

s r r

Lineimpedance,Z=25+j90

Assumingshort-linemodel,V =V +I Z

ComparingEquations(10.9)and(10.10),wehave

A=1;α=0,

B=93.4β=74.47°

P =V I cosϕ =50MW(given)

SubstitutingthesevaluesinEquation(10.8),weget

1,200.92×10 =3,755.98×10 +1.50Q ×10 +2,500.7×10 +179.98Q

1,200.92=3,755.98+1.50Q +2,500.7+179.98Q

1.50Q +179.98Q +5,055.76=0

Solvingtheaboveequation,weget

Powerfactorangleatthereceiving-endvoltage,ϕ =−41.9°

s r r

r r r r

r

r

r r

r r

r

6 6 2 6

6

2

2

∴Thepowerfactoriscosϕ =0.7442leading

∴Theratingofthesynchronousmodifier=44.87MVArs

Example10.2:A3-ϕfeederhavingaresistanceof3Ωandareactanceof10Ωsuppliesaloadof2MWat0.85p.f.lag.Thereceiving-endvoltageismaintainedat11kVbymeansofastaticcondenserdrawing2.1MVArfromtheline(Fig.10.9).Calculatethesending-endvoltageandp.f.Whatistheregulationandefficiencyofthefeeder?

Solution:

Loadcurrent,

Shuntbranchcurrent,

Receiving-endcurrent, I = I +I

= 123.5∠−31.79°+110.22∠90°

= 105−j65+j110.22

= 105+j45.22

= 114.3∠23.3°A

ThevectordiagramisshowninFig.10.10.Here,thecurrentisaleadingcurrent.

r

rL C

FromthecircuitdiagramshowninFig.10.9,

Thesending-endvoltage,V =V +I Z



=6,350.85∠0+1,193.33∠96.6=6,350.85+j0−137.16+j1,185.42

=6,213.69+j1,185.42

=6,325.74∠10.8V∴Thesending-endvoltage,

FromthephasordiagramshowninFig.10.10,

s r r

sending-endp.f.=cos(23.3—10.8)=cos12.5=0.976

Example10.3:Asingle-circuit3-ϕ220-kVlinerunsatno-load.Voltageatthereceivingendofthelineis205kV.Findthesending-endvoltageifthelinehasaresistanceof21.7Ω,areactanceof85.2Ω,andthetotalsusceptanceof5.32×10 (Fig.10.11).Thetransmissionlineistoberepresentedby∏-model.

Solution:

Thesending-endvoltageV differsfromthereceiving-endvoltageV bythevalueofvoltagedropduetochargingcurrentinthelineimpedance,asshowninFig.10.11.Withthequadrate-axiscomponentofthevoltagedropbeingneglectedasshowninFig.10.12,wefind∣V ∣


1

2

1

-4


Wecanalsofind∣V ∣fromexpression

,whereQ =−Q becausethecurrentis

leadingandP =0.

Wearegivenlinevoltageatthereceivingend,therefore,onperphase,wehave

Hence,

Sending-endvoltage,linetoline

KEYNOTES

1

2 c

2

Sourcesandsinksofareactivepoweraresynchronousmachine,overheadlines,transformers,cables,andloads.Thevoltage-controlequipmentislocatedatgeneratingstations,transformerstations,andfeeders.Thevariousmethodsforvoltagecontrolare:excitationcontrol,shuntcapacitors,seriescapacitorsbyusingtap-changingtransformers,boosters,andsynchronouscondensers.Excitationcontrol:Thismethodisusedonlyatthegeneratingstation.Duetothevoltagedropinthesynchronousreactanceofarmature,whenevertheloadonthesupplysystemchanges,theterminalvoltageofthealternatoralsochanges.Thiscanbekeptconstantbychangingthefieldcurrentofthealternatoraccordingtothechangesinload.Thisisknownasexcitationcontrol.Shuntcapacitorsandreactors:Shuntcapacitorsareusedforlaggingp.f.circuits;whereasreactorsareusedforleadingp.f.circuitssuchascreatedbylightlyloadedcables.Seriescapacitor:Itisinstalledinserieswithtransmissionlinestoreducethefrequencyofvoltagedrops.Tap-changingtransformers:Thebasicoperationofatap-changingtransformerisbychangingthetransformationratio,thevoltageinthesecondarycircuitisvaried.Boostertransformers:Theboostertransformerperformsthefunctionofboostingthevoltage.Itcanbeinstalledatasub-stationoranyintermediatepointofline.Synchronouscondensers:Itisconnectedinparallelwiththeloadatthereceivingendoftheline.Itcaneithergenerateorabsorbreactivepowerbyvaryingtheexcitationofitsfieldwinding.


1. Whatarethedifferentmethodsofvoltagecontrol?

Thefollowingmethodsaregenerallyemployedforcontrollingthereceiving-endvoltage.

1. Byexcitationcontrol.2. Byusingtap-changingtransformer.3. Auto-transformertapchanging.4. Boostertransformer.5. Inductionregulators.6. Bysynchronouscondensers.

2. Whatismeantbyexcitationvoltagecontrol?

Duetothevoltagedropsinthesynchronousreactanceofthearmature,whenevertheloadonthesupplysystemchanges,theterminalvoltageofthealternatorchangescorrespondingly.Thiscanbekeptconstantbychangingthefieldcurrentofthealternatoraccordingtothechangesinload.

3. Whatarethedisadvantagesoftap-changingtransformers?

1. Duringswitching,theimpedanceoftransformerisincreasedandtherewillbeavoltagesurge.

2. Therearetwiceasmanytappingsasthevoltagesteps.

4. Whatisthesynchronouscondenser?

Asynchronousmotortakesaleadingcurrentwhenoverexcitedandthereforebehavesasacapacitor.Anoverexcitedsynchronousmotorrunningonno-loadisknownasasynchronouscondenser.

5. Whatisaboostertransformer?

Thetransformer,whichisusedtocontrolthevoltageofthetransmissionlineatapointfarawayfromthemaintransformer,isknownasboostertransformer.

6. Howdoesashunt-capacitorbankcontrolthesystemvoltageunderlightloadsandheavyloads?

Theshunt-capacitorbankprovidedwithfixedandvariableelementsmaybeeitherremovedfromoraddedtothebanktodecreaseorincreasethecapacitanceunderno-loadandheavy-loadconditions,respectively.

7. Underwhatconditiondoesasynchronousmotortakealeadingcurrent?

Thesynchronousmotortakesleadingcurrentwhenitsfieldisoverexcitedunderhigh-loadconditions.

8. Whenistheshunt-inductorcompensationrequired?

Theshunt-inductorcompensationisrequiredwhenevertheloadingislessthanthesurgeimpedanceloading.


1. Thevoltageofthepowersupplyattheconsumer’sservicemustbeheldsubstantially_____

1. Constant.2. Smoothvariation.3. Randomvariation.4. Noneofthese.

2. Lowvoltagereducesthe_____fromincandescentlamps.

1. Poweroutput.2. Powerinput.3. Lightoutput.4. Current.

3. Motorsoperatedatbelownormalvoltagedrawabnormally_____currents.

1. Low.

2. High.3. Medium.4. Noneofthese.

4. Permissiblevoltagevariationis_____.

1. ±10%.2. ±20%.3. ±50%.4. ±5%.

5. Bydrawinghighcurrentsatlowvoltages,themotorsget_____.

1. Overheated.2. Cool.3. Constantheat.4. Noneofthese.

6. Domesticcircuits’supplyvoltageis_____.

1. 230V.2. 110V.3. 240V.4. 220V.

7. Thevoltagemaynormallyvarybetweenthelimitsof_____.

1. 210and230V.2. 230and240V.3. 230and520V.4. 210and235V.

8. Abovenormalvoltagesreducesthe_____ofthelamps.

1. Life.2. Strength.3. Lighting.4. Color.

9. Thevoltageatthebuscanbecontrolledbytheinjectionof_____powerofthecorrectsign.

1. Real.2. Reactive.3. Complex.4. Bothrealandreactive.

10. Generalmethodsofvoltagecontrolare_____.

1. Useoftap-changingtransformer.2. Synchronouscondensers.3. Staticcapacitors.4. Allofthese.

11. Useofthyristor-controlledstaticcompensatorsis_____.

1. Voltagecontrol.2. Powercontrol.3. Currentcontrol.4. Noneofthese.

12. Anoverexcitedsynchronousmachineoperatedasgeneratorormotorgenerates_____

1. kVA.

2. kVAr.3. kW.4. kI.

13. Synchronousmotorrunningatno-loadandoverexcitedloadisknownas_____

1. Synchronouscondenser.2. Shuntcapacitor.3. Seriescapacitor.4. Noneofthese.

14. Theexcitation-controlmethodisonlysuitablefor_____lines.

1. Short.2. Medium.3. Long.4. Allofthese.

15. Itis_____tomaintainthesamevoltageatbothendsoftransmissionlinesbythesynchronous-condensermethod.

1. Economical.2. Noteconomical.3. Difficult.4. Easy.

16. Shuntcapacitorsandreactorsareusedacrosslightlyloadedlinestoabsorbsomeoftheleading_____againtocontrolthevoltage.

1. VArs.2. VA.3. VBRS.4. Noneofthese.

17. Disadvantagesofshuntcapacitorsare_____.

1. Fallofvoltage.2. ReductioninVArs.3. Reductionineffectiveness.4. Allofthese.

18. _____reducestheinductivereactancebetweentheloadandthesupplypoint.

1. Shuntcapacitor.2. Shuntreactor.3. Seriescapacitor.4. Transformer.

19. Thedisadvantageofaseriescapacitoristhatitproduces_____voltageacrossthecapacitorundershort-circuitcondition.

1. Low.2. High.3. Verylow.4. Either(a)or(b).

20. Asparkgapwithahigh-speedcontactoristhe_____usedforshuntcapacitor.

1. Protectivedevice.2. Control.3. Fuse.

4. Circuitbreaker.

21. Thedifferenttypesoftap-changingtransformersare_____.

1. Off-load.2. On-load.3. Both(a)and(b).4. Either(a)or(b).

22. Thepurposeofusingboostertransformersis_____thevoltage.

1. Transforming.2. Bucking.3. Boosting.4. Buckingandboosting.

23. Moreexpensive,lessefficient,andtakemorefloorareaarethedisadvantagesofthe_____transformer.

1. Off-loadtap.2. On-load.3. Booster.4. Induction.

24. Ifasynchronousmachinegetsoverexcited,takeslaggingVArsfromthesystemwhenitisoperatedasa_____.

1. Synchronousmotor.2. Synchronousgenerator.3. Either(a)or(b).4. Synchronousphasemodifier.

25. Forasynchronousphasemodifier,theloadangleis_____.

1. 0°.2. 25°.3. 30°.4. 50°.

REVIEWQUESTIONS

1. Whyisvoltagecontrolrequiredinpowersystems?Mentionthedifferentmethodsofvoltagecontrolemployedinpowersystem.Explainonemethodofvoltagecontrolindetailgivinganeatconnectiondiagram.

2. Whyisexcitationcontrolnecessaryinanalternator?3. Describe‘off-load’and‘on-load’tap-changingtransformers.4. Explainthefunctionofasynchronousphasemodifierplacedat

thereceivingendofthetransmissionline.5. Showwiththeaidofavectordiagram,howthevoltageatthe

receivingendofatransmissionlinecanbemaintainedconstantbytheuseofasynchronousphasemodifier.

PROBLEMS

1. A3-ϕ33-kVoverheadtransmissionlinehasaresistanceof5Ω/phaseandareactanceof18Ω/phasewiththehelpofa

synchronousmodifier,thereceiving-endvoltageiskeptconstantat33kV.CalculatethekVAofthephasemodifieriftheloadatthereceivingendis60MWat0.85p.f.lagging.Whatwillbethemaximumloadthatcanbetransmitted?

2. Ifthevoltageatthesendingendistobemaintainedat66kV,determinetheMVArofthephasemodifiertobeinstalledfora3-ϕoverheadtransmissionlinehavinganimpedanceof(7+j19)Ω/phase,deliveringaloadof80MWat0.85p.f.laggingandwithvoltage66kV.

3. A3-ϕinductionmotordelivers450HPatanefficiencyof95%whentheoperatingp.f.is0.85lag.Aloadedsynchronousmotorwithapowerconsumptionof110kWisconnectedinparallelwiththeinductionmotor.CalculatethenecessarykVAandtheoperatingp.f.ofthesynchronousmotoriftheoverallp.f.istobeunity.

11

ModelingofPrimeMoversandGenerators

OBJECTIVES


developthemodelingofhydraulicandsteamturbines

discussreheatandnon-reheattypeofsteamturbineconfigurations

developthesimplifiedmodelsofasynchronousmachine

discusstheapplicationofPark’stransformationtosynchronousmachinemodeling

studytheswingequationmodelofasynchronousmachine

11.1INTRODUCTION

Forthestudyofpowersystemdynamics,thesimpleequivalentmethodsofthesynchronousgeneratorsarenotadequate.Theaccuratedescriptionofpowersystemdynamicsrequiresthedetailedmodelsofsynchronousmachines.

Inthiswork,theabove-requireddetailedmodelsofsynchronousmachinesaredevelopedfromthebasicequations.Thetime-invariantsynchronousmachineequationsaredevelopedthroughtheapplicationofPARK’Stransformationandwiththeuseofphasevariables.

Thedetailedsynchronousmachinemodelisderivedaccompaniedbyitsrepresentationusingperunitquantitiesandtheconsiderationofd-axisandq-axisequivalentcircuits.

First,themostsimplifiedmodelofthesynchronousmachineisdiscussedandlaterthedetailedmodelisdeveloped.

Theeffectofsaliencyisdiscussed.Thesteady-statemodelanddynamicmodelrepresentationsofasynchronousmachinearediscussed.Finally,themechanicalbehaviorofasynchronousmachineisstudiedintheformofderivationofswingequation.

Thespeedcontrolofaprimemoverisessentialforthefrequencyregulationofapowersystemnetwork.Thisisachievedbyprovidingaspeed-governormechanism.Theparalleloperationforgeneratorsrequiresdroopcharacteristicsincorporatedinthespeedgoverningsystemtosecurestabilityandeconomicdivisionofload.Hence,tomaintainconstantfrequency,itisnecessaryfortheprimemovercontroltoadjustthepowergenerationaccordingtoeconomicdispatchofloadamongvariousunits.

Theprimemovercontrolsareclassifiedintothreedifferentcategoriesas:

1. Primarycontrol(speed-governorcontrol).2. Secondarycontrol(loadfrequencycontrol(LFC)).3. Tertiarycontrolinvolvingeconomicdispatch.

Inthisunit,modelsofahydraulicturbinewithapenstocksystemandasteamturbinearedeveloped.Inmodelingthesteamturbine,sixcommonsystemconfigurationsintheformofnon-reheattypeandreheattypeareconsidered.

11.2HYDRAULICTURBINESYSTEM

Therepresentationofahydro-turbineishighlydependentonthetypeofprimemoverbecauseeachtypehasdifferentspeedcontrolmechanisms.

Accordingtothetypeofheadconditions,therearethreetypesofhydro-turbines.

1. Lowhead:Upto100”height,specificspeed(90−180rpm),speed

(100−400rpm).

Thesearepropellertypeofreactionturbines.

2.Mediumhead:50”−1,000”height,specificspeed(90−200rpm),speed(100−400rpm).

TheseareFrancistypeofreactionturbines.

3. Highspeed:From800”andaboveheight,specificspeed(3−7rpm),speed(120−720rpm).Theseareofimpulsetypeofturbines(Peltonwheel).

11.2.1Modelingofhydraulicturbine

Thetransientcharacteristicsofhydro-turbinesareobtainedbythedynamicsofwaterflowinthepenstock.AhydraulicturbinewithapenstocksystemisasshowninFig.11.1.

Letlbethelengthofthepenstockinm,Qthedischargeofwatertotheturbineinm /s,νthevelocityofwaterdischargeinm/s,andHtheoperatingheadinm.

Foraparticularchangeinload,letΔHbethep.u.changeinhead,ΔNthep.u.changeinspeed,ΔXthep.u.changeinturbinegateopening,ΔQthep.u.changeinwaterdischarge,andΔTthep.u.changeinturbinetorque.

Ithasbeenprovedthat

3

FIG.11.1Hydraulicturbinewithpenstocksystem

whereZisthenormalizedpenstockimpedance,τ theelasticlimitofpenstock,andsthecomplexfrequency=σ+jω.

Accordingtofirst-orderPADE’sapproximation,wehave

wherethetimeconstantt iscalledthewaterstartingtimeorwatertimeconstant.

Thechangesinflowandtorqueoftheturbineaboutasteady-stateconditioncanberepresentedbythefollowinglinearizedequations:

ΔQ=a ΔH+a ΔN+a Δx(11.3)

ΔT=a ΔH+a ΔN+a Δx(11.4)

e

w

11 12 13

21 22 23

wherea ,a ,a ,a ,a ,anda areconstantsandareexpressedas

Thechangeinturbinetorqueduetothespeedchangescanbeneglectedincomparisontootherchangessincethechangeinspeedisrelativelysmall.

∴ΔQ=a ΔH+a ΔX

ΔQ=a ΔH+a ΔX

∴ByusingEquations(11.2)andEquations(11.3),wehave

ForanidealturbinewithvalveopeningX ,

a = 0.5X

a = 0

a = 1.0

11 12 13 21 22 23

11 13

21 23

0

11 0

12

13

a = 1.5X

a = −1.0

a = 1.0

Atfull-load,X =1.0p.u.

Equations(11.5)isthetransferfunctionofclassicalhydraulicturbineofpenstockmodel.

Theapproximatelinearmodelsforhydro-turbinesareasshowninFig.11.2.

FIG.11.2(a)Linearizedmodelofhydraulicturbine;(b)linearizedmodelofan

idealhydraulicturbine

TheinputP forthehydraulicturbineisgivenfromthespeedgovernor.Itisthegateopeningexpressedinp.u.

21 0

22

23

0

GV

Thevalueofτ liesintherangeof0.5s−5.0s.thetypicalvalueofτ isaround1.0s.

11.2.1.1Calculationofwatertimeconstant(τ )

Watertimeconstantisassociatedwiththeaccelerationtimeforwaterinthepenstockbetweentheturbineinletandtheforebayorbetweentheturbineinletandthesurgetankifitexists.

Thewatertimeconstantτ isgivenby

wherelisthelengthofpenstockinm,νthevelocityofwaterflowinm/s,H thetotalheadinm,andgtheaccelerationduetogravityinm/s .

Intermsofpowergeneratedbytheplant‘P’,thewatertimeconstantτ isexpressedas

or

wherePisthegeneratedelectricalpowerinkWandisgivenas

whereAistheaveragepenstockareainm andeistheproductofefficienciesofturbineandgenerator:

i.e.,e=η ×η

ω

ω

ω

ω

T

w

turbine generator

2

2

11.3STEAMTURBINEMODELING

Thetwocommonsteamturbinesystemconfigurationsare:

1. Non-reheattype.2. Reheattype.

11.3.1Non-reheattype

Asimplenon-reheattypeturbineismodeledbyasingletimeconstant.

Thefunctionalblockdiagramrepresentationofanon-reheattypeofsteamturbineisasshowninFig.11.3.

Theapproximatelinearmodelofthenon-reheatsteamturbineisshowninFig.11.4.

Here,P representsthepoweratthegateofthevalveoutlet,τ thesteam-chesttimeconstant,andP themechanicalpowerattheturbineshaft.

FIG.11.3Blockdiagramrepresentationofanon-reheattypeofsteamturbine

FIG.11.4Approximatelinearmodelofanon-reheatsteamturbine

11.3.2Reheattype

Therearemainlytwoconfigurationsandtheyare:

GV

CH m

1. Tandemcompoundsystemconfiguration.2. Cross-compoundsystemconfiguration.

Thesetwoconfigurationsarefurtherclassifiedintothefollowingtypes:

1. Tandemcompound,singlereheattype.2. Tandemcompound,doublereheattype.3. Cross-compound,singlereheattypewithtwolow-pressure(LP)

turbines.4. Cross-compound,singlereheattypewithsingleLPturbine.5. Cross-compound,doublereheattype.

Atandemcompoundsystemhasonlyoneshaftonwhichalltheturbinesaremounted.Theturbinesareofhighpressure(HP),lowpressureLP,andintermediatepressure(IP)turbines.Sometimes,theremaybealsoaveryhighpressure(VHP)turbinemountedontheshaft.

ThefunctionalblockdiagramrepresentationsoftandemcompoundreheatsystemconfigurationsandtheirlinearmodelrepresentationsareshowninFig.11.5−Fig.11.8.

11.3.2.1Tandemcompoundsinglereheatsystem

ThetandemcompoundsinglereheatsystemisshowninFig.11.5andFig.11.6.

Allcompoundsteamturbinesusegovernor-controlledvalves,attheinlettotheHPturbine,tocontrolthesteamflow.Thesteamchest,reheater,andcross-overpipingintroducedelays.Thesetimedelaysarerepresentedby:

τ =Steam-chesttimeconstant(from0.1to0.4s)

τ =Reheattimeconstant(from4to11s)

τ =Cross-overtimeconstant(from0.3to0.5s)

Thefractionsoftotalturbinepowerarerepresentedby:

F =FractionofHPturbinepower(typicalvalueis0.3)

F =FractionIPturbinepower(typicalvalueis0.3)

CH

RH

CO

HP

IP

F =FractionofLPturbinepower(typicalvalueis0.4)

11.3.2.2Tandemcompounddoublereheatsystem

ThetandemcompounddoublereheatsystemisshowninFigs.11.7andFigs.11.8.

FIG.11.5Functionalblockdiagramrepresentation—tandemcompoundsinglereheatsystem

FIG.11.6Approximatelinearmodel—tandemcompoundsinglereheatsystem

LP

FIG.11.7Functionalblockdiagramrepresentation—tandemcompounddoublereheatsystem

FIG.11.8Approximatelinearmodel—tandemcompounddoublereheatsystem

Thetimedelaysarerepresentedby:

τ =firstreheattimeconstant

τ =secondreheattimeconstant

11.3.2.3Cross-compoundsinglereheatsystem(withtwoLPturbines)

Thecross-compoundsinglereheatsystemwithtwoLPturbinesisshowninFigs.11.9and11.10.

11.3.2.4Cross-compoundsinglereheatsystem(withsingleLPturbine)

RH1

RH2

Thecross-compoundsinglereheatsystemwithasingleLPturbineisshowninFigs.11.11and11.12.

11.3.2.5Cross-compounddoublereheattype

Thecross-compounddoublereheat-typesystemisshowninFigs.11.13and11.14.

FIG.11.9Functionalblockdiagramrepresentation

FIG.11.10Approximatelinearmodel



11.4SYNCHRONOUSMACHINES

Thesynchronousmachineisthemainorbasiccomponentoftheelectricalpowersystem.Itmayoperateeitherasageneratororasamotor.Inpowersystemoperation,thesynchronousmachineisoftenrequiredtosupplypoweratpowerfactorsotherthanunity,whichnecessitatesthesupplyorabsorptionofreactivepower.

Asynchronousmachineconsistsoftwobasicparts:thestatorandtherotor.Thetwobasicrotordesignsaresalientpoletypeandnon-salient-poletype.

11.4.1Salient-pole-typerotor

Inthistypeofrotor,thepolesprojectfromtherotorandexhibitanarrowairgapunderthepolestructureandawiderairgapbetweenthepoles.ThistypeofrotorstructureisshowninFig.11.15.

11.4.2Non-salient-pole-typerotor

ItconsistsofacylindricalrotorasshowninFig.11.16,oftenmadefromasinglesteelforging,inwhichthefieldwindingisembeddedinlongitudinalslotsmachinedinitsstructure.

Themathematicalmodelingofasynchronousmachineiscomplicatedbecauseofitsmultitudeofwindings,allcharacterizedbytime-varyingself-inductancesandmutualinductances.

11.5SIMPLIFIEDMODELOFSYNCHRONOUSMACHINE(NEGLECTINGSALIENCYANDCHANGESINFLUXLINKAGES)

Themostsimplifiedmodelofasynchronousgeneratorforthepurposeoftransientstabilitystudiesisaconstantvoltagesourcebehindproperreactance.Thevoltagesourcemaybesub-transient,transient,orsteadystateandthereactancemaybethecorrespondingreactances.

FIG.11.15Salient-pole-typerotorstructure

FIG.11.16Non-salient-pole-typerotorstructure

Inthismodel,saliencyandchangesinthefluxlinkagesareneglected.Howevertounderstandthismodel,letusconsiderasynchronousgeneratoroperatingatno-loadbeforea3-ϕshortcircuitisappliedatitsterminals.Thecurrentflowinginthesynchronousgeneratorjustafterthe3-ϕshortcircuitoccursatitsterminalsissimilartothecurrentflowsinanR–Lcircuitwhenanacvoltageissuddenlyapplied.Hence,thecurrentwillhaveboththeAC(i.e.,steadystate)componentaswellasthedc(i.e.,transient)component,whichdecaysexponentiallywiththetimeconstantL/R Ifthedccomponentisneglected,theoscillographoftheACcomponentofthecurrentthatflowsinthesynchronousgeneratorjustafterthefaultoccurswillhavetheshapeasshowninFig.11.17.

Justafterthefault,thecurrentismaximumastheairgapflux,whichgeneratesvoltage,ismaximumattheinstantthefaultoccursthanafewcycleslaterasthearmaturereactionfluxproducedduetoaverylarge

.

laggingcurrentinthearmatureprovidesnearlyademagnetizingeffect.

FromFig.11.17,letOAbethepeakvalueofsymmetricalACcurrent(neglectingDCcomponent),alsoknownaspeakvalueofthesub-transientcurrent:

∴RMSvalueofsub-transientcurrent

(11.6)

Now,ifthefirstfewcycles,wherethecurrentdecrementisveryfast,areneglectedandthecurrentenvelopeisextendeduptozerotime,theinterceptOBisobtained:

OB=Peakvalueofthetransientcurrent

∴RMSvalueoftransientcurrent (11.7)

However,thesteady-statevalueoftheshort-circuitcurrent(i.e.,sustainedvalueofshort-circuitcurrent)

(11.8)

Sincetheexcitationisaconstantfromno-loadtotheinstantwhenthe3-ϕshortcircuitoccurs,theexcitationvoltage‘E ’inthesynchronousgeneratorwillremainconstantandisknownasan‘open-circuitvoltageortheno-loadinducedemf’,andisrepresentedasshowninFig.11.18.

g

FIG.11.17Oscillographofthecurrentinthesynchronousgenerator

FIG.11.18Equivalentcircuitofthesynchronousgenerator

Thephasordiagramofanon-salient-polesynchronousgeneratorforsteady-stateanalysisisasshowninFig.11.19.

Now,themachineequationbecomes

E =V+I R +jI X (11.9)

whereE istheexcitationvoltage(or)open-circuitvoltage,

Visthefull-loadterminalvoltage,

g a a a s

g

I isthearmaturecurrent,

R isthearmatureresistance/phase,

X isthesynchronousreactance/phase,

ϕisthephaseanglebetweenVandI ,

δisthetorqueangleorpowerangle,and

θtheimpedanceangle

Itisseenthatthecurrentinthesynchronousgeneratorchangingfromsub-transientstate(I″)totransientstate(I′)andtosteadystate(I)andhencethesynchronousreactanceofthegeneratormustchange,asE isconstant,fromsub-transientreactance(X″)totransientreactance(X′)tosteady-statereactance(X).

FIG.11.19Phasordiagramofanon-salient-polesynchronousgeneratorforsteady-stateanalysis

i.e.,

a

a

s

a

g

Thearmaturereactionfluxisproducedbyalargelaggingcurrentasthiscurrentislimitedonlybyarmatureimpedance,wherewindingresistanceisnegligiblecomparedtosynchronousreactance,X .

Thisarmaturereactionfluxatthisinstantisnearlydemagnetizinginnaturebecauseitactsalongthedirectaxisofthemachine;theabovereactancesarereferredtoasdirectaxisreactances,i.e.,

directaxissub-transientreactance

directaxistransientreactance

directaxiscomponentofsteadystateor

synchronousreactance(11.11)

HereI″,I′,andIarethesub-transient,transient,andsteady-statevalueofshort-circuitcurrents,respectively,andE isexcitation(or)open-circuitvoltage(orno-loadinducedemf)inthearmature.Hence,thesimplestmodelofthesynchronousgeneratorisaconstantvoltage‘E ’inserieswiththeproperimpedance(or)reactance,i.e.,X″orX′ orX asshowninFig.11.20.

s

g

g

d

d d

FIG.11.20Simplifiedmodelofsynchronousgeneratorbyneglectingthesaliencyandfluxlinkagechanges

FIG.11.21Representationofsynchronousgenerator

Whenthesynchronousmachineconnectedtothepowersystemisoperatingatloadbeforethefaultoccurs,thesynchronousgeneratorisrepresentedbyanappropriatevoltagesourcebehindtherespectivereactancesasshowninFig.11.21.

Thismodeling(or)representationcaneasilybeobtainedforanyfaultinthepowersystemwiththehelp

ofThevenin’sequivalentcircuit,fromwhichitisclearthatthefluxlinkagesandhencetheinternalvoltageofthemachineremainconstant,butonlyitsphaseanglechanges.

Now,themachineequationsofthemodelofFig.11.21willbeexpressedas

E″ =V+jIX″

E′ =V+jIX′

E =V+jIX (11.12)

whereE “,E ′andE aresub-transient,transientorsteady-stateexcitationvoltagesofthesynchronousgenerator,respectively.Vistheterminalvoltageofthesynchronousgenerator,andIisthecurrentinthesynchronousgenerator.

Forthesynchronousmotor,Equations(11.12)mayobtainthefollowingformation.

E″ =V+jIX″

E′ =V+jIX′

E =V+jIX (11.13)

11.6EFFECTOFSALIENCY

Thesalient-polerotorisshowninFig.11.15.Ithasadirectaxisofrotor-fieldwinding,i.e.,d-axisandquadratureaxis,i.e.,theq-axisoftherotor-fieldwinding.Thed-axisandtheq-axisrevolvewiththerotor,whilethemagneticaxesofthethreestatorphasesremainfixed.

Attheinstantoftime,θistheanglefromtheaxisofPhase-atothed-axis.ThecorrespondinganglefromthePhase-baxistothed-axisisθ+240°orθ−120°.TheanglefromthePhase-caxisisθ+120°.Astherotorturns,θvarieswithtime,withaconstantrotorangularvelocity,ω,i.e.,θ=ωt.

g d

g d

g d

g g g

g d

g d

g d

Inordertoincludetheeffectofsaliency,thesimplestmodelofsynchronousmachinecanberepresentedbyafictitiousvoltage‘E ’locatedattheq-axis.Thed-axisistakenalongthemainpoleaxiswhiletheq-axislagsthed-axisby90°.ThenthevoltageE isexpressedintermsoffull-loadterminalvoltageVandfull-loadarmaturecurrentI as

E =V+I R +jIX X (11.14)

whereX isthequadratureaxissynchronousreactance.

TheequivalentcircuitandphasordiagramforthecaseofeffectofsaliencyaregiveninFigs.11.22(a)and(b).

Theexcitationvoltage(or)open-circuitvoltagewillbecalculatedas

E =V+I R +jI X +jI X (11.15)

whereX =X+X andX =X+X .X isthedirectaxisarmaturesynchronousreactance,

X thequadratureaxisarmaturesynchronousreactance,

X thed-axiscomponentofarmaturemagnetizingreactance,and

X theq-axiscomponentofthearmaturemagnetizingreactance.

q

q

a

q a a a q

q

g a a ad d aq q

d l ad q l aq d

q

ad

aq

FIG.11.22Effectofsaliency;(a)equivalentcircuit;(b)phasordiagram

X correspondstothed-axiscomponentofthearmaturereactionflux,ϕ ,andX correspondstotheq-axiscomponentofthearmaturereactionfluxϕ .

Now,thephasordiagramincludingtheeffectofsaliencyisdrawnasshowninFig.11.23.

ad

ad aq

aq

FIG.11.23Phasordiagramofsynchronousmachineincludingtheeffectofsaliency

11.7GENERALEQUATIONOFSYNCHRONOUSMACHINE

Thesynchronousmachinehasatleastfourwindings,threeonstatorcarryingACandoneonrotorwithDCexcitation.

WhenacoilhasaninstantaneousvoltageVappliedacrossitsterminalsandacurrent‘i’thatflowsfromapositiveterminalintothecoil,thegoverningequationbecomes

Hence,theinstantaneousterminalvoltageVofanywindingwillbeintheform,

V=±Σir±Σλ⋅

whereλisthefluxlinkage(itmayberepresentedbythesymbolΨ),rtheresistanceofthewinding,andithecurrentwithpositivedirectionofstatorcurrentflowingoutofthegeneratorterminal.

Forthethreestatorwindingsa,b,andc,thevoltageequationsare

andforrotor-fieldwinding,

Inpractice,r =r =r =r

∴λ=Li,

Equation(11.16)canbewrittenas

11.8DETERMINATIONOFSYNCHRONOUSMACHINEINDUCTANCES

The3-ϕsynchronousmachinewithoutdamperwindingsmaybeconsideredasasetofcoupledcircuitsformedbythe3-ϕwindingsandrotor-fieldwindingsasshowninFig.11.24.

Intheabovecircuit,thereareself-inductancesandmutualinductances,whichvaryperiodicallywiththeangularrotationofarotor.

a b c

FIG.11.24Circuitdiagramof3-ϕ,synchronousmachine

11.8.1Assumptions

Thefollowingassumptionsareusuallymadetodeterminethenatureofthemachineinductancestodevelopthedetailedmodelofthesynchronousmachines:

1. Theself-inductanceandmutualinductanceofthemachineareindependentofthemagnitudeofwindingcurrentsbecausethemagneticsaturationisneglected.Thus,themachineisassumedtobemagneticallylinear.

2. Theshapeoftheairgapandthedistributionofwindingsaresuchthatalltheaboveinductancesmayberepresentedasconstantsplussinusoidalfunctionsofelectricalrotorpositions.

3. Slottingeffectsareignored.Distributedwindingscomprisefinelyspreadconductorsofnegligiblediameter.

4. Magneticmaterialsarefreefromhysterisisandeddycurrentlosses.5. Themachinemaybeconsideredwithoutdamperwindings.Ifthe

damperwindingispresent,thenitsinfluencemaybeneglected.6. Higherordertimeandspaceharmonicsareneglected.

11.9ROTORINDUCTANCES

Inthissection,weshalldiscussrotorself-inductanceandstatortorotormutualinductancesindetail.

11.9.1Rotorself-inductance

Thestator,i.e.,armaturehasacylindricalstructure.Hence,theself-inductanceoftherotorfieldwinding‘f’willnotdependuponthepositionoftherotorandwillbeaconstantone.

i.e.,L =self-inductanceofrotor-fieldwinding=constant

11.9.2Statortorotormutualinductances

Thestatortorotormutualinductanceswillvaryperiodicallywithβ.Themutualinductancebetweenthefieldwindingandanyarmaturephaseisthegreatestwhenthed-axiscoincideswiththeaxisofthatphase.Figure11.25showstheeffectoffieldwindingonmutualinductances.

Considertheexample,themutualinductancebetweenthefieldwindingandPhase-a(M )willbemaximumatβ=0andatβ=90°andnegativemaximum(−M )atβ=180°andzeroagainatβ=270°.

Accordingly,withspacem.m.f.andfluxdistributionassumedtobesinusoidal,themutualinductancebetweenthefieldwindingandPhase-a(L )canbeexpressedas

L =L =M cosβ

Basedontheabove,thesimilarexpressionsforPhases-bandccanbeobtaineddirectlybyreplacingβwith(β−120°)and(β+120°),respectively,

i.e.,L =L =M cosβ

L =L =M cos(β−120°)

L =L =M cos(β+120°)(11.19)

ff

f

f

af

af fa f

af fa f

bf fb f

cf fc f

FIG.11.25Effectoffieldwindingonmutualinductances

11.10STATORSELF-INDUCTANCES

Theself-inductanceofanystatorphaseisalwayspositivebutvarieswiththepositionoftherotor.Itisthegreatestwhenthed-axisofthefieldcoincideswiththeaxisofthearmaturephaseandbeingleastwhentheq-axiscoincideswithit.Therewillbeasecond-harmonicsvariationbecauseofdifferentair-gapgeometryalongthedandq-axes.Forexample,theself-inductanceofPhase-a(L )willbeamaximumforβ=0andaminimumforβ=90°andmaximumagainforβ=180°andsoon.

WhenPhase‘a’isexcited,withspaceharmonicsignored,them.m.f.waveofphase‘a’willbeacosinewave(spacedistribution)centeredonthePhase-a-axisasshowninFig.11.26.

aa

FIG.11.26Them.m.f.waveofPhase-awithitsd-axisandq-axiscomponents

Thepeakamplitudeofthism.m.f.waveofPhase‘a’is

F =N i (11.20)

whereN istheeffectiveturns/phaseandi theinstantaneouscurrentinPhase‘a’.

Letusresolvethism.m.f.waveintotwo-componentsinusoidalspacedistributions,onecenteredonthed-axis(F )andtheotherontheq-axis(F ).

Thepeakamplitudesofthesetworesolvedcomponentsare

F =F casβ(11.21)

F =F cos(β+90°)=−F sinβ(11.22)

Theadvantageofresolvingm.m.f.isthattwocomponentsm.m.f.wavesactonspecificair-gapgeometryintheirrespectiveaxes.

a a a

a a

da qa

da a

qa a a

Thefundamentalair-gapfluxesperpolealongthetwoaxesare,accordingly,

whereÞ isthepermeancealongthed-axisandÞ thepermeancealongtheq-axis.

Theseareknownasmachineconstantsandtheirvaluescanbefoundfromafluxplotforspecificmachinegeometry.

Letϕ betheair-gapfluxlinkingwithPhase-aandbeexpressedas

ϕ =ϕ cosβ−ϕ sinβ(11.25)

=F (Þ cos β+Þ sin β)

Sincetheair-gapfluxlinkage,λ =ϕ N

Ifϕ representstheleakagefluxofPhase‘a’,whichdoesnotcrosstheairgap,thenthefluxlinkagesofPhase‘a’duetoleakagefluxonlyare

λ =ϕ N =F ÞN =N i Þ

whereÞ istheconstantleakagepermeanceofarmaturePhase‘a’.

d q

ga

ga ga qa

a d q

ga ga a

la

la la a a l a a a l

l

2 2

2

Therefore,thetotalfluxlinkageofPhaseacanbeexpressedas

λ =λ +λ

Sincebydefinition,theinductanceistheproportionalityfactorrelatingfluxlinkagestocurrent,theself-inductanceofPhase‘a’duetotheair-gapfluxwhenonlyPhase‘a’isexcited,willbe

⇒L =L +L cos2β(11.29)

where isaconstantterm,

istheamplitudeofsecond-harmonics

variation

ForPhase‘b’,thevariationofself-inductanceissimilar,exceptthatthemaximumvalueoccurswhenthed-axiscoincideswiththePhaseb-axis.Theself-inductancesofPhase‘b’andPhase‘c’canbeobtainedbyreplacingβby(β−120°)and(β+120°),respectively:

L =L +L cos2(β−120°)

=L +L cos(2β−240°)

=L +L cos(2β+120°)(11.30)

a ga la

aa s m

bb s m

s m

s m

L =L +L cos2(β+120°)

=L +L cos(2β+240°)

=L +L cos(2β−120°)(11.31)

i.e.,thestatorself-inductancesareobtainedas

L =L +L cos2β

L =L +L cos(2β+120°)

L =L +L cos(2β−120°)

11.11STATORMUTUALINDUCTANCES

Themutualinductancesbetweenstatorphaseswillalsoexhibitasecond-harmonicsvariationwithβbecauseoftherotorshape.Themutualinductancebetweentwophasescanbefoundbyevaluatingtheair-gapfluxlinkingonephasewhenanotherphaseisexcited.Forexample,themutualinductancebetweenPhasesaandb,L =L ,canbeobtainedbyevaluatingϕ linkingPhase‘b’whenonlyPhase‘a’isexcited.ϕ canbecomputedfromEquation(11.24)byreplacingβwith(β−120°)as

ϕ =ϕ cos(β−120°)−ϕ sin(β−120°)

=F [Þ cosβcos(β−120°)+Þ sinβsin(β−120°)]

ThemutualinductancebetweenPhase-aandPhase-bduetotheair-gapfluxisthen

L =−M +L cos(2β−120°)(11.34)

cc s m

s m

s m

aa s m

bb s m

cc s m

ab ba gba

gba

gba da qa

a d q

ab s m

where

Similarly,themutualinductancesofstatorL andLcanbeobtainedas

L =−M +L cos2β(11.35)

L =−M +L cos(2β+120°)(11.36)

i.e.,thestatormutualinductancesareexpressedas

L =−M +L cos(2β−120°)

L =−M +L cos(2β)

L =−M +L cos(2β+120°)

Here,L ,L ,andM areregardedasknownmachineconstantsandaredeterminedeitherbytestsorcalculatedbydesigned.AlltheinductancesexceptL arefunctionsofβandthustheyaretime-varying.

11.12DEVELOPMENTOFGENERALMACHINEEQUATIONS—MATRIXFORM

Fromtheknowledgeoftheabovetime-varyinginductances,thegeneralmachineequationsaredeveloped.Ifamotoringmodeisconsidered,thenforanywinding,thegeneralequationrelatingtheappliedvoltageoftheinputcurrentis

Intermsofself-inductanceandmutualinductance,thefluxlinkagesareexpressedas

λ =L i +L i +L i +L i


bc ca

bc s m

ca s m

ab s m

bc s m

ca s m

s m s

ff

a aa a ab b ac c af f

b ba a bb b bc c bf f


λ =L i +L i +L i +L i (11.37)

Equations(11.17)canbemodifiedas

Writetheaboveequationsinacompactmatrixform:

where

c ca a cb b cc c cf f

f fa a fb b fc c ff f

and

Thevectorsofthefluxlinkagesareproportionaltothecurrentswithmatrixofself-inductanceandmutualinductanceastheproportionalityfactor.

Hence,[λ]=[L][i](11.40)

where

i.e.,

TheinductanceLissymmetric,i.e.,L =L ,etc.

IftheexpressionsofinductancesofEquations(11.19)–(11.36)aresubstitutedinEquations(11.38),thesolutionofresultingequationscanbesimplifiedbyatransformation,knownasBlondel’stransformation,whichisalsogenerallycalledPark’stransformation.

11.13BLONDEL’STRANSFORMATION(OR)PARK’STRANSFORMATIONTO‘dqo’COMPONENTS

Equations(11.37)–(11.41)areasetofdifferentialequationsdescribingthebehaviorofmachines.However,thesolutionoftheseequationsiscomplicatedsincetheinductancesarethefunctionsofrotorangle‘β’,whichinturn,isafunctionoftime.

Thecomplicationingettingthesolutioncanbeavoidedbytransferringthephysicalquantitiesinthearmaturewindingsthroughalinear,time-dependent,andpower-invarianttransformationcalledPark’stransformation.

Thistransformationisbasedonthefactthattherotatingfieldproducedby3-ϕstatorcurrentsinthesynchronousmachinecanbeequallyproducedby2-ϕcurrentsina2-ϕwinding.Letusconsiderthe3-windingsforthethreephasesa,b,andcofasynchronousmachineasshowninFig.11.27.

Whenallthethreephasesareexcited,thetotalstatorm.m.f’.sthatactalongthed-axisandq-axisare

F =F cosβ+F cos(β−120°)+F cos(β+120°)

ab ba

ds a b c

=N [i cosβ+i cos(β−120°)+i cos(β+120°)(11.42)

andF =F sinβ−F sin(β−120°)+F sin(β+120°)

=N [i sinβ−i sin(β−120°)−i sin(β+120°)(11.43)

Letusconsidertwofictitiouswindings,oneplacedonthed-axisandtheotherontheq-axis,asshowninFig.11.28.

FIG.11.27Them.m.f.’sof3-ϕwindingsandtheirresultantm.m.f.’salongthed-axisandtheq-axis

a a b c

qs a b c

a a b c

FIG.11.28Fictitiouscoilsondandq-axes

Thewindingonthed-axisrotatesatthesamespeedastherotor-fieldwindingandremainsinsuchapositionthatitsaxisalwayscoincideswiththedirectaxisoftherotorfield.Hence,theinstantaneouscurrenti givesthesamem.m.f.F onthisaxisasdotheactualthreeinstantaneousarmaturephasecurrentsflowingintheactualarmaturewindings.Similarly,thecurrentiflowinginthewindingontheq-axisgivesthesamem.m.f.F onthisaxisasdotheactualthreeinstantaneousarmaturephasecurrentsflowingintheactualarmaturewindings.

Inordertotransformquantitiesinthef-a-b-caxesintothef-d-q-0axes,theconstraintsofthesystemformingthebasisofthetransformationmaybeobtainedbyviewingthecurrents,m.m.f.’s,voltages,andfluxlinkagesinthetwoaxes.ThecurrentsinviewofEquations(11.42)and(11.43)are

i =i

i =K [i cosβ+i cos(β−120°)+i cos(β+120°)]

i =K [i cos(β+90°)+i cos(β−120°+90°)+icos(β+120°+90°)]

d

ds

q

qs

f f

d d a b c

q q a b c

=−K [i sinβ+i sin(β−120°)+i sin(β+120°)](11.44)

Anewcurrent,knownaszero-sequencecurrent,whichdoesnotproduceanyrotatingfieldisintroducedandisexpressedas

whereK ,K ,andK areconstants.

Nowinthissystem,therearethreeoriginalphasecurrentsi ,i ,andi andthreefieldcurrentsi ,i ,andi .Fortheungroundedstarconnectionorforthebalanced3-ϕcondition,thesumofphasecurrentsiszeroandhencei mustalsobezero.

Equations(11.44)and(11.45)areexpressedinamatrixformas

Inamorecompactmatrixform,Equation(11.46)canbeexpressedas

ThisisknownasPARK’stransformation(orBlondel’stransformation).

where

q a b c

d q 0

a b c d q o

o

and

andisknownasPARK’stransformationmatrix(orBlondel’stransformationmatrix).

TheeffectofPark’stransformationissimplytotransformallstatorquantitiesfromphasesa,b,andcintonewvariables,theframeofreferenceofwhichmoveswiththerotor,i.e.,thePark’stransformationmatrix[P] transformsthefieldofphasorstothefieldofd-q-o-fcomponentsanditisalinear,time-dependentmatrix.

11.14INVERSEPARK’STRANSFORMATION

Theinversetransformation,whichtransformsthed-q-o-fquantitiesintothephasequantities,isexpressedas

[i] =[P] [i] (11.49)

where

dqof

abcf dqof

−1

11.15POWER-INVARIANTTRANSFORMATIONIN‘F-D-Q-O’AXES

Thetotalinstantaneouspowerdeliveredtothemotoris

P =V i +V i +V i (11.51)

Whenf-d-q-oaxes’quantitiesaresubstitutedforthephasequantitiesbyusingEquation(11.50)andbyfurthermanipulation,theseresultsgivepowerintermsofnewquantitiesas

Equation(11.52)givesthetruepowerassociatedwiththearmatureofthenewsystem.Thispowerinvarianceispreservedif

Sincetheanglebetweenthed-axisandthePhase-aaxisisβ,andtheq-axishasbeenchosenaheadofthed-axis,thevaluesofK andK mustbepositive;hence,theselectedvaluesofthethreequantitiesare

d a a b b c c

d q

Withthesevalues,Park’stransformationmatrixanditsinversebecome

Now,thetransformationiscalledunitarytransformationbecausetheinverseofthetransformationmatrixisthetransposeofthematrix,i.e.,[P] =[P ]

i.e.,Inverseofthematrix=transposeofconjugateofthematrix.

∴Thecurrentequationofasynchronousmachineequation(11.46)canbeexpressedas

T

−1 *

andintermsofinversePark’stransformation,thecurrentequationsofsynchronousmachinecanbeexpressedas

11.16FLUXLINKAGEEQUATIONS

Thefluxlinkageequationsinmatrixformare[λ]=[L][i].Intermsofself-inductanceandmutualinductance,thefluxlinkagesfromEquation(11.37)areexpressedas




a aa a ab b ac c af f

b ba a bb b bc c bf f

c ca a cb b cc c cf f


Inmatrixform,itisexpressedas

Invectorform,itcanbeexpressedas

[λ] =[L] [i] (11.57)

Thefluxlinkagesoff-d-q-ocoilsintermsoffourcurrentsareobtainedupontransformingbothsidesbyusingthetransformationmatrix[P]anditsinverse[P]asfollows:

[λ] = [P][λ] (11.58)

= [P][L] [i]

= [P][L] [P] [i] (11.59)

[∵fromEquation(11.56);[i] =[P]−1[i]

f fa a fb b fc c ff f

abcf abcf abcf

dqof abcf

abcf abcf

abcf dqof

abcf dqof

−1

−1

Bysubstituting weget

whereL =directaxissynchronousinductance=

L =quadratureaxissynchronousinductance=

L =zero-sequenceinductance=L –2M

d

q

o s s

Note:ThemutualinductanceL canalsoberepresentedbyM orM.


FromEquation(11.62),itisnoticedthattheinductancesarenotfunctionsofrotorpositionβ.Hence,itisagreateradvantageoftransformingphasequantitiestoasuitablesetofquantitiesoff-d-q-o-axes.

11.17VOLTAGEEQUATIONS

Theoriginalvoltageequationofasynchronousmachineis

i.e.,

Fortransformation,letuspre-multiplybothsidesofEquation(11.63)by[P],

af

af

Hence,

Itcanbeshownthat isamatrixwithzero

entriesexceptfor–ωinthefirstrow,secondcolumnand+ωinthesecondrow,firstcolumn,

i.e.,

Afterthetransformation,synchronousmachineequationsinmatrixformbecome

FIG.11.29(a)PhasordiagramsV versusωλ ;(b)phasordiagramsωλ

versus–V

∵R =R =R =R

whereω=angularvelocityofrotation=

FromEquation(11.67),thevoltageequationsobtainedare

q d q

d

a b c

Forsinusoidalsteady-stateconditions,thefluxphasorleadsthevoltagephasorby90°.Thismeansthatv willbeinducedbythefluxinthedirectaxis(ωλ );i.e.,v willbeinducedby(ωλ )andsimilarly−v willbeinducedby(ωλ )asshowninFigs.11.29(a)and(b).

11.18PHYSICALINTERPRETATIONOFEQUATIONS(11.62)AND(11.68)

Thetermsωλ andωλ arespeedvoltages(fluxchanges

inspace)andtheterms and aretransformer

voltages(fluxchangesintime).Usually,thesetransformervoltagesaresmallcomparedwithspeedvoltagesandmaybeneglected.TheneglectedtransformervoltagescorrespondtonegligenceoftheharmonicsandDCcomponentsintransientsolutionforstatorvoltagesandcurrents.NegligenceofharmonicsandDCcomponentsinthephasecurrentisverycommoninmachineanalysis.NeitherharmonicsnorDCcomponentshaveasignificanteffectontheaveragetorqueofthemachinesinceharmonicsareusuallysmallandDCcomponentsdieawayveryrapidly.

ThesolutionsofnetworkequationsbecomeextremelydifficultandcomplexwhentheharmonicsandDCcomponentsarepresentinelectricalquantitiesifthetransformervoltagesareincluded.Hence,itispreferabletoapproximatetheassociateddampingtorquesbyadditionaltermsintheswingequation.

q

d q

d d

q

d q

11.19GENERALIZEDIMPEDANCEMATRIX(VOLTAGE–CURRENTRELATIONS)

BycombiningEquations(11.62)and(11.68),weget

Incompactform,theabovematrixcanbeexpressedas

[V] =[Z] [i] (11.70)

Itisobservedthattheimpedancematrixissymmetricalinfandd-axes.Itconsistsoftwoterms,one

relatingtotransformervoltagesandthesecond

relatingtospeedvoltages,asgivenbelow:

dqof dqof dqof

where angularspeedofrotation

11.20TORQUEEQUATION

Thespeedvoltagesare:

1. Inthedirectaxis,λ ωand

2. Inthequadratureaxis,−λ ω

Mechanicalangularvelocity

Thetotal3-ϕpoweroutputofasynchronousmachineisgivenby

d

q

p =v i +v i +v i (11.73)

Assumebalancedbutnotnecessarilysteady-stateconditions,thusv =0andi =0.

∴p=v i +v i (11.74)

FromEquation(11.68),

Substitutingv andv expressionsinEquation(11.74),weget

out d d q q o o

0 0

d d q q

d q

Theaboveexpressionconsistsofthreetermsandtheyare:

Thefirsttermrepresentstherateofchangeofstatormagneticfieldenergy.Thesecondtermrepresentsthepowertransferredacrosstheairgap.Thethirdtermrepresentsthestatorohmiclosses.

Themachinetorqueisobtainedfromthesecondterm,

Substitutingforλ andλ fromEquation(11.62)inEquation(11.76),weget

Foracylindricalrotorsynchronousmachine,thedirectaxisandquadratureaxisinductancesareequal,

d q

i.e.,L =L :

Forasalient-polemachine,thereisasaliencytorque:

T =(L −L )i i (11.79)

Thistorqueexistsonlybecauseofnon-uniformityinthepermeanceoftheairgapalongthed-andq-axes.Thisisthereluctancetorqueofasalient-polemachineandexistsevenwhenthefieldexcitationiszero.

11.21SYNCHRONOUSMACHINE—STEADY-STATEANALYSIS

Considera3-ϕsynchronousmachinethathasthreearmature(stator)windingsa,b,andc,onefieldwinding‘f’ontherotorwithitsfluxinthedirectionofthed-axis,andonefictitiouswinding‘g’ontherotorwithitsfluxinthequadratureaxisasshowninFig.11.30.

FIG.11.30Three-phasesynchronousmachinewithstatorandrotorwindings

Thefictitiouswinding‘g’approximatestheeffectofeddycurrentscirculatingintheiron(rotorironinround-rotormachineandnegligibleinsalient-polemachine)andto

d q

saliency d q d q

someextenttheeffectofdamperwindings.Thisfictitiouswindingisshortcircuitedsinceitisnotconnectedtoanyvoltagesource.

Sincetheelectromagnetictransientsinthenetworkaremuchfasterthanthemechanicaltransients,thesteady-statephasorsolutionsonthenetworksideareperformed.

11.21.1Salient-polesynchronousmachine

Thephasordiagramofanoverexcitedsalient-polesynchronousgeneratorforlaggingp.f.isshowninFig.11.31.

FIG.11.31Phasordiagramofasalient-polesynchronousgenerator

δisthepowerangleortorqueangle

E theterminalvoltage=v

E thevoltageduetoair-gapflux

a a

ag

E thevoltageduetofluxproducedbymainrotor-fieldcurrent

I X thevoltagedropacrossd-axisarmaturemagnetizing

reactance

I X thevoltagedropacrossq-axisarmaturemagnetizing

reactance

λ thefluxlinkageduetonetair-gapflux

λ thed-axiscomponentoffluxlinkage

λ theq-axiscomponentoffluxlinkage

λ thefluxlinkageduetod-axiscomponentofI

λ thefluxlinkageduetoq-axiscomponentofI

x thed-axiscomponentofsynchronousreactance=x +x

x theq-axiscomponentofsynchronousreactance=x +x

Figure11.32representsthephasorsandtheirspeeds:

δistheanglebetweensynchronouslyrotatingreferencephasoraxisandq-axis

ω thesynchronousspeed

ωthespeedofrotor

af

ad ad

aq aq

ag

agd

agq

ad a

aq a

d l ad

q l aq

s

FIG.11.32Phasorsandtheirspeeds

FromthephasordiagramshowninFig.11.32,wehave

ByapplyingPark’stransformation,weget

Equations(11.81)canbeexpressedasaphasorequation:

I +jI =Ie

=Ῑe asῙI∠α(11.82)q d

j(a−δ)

−jδ

Forthesteady-stateanalysis,theq-axiswillbeconsideredastherealaxisandthed-axisastheimaginaryaxissincethevoltageinducedinanormalsteady-stateoperationliesontheq-axis.

Similarly,byPark’stransformation,weget

v =vcos(θ−δ)

v =vsin(θ−δ)

V =0

Incomplexnotation,

Togetthesteady-stateanalysis,wemakeuseofthefollowingassumptions:

1. Transformervoltages, and ,beingsmallandaretherefore

neglected.2. Balancednetworkcurrentsandvoltagesareassumed.

Thereasonsfortheaboveassumptionsarethatchangesinλ andλ areveryslowintimewiththeoscillationsof

angleδandhence and areverysmallcompared

withωλ andωλ .

Duetotheaboveassumptions,Equations(11.62)and(11.68)canberewrittenbydroppingthetransformervoltageterms,zero-sequencecurrents,andvoltages:

q

d

o

d q

d q

Substitutingforλ andλ intheaboveequationsofvoltages,weget

Hence,

I +jI =Ie

v +jv =ve

11.21.2Non-salient-polesynchronous(cylindricalrotor)machine

Forthiscase,X =X

∴E =E +i R +ji X +ji X

=E +i R +ji X (11.87)

d q

q d

q d

d q

af a a a a 1 a ad

a a a d d

j(α−δ)

j(α−δ)

Theequivalentcircuitofanon-salient-polesynchronousmachineisrepresentedbyasourceE (inducedemf)inserieswiththeinternalimpedanceR +jX asshowninFig.11.33.

ThephasordiagramisshowninFig.11.34.Now,thesynchronousreactanceisdefinedasX =X +X ,andifresistance‘r’isneglected,thecylindricalrotorsynchronousgeneratorisrepresentedbytheequivalentcircuitasshowninFig.11.35.

FIG.11.33Equivalentcircuitofnon-salient-polesynchronousgenerator

FIG.11.34Phasordiagramofnon-salient-polesynchronousgenerator

af

a d

s aq l

FIG.11.35Equivalentcircuit

Now,Equation(11.87)becomes

E =E ji X (11.88)

Theaboverelationshiprepresentsthemodelofthecylindricalrotor(non-salientpole)generatorundersteady-stateconditionsandofwhichaveryusefulequivalentcircuitisshowninFig.11.35.

11.22DYNAMICMODELOFSYNCHRONOUSMACHINE

Inthissection,weshalldiscussthedynamicmodelofsynchronousmachines—salientpolesynchronousgenerator,dynamicequationsofsynchronousmachine,andequivalentcircuitofsynchronousgenerator—indetail.

11.22.1Salient-polesynchronousgenerator—sub-transienteffect

Duringnormalsteady-stateconditions,thereisnotransformeractionbetweenstatorandrotorwindingsofsynchronousmachines,astheresultantfieldproducedbystatorwindingsandrotorwindingsrevolveswiththesame(synchronous)speedandinthesamedirection.However,duringdisturbances,therotorspeedisnolongerthesameasthatoftherevolvingfieldproducedbystatorwindings,whichalwaysrotateswithsynchronous

af a a s

speed.Hence,thesynchronousgeneratorbecomesatransformer.

Synchronousmachinedynamicequations

Wehave

LetX =ωL =d-axiscomponentofsynchronousreactance,

X =ωL =q-axiscomponentofsynchronousreactance:

and

d d

q q

SubstitutingtheI valueinEquation(11.90),weget

Let sub-transientvoltagealongtheq-axis

Sinceatthistime,bothwindings‘f’and‘g’arepresentalongthed-axisthen,Equation(11.91)becomes

Let transientd-axisreactance

Sinceatthistime,boththed-axiscomponentofthearmaturewindingsandthef-windingarepresent,Equation(11.92)becomes

v =−R I +X′ I +E′ (11.93)

Similarly,theequationofv isgivenby

f

q a q d d q

d

Substituting,ωL =X ,weget

Weknowthat

SubstitutingI expressioninEquation(11.94),weget

Let Transientvoltagealongthed-axis

(sinceboththeq-axiscomponentandg-windingsarepresenttogivetransientstate)andalsolet

q-axiscomponentoftransient

reactance

Hence,Equation(11.95)becomes

v =−R I −X′ I +E′ (11.96)

q q

g

d a d q q d

Let directaxisopen-circuittransienttime

constant

quadratureaxisopen-circuittransienttime

constant

andalso

i.e.,E=E +jE

SubstitutingEquation(11.97)inEquations(11.90)and(11.94),weget

v =−R I −X I +E (11.99)

FromEquations(11.93)and(11.98)

v =−R I +X′ I +E′ =−R I +X I +E

i.e.,E′ =E +(X −X′ )I (11.100)

SimilarlyfromEquations(11.96)and(11.99),wehave

v =−R I +X′ I +E′ =−R I −X I +E

i.e.,E′ =E −(X −X′ )I (11.101)

Figure11.36showsthephasordiagramofasynchronousmachineunderthetransientstate.

q d

d a d q q d

q a q d d q a q d d q

q q d d d

d a q q q d a d q q d

d d q q q

Fromthephasordiagramalsoweget

E′ =E +I (X −X′ )=voltagebehindthed-axiscomponentoftransientreactance

E′ =E −I (X −X′ )=voltagebehindtheq-axiscomponentoftransientreactance


q q d d d

d d q q q


Therefore,fromtheaboveanalysis,weget

TakingthederivativeforEquation(11.105),weget

andalsotakingthederivativeofEquation(11.106),wehave

q-axiscomponentofopen-circuittransient

reactancetimeconstant

i.e.,

Equations(11.93)and(11.96)canbewritteninthematrixformas

11.22.2Dynamicmodelofsynchronousmachineincludingdamperwinding

The‘f’and‘g’coilsintherotorwindingproducetransienteffectintermsofX′ andX′ inthesynchronousmachine.Thefieldcoil‘f’existsphysicallywhereasthe‘g’coilishypotheticalforrepresentingtherotoreddycurrentsintheq-axis.However,itisquitedifficulttocalculateg-coilinductance.

Themoreaccuraterepresentationofsynchronousmachineisobtainedbyaddingtwomorefictitiouswindingsontherotor,onealongthed-axisknownas‘K ’windingandtheotheralongtheq-axis,knownas‘K ’winding.Thesedamperwindingscanbeapproximatedbytwohypotheticalcoils,bothshort-circuitedasthereisnovoltagesourceconnectedtothem.

Thedynamicmachineequationswillnowbemodifiedtoincludethedamperwindingsbysubstitutingthescalar(λ andλ )vectors:

Nowinthismodel,thePark’stransformationwillbeappliedwiththefollowingassumptions:

1. Mutualinductancesfromstatorcoilsa,b,andc(oritscomponent‘d’and‘q’axes)tothe‘K ’coilisthesameas‘f’coil,andtotheK coil

thesameas‘g’coiland2. Mutualinductancebetween‘K ’coiland‘f’coil(and‘K ’coiland‘g’

coil)isthesameas‘d’componentofstatorcoilsandf-coils(or‘q’componentofstatorto‘g’coil).

TheresultantequationsafterPark’stransformationare

d q

d

q

f g

d g

d g

Fromtheabovematrixrepresentation,themodifiedvoltageequationswiththeinclusionofdamperwindingsare

Themodifiedfluxlinkageswiththeinclusionofdamperwindingsareobtainedintheformofmatrixas

i.e.,thefluxlinkageequationsare

11.22.3Equivalentcircuitofsynchronousgenerator—includingdamperwindingeffect

11.2.3.1Alongthed-axis

Theequivalentcircuitofthesynchronousgeneratoralongthed-axisexcludingresistancesisasshowninFig.11.37,whereX isthearmatureleakagereactance,X thearmaturemagnetizingreactance,X thefieldreactance,andX thefictitiouswindingreactance.

Initially,allthereactancesareinthecircuit(i.e.,justattheinstantwhenthefaulthasoccurred)andtherefore,initialorsub-transientreactance,arethelowest.Aftersometime,theg-winding(damperwinding)isoutofcircuitasithasaverylowtimeconstantandhencewehaveonlyfieldwindingandarmaturereactancesinparallel.Thisreactanceisknownastransientreactanceandislargerthanthepreviousone.However,aftersometime,whenthedisturbancealtogetherdisappears,fieldwindingisalsooutofcircuitandhencewehaveonlyarmaturereactance,(X =X +X )calledthesteady-statereactanceofthecircuit.

FIG.11.37Equivalentcircuitofsynchronousgenerator

Foramoreaccuraterepresentation,twomorefictitiouswindings‘K ’windingand‘K ’windingareadded.Hence,theequivalentcircuitofthesynchronousgeneratoralongthed-axiscanberepresentedasshowninFig.11.38.

TheparallelcombinationofX ,X ,andX isknownasthed-axiscomponentofsub-transientreactanceX″andisrepresentedasX″ =X /X /X .

Hence,thed-axiscomponentofthesub-transientsynchronousreactanceisgivenby

l ad

f

g

d l ad

d q

ad f kd

ad

ad ad f kd

X″ =X +X″

Thisreactanceisverysmall.Aftersometime,asthehuntingbecomesless,thewindingK isalsooutofcircuitsinceithasalowtimeconstant.

Thus,theresultantequivalentcircuitbecomesasshowninFig.11.39.

TheparallelcombinationofreactancesX andX isknownasX′ ,d-axiscomponentoftransientarmaturereactance.

Thed-axiscomponentoftransientsynchronousreactanceisX′ ==X +X′

Generally,X′ >X″

Finally,whenthedisturbanceisaltogetherover,therewillnotbehuntingoftherotorandhencetherewillnotbeanytransformeractionbetweenthestatorandtherotor.Hence,theequivalentcircuitofsynchronousgeneratorbecomesasshowninFig.11.40.

FIG.11.38Accuraterepresentationofequivalentcircuitofsynchronousgenerator

FIG.11.39Resultantequivalentcircuitofsynchronousgenerator

d 1 ad

d

ad f

ad

d 1 ad

d d

FIG.11.40Equivalentcircuit

FIG.11.41Equivalentcircuitofsynchronousgeneratoralongtheq-axis

FIG.11.42Resultantequivalentcircuitofthesynchronousgenerator

Here,X =X +X andiscalledthedirectaxiscomponentofsynchronousreactance.

ItisobviousthatX″ <X′ <X .

11.2.3.2Alongtheq-axis

Justafterthedisturbance,theequivalentcircuitofthemachinewillbecomeasshowninFig.11.41.

Here,X″ =X +(X //X //X )

whereX″ istheq-axiscomponentofsub-transientsynchronousreactance.ThevalueofX″ isverysmall.

d l ad

d d d

q 1 aq g kq

q

q

Aftersometime,huntingbecomeslessandless,bothg-windingandK ,whichhavealowtimeconstant,willbeoutofcircuitandhencetheresultantequivalentcircuitofthesynchronousgeneratorbecomesasshowninFig.11.42.

Here,X′ =X =X +X ,whereX′ istheq-axiscomponentoftransientsynchronousreactance.

11.23MODELINGOFSYNCHRONOUSMACHINE—SWINGEQUATION

Themechanicalbehaviorofasynchronousmachinecanbeestablishedbyinterconnectingtheelectricalandmechanicalsidesofasynchronousmachineintermsofelectricalandmechanicaltorque.Thisisprovidedbythedynamicequationfortheaccelerationordecelerationoftherotorofacombinedturbineandsynchronousgeneratorsystem,whichisusuallycalledtheswingequation.

Whiledevelopingaswingequationoramechanicalequation,thefollowingbasicassumptionsaretobemade:

1. Synchronousmachinerotorspeedmustbesynchronousspeed.2. Therotationalpowerlossesduetofrictionandwindageareneglected.3. Mechanicalshaftpowerissmooth,i.e.,theshaftpowerisconstant.

Letusconsiderasinglerotatingmachinewithsteady-stateangularspeed andphaseangleδ.Duetovariouselectricalormechanicaldisturbances,themachinewillbesubjectedtodifferencesinmechanicalandelectricaltorque,causingittoaccelerateordecelerate.Hence,duringdisturbance,therotorwillaccelerateordeceleratewithrespecttothesynchronouslyrotatingair-gapm.m.f.andarelativemotionbegins.

Letθbetheangularpositionoftherotoratanyinstant‘t’

ωtheangularvelocity(rad/s)

q

q q 1 aq q

αtheacceleration

δthephaseangleofarotationmachine

T thenetacceleratingtorqueinamachine

T theelectricaltorqueexertedonthemachinebythegenerator

P thenetacceleratingpower

P themechanicalpowerinput

P theelectricalpoweroutput

Jthemomentofinertiaforthemachine

M=Jω;angularmomentumofthemachineinkg-m

Jα=T

P =ωT =ω(Jα)=Mα

ConsiderasynchronousgeneratordevelopinganelectromagnetictorqueT andrunningatthesynchronousspeedω .IfT isthedrivingmechanicaltorque,thenundersteady-stateconditions,withnegligiblelosses,

T =T

Adeparturefromthesteadystateduetoadisturbanceresultsinanaccelerating(T >T )ordecelerating(T >T )torqueT ontherotor:

T =T −T

Neglectingthefrictionalanddampingtorque,fromthelawofrotation,wehave

Now,thevalueofθiscontinuouslychangingwithtime‘t’.Itisconvenienttomeasureθwithrespecttoareferenceaxis,whichisrotatingatsynchronousspeed.

net

elec

net

mech

elec

net

net net

e

s m

m e

m e e

m a

a m e

2

Ifδistheangulardisplacementofarotorinelectricaldegreefromthesynchronousrotatingreferenceaxisandω ,thesynchronousspeedinelectricaldegrees,thenθcanbeexpressedasthesumof:(i)time-varyingangleω tontherotatingaxisand(ii)thetorqueangleδoftherotorwithrespecttotherotatingreferenceaxisasshowninFig.11.43:

FIG.11.43Phasorrepresentationofrotor-fieldposition

andtherotorangularaccelerationisobtainedbydifferentiatingtheaboveequationagain:

Thetorqueactingontherotorofasynchronousgeneratorincludesthemechanicalinputtorquefromtheprimemover,torqueduetorotationallosses(i.e.,friction,windage,andcoreloss),electricaloutputtorque,

s

s

anddampingtorquesduetoprimemover,generator,andpowersystem.

Theelectricalandmechanicaltorquesactingontherotorareofoppositesignandareofaresultofelectricalinputandmechanicalload.Byneglectingdampingandrotationallosses,sothatacceleratingtorqueis

T =T −T

andmultiplyingwithω,weget

Theswingequationintermsofmomentofinertiaorangularmomentumis

∴Swingequationisalsoexpressedas

KEYNOTES

Theprimemovercontrolsareclassifiedas:

1. Primarycontrol(speedgovernorcontrol).2. Secondarycontrol(loadfrequencycontrol(LFC)).3. Tertiarycontrolinvolvingeconomicdispatch.

Thetransientcharacteristicsofhydro-turbinesareobtainedbythedynamicsofwaterflowinthepenstock.Thewaterstartingtimeorwatertimeconstantvalueliesintherangeof0.5-5.0s.Steamturbinesystemconfigurationsare:

1. Non-reheattype.

a m e

2. Reheattype.

Reheattypesteamturbinesareclassifiedas:

1. Tandemcompound,singlereheattype.2. Tandemcompound,doublereheattype.3. Cross-compound,singlereheattypewithtwoLPturbines.4. Cross-compound,singlereheattypewithsingleLPturbine.5. Cross-compound,doublereheattype.

Mostsimplifiedmodelofasynchronousgeneratorforthepurposeoftransientstabilitystudiesisaconstantvoltagesourcebehindproperreactance.Inordertoincludetheeffectofsaliency,thesimplestmodelofasynchronousmachinecanberepresentedbyafictitiousvoltage‘E ’

locatedattheq-axis.Thed-axisistakenalongthemainpoleaxiswhiletheq-axislagsthed-axisby90°.Thestatortorotormutualinductanceswillvaryperiodicallywiththeanglebetweentheq-axisandthed-axisofasynchronousmachine.Theself-inductanceofanystatorphaseisalwayspositivebutvarieswiththepositionoftherotor.Itisthegreatestwhenthed-axisofthefieldcoincideswiththeaxisofthearmaturephaseandistheleastwhentheq-axiscoincideswithit.TheeffectofPark’stransformationissimplytotransformallstatorquantitiesfromphasesa,b,andcintonewvariables,theframeofreferenceofwhichmoveswiththerotor,i.e.,Park’stransformationmatrix[P] transformsthefieldofphasorstothefieldofd-q-o-f

componentsanditisalinear,time-dependentmatrix.


1. Whatisthesignificanceofwatertimeconstant,τ ?

whereΔHisthep.u.changeinwaterhead,ΔQthep.u.changeinthewaterdischarge,τ thewatertimeconstant,τ theelastic

limitofpenstock,zthenormalizedpenstockimpedance,andτ

knownasthewatertimeconstantorwaterstartingtime.

Thevalueofτ liesintherangeof0.5–5.0s.

Thetypicalvalueofτ isaround1.0s.

2. Howarethetransientcharacteristicsofhydro-turbinesobtained?Bythedynamicsofwaterflowinthepenstock.

3. Thewatertimeconstantτ isassociatedwithwhattime?

τ isassociatedwithaccelerationtimeforwaterinthepenstock

betweentheturbineinletandtheforebayorbetweentheturbineinletandthesurgetankifitexists.

q

dqof

ω

ω e

e

ω

ω

ω

ω

4. Writetheexpressionforwatertimeconstantτ intermsof

velocityofflowofwater.

whereListhelengthofpenstockinm,vthevelocityofwaterflowinm/s,H thetotalheadinm,andgtheaccelerationdueto

gravityinm/s .

5. Writetheexpressionforwatertimeconstantτ intermsofpower

generationoftheplant.

wherePisthepowergenerationin

wheree=η ×η

6. Whatarethecommonsteamturbinesystemconfigurations?

1. Non-reheatsystem.2. reheattype.

7. Whatarethecompoundsystemconfigurationsofasteamturbine?

Tandemcompoundandcross-compoundsystemconfiguration.

8. Whatarethetypesoftandemcompoundsystemconfiguration?

Singlereheattypeanddoublereheattype.

9. Whatarethetypesofcross-compoundsystemconfiguration?

1. SinglereheattypewithtwoLPturbines.2. SinglereheattypewithsingleLPturbines.3. Doublereheattype.

10. Whatdoyoumeanbytandemcompoundreheat-typesteamturbine?

Tandemcompoundsystemconfigurationhasonlyoneshaftonwhichalltheturbine(areofHP,LP,andIP)typesaremounted.

11. Whatarethecomponentsthatintroducethetimedelaysandhowcanthesedelaysberepresented?

ω

T

ω

turbine generation

2

Steamchest,reheat,andcross-overpipingarethecomponentsthatintroducethetimedelaysintheoperationofsteamturbines.

Thetimedelayscanberepresentedby:

τ =steam-chesttimeconstant(0.1–0.4s).

τ =reheattimeconstant(4–11s).

τ =cross-pipingtimeconstant(0.3–1.5s).

12. Whatisthemostsimplifiedmodelofasynchronousgeneratorforthepurposeofatransientstabilitystudy?

Aconstantvoltagesourcebehindaproperreactance,thevoltagesourcemaybesub-transientorsteadystateandthereactancemaybecorrespondingreactance.

13. Whyisthereactanceofthesynchronousgeneratorequivalentcircuitreferredtoasdirectaxisreactancewhenthe3-ϕshort-circuitfaultoccurs?

Thearmaturereactionfluxatthatinstantisnearlydemagnetizinginnatureandbecauseitactsalongthedirectaxisofthemachine,theequivalentcircuitreactanceisreferredtoasdirectaxisreactance.

14. Themodelingofasynchronousmachineiseasilyobtainedforanyfaultinapowersystembywhichcircuit?

Thevenin’sequivalentcircuit.

15. Writetheexpressionforexcitationvoltageoropen-circuitvoltageofasynchronousmachinefortheeffectofsaliency.

E =V+I R +jI X +jI X

16. Writethevoltageequationsforthethreestatorwindingsandrotorwindingsintermsoffluxlinkages.

V =i r +dλ /dt.

V =i r +dλ /dt.

V =i r +dλ /dt.

V =i r +dλ /dt.

17. Whataretheassumptionsusuallymadetodeterminethenatureofthemachineinductancetohelpthedetailedmodelofsynchronousmachines?

CH

RH

CO

q a a ad d aq q.

a a a a

b b b b

c c c c

f f f f

1. Theself-inductanceandmutualinductanceofthemachineareindependentofthemagnitudeofwindingcurrentsbecauseofneglectingmagneticsaturation.

2. Theshapeoftheairgapandthedistributionofwindingsaresuchthatallthemachineinductancesmayberepresentedasconstantplussinusoidalfunctionsofelectricalrotorpositions.

3. Slottingsareignored.4. Magneticmaterialsarefreefromhysteresisandeddycurrentlosses.5. Themachinemaybeconsideredwithoutdamperwindings.Ifadamperwindingispresented,itsinfluencemaybeneglected.

6. Highertimeandspaceharmonicsareneglected.

18. Writetheexpressionsforstatortorotormutualinductancesofasynchronousmachine.

L =L =M cosβ

L =L =M cos(β−120°)

L =L =M cos(β+120°).

19. Writetheexpressionsforstatorself-inductancesofasynchronousmachine.

L =L +L cos2β

L =L +L cos(2β+120°)

L =L +L cos(2β−120°).

20. Writetheexpressionsforrotorself-inductancesofasynchronousmachine.

L =−M +L cos(2β−120°)

L =−M +L cos2β

L =−M +L cos(2β+120°).

21. WhatisPark’stransformationandwhatisitsrequirement?

Thebehaviorofasynchronousmachinecanbedescribedbyasetofdifferentialequations.Thesolutionofthesedifferentialequationsiscomplicatedsincetheinductancesarethefunctionsofrotorangleβ,whichinturnisafunctionoftime.Thecomplicationingettingthesolutioncanbeavoidedbytransferringthephysicalquantityinthearmaturewindingsthroughalinear,time-dependent,andpower-invarianttransformercalledPark’stransformation.

22. InPark’stransformation,whichquantitiesaretransformed?

Quantitiessuchascurrents,m.m.f.s,voltages,andfluxlinkagesinabcfaxestodqofaxesaretransformed.

af fa f

bf fb f

cf fc f

aa s m

bb s m

cc s m

ab s m

bc s m

ca s m

23. ExpressPark’stransformationmatrix.

24. WhatisthefunctionofPark’stransformationmatrix?

Park’stransformationmatrix[p]transformsthefieldofabcfcomponentstothefieldofdqofcomponents.

25. Park’stransformationmatrixiscalledunitarymatrix.Why?

BecausetheinverseofPark’stransformationmatrixisthetransposeofconjugateofthematrix.

[p] =[p*]

26. Writetheexpressionforthefluxlinkagesoffdqocoilsintermsofcoilcurrentsbyusingtransformationmatrixanditsinverse.

[λ] =[p][L] [p] [i]

27. Writetheexpressionfor3-ϕpoweroutputofasynchronousmachineintermsofrateoffluxlinkagesd-axisandq-axiscurrents.

28. Intheexpressionfor3-ϕpoweroutputofasynchronousmachineintermsofrateoffluxlinkagesd-axisandq-axiscurrents,whatparametersdothethreetermsindicate?

Firstterm: representstherateofchangeof

statormagneticfieldenergies.

Secondtermω(λ i −λ i )representsthepowertransferred

acrosstheairgap.

Thirdtermr(i +i )representsthestatorohmiclosses.

29. WhatisthesignificanceofsaliencytorqueT =(L −L )i i ?

Thesaliencytorqueexistsonlybecauseofnon-uniformityintheperformanceoftheairgapalongthed-axisandtheq-axis.Thisis

dqof abcf dqof.

d d q q

d q

e d q d q

−1 T

−1

2 2

thereluctancetorqueofasalient-polemachineandexistsevenwhenthefieldexcitationiszero.

30. Whatisthesignificanceoffictitiouswinding‘g’onrotorindetailedmodelingofasynchronousmachine?

Thefictitiouswinding‘g’ontherotorofthesynchronousmachineapproximatestheeffectsofeddycurrentscirculatingintheironcurrentsandtosomeextenttheeffectofdamperwindings.Thisfictitiouswindingisshort-circuitedsinceititnotconnectedtoanyvoltagesource.

31. Whatisthesignificanceof‘f’and‘g’coilsinrotorwindinginthemodelingofasynchronousmachine?

The‘f’and‘g’coilsintherotorwindingproducetransienteffectintermsofX′ andX′ inasynchronousmachine;thefieldcoil

‘f’existsphysically,whereas‘g’coilishypotheticalforrepresentingtherotoreddycurrentsintheq-axis.

32. Whatisthesignificanceofk windingandk windinginrotor

windinginthemodelingofasynchronousmachine?

Themoreaccuraterepresentationofasynchronousmachineisobtainedbyaddingtwomorefictitiouswindings,f-coilandg-coilwindings.Onefictitiouswindingisknownasthek winding

alongthed-axisandtheotheralongtheq-axisisknownasthek

axis.Thesetwodamperwindingscanbeapproximatedbytwohypotheticalcoilsbothshortcircuitedasthereisnovoltagesourceconnectedtothem.


1. Thetransientcharacteristicsofhydro-turbinesareobtainedby:

1. Thedynamicsofwaterinthereservoir.2. Thedynamicsofwaterinthepenstock.3. Thewaterhead.4. Noneofthese.

2. Thewatertimeconstantofhydro-turbineT isassociatedwith:

1. Theaccelerationtimeforwaterinthepenstockbetweentheturbineinletandtheforebay.

2. Theaccelerationtimeforwaterinthepenstockbetweentheturbineinletandthesurgetankifitexists.

3. Either(a)or(b).4. Noneofthese.

3. ExpressionforwatertimeconstantT intermsofpower

d q

d g

d

q

w

w

generationoftheplantPis:

1.

2.

3. Either(a)or(b).4. Noneofthese.

4. Intheexpressionofwatertimeconstant ,theterme

isgivenas:

1. e=η ×η .

2. e=η /η .

3. e=η +η .

4. e=η −η .

5. Thesteamturbinesaremainlyclassifiedinto:

1. HPturbinesandLPturbines.2. Singleanddouble-typeturbines.3. Non-reheatandreheat-typeturbines.4. Noneofthese.

6. Thesystemwhereonlyoneshaftonwhichalltheturbinesaremountedare:

1. Tandemcompoundsystem.2. Cross-compoundsystem.3. Either(a)or(b).4. Both(a)and(b).

7. Inthetandemcompoundsystem,theturbinesmountedonashaftare:

1. OnlyHPtype.2. IPtype.3. OnlyLPtype.4. Allofthese.

8. In__________typeofsteamturbines,thegovernor-controlledvaluesareusedattheinlettocontrolsteamflow.

1. Tandemcompoundsystem.2. Cross-compoundsystem.3. Either(a)or(b).4. Allcompound.

9. Incontrollingthesteamflow,thetimedelaysareintroduceddueto:

1. Steamchest.2. Reheater.3. Cross-overpiping.4. Allofthese.

turbine generator

turbine generator

turbine generator

turbine generator

10. Matchthefollowing:

A B

(a)Steam-chesttimeconstant(τ ). (i)0.3–0.5s.

(b)Reheattimeconstant(τ ). (ii)4–11s.

(c)Cross-overtimeconstant(τ ). (iii)0.1–0.4s.

(d)Watertimeconstant(τ ). (iv)0.5–5.0s.

11. Anon-reheattypesteamturbineismodeledby:

1. Asingletimeconstant.2. Twotimeconstants.3. Withouttimeconstant.4. Either(a)or(b).

12. Themostsimplifiedmodelofasynchronousgeneratoris:

1. Aconstantvoltagesourcebehindproperreactance.2. Aconstantcurrentsourcebehindproperreactance.3. Avariablevoltagesourcebehindreactance.4. Avariablecurrentsourcebehindreactance.

13. Thearmaturereactionfluxattheinstantoffaultoccursduetoaverylargelaggingcurrent,whichisnearly___________innature.

1. Magnetizing.2. Demagnetizing.3. Either(a)or(b).4. Noneofthese.

14. Thearmaturereactionfluxattheinstantoffaultoccursactsalongwhichaxisofthemachine?

1. Directaxis.2. Quadratureaxis.3. Both(a)and(b).4. Noneofthese.

15. Themostspecifiedmodelrepresentationofasynchronousgeneratorcaneasilybeobtainedforanyfaultinthepowersystemwiththehelpof___________circuit.

1. Norton’sequivalent.2. Thevenin’sequivalent.3. Maximumpowertransfertheoremequivalent.

ch

Rh

co

w

4. Noneofthese.

16. Inordertoincludetheeffectofsaliency,thesimplestmodelofasynchronousmachinecanberepresentedas:

1. AfictitiousvoltageE locatedattheq-axis.

2. AfictitiousvoltageE locatedatthed-axis.

3. AfictitiousvoltageE E alongboththeaxes.

4. Noneofthese.

17. TheexpressionforE intermsofafull-loadterminalvoltageV

andfull-loadarmaturecurrentI is:

1. E =V+jI (R +X ).

2. E =V+jI R +I X .

3. E =V−I R +jI X .

4. E =V+I R +jI X .

18. Forthesynchronousgenerator,withouttheeffectofsaliency,themachineequationcanberepresentedas:

1. E =V+jIX .

2. E =V+jIX .

3. E =V−jIX .

4. E =V−jIX .

19. Theexpressionforexcitationoropen-circuitvoltageofthesynchronousgeneratorwiththeeffectofsaliencyis:

1. E =V+jIX .

2. E =V+I R +jI X .

3. E =V+I R +jI X +jI X .

4. noneofthese.

20. Thesynchronousmachinehas:

1. threewindingsonstatorcarryingAC.2. onewindingonrotorcarryingDCexcitation.3. either(a)or(b).4. both(a)and(b).

21. Theinstantaneousterminalvoltageofsynchronousmachineofanywindingexpressedintermsoffluxlinkagesis:

1. V=ir+λ.2. V=ir+jλ.3. V=ir+dλ/dt.4. noneofthese.

22. Todevelopthedetailedmodelofsynchronousmachine,whichofthefollowingassumptionsareusuallymadetodeterminethenatureofthemachineinductance?

1. theself-inductanceandmutualinductanceofmachineareindependentofmagnitudesofwindingcurrent.

2. theself-inductanceandmutualinductancemayberepresentedasconstantsplussinusoidalfunctionsofelectricalrotorpositions.

3. slottingeffectsareignored.4. magnetizingmaterialsarefreefromhysteresisandeddycurrentlosses

1. (i)and(ii)2. allexcept(iv)

q

d

d q

q

a

q a a q

q a a a q

q a a a q

q a a a q

g q

g d

g q

g d

g d

g a a a q

q a a ad d aq q

3. allexcept(i)4. allofthese

23. Theself-inductanceofanystatorphaseofasynchronousmachineisalways___________,butvariesthepositionof___________.

1. positive,rotor.2. negative,rotor.3. positive,stator.4. negative,stator.

24. Theself-inductanceofanystatorphaseofasynchronousmachineisgreaterwhen:

1. Theq-axisoffieldcoincideswiththeaxisofarmaturephase.2. Thed-axisoffieldcoincideswiththeaxisofarmaturephase.3. either(a)or(b).4. noneofthese.

25. Theself-inductanceofanystatorphaseofasynchronousmachineisleastwhen:

1. Theq-axisoffieldcoincideswiththeaxisofarmaturephase.2. Thed-axisoffieldcoincideswiththeaxisofarmaturephase.3. either(a)or(b).4. noneofthese.

26. Theexpressionsforself-inductancesofstatorphasesofasynchronousmachineare:

1. L =L +L cos2β

L =L +L cos(2β+120°)

L =L +L cos(2β−120°).

2. L =L −L cos2β

L =L −L cos(2β+120°)

L =L −L cos(2β−120°).

3. L =L +L cos2β

L =L +L cos(2β+120°)

L =L +L cos(2β−120°).

4. L =L −L cos2β

L =L −L cos(2β+120°)

L =L −L cos(2β−120°).

27. Thestatormutualinductancesofasynchronousmachineare:

1. L =M +L cos(2β−120°)

L =M +L cos2β

L =M +L cos(2β+120°).

2. L =M +L cos2β

L =M +L cos(2β−120°)

L =M +L cos(2β+120°).

3. L =−M +L cos2β

L =−M +L cos(2β−120°)

aa m s

bb m s

cc m s

aa m s

bb m s

cc m s

aa s m

bb s m

cc s m

aa s m

bb s m

cc s m

ab s m

bc s m

ca s m

ab s m

bc s m

ca s m

ab s m

bc s m

L =−M +L cos(2β+120°).

4. L =−M +L cos(2β−120°)

L =−M +L cos2β

L =−M +L cos(2β+120°).

28. Park’stransformationmatrixis:

1. linear.2. time-dependent.3. power-invariant.4. allofthese.

29. ThefactonwhichPark’stransformationbasedis:

1. therotatingfieldproducedby3-ϕstatorcurrentsinthesynchronousmachinecanbeequallyproducedby2-ϕcurrentsin2-ϕwinding.

2. therotatingfieldproducedby3-ϕstatorcurrentsinthesynchronousmachinecanbeequallyproducedby1-ϕcurrentsin1-ϕwinding.

3. either(a)or(b).4. noneofthese.

30. ThematrixformofrepresentationofPark’stransformationis:

1. [i] ≅[P] [i] .

2. [i] ≅[P] [i] .

3. [i] ≅[P] [i] .

4. [i] ≅[P] [i] .

31. Park’stransformationmatrixis:

1.

2.

3.

4. noneofthese.

ca s m

ab s m

bc s m

ca s m

dqof dqof abcf

abcf dqof abcf

dqof abcf abcf

dqof dqof abcf

32. Park’stransformationmatrix[p] transforms:

1. fieldofstatorphasorstothefieldofd-q-o-fcomponents.2. fieldofrotorphasorstothefieldofd-q-o-fcomponents.3. fieldofstatorphasorstothefieldofa-b-c-fcomponents.4. fieldofrotorphasorstothefieldofa-b-c-fcomponents.

33. Park’stransformationmatrix[p] is:

1. alinearmatrix.2. atime-dependentmatrix.3. non-linearandtime-invariantmatrix.4. both(a)and(b).

34. Park’stransformationmatrix[p] is:

1.

2.

3.

4. noneofthese.

35. whichofthefollowingiscorrectregardingtoPark’stransformation?

dqof

dqof

dqof

−1

1. [P]=[P] .

2. [P] =[P] .

3. [P] =[P ].

4. [P] =[P ] .

36. Park’stransformationmatrixiscalledunitarytransformationsince:

1. inverseofPark’stransformationmatrixisequivalenttotransposeofthematrix.

2. Park’stransformationmatrixisequaltotransposeofthematrix.3. inverseofPark’stransformationmatrixisequaltotransposeofconjugate

ofthematrix.4. noneofthese.

37. ThefluxlinkagesoffdqocoilsareexpressedintermsofPark’stransformationmatrixanditsinverseas:

1. [λ] =[P][L] [P] [i] .

2. [λ] =[P][L] [P] [i] .

3. [λ] =[P] [L] [P][i] .

4. noneofthese.

38. Theexpressionfor3-ϕpoweroutputofasynchronousmachineis:

Whichofthefollowingiscorrect?

1. firsttermrepresentstherateofchangeofstatormagneticfieldchanges.2. secondtermrepresentsthepowertransferredacrosstheairgap.3. thirdtermrepresentsthestatorohmiclosses4. allofthese.

39. Thetorqueexpressionofasalient-polesynchronousmachineis:

1. .

2. .

3. .

4. .

40. Thetorqueexpressionofacylindricalrotorsynchronousmachineis:

dqof abcf dqof

dqof dqof abcf

dqof dqof dqof

−1

−1 T

−1 *

−1 * T

−1

−1

−1

1. .

2. .

3. .

4. .

41. Inthedynamicmodelofasynchronousmachineincludingdamperwindings,themoreaccuraterepresentationisobtainedbyaddingtwofictitiouswindingsonrotor,k windingandk

windingalongthed-axisandtheq-axis.Thesedamperwindingsareapproximatedas:

1. twohypotheticalcoilsbothopen-circuited.2. twohypotheticalcoilswithvoltagesourcesconnectedtothem.3. twohypotheticalcoilsbothshort-circuitedasthereisnovoltagesource

connectedtothem.4. noneofthese.

REVIEWQUESTIONS

1. Developthelinearizedmodelingofahydraulicturbine.2. Discussthedifferentconfigurationsofreheattypeofsteam

turbineswitharepresentationoftheirfunctionalblockdiagramsandapproximatetheirlinearmodels.

3. Explainthesimplifiedmodelofasynchronousmachine.4. Describetheeffectofsaliencyinsynchronousmachinemodeling.5. Derivetheself-inductanceandmutualinductancestatorand

rotorofsynchronousmachines.6. ExplainPark’stransformationandinversePark’stransformation.7. Developthesteady-stateanalysisofsalientandnon-salient-polesynchronousmachines.

8. Developthedynamicanalysisofsalientandnon-salient-polesynchronousmachines,withandwithoutdamperwindings.

d q

12

ModelingofSpeedGoverningandExcitationSystems

OBJECTIVES


developthemodelingofspeed-governorsystemsforsteamandhydraulicturbines

developthemodelingofspeed-governorsystemswithlimiters

studytheeffectofexcitationvariationonsynchronousmachines

discussthemethodsofprovidingexcitationofsynchronousmachines

studythestructureofageneralexcitationsystem

developthetransferfunctionsofvariouscomponentsofanexcitationsystem

12.1INTRODUCTION

Twoimportantcontrolloopsareneededfortheeconomicandreliableoperationofapowersystem.Theyare:

1. Loadfrequencycontrol(LFC)loop(p.f.controlloop)fortheregulationofsystemfrequency.

2. Automaticvoltagecontrolloop(Q–Vcontrolloop)fortheregulationofsystemvoltagemagnitude.

Thesecontrolloopsindirectlyinfluencetherealandreactivepowerbalancesinthepowersystemnetwork.

TheLFCisachievedbythespeed-governormechanism.Thebasicprincipleofthespeed-governormechanismisthataccordingtotheloadvariation,thespeedoftherotorshaftofthesynchronousmachineisvariedandhencethefrequencyofthesystemisvaried.

Thischangeinfrequencyissensedandcomparedwithareferenceandproducesafeedbacksignal.Thisfeedbacksignalmakesthevariationofgeneratedpowerofsynchronousgeneratorbyadjustingtheopeningofthesteaminletvalvetosteamturbineorwatergatesinthecaseofahydro-turbine.Hence,therealpowerbalancebetweenrealpowergenerationandrealpowerdemandisachieved.Thisisthebasicprincipleofthespeed-governormechanism.

ThespeedgovernorsareregardedasprimarycontrolelementsinanLFCsystem.

Withanincreaseinthesystemsizeduetointerconnections,innormalcases,thefrequencyvariationsbecomeverylessandLFCassumesimportance.However,theroleofspeedgovernorsinrapidcontroloffrequencycanbeunderestimated.

TheautomaticvoltagecontrolorQ–Vcontrolisachievedbyanexcitationcontrolmechanism.Themainandimportantobjectiveofanexcitationsystemistocontrolthefieldcurrentofthesynchronousmachine.Thefieldcurrentiscontrolledsoastoregulatethegeneratingvoltageofthemachine.

Asthefieldcircuittimeconstantishigh(oftheorderofafewseconds),thefastcontrolofthefieldcurrentrequires‘fieldforcing’.Thus,thefieldexcitedshouldhaveahighceilingvoltage,whichenablesittooperatetransientlywithvoltagelevelsthatarethreetofourtimesthenormalvoltages.Therateofchangeofvoltageshouldalsobefast.

Theexcitationsystemsofsynchronousmachineshaveanextremeeffectonsystemstabilityandwhenevaluatedonthebasisofanincreasedpowercarryingperincreaseinthesystemcost,theyarebyfarthemosteconomicalsourceofincreasedstabilitylimits.

Theexcitationsystemoftencontainsotherfeaturessuchasvoltagedipcompensationtocompensateforthe

voltagedropinsomeimpedancebetweenthegeneratorandtherestofthenetwork.

ThefunctioningofLFCandautomaticvoltagecontrolloopsispresentedindetailinUnit-VII(LFC-II).

Inthisunit,themodelingofspeed-governingsystemsforsteamturbinesandhydro-turbinesisdiscussed.

Theeffectofvaryingexcitationsonasynchronousgenerator,methodsofprovidingexcitation,andtheirblockdiagramrepresentationandmodelingarealsodiscussed.

12.2MODELINGOFSPEED-GOVERNINGSYSTEMS

Accordingtotheprincipleofcontrol,thespeed-governingsystemsaremainlyclassifiedintotwocategories,forbothsteamandhydraulicturbines.Theyare:

1. Mechanical-hydraulic-controlledand2. Electro-hydraulic-controlled

Inboththesetypes,hydraulicservomotorsareusedforpositioningthevalveorgate,controllingthesteamorthewaterflow.

12.3FORSTEAMTURBINES

Inthissection,weshalldiscussmechanical-hydraulic-controlledspeed-governingsystem,electro-hydraulic-controlledspeed-governingsystems,andgeneralmodelforspeed-governingsystemsforsteamturbinesindetail.

12.3.1Mechanical-hydraulic-controlledspeed-governingsystems

Forasteamturbine,themechanical-hydraulic-controlledspeed-governingsystemconsistsofaspeedgovernor,aspeedrelay,hydraulicservomotor,andgovernor-controlledvalves.

Thefunctionalblockdiagramofamechanical-hydraulic-controlledspeed-governingsystemisshowninFig.12.1.

FIG.12.1Functionalblockdiagram

FIG.12.2Approximatenon-linearmodelrepresentationwithlimits

FIG.12.3SimplifieddiagramofFig.12.2

Theapproximatenon-linearmathematicalmodelcanberepresentedbytheblockdiagramshowninFigs.12.2and12.3.

K isthegainofspeedgovernor,whichisthereciprocalofregulationordroop.Itrepresentsapositionofanassumedlinearinstantaneousindicationofspeedproducedbythespeedgovernor.

Governorspeed-changerpositionprovidesthespeedregulation(SR)signalanditisdeterminedbyasystemofautomaticgenerationcontrol.

ThesignalSRrepresentsacompositeloadandspeedreference.Itisassumedtobeconstantovertheintervalofastabilitystudy.

τ isthetimeconstantofspeedrelay.Thespeedrelayisrepresentedasanintegratorandisprovidedasdirectfeedback.

Thenon-linearpropertyofthevalveiscompensatedbymeansofprovidinganon-linearCAMinbetweenthespeedrelay,andthehydraulicservomotor.

Theservomotorcontrolsthevalve’smovementandisrepresentedasanintegratorwithtimeconstantτ andisprovidedasdirectfeedback.Ratelimitingoftheservomotormayoccurforlarge,rapid-speeddeviations,andratelimitsthatareshownattheinputtotheintegrator.Thepositionlimitsthatareindicatedcorrespondtowide-openvalvesorthesettingofaloadlimiter.

Generally,thenon-linearitiespresentinaspeedcontrolmechanismareneglectedinthestudyofpowersystemoperationandcontrollingexceptforratelimitsandthelimitsonvalveposition.

Thetypicalparametersforamechanical-hydraulicsystemare:

K =20.0

τ =0.1s=speedrelaytimeconstant

τ =0.2–0.3s=valvepositioningservomotortimeconstant

G

SR

SM

G

SR

SM

12.3.2Electro-hydraulic-controlledspeed-governingsystems

Inthistypeofspeed-governingsystems,themechanicalcomponentsinthelowerpowerportionsarereplacedbythestaticelectroniccircuitsandthusprovidesmoreflexibility.

Thefunctionalblockdiagramrepresentationofanelectro-hydraulicspeed-governingsystemisshowninFig.12.4.

Thelinearityofthesystemcanbeimprovedcomparedtomechanical-hydraulic-controlledsystembymeansofprovidingfeedbackloopsofsteamflowandtheservomotor.

TheapproximatemathematicalmodelforageneralEHCsystemisshowninFig.12.5.

Thetypicalparametersforthisblockdiagramare:

K =20.0

K =3.0withsteamflowfeedback

=1.0withoutsteamflowfeedback

τ =0.1s

G

P

SM

FIG.12.4Functionalblockdiagram

FIG.12.5Blockdiagramforapproximatemathematicalmodel

FIG.12.6Generalmodelofaspeed-governingsystemforsteamturbines

12.3.3Generalmodelforspeed-governingsystems

Asimplified,generalmodelofspeed-governingsystemsforsteamturbinesisshowninFig.12.6.

Bytheproperparameterselection,thisgeneralmodelrepresentseitheramechanical-hydraulicsystemoranelectro-hydraulicsystem.

ThismodelshowstheloadreferenceasaninitialpowerP .ThisinitialvalueiscombinedwiththeincrementsduetospeeddeviationtoobtaintotalpowerP ,subjecttothetimelagτ introducedbytheservomotormechanism.

Thetypicalvaluesoftimeconstantsare:

Foramechanical-hydraulicsystem:

τ =0.2–0.3s

τ =0

τ =0.1s

Foranelectro-hydraulicsystem:

τ =τ

τ =0.025–0.15s

Notethatwhenτ =τ ,thevalueofτ orτ hasnoeffect,asthereispole-zerocancellation.

Theratelimitsarenominally0.1p.u.persecond.Thenominalvalueofk=100/(%steady-stateSR).

12.4FORHYDRO-TURBINES

Inthissection,mechanical-hydraulic-controlledspeed-governingsystems,generalmodelforahydraulicturbinespeed-governingsystem,andEHC-controlledspeed-governingsystemsarediscussedindetail.

12.4.1Mechanical-hydraulic-controlledspeed-governingsystems

0

GV 3

1

2

3

1 2

3

1 2 1 2

Itconsistsofaspeedgovernor,aunitofpilotvalveandservomotor,aunitofdistributorvalve,andgateservomotorandgovernor-controlledgates.

Thefunctionalblockdiagramofamechanical-hydraulic-controlledspeed-governingsystemisshowninFig.12.7.

Thespeed-governingrequirementsforhydro-turbinesarestronglyinfluencedbytheeffectsofwaterinertia.

FIG.12.7Functionalblockdiagramrepresentationofmechanical-hydraulic-controlledspeed-governingsystem

FIG.12.8Blockdiagramforapproximatenon-linearmodel

Toachievethestableperformanceofaspeed-governingsystem,thedashpotfeedbackisrequired.

Anapproximatenon-linearmodelfortheabovesystemisshowninFig.12.8.

Thegateservomotormayberate-limitedforlargerapid-speedexcursions.However,transientdroopfeedbackreducesthelikelihoodrate-limitinginstabilityanalysis.Positionlimitsexistcorrespondingtotheextremesofgateopening.

Thetypicalparametersofaspeed-governingsystemforhydro-turbinesandtheirvaluesandtheirrangesaregiveninTable12.1,whereτ isthetimeconstantofdashpot,τ thegatetimeconstantofgateservomotor,τthetimeconstantofpilotvalve,δthetransientspeeddroopcoefficient,andσthepermanentspeeddroopcoefficient.

Typically,τ andδarecomputedas

τ =5τ

TABLE12.1Typicalparametersofaspeed-governingsystemforhydro-turbines

Parameters Typicalvalue Range

τ 5.00 2.5–25.0

τ 0.20 0.2–0.40

τ 0.04 0.03–0.05

δ 0.30 0.2–1.00

σ 0.05 0.03–0.06

and

R

G P

R

R w

R

G

P

whereτ isthewaterstartingtimeandHtheturbine-generatorinertiaconstant.

12.4.1.1GeneralModelforHydraulicTurbineSpeed-GoverningSystem

Thegeneralmodelforahydraulicturbinespeed-governingsystemisshowninFig.12.9.

Let

Thenτ andτ ofFig.12.9canbeexpressedapproximatelyas

AlsofromFig.12.9, τ =0

P istheinitialpower(loadreferencedeterminedfromautomaticgenerationcontrol).

P istheoutputofthegovernorandisexpressedaspowerreferenceinp.u.ItisalsotobenotedthatKisthereciprocalofσ(steady-stateSRinp.u.).

12.4.2Electric-hydraulic-controlledspeed-governingsystem

Thelow-powerfunctionsassociatedwithspeedsensinganddroopcompensationinamodernspeed-governingsystemforhydro-turbinescanbeperformedbyanelectronicapparatus,whichresultsinthebetter

w

1 3

2

0

GV

performanceandgreaterflexibilityinbothdeadbandanddeadtime.Forinterconnectedsystemoperation,however,thedynamicperformanceoftheelectricgovernorisnecessarilyadjustedtobeessentiallythesameasthatforthemechanicalgovernor,sothataseparatemodelisnotneeded.

FIG.12.9Generalmodelforaspeed-governingsystemforhydro-turbines

12.5MODELINGWITHLIMITS

Therearetwotypesoflimitersthataredifferentintermsofbehavior:

1. Wind-uplimiter.2. Non-wind-uplimiter.

12.5.1Wind-uplimiter

Theblockdiagramrepresentationofawind-uplimiterisshowninFig.12.10.

Inthiscaseoflimiter,theoutputvariable(y)ofthetransferfunctionG(s)isnotlimitedandisfreetovary.Hence,thelimitercanbetreatedasaseparateblockwhoseinputis‘y’andtheoutputis‘z’.

If theequationswiththewind-uplimiter

are:

IfL≤y≤H,thenz=y

y>H,thenz=H

y<L,thenz=L

whereListhelowerlimitofoutputzandHtheupperlimitofoutputz.

12.5.2Non-wind-uplimiter

Inthiscase,theoutputofthetransferfunctionG(s)islimitedandthereisnoseparateblockforthelimiter.

Theequationsare:

f=(u–y)τ

Ify=Handf>0,

y=LandF<0

then

Otherwise,

andL≤y≤H

FIG.12.10Blockdiagramrepresentationofawind-uplimiter

FIG.12.11Blockdiagramrepresentationofanon-wind-uplimiter

Theblockdiagramrepresentationofanon-wind-uplimiterisshowninFig.12.11.

Note:

Astheoutputzofthelimiterdoesnotchangeuntilycomeswithinthelimits,thewind-uplimitercanchangeintermsofslowresponse.Generally,allintegratorblockshavenon-wind-uplimits.

12.6MODELINGOFASTEAM-GOVERNORTURBINESYSTEM

Whilemodelingthesteamgenerators,ateveryinstanttheboilercontrolsandon-linefrequencycontrolequipmentaretobeignoredduetotheirsloweroperations.

Aboilercontainsacertainamountofheatstoredinitshotmetalandthisisusuallysufficienttoguaranteethatthedemandsforextrasteamduringsystemdisturbancescanbemet.

Duringlong-termoperations,wemustconsidertherateofdeteriorationofsteamconditionsasaboilerhavingsufficientcapacityofproducingindefinitelyonlyagivenamountofextraenergyateachlevelofoutput.Demandsforthelargerincreasewillbemetforashorttime(from30sto5min);butafterthat,thesteamconditionswilldeteriorateandtheturbineoutputwilldecline.Itisextremelydifficulttoexaminethisproblemrigorouslyatpresentbecauseboilerturbinemodelsarenotcomprehensiveenough.

12.6.1Reheatsystemunit

ThebasiccomponentsofareheatsystemunitareshowninFig.12.12.

FIG.12.12Basiccomponentsofareheatsystemunit

Aisaprimarygoverningsystem

Bisananticipatorygovernorsystem

Cisthemaingoverningvalveorthrottleblade

Disthecombinedstopandemergencyvalve

Eistheinterceptorgovernorvalve

Fisthecombinedstopandemergencyvalve

12.6.1.1PrimaryGoverningSystem

Itrespondstothespeedofthemainshaft.Itcontrolseitherthemaingovernorvalveorthrottleblades.

12.6.1.2SecondaryGoverningSystem

Theinterceptorgoverningsystemwillactasasecondarygoverningsystemanditrespondstothefrequencyofturbines.ItcontrolstheinterceptorvalvesbetweentheHPstageandthereheater.Itisusuallysetsothattheinterceptorvalveisclosedanditisabout25%to50%openbeforethemaingoverningvalvescommenceto

open.Consequently,thisgoverningsystemisusuallyignored.

12.6.1.3AnticipatoryGoverningSystem

Itrespondstotheacceleratingpoweroftheunitanditisusuallynotsettooperateifeither:

1. Thegeneratoroutputismorethanacertainvalue(i.e.,25%ofmaximumoutput)or

2. Theturbinemechanicalpoweroutput(P )islessthanacertainvalue

(i.e.,80%ofthemaximumcapacity).

Fortheactivationofagovernor,boththeseconditionsshouldbeviolated.

Thisgoverningsystemisactivatedonlywhentheunitsufferslossofalargepercentageofitsloadandonsensingthiscondition,theemergencystopvalvesareclosedveryrapidlytopreventdangerousoverspeed.Theemergencystopvalvesarelocatedveryadjacenttothemaingoverningvalves.

Aresettimedelayisincludedsothatwhenbothelectricalandmechanicalpowersreverttowithinsettings,theemergencystopvalveswillopenafteracertaintime.

Thisgoverningsystemisgenerallyappliedonlyonsomemodernlargesteamunits.

12.6.1.4EmergencyOverspeedGovernorTrip

Whentheshaftvelocityexceedsapre-setvalue,thenthisgovernorwillclosethecombinedstopandemergencyvalvesandshutthesetdown.Startingupisalengthyprocess,andusuallythissetwouldnotfigureanyfurtherinthestabilitycalculations.

12.6.2Blockdiagramrepresentation

TheblockdiagramrepresentationofmodelingareheatsystemunitwithareductioninelementsofFig.12.12isshowninFig.12.13.

m

r=Steady-statedroopsystemsettinginrad/s/MWturbinepoweroutput

τ =Timeconstantofagoverningsystem

R =MaximumclosingrateofagoverningvalveinMW/s

FIG.12.13Blockdiagramrepresentationofmodelingasimplereheatsteamturbineunit

R =MaximumopeningrateofthegoverningvalveinMW/s

POS =MaximumpoweroutputoftheturbineinMW(maximumgovernorvalveopening)

POS =MinimumpoweroutputoftheturbineinMW(governingvalvemaybeclosed)

τ =EquivalenttimeconstantofsteamentrainedintheturbineHPstage

τ =Equivalenttimeconstantofthereheaterandtheassociatedpiping

ω =Referencespeedsettingofthegovernorinrad/s

ω =Actualangularrotorvelocityinrad/s

Delaysanddeadbandsarepresentintheoperationsof:

1. Thespeed-sensingmechanism,friction,andblacklash.2. Overcappingofoilportsintheservosystemaswellasfriction.3. Frictioninthemaingoverningvalve.

g

max

min

s

rh

ref

shaft

−

+

12.6.3Transferfunctionofthesteam-governorturbinemodeling

Let GP = PowerdevelopedintheturbineattheHPstage

τ = TimeconstantassociatedwithentrainedsteamintheHPstage

GP = Powerdevelopedinthesubsequentstageoftheturbine

τ = Timeconstantassociatedwithentrainedsteaminthereheaterandconnectedpipework

τ = TimeconstantassociatedwithentrainedsteaminIPandLPstagesoftheturbine.

TheexpressionfortheturbineshaftpowerP asafunctionofthegovernorvalveopeningGis:

Sincethereheatertimeconstantislower,

whereCisthefractionofpowerdevelopedinHPstage.Theturbinepowerexpressionbecomes

1

h

2

r

I

t

Forrepresentinganon-reheatsystemofturbine,simplyreplaceτ byτ inEquation(12.3)andweget

12.7MODELINGOFAHYDRO-TURBINE-SPEEDGOVERNOR

Theblockdiagramrepresentationofasimple,generalhydro-turbine-speed-governormodelingisshowninFig.12.14.

r=Steady-statedroopsettinginrad/s/MWturbineoutputpower

R=Transientdroopsettinginrad/s/MWturbineoutputpower

τ =Recoverytimeconstantoftemperaturedroopdashpot

τ =Equivalentgovernorsystemtimeconstant

R =MaximumclosingrateofthegovernorvalveinMW/s

R =MaximumopeningrateofthegovernorvalveinMW/s

POS =MaximumpoweroutputoftheturbineinMW(maximumgovernorvalveopening)

POS =MinimumpoweroutputoftheturbineinMW(usuallythegovernorvalveisfullyclosed)

TheabovemodelingisbasedonsomeassumptionsaccordingtoKirchmayer:

1. Neglectingdeadband,delays,andnon-linearperformanceinthegoverningsystem.

2. Neglectingthevariationinheadofthesetwithdailyuse(orseasonaluse).

3. Assumingaconstantequivalentwaterstartingtimeconstant.

r I

r

g

max

min

−

+

FIG.12.14Blockdiagramrepresentationofahydraulicturbine-speedgovernor

Atransientdroopsetting‘r’anddashpot(i.e.,damping)recoverytimeconstant‘τ’arequiteimportantinmoststabilitystudies,asthesteadytimeisusuallytooshortfortheeffectivedrooptoreducethesteady-statevalue.

Representationofthewatercolumninertiaisimportantasthereisaninitialtendencyfortheturbinetorquetochangeintheoppositedirectiontothatfinallyproducedwhenthereisachangeinthewicketgateinthecaseofareactionturbineoranorificeopeninginthecaseofanimpulseturbine.

12.8EXCITATIONSYSTEMS

Theexcitationsystemconsistsofanexciterandanautomaticvoltageregulator(AVR).Anexciterprovidestherequiredfieldcurrenttotherotorwindingofthealternator.Thesimplestformofanexcitationsystemisanexciteronly.Whenthetaskofthesystembecomesmaintainingtheconstantterminalvoltageofanalternatorduringvariableloadconditions,itincorporatesthevoltageregulator.

Thevoltageregulatorsensestherequirementfromtheterminalvoltageofthealternatorandactuatestheexciterforthenecessaryincreasingordecreasingofthevoltageacrossthealternatorfield.

Anexcitationsystemwithbetterreliabilitiesispreferable,eveniftheinitialcostismorebecauseofthefactthatthecostofanexcitationsystemisverysmallascomparedtothecostofthealternator.

12.9EFFECTOFVARYINGEXCITATIONOFASYNCHRONOUSGENERATOR

Considerasynchronousgeneratorsupplyingconstantpowertoaninfinitebusthroughatransmissionlineofreactance‘X’ΩasshowninFig.12.15.

Theoutputpowerofasynchronousgeneratorisexpressedas:

P =|V||I|cosϕ(12.4)

where|V|isthemagnitudeofterminalvoltage,|I|themagnitudeofcurrent,andcosϕthep.f.Itmaybeexpressedintermsoftorqueangleδas

where|E|isthemagnitudeofexcitationvoltage,|V|themagnitudeofvoltageatthebus,andδthetorqueangle.

FIG.12.15Synchronousgeneratorconnectedtoaninfinitebus

G

FIG.12.16Effectofvaryingexcitationofasynchronousgenerator

ThepoweroutputP andvoltagemagnitude|V|ataninfinitebusareconstant;therefore,fromEquations(12.4)and(12.5),wehave

|I|cosϕ=K(12.6)

|E|sinδ=K′[∴Xisalsoaconstantforthisproblem](12.7)

whereKandK′areconstants.

Equations(12.6)and(12.7)areclearlyexplainedbythephasordiagramgiveninFig.12.16.

AccordingtothephasordiagramshowninFig.12.16,forthevariationofexcitation,thetipoftheexcitationvoltagevector‘E’isrestrictedtomovealongthehorizontaldottedline,andthetipofthecurrentvector‘I’isrestrictedtomovealongtheverticaldottedline.

ItisobservedfromFig.12.16thatwhentheexcitationincreasesthetorqueangle‘δ’reduces(fromδtoδ′),thecurrentincreases,andpowerangleincreasesfromϕtoϕ′andhencebecomesmorelaggingwithrespecttotheterminalvoltage‘V’.

G

Hence,thetorqueangle‘δ’isreducedwithanincreaseinexcitation,whichresultsinanincreaseinstiffnessofthemachine,i.e.,thecouplingsofthegeneratorrotorandrotatingarmaturefluxbecomemoretight.Inotherwords,withtheincreaseinexcitation,thestabilityofthemachinewillbecomeenhanced.

12.9.1Explanation

Fortheincrementinexcitationvoltage,thetorqueangleδreduces.

Letusassumethatacylindricalrotor(woundrotor)synchronousgeneratorconnectedtoaninfinitebusisinitiallyoperatingattorqueangleδ andsupplyingapower .

Thegeneratoroutputisequaltotheturbinepower,.

Now,drawthepoweranglecharacteristicsofthegeneratorasshowninFig.12.17(a).

Wehavethepoweroutputofthegeneratoras:

Forastableregion,

Atδ=90°, [∵sinδ=0]

Foranunstableregion,

Withthedecreaseinexcitation,thetorqueangleincreases,andhencethestiffnessofthemachinedecreases.

0

FromthepoweranglecharacteristicsshowninFig.12.17(a),agraphbetween|E|andδcanbedrawnasshowninFig.12.17(b).

Itisobservedthatbydecreasingtheexcitation,pointDisreachedandtheinstabilitywilltakeplaceat

.Theoccurrenceofinstabilityisshownasa

driftinthecurve.Atanypointonthelowerportionofthecurve,δ=f(|E|),thestablestateismaintainedsince

itcorrespondstothepoint onP =f(δ)curve.G

FIG.12.17(a)Poweranglecharacteristicsofasynchronousgeneratoratdifferentexcitations;(b)thevalueofδasafunctionof|E|

∴Thestabilitycriterionofthesystemcanbemathematicallyformulatedas

Anditscriticalpointisgivenby .

Differentiatingtheaboveequationwithrespectto|E|,weget

Instabilitywilloccurwhenδ→90°,

FromFig.12.17,itisconcludedthatthesteady-statestabilityofthesynchronousgeneratorisimprovedbyincreasingitsexcitation.

12.9.2limitationsofincreaseinexcitation

Theincreaseinexcitationislimitedbythefollowingfactors:

Maximumoutputvoltageoftheexcitersupplyingthefieldcurrent.Resistanceofthefieldcircuit.Saturationofthemagneticcircuitandrotorheating.

12.10METHODSOFPROVIDINGEXCITATION

Theexcitationisprovidedbythefollowingtwomethods:

1. Commonexcitationbusmethod.2. Individualexcitationmethod

12.10.1Commonexcitationbusmethod

Itisalsoknownasthecentralizedexcitationmethod.Inthismethod,twoormorenumberofexcitersfeedacommonbus,whichsuppliesanexcitationtothefieldsofallgeneratorsintheplant.

12.10.2Individualexcitationmethod

Itisalsoknownastheunit-excitermethod.Inthismethod,eachgeneratorisfedfromitsownexciter,whichisusuallydirectconnectedtothegeneratorshaft,butsometimesitisdrivenbyamotororasmallprimemoverorboth.

Theindividualexcitation(or)unit-excitermethodismorepreferablebecauseafaultinanyoneexciteraffectstheentireexcitationsystem.

12.10.2.1Meritsofindividualexcitationmethods

1. Simplicity:Sinceeachalternatorhasitsownexciter,thismethodofexcitationresultsinasimplelayoutofthestation.Theexcitersaresoselectedaccordingtotherequirementofindividualgeneratorsthatthemainfieldrheostatsandhigh-capacityswitchgeararenotrequired,whicharenecessaryinthecaseofthecommonbusexcitationmethod.

2. Lessohmiclosses:Theohmiclossesareverylessbecausenorheostatsarerequiredinthegeneratorfieldcircuit,andtheexciterfieldrheostatsareoperatedatamuchlowerpower.

3. Higherreliability:Asanyfaultthatoccursinexciteraffectsonlythegeneratortowhichitisconnected,theunit-excitermethodhashigherreliabilitythancommonexcitermethod.

4. Incorporationautomaticregulators:TheAVRsareincorporatedinanindividual(or)unitexcitationsystemforreliablesharingofreactivepowertomaintainconstantterminalvoltagewhilethegeneratorsarerunninginparallel.

5. Lessmaintenance:Sincetheunitexcitationsystemhasnomainfieldrheostatsandhigh-capacityswitchgear,theindividualexcitationsystemrequireslessmaintenanceandduetothisithaslessmaintenancecost.

Itisimportanttonotethatanexcitationsystemwithbetterreliabilityispreferredeventhoughitsinitialcostismorebecauseofthefactthatthecostofanexcitationsystemisverylessascomparedtothecostofagenerator.

12.10.3Blockdiagramrepresentationofstructureofageneralexcitationsystem

AblockdiagramrepresentationofthestructureofageneralexcitationsystemisshowninFig.12.18.

Themaincomponentspresentintheblockdiagramare:

12.10.3.1Synchronousgenerator

Itmaybethetypeofhigh-speedturbo-alternatorrunbyasteamturbineoralow-speedACgeneratorrunbyahydro-turbine.Withthehelpofanexcitationsystem,theterminalvoltageofanalternatororasynchronousgeneratorshouldbemaintainedconstantduringvariableloadsituations.

12.10.3.2Exciter

Itsuppliesthefieldcurrenttotherotorfieldcircuitofthesynchronousgenerator.Itmayeitherbeaself-excitedtypeorseparatelyexcitertypeofDCgenerator.

FIG.12.18Blockdiagramrepresentationofageneralexcitationsystem

FIG.12.19Responseofanexciterwhenseparatelyexcitedandself-excited

Inaself-excitedexciter,afewturnsareaddedforcompoundingandinter-polesareused.Inseparatelyexcitedexciters,anexciterfieldissuppliedfromasmallDCgeneratorknownasthepilotexciter.Apilotexciterisalevelcompoundgeneratorandmaintainsconstantvoltageexcitationforthemainexciter.

Theresponseofanexciterwhencomparedtoseparatelyexcitedwiththatwhenself-excitedisshowninFig.12.19.

Usually,theseparatelyexcitedexciter,knownasmainexciter,isprovidedwithtwoormorethantwofieldwindings,asshowninFig.12.20.Duetothisarrangementoffield,aneasierautomaticvoltageregulationispermitted.

Thevoltageofthemainexcitershouldbecontrolledfromzerotoceilingvoltage,themaximumvoltagethatmaybeattainedbytheexciterunderspecifiedconditions,toobtainrapidcorrectionofexcitervoltageafterdisturbanceorfault.ThefaultsorsystemdisturbancescauseanAVRtoforceanexcitationup.Afterpost-fault,rapidreductionoffieldisnecessarytoadjusttheexcitationtothecorrectvalue.Thisiseasilyachievedwithanegativefield.Themainpositivefieldisarrangedintwoparallelsectionswithrheostatsforadjustingthefieldcurrentsasrequired.

Hence,thepositiveandnegativefieldwindingsofthemainexciterwiththeadjustmentsofcurrentsaccordingtotheloadontheexcitermaintaintheexcitervoltageandexcitationasrequired.

FIG.12.20Exciterfieldarrangements

Duetotheseveralparallelconnectedfieldwindings,thefastresponseoftheexciterisachieved,becauseoflowtimeconstantofthewholefieldcircuit.

Forsmall-sizedturbo-generators,theexcitersareusuallydirectlycoupledtothegeneratorshaft.Formedium-andlarge-sizedturbo-generators,theexcitersarecoupledtothemainshaftthroughthegearandaregenerallydrivenat1,000rpm.

Forsmallergenerators(i.e.,ratedupto25MVAorso),self-excitedexcitersmustbeusedandforlarge-sizedgeneratorsofabove25MVA,separatelyexcitedexcitersareused.Theexcitervoltageofthemainexciterisusually230V.Insomecases,anominalvoltageof440Visused.Themainexciterloadintheresistanceisthealternatorfieldwindingandthisisgenerallybetween0.25and1.0Ω.Therotorcurrentisabout10AperMVAofalternatorrating.

12.10.3.3Useofamplidyne

Insomecases,theDCexcitationsystemisequippedwithanamplidyne.Anamplidyneprovideslargecurrentstothefieldwindingofthemainexciter.

Itisahigh-responsecross-fieldgeneratorandhasanumberofcontrolwindings,whichcanbesuppliedfromthepilotexciterandanumberoffeedbackcircuitsofanAVRandmagneticamplifier,etc.forcontrolpurposes.Anamplidynehasaveryhighamplificationfactorof10orevenmoreandneedsverysmallcontrolpower.

12.10.3.4AVR

AnAVRinconjunctionwiththeexcitertriestomaintainconstantterminalvoltageofonanACgenerator.Thevoltageregulator,infact,couplestheoutputvariablesofthesynchronousgeneratortotheinputoftheexciterthroughfeedbackandforwardingelementsforthepurposeofregulatingthesynchronousmachineoutputvariables.Thus,thevoltageregulatormaybeassumedtoconsistofanerrordetector,pre-amplifier,poweramplifier,stabilizers,compensators,auxiliaryinputs,andlimiters.Thevoltageregulatoristreatedastheheartofanexcitationsystem.Exciterandregulatorconstituteanexcitationsystem.Exciter,regulator,andsynchronousgeneratorconstituteasystemknownastheexcitationcontrolsystem.

12.11EXCITATIONCONTROLSCHEME

AtypicalexcitationcontrolschemeisshowninFig.12.21.

Thefieldwindingofanalternatorisconnectedtotheexciter.Thealternatorterminalvoltageisrectifiedbymeansofapotentialtransformer(PT)andrectifier,andisfedtoavoltageregulator.AvoltageregulatorcomparestherectifiedoutputvoltagewithareferencevoltageV .TheerrorsignaloutputV =|V –V |fromthevoltageregulatorisamplifiedbyanamplifierandtheamplifieroutputvoltageisfedtotheexciterfieldwinding.

ref e ref dc

6

FIG.12.21Atypicalexcitationcontrolscheme

ThereisnoerrorsignaloutputfromtheregulatorandthefieldwindingcurrentofexciterI isconstantwhentheoutputvoltage(terminalvoltage)ofanalternatorisatanominalvalue.

Whentheloadonthealternatorvaries,theterminalvoltagealsovaries.Hence,theerrorsignalcanbeproducedbytheregulator,amplified,andfedtothefieldwindingoftheexciter.Thefieldwindingcurrentoftheexciterisvariedandhencetheterminalvoltagereachestherequiredlevel.

12.12EXCITATIONSYSTEMS—CLASSIFICATION

Theexcitationsystemsarebroadlyclassifiedintothefollowingcategories:

1. DCexcitationsystem.2. ACexcitationsystem.3. Staticexcitationsystem.

12.12.1DCexcitationsystem

Itconsistsofdifferentconfigurationsofrotatingexciterslike:

1. Self-excitedexciterwithadirect-actingrheostatic-typevoltage

e

regulator.2. Mainandpilotexciterswithanindirect-actingrheostatic-typevoltage

regulator.3. Mainexciter,amplidyne,andstaticvoltageregulator.4. Mainexciter,magneticamplifier,andstaticvoltageregulator.

ThemaindrawbacksofaDCexcitationsystemare:

Complexityismoreduetorotatingexciters,voltageregulators,andmovingcontactslikeslipringsandbrushes.Timeconstantsofexciter,voltageregulator,amplidyne,andmagneticamplifierarelarge(about3s).Difficultiesofcommutation.Smoothlessoperationneedscontinuousmaintenance.Reliabilityisless.Noiselevelismoreduetorotatingexciters.

12.12.2ACexcitationsystem

ItconsistsofanACgeneratorandathyristor(rectifier)bridgecircuitdirectlyconnectedtothealternatorshaft.Themainadvantageofthismethodofexcitationisthatthemovingcontactssuchasslipringsandbrushesarecompletelyeliminatedthusofferingsmoothandmaintenance-freeoperation.Suchasystemisknownasabrushlessexcitationsystem.Inthissystem,therearenocommutationproblems.

12.12.3Staticexcitationsystem

Itconsistsofastep-downtransformerandarectifiersystemusingmercuryarcrectifiersofsilicon-controlledrectifiers(SCRs).Therotatingamplifiersandrotatingexcitersarereplacedbythestaticdevicesofarectifiersystem.

Theadvantagesofastaticexcitationsystemare:

1. Noise-freeoperationintheplantisobtainedastherotatingexcitersarereplacedwithstaticdevicesofrectifiers.

2. Sincethestaticexcitationequipmentmaybemountedorplacedseparatelyataconvenientplace,thecomplexityoftheexcitationsystemisreduced.

3. Duetostaticdevices,theoveralllengthofthegeneratorshaftisreduced,whichsimplifiesthetorsionproblemandtheproblemofcriticalspeed.Thegeneratorrotoriseasilywithdrawnfor

maintenancepurpose.4. Highreliabilitycomparedtootherexcitationsystemsbecauseof

havingmorereliablestaticdevicesand5. Staticdevicesareprovidedwithlow-speedhydro-alternators,where

large-sizedrotatingexcitersareneeded.

12.13VARIOUSCOMPONENTSANDTHEIRTRANSFERFUNCTIONSOFEXCITATIONSYSTEMS

Inthissection,weshalldiscussPTandrectifier,voltagecomparators,andamplifiersindetail.

12.13.1PTandrectifier

OnepossiblearrangementofaPTandarectifierisshowninFig.12.22.TheterminalvoltageofthealternatorissteppeddownbythePTandrectifiedtoformV ,whichisproportionaltotheaverageRMSvalueoftheterminalvoltageV .

FIG.12.22ConnectionsofPTandrectifier

Transferfunctionofthearrangementisrepresentedas:

DC

t

whereK istheproportionalityconstant(or)gainofthePTandrectifierassemblyandτ thetimeconstantoftheassemblyduetofilteringintheassemblyarrangement.

12.13.2VoltageComparator

AvoltagecomparatorcomparestherectifiedDCvoltageofthegeneratorwithareferencevoltageV andproducesanoutputintheformofanerrorsignalV .Figure12.23showsanelectronicdifferenceamplifierasacomparator.

TheoutputvoltageoranerrorvoltageV isexpressedas

V (S)=K(V (S)–V (S))

12.13.3Amplifiers

Amongthevarioustypesofamplifiersusedinanexcitationsystem,theamplidyneandmagneticamplifierhavehighamplificationfactors.

12.13.3.1Amplidyne

Basically,across-fieldDCgeneratorisoperatedasanamplidyne.AnamplidyneconfigurationisshowninFig.12.24.

Theoperationofanamplidyneconsistsoftwostagesofamplification.

FirstStageofAmplification

Anamplidyneconsistsofbrushesalongthed-axisandtheq-axis.Thebrushesalongtheq-axisareshort-circuited.Asthearmatureresistanceisverysmall,asmallamountoffieldm.m.f.resultsinalargeq-axis

R

R

ref

e

e

e ref dc

current.Thisproductionofthelargeq-axiscurrentistreatedasthefirststageofamplification.

SecondStageofAmplification

Thelargeq-axiscurrentproducesafluxintimeandspacephasewithitself.Correspondingtothisq-axisflux,thevoltagewillbedevelopedacrossthed-axisbrushes.Thedevelopmentofvoltageacrossd-axisbrushesduetotheq-axiscurrentistermedasthesecondstageofamplification.

FIG.12.23Electronicdifferenceamplifierasacomparator

FIG.12.24Amplidyneconfiguration

Thed-axisbrushesareconnectedtotheloadalongwithacompensatingwinding,whichprovidesnoresultantexcitationduetoloadcurrentsinceitmayreducetheoriginalfieldexcitation.(IthasanumberofcontrolwindingssuppliedfromthepilotexciterandhasanumberoffeedbackcircuitsofAVRandmagneticamplifier,etc.forcontrolpurposes.)

Theadvantagesofthesecondstageofamplificationareasfollows:

Powerrequiredforcontrolpurposeisverysmall.Astheresponsetimeisveryless,ithasafastresponse.Amplificationfactorisof10 orevenmore.

Thetransferfunctionofanamblidyneunderno-loadconditioncanbeexpressedas

whereτ isthetimeconstantofarmature,τ thetimeconstantoffield,andK thevoltageamplificationfactor.

12.13.3.2Magneticamplifier

MagneticamplifierconfigurationisshownisFig.12.25.Itconsistsofasaturablecore,controlwinding,andarectifiercircuit.

ThesaturablecorereactorcanbedesignedsothatwhennoDCcurrentisflowingthroughtheDCcontrolwinding,theinductivereactanceofACcoilsisveryhighandlimitstheflowofACcurrenttoasmallvalue.

InamagneticamplifierwhenlargeDCcurrentflowsthroughthecontrolwinding,thecoregetssaturated.ThisresultsinthedecrementinpermeabilityandhencethereactanceofACcoilsdecreases.Therefore,moreACcurrentflows.ThisACcurrentisrectifiedandfedtotheload.

a r

A

6

FIG.12.25Configurationofamagneticamplifier

Thecontrollingofalargeoutputcurrentbymeansofasmallcontrolcurrentisthemainprincipleofamagneticamplifier.

Thetransferfunctionofamagneticamplifiercanbeexpressedas

whereV isanerrorsignalinputappliedtocontrolwinding,V istheoutputvoltageandisgovernedbythelimits,i.e.,V ≤V ≤V ,K istheamplificationfactor,andτ isthetimeconstantamplifier.

12.14SELF-EXCITEDEXCITERANDAMPLIDYNE

e

R

Rmin R Rmax A

A

TheamplidyneisconnectedinserieswiththeshuntfieldofthemainexciterasshowninFig.12.26.

LetV bethearmatureemfoftheamplidyne,

e thearmatureemfofmainexciter,

N thenumberoffieldturnsofmainexciterunderno-loadcondition,

ϕ thefluxofthemainexciterunderno-loadcondition,

r thefieldcircuitresistanceoftheexciterunderno-loadcondition,and

i thefieldcurrentoftheexciterunderno-loadcondition.

FIG.12.26Circuitdiagramofseriescombinationofanamplidynewiththeshuntfieldofthemainexciter

Foranexciterfieldcircuit,

Sincee isafunctionofϕ ,effectivefluxofthemainexciter

e =Kϕ

R

fd

e

e

e

e

fd e

fd e

where ,aconstantforthearmature

Differenceofϕ andϕ isanaccountofleakagesfluxϕ ,proportionaltofieldcurrenti ,andmaybewrittenas

ϕ =ϕ +ϕ

andϕ =C ϕ

whereC istheproportionalityconstant:

ϕ =ϕ +C ϕ

=(1+C )ϕ

ϕ =σϕ

whereσisknownasthecoefficientofdispersionhavingavalueintherangeof1.1–1.2.

whereτ isknownastimeconstantoftheexciter.

However,theeffectofsaturationoftheexcitervoltagee istakenintoaccountwhilesolvingtheaboveequation.

TheexcitercharacteristicsareshowninFig.12.27.

ItisevidentthatthesaturationoftheexciterS isanon-linearfunctionoftheexcitervoltagee andisgivenas

fe e l

e

fe e 1

1 1 e

1

fe e 1 e

1 e

fe e

E

fd

E

fd

FIG.12.27Excitercharacteristics

Fromtheabove,wecanwrite

Iftheslopeoftheair-gaplineis

ori =G(1+S )e

Substitutingi inequation wehave

e E fd

e

TakingLaplacetransformoftheaboveequation,weget

V (S)E (S)=r G(1+S )E (S)+sτ E (S)

Fromtheabove,wecanget

whereK =r G−1

12.15DEVELOPMENTOFEXCITATIONSYSTEMBLOCKDIAGRAM

ThesimplifieddiagramofanexcitationsystemwithfundamentalcomponentsisasshowninFig.12.28.

FIG.12.28Simplifieddiagramofabuck-boostexcitationsystem

Forthecompleteanalysisoftheexcitationsystem,itisnecessarytodevelopthetransferfunctionofeach

R fd e E fd B fd

E e

componentandthenthetransferfunctionofanoverallexcitationsystem.

TransferfunctionofPTandrectifieris

Transferfunctionofanamplifieris

and

Transferfunctionoftheexciteris

Here,

Ifthesaturationisneglectedi.e.,S =0,

∴Transferfunctionoftheexcitercanbewrittenas

12.15.1Transferfunctionofthestabilizingtransformer

AnequivalentcircuitofastabilizingtransformerisshowninFig.12.29.

Theexcitationsystemthatwasdescribedearlierhasadynamicresponsethatispronetoexcessiveovershoot

E

andstabilityproblems.Theseproblemsareovercomebyaddingastabilizingtransformer.Forthestabilizingtransformer,theinputbecomesE andtheoutputisV .

FIG.12.29Equivalentcircuitofastabilizingtransformer

TheoutputV issubtractedfromV,i.e.,V−V toprovidetheinputtotheamplifier:

Input,

InLaplacetransform,

E (s)=(R +sL )I (s)

andoutput

i.e.,V (s)=sMI (s)

∴Transferfunctionofastabilizingtransformeris

where =Transformergain

fd ST

ST ST

fd 1 1 1

ST 1

and =Transformertimeconstant.

12.15.2Transferfunctionofsynchronousgenerator

whereK isthegainofthegeneratorandτ thetimeconstantofarotorfield.

12.15.3IEEEtype-1excitationsystem

MostoftheexcitationsystemsaremodeledbasedonIEEEtype-1excitationsystem,whichwasproducedbythereportofthefirstIEEEcommitteein1968.ThecompleteblockdiagramofanIEEEtype-1excitationsystemisasshowninFig.12.30byinterconnectingallthecomponentsintheforwardpathandthefeedbackcontrolloop.

FIG.12.30IEEEtype-1excitationsystem

E = Exciteroutputvoltage(appliedtogeneratorfield)

G G

FD

I = Generatorfieldcurrent

I = Generatorfieldterminalcurrent

K = Regulatorgain

K = Exciterconstantrelatedtoself-excitedfield

K = Excitersaturationfunction

τ = Regulatoramplifiertimeconstant

τ = Excitertimeconstant

τ = Regulatedstabilizingcircuittimeconstant(τ andτ )

τ = Regulatedinputfiltertimeconstant

V = Regulatoroutputvoltage

V = Terminalvoltageofthegeneratorappliedtotheregulatorinput

K = Gainofthegenerator

τ = Timeconstantofthegeneratorrotorfield

FD

T

A

E

F

A

E

FF1 F2

R

R

t

G

G

Whenthegeneratorisoperatingatanequilibriumstate,i.e.,atratedvoltage,thevoltageoftherotatingamplifierV becomeszero.IfthegeneratorloadisincreasedsuchthatthesensingcircuitshowninFig.12.30detectsthisfallinterminalvoltage,itcausestheamplidynetoincreasethefieldcurrentI intheexciterfield.Hence,theexcitervoltageincreasesandinturnincreasesthegeneratorfieldcurrentI thatultimatelyshouldrisetheterminalvoltageofgenerator,V .

Understeady-stateconditions,E=E .

Undertransientconditions,anymismatchbetweenEandE willcausethevoltage tochangeaftersomedelay.

Mathematically,

where istheopen-circuitgenerator,directaxistransienttimeconstant.

12.15.4TransferFunctionofOverallExcitationSystem

Thetransferfunctionofanoverallexcitationsystem,showninFig.12.31,canbeobtainedbyeitherusingtheblockdiagramreductiontechniqueorthesignalflowgraphmethod.

First,neglecttheeffectofsaturation:

i.e.,S =0

andremovethestabilizingtransformerfromtheblockdiagramshowninFig.12.30.

∴Thetransferfunctionofthesystemisoftheform:

R

e

f

t

fd

fd

E

where andisknownasthe

feed-forwardtransferfunctionand isknown

asthefeedbacktransferfunction.

∴Thetransferfunctionoftheexcitationsystemisexpressedas

Inthetransferfunction ,τ isasimpletime

constantrepresentingregulatorinputfiltering.Itisverysmallandmaybeconsideredtobezeroformanysystems.

Thefirstsummingpointcomparestheregulatorreferencewiththeoutputoftheinputfiltertodeterminethevoltageerrorinputtotheregulatoramplifier.

TheAVRusuallycomprisesseveralcontrolloopsanda

simplereductionisnecessarytotheform .Voltage

regulatorgain(K )hasanimportanteffectonpowersystemperformancewhilethetimeconstantτ hasamuchsmallerinfluenceowingtolargeτ inseries.

R

A

A

e

Becauseofhighgaininanexcitationsystem(100–400),errorsinforward-pathgainK aremoreimportantthanerrorsinmostotherparameters(includinggeneratorandnetwork).

Thesecondsummingpointcombinesthevoltageerrorinputwiththeexcitationdampingloopsignal.K andτrepresentthemainregulatorgainanditstransferfunction.Theminimumandmaximumlimitsoftheregulatorareimposedsothatlargeinputerrorsignalsmaynotproduceanegativeoutput,whichexceedsthepracticallimit.

ThenextsummingpointsubtractsasignalthatrepresentsthesaturationfunctionS =f(E )oftheexciter.Thatis,theexciteroutputvoltage(orgeneratorfieldvoltageE )ismultipliedbyanon-linearsaturationfunctionandsubtractedfromtheregulatoroutputsignal.Theresultantisappliedtotheexcitertransferfunction

.

Majorloopdampingisprovidedbythefeedback

transferfunction, fromtheexciteroutputE to

thefirstsummingpoint.

Ifthestabilizingloopisomitted,theexcitersystemandthemaingeneratorwillbeunstableformostpracticalvaluesofK .Itcanonlybeomittedwhenthereareadditionalinputsignalstotheexcitationsystemsuchasfrequencyderivatives,etc.

TheusefulvalueofK isfrom0.1to0.15andτ variesintherange0.5–2.0s.

V =Regulatorreferencevoltagesetting

A

A A

E FD

FD

FD

A

F F

ref

V =Fieldrheostatsetting

V =Generatorterminalvoltage

ΔV =Generatorterminalvoltageerror

NotethatthereisaninterrelationbetweentheexciterceilingvoltageE ,regulatorceilingE ,exciter

saturation,S andK .

Understeady-statecondition,

V –(K +S )E =0;E ≤E ≤E

AttheceilingorE αE ,theaboveequation

becomes

V −(K +S )E =0

TheexcitersaturationfunctionisdefinedasthemultiplieroftheexciteroutputE torepresenttheincreaseinexciterexcitationrequirementbecauseofsaturation.

Theexcitertimeconstantτ isadominanttimeconstantintheexcitationsystem.Ifitisnotpossibletoobtaindataforthemainexcitersaturationfunction,thenausefulapproximationistoincreaseτ by20%anddecreasetheexciterceilingvoltageby20%.

12.16GENERALFUNCTIONALBLOCKDIAGRAMOFANEXCITATIONSYSTEM

ThegeneralfunctionalblockdiagramofanexcitationsystemisshowninFig.12.31.

12.16.1Terminalvoltagetransducerandloadcompensation

TheterminalvoltageofthealternatorissensedandrectifiedintoaDCvoltagebymeansofaterminalvoltagetransducer.

RH

T

T

FDmax Rmax

E E

R E E FD FDmin FD FDmax

FD FDmax

Rmax E Emax FDmax

FD

e

e

Theloadcompensationsynthesizesavoltage,whichdiffersfromtheterminalvoltagebythevoltagedropinanimpedance(R +jX ).BothvoltageandcurrentphasorsmustbeusedincomputingthecompensatingvoltageV .

12.16.1.1Objectivesofloadcompensation

Sharingofreactivepoweramongtheunits,whicharebussedtogetherwithzeroimpedancebetweenthem.Forthis,R andX arepositive

andthevoltageisregulatedatapointinternaltothegenerator.Whentheunitsareoperatinginparallelthroughunittransformers,itisdesirabletoregulatevoltageatapointbeyondthemachineterminalstocompensateforaportionoftransformerimpedance.Forthiscase,bothR andX arenegativevalues.R isneglectedin

mostofthecases.

FIG.12.31Functionalblockdiagramofanexcitationcontrolsystem

FIG.12.32Modelingoftransducerandloadcompensation

ThemodelingofterminalvoltagetransducersandloadcompensationisasshowninFig.12.32.

Thevoltagetransducerisusuallymodeledasasingletimeconstantτ anditisverysmallandassumedtobezeroforsimplicityinmanycases.

LC LC

C

LC LC

LC LC LC

R

12.16.2Excitersandvoltageregulators

TheAVRsofmoderntypearecontinuouslyactingelectronicregulatorswithhighgainandlowertimeconstants.

12.16.2.1Typesofexciters

ThetypesofexcitersareshowninFig.12.33(a).TheblockdiagramrepresentationofanexciterandaregulatorisshowninFig.12.33(b).

InFig.12.33(b),V istheoutputoftheregulator,whichislimited:

τ —singletimeconstantofregulator

K —positivegain

SaturationfunctionS =f(E )representsthesaturationoftheexciter.

Note:ThelimitsonV canbefoundfromsteady-stateequation:

V –(K +S )E =0

ThisimplieslimitsonE suchthat:

E ≤E ≤E

Withthespecificationofparameters,K =1,τ =0,S=0,andV =K V ,IEEEtype-1systemrepresentsthe

staticexcitationsystemwithpotentialsource-controlledrectifiertype.

12.16.3Excitationsystemstabilizerandtransientgainreduction

Thissystemisusedtoincreasethestabilityregionofoperationoftheexcitationsystemandalsopermithigherregulatorgains.

ThefeedbacktransferfunctionoftheESSisshowninFig.12.34.

R

A

A

E FD

R

R E E RD

FD

FDmin FD FDmax

E E E

Rmax p T

TheESSisrealizedbyanidealtransformerwhosesecondaryisconnectedtohighimpedanceasshowninFig.12.35.

Theturnsratioofthetransformer(n)andthetime

constantofthecircuit determineK andτ suchas

and

τ isusually1s.

Aseries-connectedloadorlagcircuitcanalsobeusedinsteadoffeedbackcompensationcircuitforESSasshowninFig.12.36.

F F

F

FIG.12.33(a)Classificationofexciters;(b)blockdiagramrepresentationofexciterandvoltageregulator

FIG.12.34ESStransferfunction

FIG.12.35RealizationofESS

FIG.12.36TGR

Here,τ >τ andstabilizationistermedasTGR.Reducingthetransientgain(orgainathigherfrequencies),therebyminimizingthenegativecontributionoftheregulatortosystemdamping,isthemainobjectiveofTGR.TheTGRmaynotberequired,ifpowersystemstabilizer(PSS)isspecificallyusedtoenhancesystemdamping.

C B

Usually,TGRfactor

12.16.4Powersystemstabilizer

Duringthetransientdisturbance,therotoroscillationsoffrequency0.2–2.0HzaredampedoutbyprovidingthePSSs.Thedampingofrotoroscillatationscanbeimpairedbytheprovisionofhigh-gainAVR,particularlyathighloadingconditionswhenageneratorisconnectedthroughahighexternalimpedance(duetoweaktransmissionnetwork).

TheinputsignaltoPSSisderivedfromspeedorfrequencyoracceleratingpoweroracombinationofthesesignals.

TheoutputofPSS,V ,isaddedtotheterminalvoltageerrorsignal.

12.17STANDARDBLOCKDIAGRAMREPRESENTATIONSOFDIFFERENTEXCITATIONSYSTEMS

ThestandardblockdiagramsofdifferentexcitationsystemsbasedonsupplywerepublishedbythesecondIEEEcommitteereportintheyear1981.

12.17.1DCexcitationsystem

Figure12.37showsthetypeDC-1excitationsystem.ItconsistsofaDCcommutatorexciterwithacontinuouslyactingvoltageregulator.ThisissimilartotheIEEEtype-1excitationsystem.

TheTGRcanberepresentedbythetransferfunction

withτ >τ .IthasthesimilarfunctionasESSin

thefeedbackpath.

S

B C

EitherTGRintheforwardpathorESSinthefeedbackpathisshownintheblockdiagramrepresentation.

Withτ =τ ,theTGRcanbeavoidedandsimilarlywithK =0,ESScanbeneglected.

12.17.1.1Derivationoftransferfunction

(i)ForseparatelyexcitedDCgenerator(exciter)

Figure12.38showstheseparatelyexcitedDCgenerator.FromFig.12.38,

Thegenerator(exciter)outputvoltageE isanon-linearfunctionofI asshowninFig.12.39.

FIG.12.37TypeDC1-DCcommutatorexciter

FIG.12.38SeparatelyexcitedDCgenerator

B C

F

g

f

FIG.12.39Exciterloadsaturationcurve

Assumethespeedoftheexcitertobeconstant.FromFig.12.39,wehavethefollowing:

whereR istheslopeofthesaturationcurvenearE =0.ExpressI inp.u.ofI :

whereE istheratedvoltagethatisdefinedasthevoltage,whichproducesratedopen-circuitvoltageinthegenerator-neglectingsaturation:

whereS′ =R S

g g

f fb

gb

E g e

TheblockdiagramofaseparatelyexcitedgeneratorisshowninFig.12.40.

(ii)Self-excitedDCgenerators

Theschematicdiagramofaself-excitedexciterisshowninFig.12.41.

E representsthevoltageoftheamplifierinserieswiththeexcitershuntfield.

FIG.12.40BlockdiagramofaseparatelyexcitedDCgenerator

FIG.12.41Schematicdiagramofaself-excitedexciter

a

TheblockdiagramofFig.12.41withE =V canbe

reducedsuchthat

TheR isperiodicallyadjustedtomaintainV =0inthesteadystate,forthisK =–S whereS isthevalueof

saturationfunctionS attheinitialoperatingpointandK isgenerallynegativeforaself-excitedexciter.

12.17.2ACexcitationsystem

TheblockdiagramofatypeAC-1excitationsystemisshowninFig.12.42.Thisrepresentsthefield-controlledalternatorrectifierwithnon-controlledrectifier-typeACexcitationsystem.

ThetermK I representsarmaturereactionofthealternatorandF representsrectifierregulation.

ConstantK isafunctionofthesynchronousalternator,andtransientreactanceconstantK isafunctionofthecommutatingreactance.

ThefunctionF isgivenas

a R

f R

E Eo Eo

E

E

D FD

EX

D

C

BX

FIG.12.42BlockdiagramoftypeAC-1excitationsystem

ThesignalV isproportionaltotheexciterfieldcurrentandisusedasaninputtoESS.

12.17.3Staticexcitationsystem

Therearetwotypesofstaticexcitationsystems:

1.Withapotentialsource-controlledrectifier—Inthis,theexcitationpowerissuppliedthroughaPTconnectedtogeneratorterminals.

2.Withacompoundsource-controlledrectifier—Inthis,bothcurrenttransformer(CT)andPTareusedatgeneratorterminals.

Theblockdiagramofthepotentialsource-controlledrectifierexcitationsystemisshowninFig.12.43.

Inthisblockdiagram,theinternallimiterfollowingthesummingjunctioncanbeneglected,butfieldvoltagelimitsthataredependentonbothV andI mustbeconsidered.

Fortransformer-fedsystems,K issmallandcanbeneglected.

Inthesesystems,transformersareusedtoconvertvoltage(andalsocurrentincompoundedsystems)totherequiredleveloffieldvoltagewiththeaidofcontrolledoruncontrolledrectifiers.Astheexciterceilingvoltagetendstobehighinstaticexciters,fieldcurrentlimitersareusedtoprotecttheexciterandfieldcircuit.

FE

T FD

C

FIG.12.43Blockdiagram-typeST1-potentialsource-controlledrectifierexcitationsystem

KEYNOTES

Accordingtothecontrol,speed-governingsystemsareclassifiedas:

1. Mechanical-hydraulic-controlled.2. Electro–hydraulic-controlled.

Thesignificanceofrateandpositionlimitsinaspeed-governingsystemare:

1. Ratelimitingofservomotormayoccurforlarge,rapid-speeddeviations,andratelimitsareshownattheinputtotheintegrator.

2. Positionlimitsareindicatedthatcorrespondtowide-openvalvesorthesettingofaloadlimiter.

Forwind-uplimiter,theoutputvariableofthetransferfunctionG(s)isnotlimitedandisfreetovary.Hence,thewind-upcanbetreatedasaseparateblockinthemodelingofaspeed-governingsystem.Fornon-wind-uplimiter,theoutputvariableofthetransferfunctionG(s)islimitedandthereisnoseparateblockinthemodelingofaspeed-governingsystem.SecondarygoverningsystemrespondstothefrequencyofturbinesanditcontrolstheinterceptorvalvesbetweentheHPstateandthereheater.Exciterprovidestherequiredfieldcurrenttotherotorwindingofasynchronousgenerator.Itmayeitherbeself-excitedtypeorseparatelyexcitedtype.Intheunitexcitationmethod,eachgeneratorisfedfromitsexciter,whichisusuallydirectlyconnectedtothegeneratorshaft.Amplidyneisahigh-responsecross-fieldgenerator,whichhasanumberofcontrolwindingsthatcanbesuppliedfromapilotexciterandanumberoffeedbackcircuitsofAVRandmagneticamplifierforcontrolpurposes.AnACexcitationsystemconsistsofanACgeneratorandathyristorbridgecircuitdirectlyconnectedtothegeneratorshaftByprovidingPSS,therotoroscillationsaredampedoutduringthetransientdisturbance.


1. Whatistheclassificationofspeed-governingsystemsaccordingtothecontrol?

1. Mechanical-hydraulic-controlled.2. Electro-hydraulic-controlled.

2. Whatisthefunctionofhydraulicservomotorsusedinmechanical-hydraulic-controlledandelectro–hydraulic-controlledspeed-governingsystems?

Forpositioningvalveorgatecontrollingsystemorwaterflow.

3. Whatarethecomponentsofmechanical-hydraulic-controlledspeed-governingsystemsusedforsteamturbines?

Speedgovernor,speedrelay,hydraulicservomotor,andspeed-governor-controlledsystems.

4. Whydotheelectro-hydraulic-controlledspeed-governingsystemsprovidemoreflexibilitythanmechanical-hydraulic-controlledspeed-governingsystems?

Inelectro-hydraulic-controlledspeed-governingsystems,mechanicalcomponentsinthelowerpowerportionsarereplacedbythestaticelectroniccircuits.

5. Howdoesthelinearityofelectro-hydraulic-controlledspeed-governingsystemsimprove?

Byprovidingfeedbackloopofsteamflowandtheservomotor.

6. Whatarethecomponentsofmechanical-hydraulic-controlledspeed-governingsystemsforhydro-turbines?

Aspeedgovernor,apilotvalve,andservomotor,adistributorvalveandgateservomotor,andgovernor-controlledgates.

7. Whatisrequiredtoachievethestableperformanceofspeed-governingsystemforhydro-turbines?

Dashpotfeedbackisrequiredtoachievethestableperformanceofspeed-governingsystemforhydro-turbines.

8. Howisthespeedrelayrepresentedintheapproximatenon-linearmodelingofaspeed-governingsystem?

Asanintegratorandprovidedasadirectfeedback.

9. Howisthenon-linearpropertyofthespeed-governingvalvecompensated?

Byprovidinganon-linearCAMinbetweenthespeedrelayandthehydraulicservomotor.

10. Whatisthesignificanceofaservomotorinthespeed-governingsystem?

Theservomotorcontrolsthemovementsofvalvesanditisrepresentedasanintegratorwithtimeconstantτ andit

providesasdirectfeedback.

11. Whatisthesignificanceofratelimitsandpositionlimitsinapproximatenon-linearmodelingofaspeed-governingsystem?

sm

1. Ratelimitingofservomotormayoccurforlarge,rapid-speeddeviations,andratelimitsareshownattheinputtotheintegrator.

2. Positionlimitsareindicatedastherecorrespondingtowide-openvalvesorthesettingofaloadlimiter.

12. Whatdoyoumeanbywind-uplimiterandnon-wind-uplimiter?

Inthecaseofawind-uplimiter,theoutputvariableofthetransferfunctionG(s)isnotlimitedandisfreetovary.Hence,thewind-upcanbetreatedasaseparateblockinthemodelingofaspeed-governingsystem.

Inthecaseofanon-wind-uplimiter,theoutputvariableofthetransferfunctionG(s)islimitedandthereisnoseparateblockinthemodelingofthespeed-governingsystem.

13. Whenmodelingsteam-turbinegenerators,whichequipmentistobeignoredateveryinstant?

Theboilercontrolsandon-linefrequencycontrolequipmentshouldbeignoredateveryinstantduetotheirsloweroperations.

14. Whatarethepartsofaspeed-governingsysteminareheatsystemunit?

Primarygoverningsystem,secondarygoverningsystem,andanticipatorygoverningsystemarethepartsofaspeed-governingsystem.

15. Whatistheprimarygoverningsystemofareheatsystemunit?

Primarygoverningsystemrespondstothespeedofamainshaft.Itcontrolsthemaingovernorvalveorthrottleblades.

16. Whatisthesecondarygoverningsystemofareheatsystemunit?

SecondarygoverningsystemrespondstothefrequencyofturbinesanditcontrolstheinterceptorvalvesbetweentheHPstateandthereheater.Theinterceptorgoverningsystemwillactasthesecondarygoverningsystem.

17. Whatisanticipatorygoverningsystemofareheatsystemunit?

Ananticipatorygoverningsystemrespondstotheacceleratingpowerandisusuallynotsettooperateifeitherthegeneratoroutputismorethanacertainvalue(25%ofmaximumoutput)ortheturbinemechanicalpoweroutputislessthanacertainvalue(i.e.,80%ofmaximumcapacity).

18. Whenwilltheanticipatorygoverningsystembeactivated?

Onlywhenthereheatsystemunitsufferslossofalargepercentageofitsloadandonservingthiscondition,theemergencystopvalvesareclosedveryrapidlytopreventdangerousoverspeeds.

19. Whenwilltheemergencyoverspeedgovernortrip?

Whenthevelocityofshaftexceedsapre-setvalue,thenemergencyoverspeedgovernorwillclosethecombinedstopandemergencyvalvesandshutthesetdown.

20. Onwhatassumptionsisthemodelingofahydro-turbinebased?

AccordingtoKirchmayer,themodelingofsingle,generalhydro-turbinesisbasedonthefollowingassumptions:

1. Neglectingdeadband,delaysandnon-linearperformanceinthegoverningsystems.

2. Neglectingthevariationsinheadofthesetofhydro-turbineunitwithdailyuseorseasonaluse.

3. Assumingaconstantequipmentwaterstartingtime(τ ).

21. Whatisthefunctionofanexciterinanexcitationsystem?

Itprovidestherequiredfieldcurrenttotherotorwindingofasynchronousgenerator.Itmaybeeitherself-excitedtypeorseparatelyexcitedtype.

22. Whataretheeffectsofincreaseinexcitation?

1. Thetorqueangleδreducesand2. Thecurrentincreasesandthepoweranglealsoincreasesandhence

becomesmorelaggingwithrespecttoterminalvoltage.

23. Whatistheeffectofincreaseinexcitationonstabilityofthesynchronousmachine?

Whentheexcitationincreases,thetorqueangleδreduces,whichresultsinanincreaseinstiffnessofthemachine.Inotherwords,withanincreaseinexcitation,thestabilityofthemachinewillimprove.

24. Whatarethefactorsbywhichtheincreaseinexcitationislimited?

1. Maximumoutputvoltageoftheexciter.2. Resistanceofthefieldcurrent.3. Saturationofthemagneticcircuit.4. Rotorheating.

25. Whatarethemethodsofprovidingexcitation?

1. Commonorcentralizedexcitationmethod.2. Individualorunitexcitationmethod.

26. Whatdoyoumeanbyindividualorunitexcitationmethod?

Inindividualorunitexcitationmethod,eachgeneratorisfedfromitsownexciter,whichisusuallydirectlyconnectedtothegeneratorshaft.

27. Whatisthemeaningofcommonorcentralizedexcitationmethod?

Incommonorcentralizedexcitationmethod,twoormorenumberofexcitersfeedacommonbus,whichsuppliesexcitationtothefieldsofallgeneratorsintheplant.

28. Whichmethodofprovidingexcitationismorepreferable?

Unitexciterorcommonexcitationmethodismorepreferable.

29. Whatarethemeritsofindividualorunitexcitationmethod?

w

Simplicity,lessohmiclosses,higherreliability,lessmaintenance,andthepossibilityofincorporationofautomaticregulators.

30. Whyareautomaticregulatorsincorporatedinindividualorunitexcitationmethod?

Forthereliablesharingofreactivepowertomaintainconstantterminalvoltagewhilegeneratorsarerunninginparallel.

31. Whatispilotexciter?

Inseparatelyexcited-typeexciters,exciterfieldissuppliedfromasmallDCgeneratorknownasapilotexciter,whichisalevelcompoundgeneratorandmaintainsconstantvoltageexcitationforthemainexciter.

32. WhatistheuseofanamplidyneinaDCexcitationsystem?

Anamplidyneprovideslargecurrentstothefieldwindingofamainexciter.

33. Whatisanamplidyne?

Amplidyneishigh-responsecross-fieldgenerator,whichhasanumberofcontrolwindingsthatcanbesuppliedfrompilotexciterandanumberoffeedbackcircuitsofanAVRandamagneticamplifierforcontrolpurposes.

34. WhatisanAVR?Whatareitscomponents?

TheAVRinconjunctionwiththeexcitationtriestomaintainaconstantterminalvoltageofasynchronousgenerator.Itconsistsofanerrordetector,pre-amplifier,poweramplifier,stabilizer,compensators,auxiliaryinputs,andlimiters.

35. Whichistreatedastheheartofanexcitedsystem?

TheheartofanexcitationsystemistheAVR.

36. Whatisanexcitationcontrolsystem?

Anexciter,voltageregulator,andsynchronousgeneratorconstituteasystemknownasanexcitationcontrolsystem.

37. Whataretheclassificationsofexcitationsystems?

1. DCexcitationsystem.2. ACexcitationsystem.3. Staticexcitationsystem.

38. WhatarethedrawbacksofaDCexcitedsystem?

1. Morecomplexity.2. Largertimeconstantsofanexciter,voltageregulatorandamplidyne,and

magneticamplifier.3. Difficultiesofcommutation.4. Smoothlessoperation.5. Lessreliability.6. Morenoiselevelduetorotatingexciter.

39. WhatisanACexcitationsystem?

AnACexcitationsystemconsistsofanACgeneratorandathyristorbridgecircuitdirectlyconnectedtothegeneratorshaft.

40. WhatarethemeritsofanACexcitationsystem?

1. Movingcontactsarecompletelyeliminated.2. Offeringsmoothandmaintenance-freeoperation.3. Therearenocommutationproblems.

41. Whichsystemofexcitationisbrushlessexcitation?

AnACexcitationsystemisabrushlessexcitationsystem.

42. Whatarethemeritsofstaticexcitationsystems?

1. Noise-freeoperation.2. Lesscomplexity.3. Highreliabilitycomparedtootherexcitationsystems.4. Theoveralllengthofthegeneratorshaftisreduced,whichsimplifies

torsionproblemandcriticalspeedoperation.

43. Writethetransferfunctionofapotentialtransformerandrectifierofanexcitationsystem.

whereK isthegainofPTandrectifierassembly

τ isthetimeconstantoftheassembly

44. Whatarethestagesofoperationofanamplidyne?

Firststageofamplificationandsecondstageofamplificationarethestagesofoperationofanamplidyne.

45. Whatisthefirststageofamplificationfortheoperationofanamplidyne?

Anamplidyneconsistsofbrushesalongtheq-axisandthed-axis.Thebrushesalongtheq-axisareshort-circuited.Asthearmatureresistanceissmall,asmallamountoffieldm.m.f.resultsinalargeq-axiscurrent.Thisisthefirststageofamplification.

46. Whatisthesecondstageofamplificationofoperationofanamplidyne?

Thedevelopmentofvoltageacrossd-axisbrushesduetoq-axiscurrentistermedasthesecondstageofamplification.

47. Writethetransferfunctionofanamplidyneunderno-loadcondition.

whereτ isthetimeconstantofthearmature,τ thetimeconstant

offield,andK thevoltageamplificationfactor.

48. Whatarethecomponentsofamagneticamplifier?

Asaturablecore,controlwinding,andarectifiercircuitarethecomponentsofamagneticamplifier.

r

r

a f

A

49. Whatisthemainprincipleofamagneticamplifier?

Thecontrollingofalargeoutputcurrentbymeansofsmallcontrolcurrentisthemainprincipleofamagneticamplifier.

50. Writethetransferfunctionofamagneticamplifier.

whereτ isthetimeconstantofthemagneticamplifier,V the

outputvoltage,K thevoltageamplificationfactor,andV the

errorsignalinput.

51. Writethetransferfunctionofanexciter.

whereτ isthetimeconstantoftheexciter,K thegainofthe

exciter,V thearmatureemfofanamplidyne,E thearmature

emfofthemainexciter.

52. Whatisthefunctionofastabilizingtransformer?

Theexcitationsystemhasdynamicresponsewhichispronetoexcessiveovershootandstabilityproblems.Theseproblemsareovercomebyaddingastabilizingtransformertotheexcitationsystem.

53. Whatisthefunctionofaterminalvoltagetransducerrepresentedinafunctionalblockdiagramofanexcitationsystem?

TheterminalvoltageofthealternatorissensedandrectifiedintoaproportionateDCsignalbyusingaterminalvoltagetransducer.

54. Whatisthefunctionofloadcompensationblockinanexcitationsystemblockdiagram?

Loadcompensationsynthesizesavoltagethatdiffersfromtheterminalvoltagebythevoltagedropintheimpedance.

55. WhatisthesignificanceofasaturationfunctionS ,whichis

representedinanexcitationsystemfunctionalblockdiagram?

ThesaturationfunctionS =f(E )representsthesaturationof

theexciter.

56. Whatarethemainclassificationsofanexciter?

Rotating-typeandstatic-typeexcitersarethemainclassificationsofanexciter.

57. Whatisthefunctionofanexcitationsystemstabilizertransientgainregulatorblock?

Toincreasethestabilityregionofoperationoftheexcitationsystemandalsotopermitahigherregulatinggain.

A r

A e

E E

E fd

E

E fd

58. Howistheexcitationsystemstabilizerrealized?

Anexcitationsystemstabilizerisrealizedbyanidealtransformerwhosesecondaryisconnectedtoahighimpedance.

59. WhatisthefunctionofPSS?

Duringthetransientdisturbance,therotoroscillations(offrequency0.2–2Hz)aredampedoutbyprovidingthePSS.

60. Whatisapotentialsource-controlled-typestaticexcitationssystem?

Inapotentialsource-controlled-typestaticexcitationssystem,theexcitationpowerspecifiedissuppliedthroughaPTconnectedtogeneratorterminals.

61. Whatisastaticexcitationsystemwithacompoundsource-controlledrectifier?

TheexcitationpowerissuppliedthroughbothCTsandPTsconnectedtogeneratorterminals.

62. WhatistheinputsignaltoPSS?

TheinputsignaltoPSSisderivedfromspeedorfrequencyoroscillatingpoweroracombinationofthesesignals.


1. Hydraulicservomotorsareusedin__________typeofspeed-governingsystems.

1. Mechanical-hydraulic-controlled.2. Electro-hydraulic-controlled.3. Either(a)or(b).4. Both(a)and(b).

2. Inhydraulic-controlledspeed-governingsystems,thehydraulicservomotorsareusedfor:

1. Positioningthevalveorgate,controllingsteamorwaterflow.2. Removingthevalveorgate,controllingsteamorwaterflow.3. Improvingthewaterhead.4. Improvingthesteampressureandtemperature.

3. Forasteamturbine,themechanical-hydraulic-controlledspeed-governingsystemconsistsofwhichofthefollowing?

1. Aspeedgovernor.2. Aspeedrelay.3. Ahydraulicservomotor.4. Governor-controlledvalves

1. (i)and(iv)2. (iii)and(iv)3. Allexcept(ii)4. Allofthese.

4. Intheapproximatenon-linearmathematicalmodelofamechanical-hydraulic-controlledspeed-governingsystem,the

termK representsthegainofspeed-governorsystem,whichis

___________.

1. Theregulationordroopofcharacteristics.2. Thereciprocalofregulationordroopofcharacteristics.3. Notthefunctionofregulationordroop.4. Noneofthese.

5. Thegainofaspeed-governorK represents:

1. Apositionofanassumedlinearinstantaneousindicationofaspeedproducedbythespeedgovernor.

2. Theregulationordroopofspeed-governorcharacteristics.3. Thegovernorspeed-changerposition.4. Noneofthese.

6. Thegovernorspeed-changerpositionprovidestherelaysignalandisdeterminedby:

1. Asystemofspeedgoverning.2. Asystemofautomaticgenerationcontrol.3. Asystemofhydraulicservomotorcontrol.4. Noneofthese.

7. Thespeedrelaysignalinmechanical-hydraulicspeed-governingsystemrepresentsacompositeloadandspeedreferenceandisassumed___________overtheintervalofastabilitystudy.

1. Variable.2. Constant.3. Either(a)or(b).4. Noneofthese.

8. Thespeedrelayinamechanical-hydraulicspeed-governingsystemisrepresentedas:

1. Anintegrator.2. Adifferentiator.3. Anamplifier.4. Noneofthese.

9. Thespeedrelayinamechanical-hydraulicspeed-governingsystemprovides:

1. Anindirectfeedback.2. Adirectfeedback.3. Nofeedback.4. Noneofthese.

10. Thenon-linearitypropertyofthevalveiscompensatedbymeansofproviding:

1. AlinearCAM.2. Anon-linearCAM.3. Aspeedrelay.4. Ahydraulicservomotor.

11. Anon-linearCAMisprovidedtocompensatethenon-linearpropertyofthevalveinbetween:

1. Thehydraulicservomotorandgovernor-controlledvalve.2. Thespeedgovernorandthespeedrelay.3. Thespeedrelayandthehydraulicservomotor.

G

G

4. Thehydraulicservomotorandthespeedgovernor.

12. Thehydraulicservomotor:

1. Controlsthemomentsofvalves.2. Isrepresentedasanintegratorintheapproximatelinearmodel.3. Forwhichtheratetimingmayoccurforlarge-andrapid-speeddeviations.4. Allofthese.

13. Thehydraulicservomotorcontrolprovides:

1. Adirectfeedback.2. Anindirectfeedback.3. Nofeedback.4. Noneofthese.

14. Thepositionlimitsofthehydraulicservomotorthatareindicatedcorrespondto:

1. Wide-openvalves.2. Thesettingofaloadlimiter.3. Either(a)or(b).4. Noneofthese.

15. Thenon-linearitiespresentinspeedcontrolmechanismarenotneglectedfor:

1. Ratelimitsofservomotor.2. Positionlimitsofvalve.3. Studyofpowersystemcomponents.4. Both(a)and(b).

16. Themechanicalcomponentsinthelowerpowerportionsarereplacedbythestaticelectroniccircuitsin:

1. Mechanical-hydraulicspeed-governingsystem.2. Electro-hydraulicspeed-governingsystem.3. Both(a)and(b).4. Noneofthese.

17. Theflexibilityismoreinwhichtypeofspeed-governingsystem?

1. Mechanical-hydraulicspeed-governingsystem.2. Electro-hydraulicspeed-governingsystem.3. Both(a)and(b).4. Noneofthese.

18. Thelinearityoftheelectro-hydraulic-controlledtypespeed-governingsystemcanbeimprovedbymeansof:

1. Providingspeedrelayandhydraulicservomotor.2. Providingexcitationcontrolsignals.3. Providingfeedbackloopsofsteamflowandservomotors.4. Providinglinearcomponents.

19. Forhydraulicturbines,themechanical-hydraulic-controlledspeed-governingsystemconsistsof:

1. Aspeedgovernor.2. Apilotvalveandservomotor.3. Adistributionvalveandgateservomotor.4. Governor-controlledgates

1. (i)and(ii)2. (i)and(iii)

3. (i)and(iv)4. Allofthese.

20. Thedashpotfeedbacksystemisrequiredtoachievethestableperformanceofaspeed-governingsystemof:

1. Hydro-turbines.2. Steamturbines.3. Both(a)and(b).4. Either(a)or(b).

21. Inthespeed-governingsystems,thegateservomotorrateislimitedfor:

1. Large,rapid-speedexcursions.2. Extremesofgateopening.3. Either(a)or(b).4. Both(a)and(b).

22. Inspeed-governingsystems,thepositionlimitsexistcorrespondingto:

1. Large,rapid-speedexcursions.2. Extremesofgateopening.3. Either(a)or(b).4. Both(a)and(b).

23. Thespeed-governingrequirementsforhydro-turbinesarestronglyinfluencedbytheeffectsof:

1. Thepositionofpenstock.2. Headofwater.3. Waterinertia.4. Allofthese.

24. Accordingtothebehavior,theoutputvariableofthetransferfunctionisnotlimitedandisfreetovaryinthecaseof:

1. Wind-uplimiter.2. Non-wind-uplimiter.3. Ratelimiter.4. Positionlimiter.

25. Aseparateblockisneededtorepresentintheblockdiagraminthecaseof:


26. Accordingtothebehavior,theoutputvariableofthetransferfunctionislimitedandnoseparateblockisneededforthelimiterinthecaseof:


27. Generally,theintegratorblockshave:

1. Wind-uplimiter.2. Non-wind-uplimiter.

3. Ratelimiter.4. Positionlimiter.

28. Whilemodelingsteamgenerators,thefollowingequipmentisignoredateveryinstantduetotheirslowoperations:

1. Theboilercontrolsequipment.2. Onlinefrequencycontrolsequipment.3. Both(a)and(b).4. Either(a)or(b).

29. Primarygoverningsystemofareheatsystemunitrespondstothe:

1. Speedofthemainshaft.2. Frequencyoftheturbine.3. Either(a)or(b).4. Both(a)and(b).

30. Thesecondarygoverningsystemofareheatunitrespondsto:

1. Speedofthemainshaft.2. Frequencyoftheturbine.3. Either(a)or(b).4. Both(a)and(b).

31. Thegoverningsystemthatcontrolseithermaingovernorvalveorthrottlebladesis:

1. Primarygoverningsystem.2. Secondarygoverningsystem.3. Anticipatorygoverningsystem.4. Noneofthese.

32. ThegoverningsystemthatcontrolstheinterceptorvalvesbetweentheHPstateandthereheateris:


33. Anticipatorygoverningsystemofareheatunitrespondsto:

1. Speedofthemainshaft.2. Frequencyoftheturbine.3. Theacceleratingpoweroftheunit.4. Noneofthese.

34. Theanticipatorygoverningsystemisusuallysetnottooperateif:

1. Thegovernoroutputismorethanacertainvalue.2. Theturbinemechanicalpoweroutputislessthanacertainvalue.3. Either(a)or(b).4. Both(a)and(b).

35. Thespeed-governingsysteminwhich,onsensingthelossofalargepercentageofitsload,theemergencystopvalvesareclosedveryrapidlytopreventdangerousoverspeed:


36. Inananticipatoryspeed-governingsystem,theemergencystopvalvesarelocatedveryadjacentto:

1. Theservomotor.2. Themaingoverningvalves.3. Thespeedrelays.4. Allofthese.

37. Emergencyoverspeedgovernorwilltripwhen:

1. Thevelocityoftheshaftexceedsapre-setvalue.2. Thefrequencyofthesystemismaintainedconstant.3. Theoverallefficiencyofthespeed-governingsystemandturbinereduces.4. Thevelocityofshaftexceedsthefrequency.

38. Theinterceptorgoverningsystemofareheatunitwillactas:


39. Thesimplemodelingofahydro-turbineunitisbasedontheassumptionaccordingtoKirchmayer:

1. Neglectingdeadband,delays.2. Neglectingthevariationinheadofwater.3. Both(a)and(b).4. Noneofthese.

40. Inmodelingahydro-turbineunit,whichofthefollowingisimportant?

1. Representationofthewatercolumncriteria.2. Representationofthewaterhead.3. Representationofthespeed.4. Allofthese.

41. Whentheexcitationincreases,thetorqueangle‘δ’___________.

1. Increases.2. Reduces.3. Noeffect.4. Noneofthese.

42. Whentheexcitationincreases,thecurrent___________andthepowerangle___________.

1. Increases,increases.2. Decreases,increases.3. Increases,decreases.4. Decreases,decreases.

43. Whentheexcitationincreases,thepoweranglebecomes___________withrespecttoterminalvoltage.

1. Moreleading.2. Morelagging.3. Zero.4. 90°.

44. Withanincreaseinexcitation,

1. Thetorqueangleδreduces.2. Thestiffnessofthemachineincreases.3. Thecouplingofgeneratorandrotatingarmaturefluxbecomesmoretight.4. Allofthese.

45. Withanincreaseinexcitation,

1. Thestabilityofthemachinewillimprove.2. Thestabilityofthemachinewilldecrease.3. Thereisnoeffectonthestabilityofthemachine.4. Noneofthese.

46. Theincreaseinexcitationislimitedbywhichofthefollowingfactors?

1. Resistanceoffieldcircuit.2. Saturationofmagneticcircuit.3. Rotorheating.4. Maximumoutputvoltageofexcitation

1. (i)and(ii)2. Allexcept(iii)3. Allexcept(i)4. Allofthese.

47. Theexcitationsystemconsistsof:

1. Anexciter.2. AnAVR.3. Both(a)and(b).4. Noneofthese.

48. Incommonexcitationbusmethod,

1. Twoormorenumberofexcitersfeedacommonbus.2. Eachgeneratorisfedfromitsownexciter.3. Either(a)or(b).4. Noneofthese.

49. Intheindividualexcitationmethod,

1. Twoormorenumberofexcitersfeedacommonbus.2. Eachgeneratorisfedfromitsownexciter.3. Either(a)or(b).4. Noneofthese.

50. Unit-excitermethodisnothingbut:

1. Commonexcitationbusmethod.2. Centralizedexcitationmethod.3. Individualexcitationmethod.4. Allofthese.

51. Whichofthefollowingmethodsismorepreferable?

1. Commonexcitationbusmethod.2. Centralizedexcitationmethod.3. Individualexcitationmethod.4. Allofthese.

52. Themeritsofaunitexciterare:

1. Simplicity.2. Lessmaintenance.3. Lessohmiclossandhighreliability.4. Allofthese.

53. Thefunctionofexciteristhestructureofexcitation:

1. Tosupplyterminalvoltagetotherotorcircuit.2. Tosupplycurrenttotherotorfieldcircuitofasynchronousgenerator.3. Tosupplycurrenttothestatorcircuitofasynchronousgenerator.4. Allofthese.

54. Apilotexciteris:

1. AlevelcompoundsmallDCgenerator.2. Asmallservotypesynchronousgenerator.3. Amainsynchronousgenerator.4. Amainexciter.

55. Thefunctionofapilotexciteris:

1. Tosupplycurrenttotherotorcircuit.2. Tomaintainconstantvoltageexcitationforthemainexciter.3. Tosupplyvariableexcitationforthemainexciter.4. Noneofthese.

56. Themainexciteris:

1. Alevelcompoundsmallgenerator.2. Amainsynchronousgenerator.3. Aseparatelyexcitedexciter.4. Apilotexecuter.

57. Thefastresponseoftheexciterisobtaineddueto:

1. Severalseries-connectedfieldwindings.2. Severalparallel-connectedfieldwindings.3. Combinationofseries-connectedandparallel-connectedfieldwindings.4. Noneofthese.

58. Thefunctionofanamplidyneis:

1. Toprovidetheconstantexcitationtoasynchronousgenerator.2. Toprovidelargecurrentstothefieldwindingsofamainexciter.3. Toprovidesupplytothesynchronousmachine.4. Noneofthese.

59. Whichofthefollowingiscorrectregardingtheamplidyne?

1. Amplidyneisahighresponsecross-fieldgenerator.2. Amplidynehasanumberofcontrolwindingssuppliedfrompilotexciter.3. AmplidynehasanumberoffeedbackcircuitsofanAVRandmagnetic

amplifier.4. Allofthese.

60. Whichistreatedastheheartoftheexcitedsystem?

1. Mainexciter.2. Pilotexciter.3. Rotorfieldexciter.4. AVR.

61. Excitationfieldcontrolsystemisthesystemthatconsistsof:

1. Exciterandregulator.2. Exciterandfieldsystem.3. Exciterregulatorandsynchronousgenerator.4. Noneofthese.

62. ThedrawbackofDCexcitationsystemis:

1. Morecomplexity.2. Largertimeconstants.3. Lessreliability.4. Morenoiselevel.5. Noneofthese.

63. ThemainadvantageofanACexcitationsystemis:

1. Movingcontactsarecompletelyeliminated.2. Smoothoperation.3. Maintenance-freeoperation.4. Allofthese.

64. Thebrushlessexcitationsystemis:

1. DCexcitationsystem.2. ACexcitationsystem.3. Staticexcitationsystem.4. Noneofthese.

65. Nocommutationproblemsoccurin:

1. DCexcitationsystem.2. ACexcitationsystem.3. Staticexcitationsystem.4. Noneofthese.

66. Theadvantageofastaticexcitationsystemis:

1. Noise-freeoperation.2. Highreliabilityduetomorereliablestaticdevices.3. Overalllengthofthegeneratorshaftisreduced,whichsimplifiesthe

torsionandcriticalspeedproblems.4. Allofthese.

67. Inthefirststageofamplificationofoperationofamplitude,

1. Asmallamountoffieldm.m.f.resultsinlargeq-axiscurrent.2. Thevoltagewillbedevelopedacrossd-axisbrushesduetoq-axiscurrent.3. Thebrushesinq-axisareshort-circuited

1. (i)and(iii)2. (ii)and(iii)3. Only(i)4. Allofthese.

68. Inthesecondstageofamplificationofoperationofamplidyne,whichofthefollowingoccurs?

1. Asmallamountoffieldm.m.f.resultsinlargeq-axiscurrent.2. Thevoltagewillbedevelopedacrossd-axisbrushesduetoq-axiscurrent.3. Thed-axisbrushesareconnectedtoloadalongwithacompensating

winding

1. (i)and(iii)2. (ii)and(iii)3. Only(i)4. Allofthese.

69. Themainprincipleofamagneticamplifieris:

1. ThemagneticcoregetssaturatedwhenlargeACcurrentflowsthroughcontrolwindingandresultsinthedecrementofpermeability.

2. ReactanceofACcoilsincreasesduetothedecrementinpermeability.3. Thecontrollingofalargeoutputcurrentbymeansofasmallcontrol

current.

4. Noneofthese.

70. Theamplidyneisconnectedin___________withtheshuntfieldofthemainexciter.

1. Series.2. Parallel.3. Seriesforsometimeandparallelforsometime.4. Noneofthese.

71. Advantageofastabilizingtransformeris:

1. Theproblemofdynamicresponsethatispronetoexclusiveovershootisovercome.

2. Stabilityproblemsareovercome.3. Both(a)and(b).4. Noneofthese.

72. Mostoftheexcitationsystemsaremodeledbasedon:

1. ACexcitationsystem.2. DCexcitationsystem.3. Staticexcitationsystem.4. IEEEtype-1excitationsystem.

73. Theobjectiveofloadcompensationis:

1. Sharingofreactivepoweramongtheunits,whicharebussedtogetherwithzeroimpedancebetweenthem.

2. Whentheunitsareoperatinginparallelthroughaunittransformer,itisdesirabletoregulatethevoltageatapointbeyondthemachineterminalstocompensateforaportionoftransformerimpedance.


74. Thefunctionofterminalvoltagetransduceris:

1. TosensetheterminalvoltageofanalternatorandrectifyitintoaproportionalDCvoltage.

2. Tosynchronizeavoltagethatdiffersfromtheterminalvoltagebythevoltagedrop.


75. Thefunctionofanexcitationsystemstabilizerandtransientgainregulatoristo___________thestabilityandpermit___________regulatorgains

1. Increase,lower.2. Decrease,lower.3. Increase,higher.4. Decrease,higher.

76. Excitationsystemstabilizerisrealizedby:

1. Apracticaltransformerwhosesecondaryisconnectedtoahighimpedance.2. Apracticaltransformerwhosesecondaryisconnectedtoalowimpedance.3. Anidealtransformerwhosesecondaryisconnectedtoalowimpedance.4. Anidealtransformerwhosesecondaryisconnectedtoahighimpedance.

77. Reducingthetransientgainorgainathigherfrequency,therebyminimizingthenegativecontributionoftheregulatortosystemdampingisthemainaimof:

1. Powersystemstabilizer(PSS).2. Excitationsystemstabilizer(ESS).3. Transientgainregulator(TGR).4. Mainexciter.

78. Duringthetransientdisturbance,therotoroscillationsoffrequency0.2–2Hzaredampedoutbyproviding___________.

1. Powersystemstabilizer.2. Excitationsystemstabilizer.3. Transientgainregulator.4. Mainexciter.

79. Theinputsignaltopowersystemstabilizerisderivedfrom:

1. Speed.2. Frequency.3. Acceleratingpower.4. Combinationofsignalsof(i),(ii),and(iii)

1. (i)and(iii)2. (ii)and(iii)3. Only(iv)4. Anyoftheabove.

80. Inapotentialsource-controlledrectifiertypeofstaticexcitationsystem,

1. TheexcitationpowerissuppliedthroughaPTconnectedtogeneratorterminals.

2. TheexcitationpowerissuppliedthroughbothPTandCTconnectedtogeneratorterminals.

3. TheexcitationpowerissuppliedwithoutconnectingPTandCTconnectedtogeneratorterminals.

4. Noneofthese.

81. Inacompoundsource-controlledrectifiertypeofstaticexcitationsystem,

1. TheexcitationpowerissuppliedthroughaPTconnectedtogeneratorterminals.

2. TheexcitationpowerissuppliedthroughbothPTandCTconnectedtogeneratorterminals.

3. TheexcitationpowerissuppliedwithoutconnectingPTandCTconnectedtogeneratorterminals.

4. Noneofthese.

82. Theoutputofapowersystemstabilizeris:

1. V isaddedtotheterminalvoltage.

2. V isaddedtotheterminalvoltageerrorsignal.

3. V issubtractedfromtheterminalvoltageerrorsignal.

4. Noneofthese.

REVIEWQUESTIONS

1. Discussthemechanical-hydrauliccontrolandelectro-hydrauliccontrolspeed-governingsystemofsteamturbines.

2. Discussthemechanical-hydrauliccontrolandelectro-hydrauliccontrolspeed-governingsystemofhydraulicturbines.

s

s

s

3. Explainthedifferenttypesoflimitersandtheirroleinspeed-governingsystemmodeling.

4. Explaintheeffectofvaryingexcitationofasynchronousgenerator.

5. Explainthemethodsofprovidingexcitationsystems.6. Explainthevariouscomponentsofablockdiagram

representationofageneralexcitationsystem.7. Explaintheclassificationofexcitationsystems.8. Derivethetransferfunctionofanoverallexcitationsystem.

13

PowerSystemSecurityandStateEstimation

OBJECTIVES


1. knowthemeaningofsecuritycontrolsystemanditsimportance

2. discusstheapplicationsofplanningofsecurityanalysis

3. developthemathematicalmodelingofsecurity-constrainedoptimizationproblem

4. studythevarioustechniquesusedforsteady-stateandtransient-statesecurityanalysis

5. knowtheneedofstateestimationinpowersystems

6. discusstheapplicationsofstateestimation

13.1INTRODUCTION

Theconceptofcontrolisfundamentaltotheproperfunctioningofanysystem.Irrespectiveofwhetheritisanengineeringsystemoraneconomicsystemorasocialsystem,itisessentialtoexertsomekindofcontrol,suchasqualitycontrol,inventorycontrol,orpopulationcontrol,toachievecertainobjectiveslikebetterqualityofoutputorbettereconomics,etc.Itisonlynaturalthatpowersystem,whichisoneofthemostcomplexman-madesystems,callsfortheimplementationofanumberofcontrolsforsatisfactoryoperation.Powersystemcontrolhasgonethroughalotofchangesoverthepastthreedecades.Beginningwithsimplegovernorcontrolatthemachinelevel,ithasnowgrownintoasophisticatedmultilevelcontrolneeding,areal-timecomputerprocess,andsystem-wideinstrumentation.

Theultimateobjectiveofpowersystemcontrolistomaintaincontinuouselectricsupplyofacceptablequalitybytakingsuitablemeasuresagainstthevariousdisturbancesthatoccurinthesystem.Thesedisturbancescanbeclassifiedintotwomajorheads,namely,small-scaledisturbancesandlarge-scaledisturbances.Small-scaledisturbancescompriseslowlyvaryingsmall-magnitudechangesoccurringintheactiveandreactivedemandsofthesystem.Large-scaledisturbancesaresudden,large-magnitudechangesinsystemoperatingconditionssuchasfaultsontransmissionnetwork,trippingofalargegeneratingunitorsuddenconnectionorremovaloflargeblocksofdemand.Whilethesmall-scaledisturbancescanbeovercomebyregulatorycontrolsusinggovernorsandexciters,thelarge-scaledisturbancescanonlybeovercomebyproperplanningandadoptingemergencyswitchingcontrols.

13.2THECONCEPTOFSYSTEMSECURITY

‘Securitycontrol’ora‘securitycontrolsystem’maybedefinedasasystemofintegratedautomaticandmanualcontrolsforthemaintenanceofelectricpowerserviceunderallconditionsofoperation.Itmaybenotedfromthisdefinitionthatsecuritycontrolisasignificantdeparturefromtheconventionalgenerationcontrolorsupervisorycontrolsystems.First,theproperintegrationofallthenecessaryautomaticandmanualcontrolfunctionsrequiresatotalsystemsapproachwiththehumanoperatorbeinganintegralpartofthecontrolsystemdesign.Second,themissionofsecuritycontrolisall-encompassing,recognizingthatcontroldecisionsbythemaincomputersystemmustbemadenotonlywhenthepowersystemisoperatingnormallybutalsowhenitisoperatingunderabnormalconditions.Aspowersystemshavebecomemoretightlycoupled,theproblemofmakingtheoperatingdecisionsundervaryingconditionshasbecomeextremelydifficult.

Tokeepthesystemalwayssecure,itisnecessarytoperformanumberofsecurity-relatedstudies,whichcanbegroupedintothreemajorareas,namely,long-termplanning,operationalplanning,andon-lineoperation.

Certainsignificantapplicationsineachoftheseareasarelistedasfollows:

13.2.1Long-termplanning

Evaluationofgenerationcapacityrequirements.Evaluationofinterconnectedsystempowerexchangecapabilities.Evaluationoftransmissionsystemadequacy.

13.2.2Operationalplanning

Determinationofspinningreserverequirementsintheunitcommitmentprocess.Schedulingofhourlygenerationaswellasinterchangeschedulingamongneighboringsystems.Outagedispatchingoftransmissionlinesandtransformersformaintenanceandsystemoperation.

13.2.3On-lineoperation

Monitoringandestimationoftheoperatingstateofthesystem.Evaluationofsteady-state,transient,anddynamicsecurities.Quantitativeassessmentofsecurityindices.Securityenhancementthroughconstrainedoptimization.

13.3SECURITYANALYSIS

Securityanalysisisthedeterminationoftheasecurityofthesystembasedonanext-contingencyset.Thisinvolvesverifyingtheexistenceandnormalcyofthepost-contingencystates.Ifallthepost-contingencystatesexistandarefoundtobenormal,thestateissecure.Ontheotherhand,thenon-existenceofevenoneofthepost-contingencystatesoremergencynatureofanexistingpost-contingencystateindicatesthatthecurrentstateisinsecure.

Thoughitmaybetheoreticallypossibletoconductasecurityanalysisforboththesteady-stateemergencyand

dynamicinstability,thetrendhasbeentohaveaseparateanalysisforeachofthesetwotypesofemergency.Themainreasonforthisistheextremedifficultyinimplementingadynamic-securityanalysiswiththepresentmethodsofstabilityanalysis.Ontheotherhand,forthesteady-statesecurity(SSS)analysis,severalapproachesarepossibleandareinuse.Basically,theseapproachesstartwithaknowledgeofthepresentstateofthesystemasobtainedfromthesecuritymonitoringfunction.Thesystemisthentestedforvariousnext-contingenciesby,ineffect,solvingforthechangesinthesystemconditionsforagivencontingencyandcheckingthenewvaluesagainsttheoperatingconstraints.

‘Transientsecurityanalysis’referstoanon-lineprocedurewhoseobjectiveistodeterminewhetherornotapostulateddisturbancewillcausetransientinstabilityofthepowersystem.Atransientinstabilityconditionimpliesthelossofsynchronismoroscillations,whichincreaseinamplitude,leadingtocascadingoutagesandsubsequentsystembreakup.AsagainsttheSSSanalysiswherethenext-contingenciestobeconsideredareonlyoutagesoflines/transformersorgenerators,inthecaseoftransientsecurityanalysis,amuchwiderrangeofpossiblecontingenciesmustbeconsideredsuchas:

Single-phase,two-phase,andthree-phasefaultconditions.Faultswithorwithoutreclosing.Properoperationorfailureofprotectiverelays.Circuitbreakeroperationorfailuretoclearthefault.Lossofgenerationoralargeblockofload.

Directmethodsfortransientsecurityanalysishavebeensuggested,butnoneofthesehaveyetpassedtheexperimentalstage.Thecurrentindustrypracticeistoexpressthesecurityconstraintsassociatedwithtransientstabilityassteady-stateoperatinglimitsonpowertransferorphase-angledifferenceacrossselectedtransmissionlines.Thegeneralapproachforimposing

transientsecurityconstraintsonanoperatingpowersystemconsistsofthefollowingsteps:

1. Performextensiveoff-linetransientstabilitystudiesforarangeofoperatingconditionsandpostulatedcontingencies.

2. Onthebasisofthesestudiesandpre-determinedreliabilitycriteria(e.g.,thesystemmustwithstandthree-phasefaultswithnormalclearing),establishsteady-stateoperatinglimitsforlinepowerflowsorlinephase-angledifferences.

3. Operatethesystemwithintheconstraintsdeterminedinthepreviousstep.

Transientstabilityanalysisofalargesystem,thoughdoneusingacrudemachinemodelofconstantemfbehindtransientreactance,requiresquitealotofcomputertimebecauseofthelargenumberofdifferentialalgebraicequationsinvolved.Whenonegoesforamodelofhigherdegreeofcomplexity,whichmayincludeanexcitationsystemmodel,detailedgeneratorelectricalmodel,governorcontrolmodel,andturbinemodelandifoneconsiderssimulationforeachofthecontingenciesinthenext-contingencyset,thenthecomputationtimeneededbecomesprohibitive.Inrecentyears,considerableamountofresearchhasbeendevotedtodevelopingefficientandeffectivetechniquesforon-linetransientstabilityanalysis.Thesuggestedtechniquescanbeclassifiedaccordingtothefollowingbasicapproaches:

1. Digitalsimulation.2. Hybridcomputersimulation.3. Lyapunovmethods.4. Patternrecognition.

13.3.1Digitalsimulation

Digitalsimulationtechniques,thoughveryadaptableandflexible,areslowinspeedandhenceitappearsthattheywilluseon-lineanalysisinacomplementaryrolewithfaster,butpossiblylessaccurate,techniques.Recentadvancesindigitalmethodshavebeendirectedtowardimplicitintegrationtechniquesandthesimultaneoussolutionofthewholesetofdifferential-algebraic

equationsusingsparsitytechniques.Itisclaimedthatamethodlike‘variableintegrationsteptransientanalysis’(VISTA)canreducethesimulationtimeasmuchasfivetimesascomparedwiththeconventionalexplicitintegrationmethods.

13.3.2Hybridcomputersimulation

Hybridcomputersimulationofatransientstabilityproblemcouldbemademanytimesfasterthanrealtime.Althoughhybridcomputershavebeenabletoprovidematchlesssolutionspeeds,theirapplicationtopowersystemoperationislimitedbydisadvantagessuchasverylargeinitialinvestmentinthecaseoflargesystems;applicableonlytoalimitednumberofselecton-linecomputationfunctionsandlimitedflexibilityduetothenormalpatchingoftheanalogcomputers.

13.3.3Lyapunovmethods

ThesecondmethodofLyapunovhasreceivedaconsiderableamountofattentionfordeterminingpowersystemtransientstability,particularlyforon-lineapplication.ThismethodinvolvesthederivationofascalarLyapunovV(X),whereXisthedynamic-statevectorofthesystemsetofdifferentialequations,whichhasthefollowingproperties:

V(0)=0,i.e.,X=0istheequilibriumstate

V(X)>0,X∈Ω,X≠0V(X)≤0,X∈Ω

whereΩisaregionaroundthestablepointX=0,whichiscalledtheregionofstability.Whilethismethodcanofferconsiderablegainincomputationalspeed,thedrawbacksofthismethodasfollowsare:

Tooconservative,especiallyforsystemswithmorethanthreeorfourmachines.Computationalrequirementshavemadethestudyoflarge-scalepowersystemsinfeasible.

Requiresasimplifiedsystemmodel.

Thelastlimitationisnotassevereasthefirsttwosincemuchusefulinformationcanbeobtainedfromanalyticalstudieswiththesimplifiedmodels.Veryrecentdevelopmentsindicatethatabreakthroughinovercomingthefirsttwoproblemsisnowpossible.First,anefficientmethodofcalculatingtheunstableequilibriumpointhasbeendevelopedusingamodifiedNewton-Raphsonloadflow.Second,thereisnowanincreasedamountofawarenessastowhythesecondmethodofLyapunovishighlyconservative.

13.3.4Patternrecognition

Patternrecognitionisanotherapproachaimedatovercomingthehighcomputationalrequirementsofon-linetransientstabilitystudies.Alargenumberofoff-linestabilitystudiesareperformedtoforma‘trainingset’andcertainimportantfeaturesareselected.Anon-lineclassifiercomparestheactualoperatingconditionswiththetrainingsetand,onthebasisofthiscomparison,classifiestheexistingstateaseithersecureorinsecure.Thismethodisveryappealingforon-lineassessmentbecauseofitstremendousspeedandtheminimumon-linedatathatitrequires.

However,thedisadvantagesofthismethodare:

1. Theaccuracyoftheclassificationmethodisnotasgoodasthatofthedirectsolutionmethodssinceitisbasicallyaninterpolationtechnique.

2. Averylargenumberofsamples(andhencesimulation)mayberequiredfortheformationofanadequatetrainingset.

3. Ithasdifficultyinhandlingabnormalconditions,whichmayariseduetounusualloadpatternsand/ornetworkconfigurations.

Ifthedurationoftheanalysistobeconductedislongerthan1–3s,thedynamicsoftheboilers,turbines,andotherpowerplantcomponentscannotbeignored.Inadditiontothis,thedynamicsofAGCandSVCshouldbetakenintoaccountalongwiththecontrolactionof

impedanceandunder-frequencyload-sheddingrelays.Asaresult,theeffectofafault-initiateddisturbancemaycontinuepastthetransientstabilityphasetotheso-calledlong-termdynamicstabilityphase,whichcanbeoftheorderof10–20minormore.Theobjectivesofalong-termdynamicresponseassessmentare:

1. Evaluationofdynamicreserveresponsecharacteristicsincludingthedistributionofreservesandeffectoffast-startingunits.

2. Evaluationofemergencycontrolstrategieslikeload-sheddingbyunder-frequencyrelays,fastvaluing,dynamicbraking,andothers.

Theseobjectivesfallprimarilyundersystemplanning,controlsystemdesign,aswellaspost-disturbanceanalysis.However,theoperatingimplicationscannotbeneglected,inviewofthefactthatseriousblackoutsthathaveoccurredoverthepast15–20yearsweregenerallytheresultoflong-terminstabilityandsequencesofcascadingevents.

13.4SECURITYENHANCEMENT

Securityenhancementisalogicaladjuncttosecurityanalysisanditinvolveson-linedecisionsaimedatimproving(ormaintaining)thelevelofsecurityofapowersysteminoperation.Securityenhancementincludesacollectionofcontrolactions,eachaimedattheeliminationofsecurityconstraintviolations.Thesecontrolsmaybeclassifiedas:

1. Preventivecontrolsinthenormaloperatingstate,whenon-linesecurityanalysishasdetectedaninsecureconditionwithrespecttoapostulatednext-contingency.

2. Correctableemergencycontrols(simplycalled‘correctivecontrols’)inanemergencystate,whenanout-of-boundoperatingconditionalreadyexistsbutmaybetoleratedforalimitedtimeperiod.

Ineithercase,theprimaryobjectiveistofindfeasibleandpracticalwaystoremedyapotentiallydangerousoperatingconditiononcethesecurityanalysisprogramrevealstheexistenceofsuchacondition.

Securityenhancementimpliestheutilizationofavailablegenerationandtransmissioncapacityto

improvethesecurityofapowersystem.Therearefivegenericapproachestotheuseofavailablesystemresourcesforsecurityenhancement,namely:

1. Manipulationofreal-powerflowsincertainpartsofthesystemthroughreschedulingofgenerationalongwithothercontrolvariablessuchasphase-shifterratios.

2. Manipulationofreactive-powerflowsinthesystemtomaintainagood‘voltageprofile’throughexcitationcontrolofgeneratorsalongwithothercontrolvariablessuchasshuntcapacitororreactorswitching,off-nominaltapratiosoftransformers,etc.

3. Utilizingheatcapacityofcomponentsliketransformersandundergroundcablestopermitshort-termoverloadingofcertainpiecesofequipment.

4. Changingthenetworktopologyviaswitchingactions.5. Modifyingthesettingsofprotectiverelaysorcontrollogic.

Allthefiveoptionsgivenaboveinvolvesometrade-offsbetweentheeconomyandthesecurityofpowersystemoperation.Forexample,generationshiftingorreschedulingpowertransactionsusuallyresultinhigheroperatingcosts.Hence,forthosepreventivecontrolactionsthatdrasticallyaffecttheeconomyofoperation,theoperatormaydecidenottoexecutetherecommendedcontrolactionsuntilthepostulatedcontingencyactuallytakesplace,dependingonthegeneraloperatingphilosophyoftheparticularsystemandthenatureofthepredicatedconstraintviolations.

Security-constrainedoptimizationmaybeusedasaconvenientframeworkfordiscussingapproachestosystemsecurityenhancement,especiallyforSSS.Theconstrainedoptimizationproblemofobtainingthe‘best’operatingconditionthatsatisfiesnotonlytheloadconstraintsandtheoperatingconstraintsbutalsothesecurityconstraintsmaybestatedasfollows:

Minimize

f(X,U)objectivefunction

Subjectto:

G(X,U)=0,loadconstraints

H(X,U)≥0,operatingconstraints

S(X,U)≤0,securityconstraints

wherefisascalar-valuedfunction.

Thesecurityconstraintsreflectalltheoperatingandloadconstraintsassociatedwiththepostulatedpost-contingencystates;these‘logicalconstraints’canberigorouslyformulatedandexpressedasasetofinequalityconstraintsasindicatedabove.Thesefunctionalconstraints,toolargeinnumber,maketheproblemverycomplex.Twonon-linearprogrammingtechniques,namelythepenaltyfunctiontechniqueandthegeneralizedreducedgradienttechnique,havebeenidentifiedasmostsuitableforsolvingtheconstrainedoptimizationproblem.Foraquickon-linesolution,theduallinearprogrammingtechniqueusingthelinearmodelaswellasthesuccessivelinearprogrammingtechniqueusinglinearizedmodelshavebeenfoundtobemostuseful.

Onlyalimitedamountofresearchhasbeendirectedatthedevelopmentofcontrolalgorithmsfortransientsecurityenhancement.Sinceon-lineimplementationofcontrolalgorithmstoenhancesystemsecurityisverydifficulttoachieve,onecanconsidertheintermediatestepofcomputingandpresentingsuitablesecurityindicestotheoperatorswhowillinturntakecontroldecisions.Anumberofsecurityindices,bothforSSSaswellasfortransientsecurity,havebeenproposedalongwithasuitabletechniqueofobtainingthemfromtheon-linesecurityanalysis.

13.5SSSANALYSIS

ThoughSSSanalysisisonlyapartoftheoverallsecurityassessmentprocess,itsimportanceshouldnotbeunderestimated.Thereasonsforitsprominenceare:first,itistheonlysimplesimulationprocessthatcanbeimplementedon-line.Itshouldbenotedthatatthistimeofwritingwhenon-linesecurityanalysisisnotyetincommonuse,exceptforafewpioneeringapplications,

theSSSanalysisiseitherintheearlystagesofon-lineimplementationorplannedforseveralnewenergycontrolcentersindevelopedcountries.Second,itisadvantageoustoknowwhether(ornot)thepost-contingencystateofthesystemwouldbeacceptablefromsteady-stateconsiderations,evenbeforeinvestigatingthetransientanddynamicperformance.Third,anapproximatecheckoftransientstabilitycouldalsobeincorporatedbyimposingonthepost-contingencysteadystates,appropriatepowerflow,orotherconstraintsderivedfromoff-linetransientstabilitystudies.Hence,itislogicaltodevotemoreattentiononthevariousaspectsoftheSSSanalysis.

TheobjectiveoftheSSSanalysisistodeterminewhether,followingapostulateddisturbance,thereexistsanewsteady-stateoperatingpointwheretheperturbedpowersystemwillsettleafterthepost-faultdynamicoscillationshavebeendampedout.Anon-linealgorithmsimulatesthepredictedsteady-stateconditionsforaspecifiedsetofnext-contingenciesandchecksforoperatingconstraintviolations.Ifthenormalsystemfailstopassanyoneofthecontingencytests,itisdeclaredtobe‘steady-stateinsecure’andtheparticularcontingencieswiththeattendantlimitexcursionsarenoted.Moreprecisely,SSSisdefinedastheabilityofthesystemtooperatesteady-state-wisewithinthespecifiedlimitsofsafetyandsupplyqualityfollowingacontingency,inthetimeperiodafterthefast-actingautomaticcontroldeviceshaverestoredthesystemloadbalance,butbeforetheslow-actingcontrols,e.g.,transformertapingsandhumandecisions,haveresponded.

13.5.1RequirementsofanSSSassessor

TheSSSassessorisdefinedasanon-lineprocessusingreal-timedataforconductingSSSanalysisonthecurrentstateofthesystem.Eachcontingencyissolved

approximatelyasasteady-stateACpowerflowproblem.Exceptforsimulatedoutages,thenetworkisthesameastheactualoperatingsystemandthebuspowerinjection(definedasgenerationminusload)schedulecorrespondstothecurrentlyestimatedstateofthesystem.Theresultsofeachsolutionarecheckedagainstpre-determinedconstraints.Ifacontingencycausesaconstraintviolationorifasolutionforacontingencyisimpossible,thisinformationistransferredfromtheSSSassessortoanotherfunctioninthecontrolcenterinwhichappropriatecontrolactionswillbetakentoenhancethesecurityofthesystem.

TheSSSassessorwillbeoneofseveralinterrelatedprogramsinanautomateddispatchcenter.Thewaysinwhichitcanbeintegratedwithotherfunctionsarenotconsideredhere.However,asimplifiedschematicdiagramfortheflowofinformationisshowninFig.13.1.Themostcriticalinputisthestatevectorfromthestateestimator.ThisestimateistransmittedintheformofthestatevectorVconsistingofcomplexvoltagesateachnodeofthemonitoredsystem.

Otheressentialinputsareexplainedasfollows.

13.5.1.1NetworkData

Thepassivenetworkismodeledbythebusadmittancematrix[ ],whichisdevelopedfromadetailedlistofbasicnetworkcomponentsincludingtransmissionlines,transformers,capacitors,andreactors.Itisessentialtohavereal-timeinformationonthestatusofthesecomponentsatthebeginningofeachsolutioncycle.Asolutioncycleisthesolutionandcheckingofresultsforallcontingentoutagesinaspecifiedcontingencylist.Thechangeofstatusofeverynetworkcomponentistransmittedtothecontrolcomputerandwheneveracomponentisswitchedinorout,itseffectisreflectedbyachangeintheadmittancematrix.

FIG.13.1Flowofinformationinasecurityassessor

13.5.1.2BusPowerInjections

Foralineoutage,theinjectionsshouldcorrespondtotheactualstateofthesystem.Theinjectionscheduleiscomputedonceatthebeginningofeachsolutioncyclebasedonthenetworkadmittancesandthestatevector.Theinjectionateverybuskiscomputedas

whereP andQ aretherealandreactivepowers, andareelementsofthestatevector isanelementof

thebusadmittancematrix[ ],andα isthesetofallnodesadjacenttonodek.

Forphysicalaswellasmathematicalreasons,itisnecessarytofixthevoltageangleattheslackbusandallowthevariationinlossestobesuppliedbytheinjectionatthisbus.

k k

k

13.5.1.3SecurityConstraints

Theconstraintsaretransmissionlinepowerflows,busvoltages,andreactivelimits.Theseconstraintsmayoriginatefromcustomerrequirements,relaysettings,insulationlevels,equipmentratings,orothersources.Constraintscanalsobeestablishedbyoff-linesimulationstudies.Lineflowconstraintsareusuallyexpressedeitherintermsofmaximumcontinuouscurrentorpowerratings(normallyforshorterlines),orintermsoftheallowablemaximumsteady-statephase-singledifferencesbetweenconnectedbuses(normallyforlongerlines).Asstatedearlier,SSSconstraintscanbederivedtosuittransientstabilityrequirements.However,theseconstraintsaredifficulttospecify,sincethetransientstabilitypropertiesofalinedependon:thegeneration/loadpatternthroughouttheentirenetwork,theprecisenatureofthecontingency,theconfigurationofthepost-faultsystem,etc.

Insomesystems,itmaybedesirabletoaltertheconstraintsaccordingtothestateofthesystem,withdifferentconstraintsapplyingunderdifferentoperatingconditionsorcontingencies.EstablishingappropriateconstraintsfortheSSSassessorisanimportantsub-problemthatrequiresmoreinvestigation.

13.5.1.4ContingencyList

ForthepurposeofSSSanalysis,thefollowingcontingenciesshouldbeconsidered:

1. Lossofageneratingunit.2. Suddenlossofaload.3. Suddenchangeinflowinaninter-tie.4. Outageofatransmissionline.5. Outageofatransformer.6. Outageofashuntcapacitororreactor.

Theseoutagescanbegroupedintotwocategories:‘networkoutage’and‘poweroutage’.Anetworkoutageinvolvesonlychangesinthenetworkadmittanceparametersandincludesitems(iv)to(vi)givenabove.A

poweroutageinvolvesonlychangesinbuspowerinjectionsandhenceincludesitems(i)to(iii)givenabove.

TheusualpracticeinSSSanalysisistoassumethatthenetworkconfigurationandtheinjectionscheduleatthecontingencystateremainthesameasinthebasecasestateexceptforthesimulatedoutages.However,indealingwithpoweroutagesthatinvolvethelossofcertaingeneratingunits,theinjectionscheduleatthecontingencystateshouldtakeintoaccounttheredistributionoflostgenerationtotheremaininggeneratorsinservice.Thismaybedonewiththehelpofagenerationallocationfunction.Sincetheanalysisisconcernedwiththenewstudyaftertheoutagetransientshavesettled,thegenerationallocationwillbedeterminedbythenaturalgovernorcharacteristicsoftheavailableunitsinthesystem.However,ifitisrequiredtocheckthepowerflowsimmediatelyaftertheoutage,thenalltheremaininggeneratorsinthesystemwillshare,temporarily,thelostgenerationinproportiontotheirinertias.

13.6TRANSIENTSECURITYANALYSIS

Inrecentyears,aconsiderableamountofresearchhasbeendevotedtodevelopingefficientandeffectivetechniquesforon-linetransientstabilityanalysis.Transientstabilityassessmentconsistsofdeterminingifthesystem’soscillationsfollowingashort-circuitfaultwillcauselossofsynchronismamonggenerators.Theprimaryphysicalphenomenoninvolvedhereisthatofinertialinteractionamongthegeneratorsasgovernedbythetransmissionnetworkandbusloads.Thisphenomenonisofshortduration(1–3s)ingeneral.Forlongerdurations,thedynamicsofboilers,turbines,andotherpowerplantcomponentscannotbeignored.Thesuggestedtechniquestosolvetransientstabilityproblemscanbeclassifiedaccordingtothefollowingbasicapproaches:

1. Digitalsimulation.2. Patternrecognition.3. Lyapunovmethod.4. Hybridcomputersimulation.

13.6.1Digitalsimulation

Severalnumericalintegrationapproacheshavebeenproposedandused.Asinallintegrationschemes,theusuallimitingfactoristhesmallesttimeconstantofthesystem,whichisnormallycausedbysynchronizingoscillations.Theuseofimplicitpredictor-correctormethodshasgenerallyallowedlargerstepsizeswhilemaintainingahighlevelofnumericalstability.Normally,thetransientstabilityprogramwillalternatebetweenanintegrationstepandaloadflowsolutiontosolvethenetworkequations.Thus,sparsematrixmethodscanbequiteeffectiveandusefulinthiscontext.

13.6.2Patternrecognition

Itisanotherapproachaimedatovercomingthehighcomputationalrequirementsofonlinetransientstabilitysolutions.Alargenumberofoff-linestabilitystudiesareperformedtoforma‘trainingset’andcertainimportantfeaturesareselected.Anon-lineclassifiercomparestheactualoperatingconditionswiththetrainingsetand,onthebasisofthiscomparison,classifiestheexistingsystemaseithersecureorinsecure.Consequently,thebulkofthecomputationloadistransferredtotheoff-linestudies’timeframe.Thismethodleadstothegenerationofafunctionknownasa‘securityfunction’,whichisusedtoassessthesecurityofthesystem.

13.6.3Lyapunovmethod

ThesecondmethodofLyapunovhasreceivedconsiderableattentionfordeterminingpowersystemtransientstability,particularlyforon-lineapplication.However,theresultsofthisresearchhavebeenoflittlepracticalvaluetodate,duetothreebasicproblems.The

classicalLyapunovmethodyieldssufficientbutnotnecessaryconditionsforstability;theseconditionsarediscussedindetailinthefollowingsections.

13.7STATEESTIMATION

Thestateofapowersystemisdefinedintermsofthevoltagemagnitudeandphaseangleofeverybusinthepowersystem.Thestateestimationplaysaveryvitalroleinpowersystemoperation,monitoringandcontrolintermsofavoidingsystemfailuresandregionalblackouts.Themainobjectiveofstateestimationistoobtainthebestpossiblevaluesofthemagnitudesofbusvoltagesandtheiranglesanditrequiresthemeasurementofelectricalquantities,suchasrealandreactive-powerflowsintransmissionlinesandrealandreactive-powerinjectionsatthebuses.

Stateestimationisanavailabledataprocessingschemetofindthebeststatevectors,usingtheweightedleastsquaremethodtofitascatterofdata.Inordertoobtainahigherdegreeofaccuracyofthesolutionofthestateestimationtechnique,twomodificationsareintroduced.First,itisrecognizedthatthenumericalvaluesoftheavailabledatatobeprocessesforthestateestimationaregenerallynoisyduetothepresenceoferrors.Second,itisnotedthattherearealargenumberofvariablesinthesystem(activeandreactive-powerlineflows),whichcanbemeasuredbutnotutilizedintheloadflowanalysis.Thus,theprocessinvolvesimperfectmeasurementsthatareredundantandtheprocessofsystemstateestimationisbasedonastatisticalcriterionthatestimatesthetruevaluesofthestatevariableseithertominimizeormaximizetheselectedcriterion.Acommonlyusedcriterionisthatofminimizingthesumofthesquaresofthedifferencesbetweentheestimatedandmeasuredtruevaluesofafunction.

Allthesysteminformationiscollectedbythecentralizedautomationcontrolofpowersystemdispatch

throughremoteterminalunits(RTUs).TheRTUssampletheanalogvariablesandconvertthemintoadigitalform.Thesedigitalsignalsareinterrogatedperiodicallyforthelatestvaluesandaretransmittedbytelephoneandmicrowavecommunicationlinktothecontrolcenter.

Thecontrolcenteroperationmustdependonmeasurementsthatareincomplete,inaccurate,delayed,andunreliable.Thestateestimationtechniqueisusedtoprocessalltheavailabledataandhencethebestpossibleestimateofthetruevalueofthesystemisfound.

13.7.1Stateestimator

Itprocessesreal-timesystemdata,whichisredundantandcomputesthemagnitudesofbusvoltagesandbusvoltagephaseangleswiththehelpofacomputerprogram.Theinputstoanestimatorareimperfect(noisy)powersystemmeasurements.Itisdesignedtogivethebestestimateofsystemstatevariables(i.e.,busvoltagemagnitudesandphaseangles).

Thestateestimatordetectsbadorinaccuratedatabyusingstatisticaltechniques.Forthis,stateestimatorsaredesignedsuchthattheyhavewell-definederrorlimitsandarebasedonthenumber,types,andaccuracyofmeasurements.

ThestateestimatorapproximatesthepowerflowsandvoltagesatabuswhosemeasurementsarenotavailablebecauseofRTUfailureorbreakdownoftelephoneoracommunicationlink.Undersuchacondition,thestateestimatorisrequiredtomakeavailableasetofmeasurementstoreplacemissingordefectivedata.

13.7.2Static-stateestimation

Therearetwodifferentmodesofstateestimationasappliedtopowersystems:

1. Static-stateestimation.

2. Dynamic-stateestimation.

Static-stateestimationpertainstotheestimationofasystemstatefrozenataparticularpointintime.Figurativelyspeaking,itisasnapshotofthesystem.Inthesteady-stateoperationofasystem(e.g.,thesuddenopeningofoneofthephasesofatransmissionlineisreflectedinthepowerflowinthetwohealthyphasesmuchlesserthantheaveragepowerflowindicatedbythelast-stateestimation),thestateestimatorisrequiredtodetectachangeinnetworkconfigurationandconveyasignalindicatingthechangeincircuitconfigurationandtopreparetheoperatorforcorrectiveactiononthefirstdatascan.Ontheotherhand,dynamic-stateestimationisacontinuousprocess,whichtakesintoaccountthedynamicsofthesystemandgivesanestimateofthesystemstateasitevolvesintime.Atthepresentmoment,mostofthestateestimatorsinpowersystems,whichareoperational,belongtothefirstcategory.

Onthefaceofit,itmayappearasifthereisnotmuchofadifferencebetweenloadflowcalculationsandstatic-stateestimation.But,thisisasuperficialpointofview.Inloadflowstudies,itistakenforgrantedthatthedateonwhichcalculationsarebasedareabsolutelyfreefromerror.Ontheotherhand,instate-estimationmethods,accuracyofmeasurementonmodelingerrorsaretakenintoaccountbyensuringredundancyofinputdata.Thismeansthatthenumberofinputdata‘m’onwhichcalculationsarebasedaremuchmorethanthenumberofunknownvariables‘n’whoseknowledgecompletelyspecifiesthesystem.Themoretheredundancy,thebetteritisfromanestimationpointofview.Butredundancyhasapricetopayintermsofinstallationofadditionalmeasuringequipmentandcommunicationfacilities.

13.7.3Modelingofuncertainty

Fromamathematicalviewpoint,thesimplestwayofdescribingarandomvector‘v’isbyassigningaGaussiandistributiontoit.Theprobabilitydensityfunctionfor‘v’isthengivenby

Here,theexpectedvalueofvisassumedtobezeroandRdenotesthecovariancematrixofv.Therandomvectorvrepresentsthefollowingerrors:

1. Instrumentationerrors(metererrors,incompleteinstrumentation,andbaddata).

2. Operationaluncertainties(unexpectedsystemchanges,measurementdelay).

3. Incompletenessofthemathematicalmodel(modelingerrors,inaccuracyinnetworkparameters).

13.7.4Somebasicfactsofstateestimation

Therearethreeimportantquantitiesofinterestinstateestimation.Theyare:

1. Thevariabletobeestimated.2. Theobservations.3. Themathematicalmodelshowinghowtheobservationsarerelatedto

thevariablesofinterest(whicharetobeestimated)andtheever-presentuncertainties.

Thevariablestobeestimatedarethestatevariablesx,theobservationsarerepresentedbyz,andthemathematicalmodelisgivenby

z=h(x)+v(13.3)

InEquation(13.3),‘h’representsaknownnon-linearrelationconnectingzandx.Forpedagogicalreasons,theabovequantitiesarerepresentedspecificallyas

x =truevalueofstatex

z =actualvalueofobservation

v =actualvalueofobservationuncertainty

Further,forsimplicityofexplanation,letusassumethatthenon-linearrelationinEquation(13.3)isreplacedbyalinearrelationviz.,

z =h(x) +v (13.4)

where

InEquation(13.4),weknowthat

Wenotethateventhoughx andv arenotknown,themathematicalmodelconveyssomeinformationontheirvalues,i.e.,thereisamodelfortheiruncertainty.Nowdefine:

:estimateofvaluex

Theestimate dependsonthevaluezandthemathematicalmodel(andtheuncertaintymodelsforxandv ).Usually,itisdesirabletoviewtheestimateassomespecifiedfunctionoftheobservationz .Thisfunctioniscalledanestimator.Thenatureofthis

true

actual

actual

actual true actual

true actual

true

true

actual

actual

estimatorcanbedeterminedfromhandthemodelsofx andv .Itcanthereforebespecifiedbeforeobservationsareactuallymade.TheestimatorforlinearsystemsisoftenalinearmatrixoperatorW.

Thus, :Wz (13.5)

Ingeneral, isnotequaltox .Hence,thefirstproblemistochoosethebestestimator(thebestW),whichminimizes,insomesense,theerror(x − ).AssumingthatsuchaWhasbeenchosen,thesecondproblemistodeterminehowclose istox .Sincethenumericalvalueoftheerror(x − )isnotknown,theproblemistodevelopanuncertaintymodelforthesame.Theuncertaintyin(x − )dependsuponh,theuncertaintyinx andv andofcoursetheestimatorW.Hence,ingeneralterms,thebasicestimationprobleminvolvesthefollowingsteps:

1. FindtheestimatorW

suchthat isasclosetox aspossible.

2. Determinethemodelfortheuncertaintyin(x − ).Thismodel

dependsonthechosenW.

Therearetwomodelsfortheuncertaintyx andtheyare:

1. Apriorimodel:Themodelforx ,whichmodelstheuncertainty

beforetheobservationismade.2. Aposteriorimodel:Itisthemodelfor(x − ),whichmodelsthe

uncertaintyinx aftertheobservationhasbeenmadeand

processedtoyield .

ThechoiceofestimatorsuchasWdependsontheapriorimodel.TheaposteriorimodeldependsonwhichestimatorWischosen.Inwhatistofollow,wedropthenotationx andz infavorofxandz.

true actual

actual

true

true

true

true

true

true actual

true

ture

true

true

true

true

true actual

Therearemanywaysofmodelingtheuncertaintyofxandv.Someofthemoreimportantwaysare:

(i)Bayesianmodel : xandvarerandomvectors.

(ii)Fishermodel : xiscompletelyunknown;‘v’israndomvector.

(iii)Weightedleastsquares

: Nomodelsforxandv.

(iv)Unknownbutbounded

: xandvareconstrainedtolieinspecifiedsets.

13.7.5Leastsquaresestimation

Considertherelation:

z=h(x)+v

orv=[z−h(x)](13.6)

andfromEquation(13.2),

Theoptimalestimate isgivenbythatvalueofxforwhichthescalarfunctionoftheweightedsquares:

J=v′R v=[z−h(x)]′R [z−h(x)](13.7)

hasaminimumvalue.TheweightingmatrixR istheinverseofthecovariancematrixoftheobservationnoisev.

−1 −1

−1

Applyingthefirst-ordernecessaryconditionsforminimizingJ,wehave

ThesecondpartialderivativeofJwithresectto‘x’viz.,

isamatrixknownastheHessianmatrixandis

denotedherebyG(x):

Thesecond-ordersufficiencyconditiondemandsthatG(x)bepositive,definiteattheminimum.

Asusualinsuchproblems,wefollowtheiterativeproceduretosuccessivelycloseinontheminimumpoint,whichinthiscase,istheleastsquareestimate.

Therefore,assumetheiterativeform:

x =x −A g(x )(13.10)

Asktendstoinfinity,hopefullyx →x andA g(x )→0,whichimpliesfornon-singularA thatg(x )=0.ThisispreciselytheconditiontobesatisfiedbyEquation(13.8)andhencethedesiredresultisobtained.

ThereareseveralmethodstoarriveatthematrixA .Aisascalarmultipleofunitmatrixinthesteepestdescentmethod;itistheinverseoftheHessianmatrixG(x )inNewton’smethod.ItispossibletochooseA bytakingTaylor’sseriesexpansionofh(x)aboutainitialpointx :

k+1 k k k

k k +1 k k

k k

k k

k

k

0

i.e.,h(x)=h(x )+h(x )(x−x )+higherorderterms(13.11)

SubstitutingthisapproximatevaluefromEquation(13.11)afterneglectinghigherordertermsintheobjectivefunctionJgivenbyEquation(13.7),weget

J =[z−h(x )−H(x )(x−x )]′R [z−h(x )−H(x )(x−x )]

Here,

H′R [z−h(x )−HΔx]

whereΔx=(x−x )

Usingtheoptimalitycondition weget

h′R [z−h(x )−hΔx]=0

Hence,Δx=[H′R H] H′R [z−h(x )](13.13)

Thevectorx=(x +Δx)yieldstheabsoluteminimumofJ ,butdoesnotyieldtheminimumforthefunctionJ.Thiscallsforfurtheriterationstillthevalue|x –x |iswithinprescribedbounds.

Specifically,

x =x +Δx

=x [H′R H ]H′R [z−h(x )]+x (13.14)

ButbyEquation(13.10),

0 0 0

1 0 0 0 0 0

0

0

0

−1 0

0

0

1

k k + 1

k+1 k k

k k k

−1

−1

−1 −1 −1

−1 −1 −

Wemayalsoidentify withA ofEquation(13.10).

Hence,x =x –A g(x ),whichisthegeneralformoriginallypostulated.

Tostartwith,weassumeasuitablevalueforx .Thismaybeobtainedeitherfromapreviousloadflowstudyormaybearbitrarilychosen,e.g.,chooseV =e +jf withe =1andf =0forallirangingfrom1toN.ThealgorithmgivenbyEquation(13.12)isnotaneasilyimplementedtableforthefollowingtworeasons:

1. TheJacobianHhastobeevaluatedforeveryiteration.2. Eachiterationrequiresamatrixinversion.

Forexample,considerablesimplificationmaybeachievedifthematrix[H′R H] ofEquation(13.14)isevaluatedonlyoncefortheinitialstatex .

Let

ThenEquation(13.14)becomes

x =x +P H R [z−h(x )](13.16)

Thissimplification,nodoubt,reducestheconvergencespeedascomparedtoEquation(13.14)butthisisoffsetbythegreatlyreducedcomputingtime.

13.7.6Applicationsofstateestimation

Static-stateestimationmaybesuccessfullyusedinestimatingthestatusofthecircuitbreakersandotherswitchesinthesystem.Inacomplexpowersystem,the

k

k +1 k k k

0

i i i

i i

0

k+1 k 0 0 k

− 1 −1

−1

networktopologycontinuouslychanges.Thedataregardingthewronginformationofswitchpositionsmaybeeasilycheckedbycomparingestimationrunsobtainedatdifferentinstants.ItisalsopossibletodecideonthequantumofadditionalinstrumentationbymerelycomparingtheminimumvaluesoftheobjectivefunctionJ(x)fordifferentinstrumentationconfigurations,theusestowhichstateestimationmaybe:

1. Dataprocessinganddisplay[baddatadetection,samplingrate].2. Securitymonitoring[overloadlimits,rescheduling,switching,and

loadshedding].3. Optimalcontrol[loadfrequencycontrol(LFC),economicload

dispatch].

KEYNOTES

‘Securitycontrol’ora‘securitycontrolsystem’maybedefinedasasystemofintegratedautomaticandmanualcontrolsforthemaintenanceofelectricpowerserviceunderallconditionsofoperation.Tokeepthesystemalwayssecure,itisnecessarytoperformanumberofsecurity-relatedstudies,whichcanbegroupedintothreemajorareas,namely:long-termplanning,operationalplanning,andon-lineoperation.Securityanalysisisthedeterminationofthesecurityofthesystembasedonanext-contingencyset.Thisinvolvesverifyingtheexistenceandnormalcyofthepost-contingencystates.Thepossiblecontingenciesconsideredintransientsecurityanalysisare:

1. Single-phase,two-phase,andthree-phasefaultconditions.2. Faultswithorwithoutreclosing.3. Properoperationorfailureofprotectiverelays.4. Circuitbreakeroperationorfailuretoclearthefault.5. Lossofgenerationoralargeblockofload.

Transientstabilityanalysistechniquesarebasedon:


TheobjectiveofanSSSanalysisistodeterminewhether,followingapostulateddisturbance,thereexistsanewsteady-stateoperatingpointwheretheperturbedpowersystemwillsettleafterthepost-faultdynamicoscillationshavebeendampedout.Themainobjectiveofstateestimationistoobtainthebestpossiblevaluesofthemagnitudesofbusvoltagesandtheiranglesanditrequiresthemeasurementofelectricalquantities,suchasrealandreactive-powerflowsintransmissionlinesandrealandreactive-

powerinjectionsatthebuses.Functionsofastateestimatorare:

1. Itprocessesreal-timesystemdata,whichareredundantandcomputethemagnitudesofbusvoltagesandbusvoltagephaseangleswiththehelpofacomputerprogram.

2. Itdetectsbadorinaccuratedatabyusingstatisticaltechniques.

Theapplicationsofstateestimationare:

1. Dataprocessinganddisplay.2. Securitymonitoring.3. Optimalcontrol.


1. Howisthesecuritycontrolsystemdefined?

‘Securitycontrol’ora‘securitycontrolsystem’maybedefinedasasystemofintegratedautomaticandmanualcontrolsforthemaintenanceofelectricpowerserviceunderallconditionsofoperation.

2. Howisthesecuritycontrolconsideredasasignificancedeparturefromconventionalgenerationcontrolorsupervisorycontrol?

First,theproperintegrationofallthenecessaryautomaticandmanualcontrolfunctionsrequiresatotalsystemsapproachwiththehumanoperatorbeinganintegralpartofthecontrolsystemdesign.Second,themissionofsecuritycontrolisall-encompassing,recognizingthatcontroldecisionsbythehumancomputersystemmustbemadenotonlywhenthepowersystemisoperatingnormallybutalsowhenitisoperatingunderabnormalconditions.

3. Whatarethethreemajorareasofsecurity-relatedstudies?

Tokeepthesystemalwayssecure,itisnecessarytoperformanumberofsecurity-relatedstudies,whichcanbegroupedintothreemajorareas,namelylong-termplanning,operationalplanning,andon-lineoperation.

4. Whataretheapplicationsoflong-termplanning?

Theapplicationsoflong-termplanningare:

1. Evaluationofgenerationcapacityrequirements2. Evaluationofinterconnectedsystempowerexchangecapabilities.3. Evaluationoftransmissionsystemadequacy.

5. Whataretheapplicationsofoperationalplanning?

Theapplicationsofoperationalplanningare:

1. Determinationofspinningreserverequirementsintheunitcommitmentprocess.

2. Schedulingofhourlygenerationaswellasinterchangeschedulingamongneighboringsystems.

3. Outagedispatchingoftransmissionlinesandtransformersformaintenanceandsystemoperation.

6. Whataretheapplicationsofon-lineplanning?

Theapplicationsofon-lineplanningareasfollows:

1. Monitoringandestimationoftheoperatingstateofthesystem.2. Evaluationofsteady-state,transient,anddynamicsecurities.3. Quantitativeassessmentofsecurityindices.4. Securityenhancementthroughconstrainedoptimization.

7. Whatissecurityanalysis?

Securityanalysisisthedeterminationofthesecurityofthesystembasedonanext-contingencyset.Thisinvolvesverifyingtheexistenceandnormalcyofthepost-contingencystates.

8. Whatindicatestheinsecurityofacurrentstate?

Thenon-existenceofevenoneofthepost-contingencystatesoremergencynatureofanexistingpost-contingencystateindicatesthatthecurrentstateisinsecure.

9. Whatistheobjectiveoftransientsecurityanalysis?

‘Transientsecurityanalysis’referstoanonlineprocedurewhoseobjectiveistodeterminewhetherornotapostulateddisturbancewillcausetransientinstabilityofthepowersystem.

10. Whatarethepossiblecontingenciesconsideredintransientsecurityanalysis?

Thepossiblecontingenciesconsideredintransientsecurityanalysisare:

1. Single-phase,two-phase,andthree-phasefaultconditions.2. Faultswithorwithoutreclosing.3. Properoperationorfailureofprotectiverelays.4. Circuitbreakeroperationorfailuretoclearthefault.5. Lossofgenerationoralargeblockofload.

11. Whatarethestepsofgeneralapproachforimportingtransientsecurityconstraintsonanoperatingpowersystem?

Thegeneralapproachforimposingtransientsecurityconstraintsonanoperatingpowersystemconsistsofthefollowingsteps:

1. Performextensiveoff-linetransientstabilitystudiesforarangeofoperatingconditionsandpostulatedcontingencies.

2. Onthebasisofthesestudiesandpre-determinedreliabilitycriteria(e.g.,thesystemmustwithstandthree-phasefaultswithnormalclearing),establishsteady-stateoperatinglimitsforlinepowerflowsorlinephase-angledifferences.

3. Operatethesystemwithintheconstraintsdeterminedinthepreviousstep.

12. Whatarethesuggestedtechniquestobecarriedoutinthetransientstabilityanalysis?

Thesuggestedtechniquescanbeclassifiedaccordingtothefollowingbasicapproaches:


13. Whatissecurityenhancement?

Securityenhancementisalogicaladjuncttosecurityanalysisanditinvolveson-linedecisionsaimedatimproving(ormaintaining)thelevelofsecurityofapowersysteminoperation.Securityenhancementincludesacollectionofcontrolactionseachaimedattheeliminationofsecurityconstraintviolations.

14. Whatarethetwocontrolsusedforsecurityenhancement?

Thecontrolsusedforsecurityenhancementareclassifiedas:

1. Preventivecontrolsinthenormaloperatingstate,whenon-linesecurityanalysishasdetectedaninsecureconditionwithrespecttoapostulatednext-contingency.

2. Correctableemergencycontrols(simplycalled‘correctivecontrols’)inanemergencystate,whenanout-of-boundoperatingconditionalreadyexistsbutmaybetoleratedforalimitedtimeperiod.

15. Whatarethetechniquesusedforsolvingthesecurity-constrainedoptimizationproblem?

Twonon-linearprogrammingtechniques,namelythepenaltyfunctiontechniqueandthegeneralizedreducedgradienttechniquehavebeenidentifiedasthemostsuitableonesforsolvingtheconstrainedoptimizationproblem.Foraquickon-linesolution,theduallinearprogrammingtechniqueusinglinearmodelaswellasthesuccessivelinearprogrammingtechniqueusinglinearizedmodelshavebeenfoundtobemostuseful.

16. DefineSSS.

SSSisdefinedastheabilityofthesystemtooperatesteady-state-wisewithinthespecifiedlimitsofsafetyandsupplyqualityfollowingacontingency,inthetimeperiodafterthefast-actingautomaticcontroldeviceshaverestoredthesystemloadbalance,butbeforetheslow-actingcontrols,e.g.,transformertapingsandhumandecisions,haveresponded.

17. WhataretheobjectivesofSSSanalysis?

TheobjectiveofSSSanalysisistodeterminewhether,followingapostulateddisturbance,thereexistsanewsteady-stateoperatingpointwheretheperturbedpowersystemwillsettleafterthepost-faultdynamicoscillationshavebeendampedout.

18. Whatarethesecurityconstraints?

Theconstraintsaretransmissionlinepowerflows,busvoltages,andreactivelimits.

19. WhatarethecontingenciesthatshouldbeconsideredforSSSanalysis?

ForthepurposeofSSSanalysis,thefollowingcontingenciesshouldbeconsidered:

1. Lossofageneratingunit.2. Suddenlossofaload.3. Suddenchangeinflowinaninter-tie.4. Outageofatransmissionline.

5. Outageofatransformer.6. Outageofashuntcapacitororreactor.

20. Whatisthemainobjectiveofstateestimation?

Themainobjectiveofstateestimationistoobtainthebestpossiblevaluesofthemagnitudesofbusvoltagesandtheiranglesanditrequiresthemeasurementofelectricalquantities,suchasrealandreactive-powerflowsintransmissionlinesandrealandreactive-powerinjectionsatthebuses.

21. Whatarethetwomodificationsintroducedtoobtainahigherdegreeofaccuracyofthesolutiontothestateestimationtechnique?

Inordertoobtainahigherdegreeofaccuracyofthesolutiontothestateestimationtechnique,twomodificationsareintroduced.First,itisrecognizedthatthenumericalvaluesoftheavailabledatatobeprocessedforthestateestimationaregenerallynoisyduetothepresenceoferrors.Second,itisnotedthattherearealargenumberofvariablesinthesystem(activeandreactive-powerlineflows),whichcanbemeasuredbutnotutilizedintheloadflowanalysis.

22. Whatisthefunctionofastateestimator?

1. Itprocessesreal-timesystemdata,whichareredundantandcomputethemagnitudesofbusvoltagesandbusvoltagephaseangleswiththehelpofacomputerprogram.

2. Itdetectsbadorinaccuratedatabyusingstatisticaltechniques.

23. Whatdoyoumeanbystatic-stateanddynamic-state-estimationmodes?

Static-stateestimationpertainstotheestimationofasystemstatefrozenataparticularpointintime.Dynamic-stateestimationisacontinuousprocess,whichtakesintoaccountthedynamicsofthesystemandgivesanestimateofthesystemstateasitevolvesintime.

24. Whataretheapplicationsofstateestimation?

Theapplicationsofstateestimationare:

1. Dataprocessinganddisplay.2. Securitymonitoring.3. Optimalcontrol.


1. Securitycontrolsystemisasystemof:

1. manualcontrol.2. integratedautomaticcontrol.3. conventionalgenerationcontrol.4. both(a)and(b).

2. Evaluationofgenerationcapacityrequirementsisa:

1. long-termplanningofsystemsecurity.2. operationalplanningofsystemsecurity.3. on-lineoperationapplicationofsystemsecurity.4. allofthese.

3. Theoperationalplanningofsystemsecuritycontrolincludes:

1. spinningreserverequirementsdetermination.2. schedulingofhourlygenerationaswellasinterchangescheduling.3. outagedispatchingoftransmissionlinesandtransformers.4. allofthese.

4. ThemonitoringandestimationofoperatingstateofthesystemandevaluationofSSSstate,transient,anddynamicsecuritiesaretheapplicationsof:

1. on-lineoperationofsecuritycontrolsystem.2. operationalplanningofsecuritycontrolsystem.3. long-termplanningofsecuritycontrolsystem.4. allofthese.

5. Securityanalysisisthedeterminationofthesecurityofasystem.

1. basedonanext-contingencyset.2. involvesverifyingtheexistenceofpost-contingencystates.3. involvesverifyingthenormalcyofpost-contingencystates.4. allofthese.

6. Non-existenceofevenoneofthepost-contingencystatesoremergencynatureofanexistingpost-contingencystateindicates:

1. securityofcurrentstate.2. securityofpreviousstate.3. insecurityofcurrentstate.4. insecurityofpreviousstate.

7. InSSSanalysis,thenextcontingenciestobeconsideredare:

1. outagesoflinesortransformersorgenerators.2. faultswithorwithoutreclosing.3. circuitbreakeroperationorfailuretoclearthefault.4. lossofgeneration.

8. Securityenhancementinvolves:

1. on-linedecisionsaimedatmaintainingthelevelofsecurity.2. acollectionofcontrolactionsaimedattheeliminationofsecurity

constraintviolations.3. failureofevenoneofpost-contingencies.4. both(a)and(b).

9. Forgettingquickon-linesolutiontoasecurity-constrainedoptimizationproblem,thetechniqueusedis:

1. duallinearprogrammingtechniqueusinglinearizedmodel.2. successivelinearprogrammingtechniqueusinglinearizedmodel.3. both(a)and(b).4. noneofthese.

10. Ifthenormalsystemfailstopassanyoneofthecontingencytests,itisdeclaredtobe:

1. Steady-statesecure.2. steady-stateinsecure.

3. transient-statesecure.4. transient-stateinsecure.

11. TheSSSassessorisanon-lineprocessusingreal-timedataforconductingSSSanalysison:

1. thepreviousstateofthesystem.2. thecurrentstateofthesystem.3. thepost-stateofthesystem.4. allofthese.

12. Anetworkoutageinvolves:

1. onlychangesinthenetworkadmittanceparameters.2. outagesoftransmissionlineortransformerorshuntcapacitororreactor.3. onlychangesinbuspowerinjections.4. both(a)and(b).

13. Apoweroutageinvolves:

1. onlychangesinnetworkadmittanceparameters.2. onlychangeinbuspowerinjections.3. lossofageneratingunitorsuddenlossofload.4. both(b)and(c).

14. Themainobjectiveofstateestimationis:

1. toobtainthebestvaluesofthemagnitudesofbusvoltagesandangles.2. tomaintainconstantfrequency.3. toreducetheloadlevels.4. toincreasethepowergenerationcapacity.

15. Stateestimationprocessrequiresthemeasurementof:

1. realandreactive-powerflowsintransmissionlines.2. realandreactive-powerinjectionsatthebuses.3. onlyreactivepowerabsorbedbyload.4. both(a)and(b).

16. Stateestimationis:

1. anavailabledata-sharingscheme.2. anavailabledata-measuringscheme.3. anavailabledata-processingscheme.4. anavailabledata-sendingscheme.

17. Stateestimationschemeuses:

1. Lagrangianfunctionmethod.2. Negativegradientmethod.3. Lyapunovmethod.4. weightedleastsquaremethod.

18. Inthestateestimationscheme,allthesysteminformationiscollectedbythecentralizedautomationcontrolofpowersystemdispatchthrough:

1. remoteterminalunits.2. transmitters.3. digitalsignalprocessors.4. allofthese.

19. Theinputstostateestimationare:

1. perfectpowersystemmeasurements.

2. imperfectpowersystemmeasurements.3. dependsonloadconnectedtopowersystem.4. allofthese.

20. Mostofthestateestimatorsinpowersystemsatpresentbelongto:

1. static-stateestimators.2. dynamic-stateestimators.3. either(a)or(b).4. both(a)and(b).

REVIEWQUESTIONS

1. Explaintheconceptofsystemsecurity.2. Discussthesignificanceapplicationsofsystemsecurity.3. Explainthetechniquesusedfortransientsecurityanalysis.4. Explainthesecurityenhancement.5. Explainthemathematicalmodelingofsecurity-constrained

optimizationproblem.6. ExplaintheSSSanalysis.7. Discusstheneedofstateestimation.8. Explainthefunctionofastateestimator.9. Discussthedifferencebetweenstatic-stateestimationand

dynamic-stateestimation.10. Explaintheleastsquareestimationprocess.11. Explaintheapplicationsofstateestimationprocess.

AppendixA

Chapter1

(1)a (12)b (23)b (34)b

(2)c (13)c (24)a (35)d

(3)b (14)a (25)d (36)a

(4)a (15)a (26)c (37)a

(5)a (16)c (27)c (38)c

(6)d (17)b (28)c (39)c

(7)b (18)d (29)a (40)d

(8)b (19)b (30)d (41)d

(9)e (20)a (31)c (42)d

(10)c (21)c (32)c

(11)a (22)d (33)a

Chapter2

(1)a (10)a (19)a (28)a

(2)a (11)a (20)d (29)c

(3)b (12)a (21)c (30)d

(4)a (13)c (22)d (31)d

(5)c (14)d (23)b (32)a

(6)d (15)c (24)b (33)a

(7)d (16)b (25)c

(8)d (17)c (26)b

(9)c (18)d (27)a

Chapter3

(1)c (10)d (19)b (28)c

(2)b (11)a (20)a (29)c

(3)d (12)a (21)c (30)d

(4)d (13)d (22)a (31)c

(5)a (14)a (23)c (32)c

(6)c (15)a (24)b (33)d

(7)c (16)b (25)a (34)d

(8)c (17)c (26)a

(9)a (18)d (27)d

Chapter4

(1)c (14)d (27)b (40)b

(2)c (15)d (28)a (41)a

(3)a (16)a (29)b (42)b

(4)d (17)c (30)a (43)b

(5)a (18)a (31)a (44)a

(6)b (19)a (32)b (45)b

(7)a (20)b (33)c (46)b

(8)d (21)b (34)a (47)b

(9)d (22)d (35)c (48)b

(10)a (23)b (36)a (49)a

(11)d (24)a (37)d

(12)d (25)b (38)b

(13)d (26)a (39)b

Chapter5

(1)d (9)b (17)b (25)b

(2)c (10)a (18)a (26)c

(3)b (11)c (19)Rigid,soft (27)b

(4)a (12)a (20)b (28)d

(5)b (13)d (21)a (29)a

(6)c (14)a (22)b (30)d

(7)d (15)a (23)c

(8)b (16)c (24)b

Chapter6

(1)a (12)a (23)a (34)d

(2)b (13)d (24)a (35)b

(3)a (14)d (25)b (36)b

(4)b (15)c (26)c (37)a

(5)b (16)d (27)b (38)a

(6)c (17)a (28)c (39)d

(7)a (18)b (29)d (40)d

(8)c (19)b (30)c (41)c

(9)d (20)c (31)a

(10)c (21)c (32)b

(11)d (22)d (33)d

Chapter7

(1)a (14)b (27)d (40)b

(2)c (15)b (28)d (41)c

(3)c (16)b (29)a (42)c

(4)a (17)b (30)a (43)c

(5)d (18)c (31)c (44)d

(6)b (19)b (32)d (45)a

(7)c (20)c (33)c (46)a

(8)c (21)a (34)b (47)b

(9)b (22)b (35)a (48)a

(10)c (23)a (36)a (49)d

(11)c (24)c (37)b (50)c

(12)c (25)b (38)d

(13)c (26)d (39)c

Chapter8

(1)b (6)a (11)c (16)b

(2)c (7)d (12)b (17)c

(3)b (8)c (13)c (18)a

(4)b (9)d (14)a (19)b

(5)c (10)d (15)a (20)c

Chapter9

(1)a (17)c (33)c (49)d

(2)a (18)a (34)c (50)d

(3)b (19)a (35)c (51)a

(4)b (20)a (36)c (52)a

(5)b (21)d (37)c (53)b

(6)b (22)a (38)c (54)b

(7)b (23)b (39)a (55)b

(8)b (24)b (40)c (56)a

(9)d (25)d (41)c (57)d

(10)a (26)c (42)c (58)a

(11)a (27)d (43)c (59)c

(12)c (28)b (44)d (60)c

(13)a (29)c (45)c (61)c

(14)d (30)c (46)b (62)d

(15)c (31)c (47)b (63)c

(16)b (32)c (48)d (64)a

Chapter10

(1)a (8)a (15)b (22)d

(2)c (9)b (16)a (23)c

(3)b (10)d (17)d (24)d

(4)d (11)a (18)c (25)a

(5)a (12)b (19)b

(6)d (13)a (20)a

(7)a (14)a (21)c

Chapter11

(1)b (8)d (13)b (20)d

(2)c (9)d (14)a (21)c

(3)c (10)(i)-(c),(ii)-(b),

(15)b (22)d

(4)a (iii)-(a),and (16)a (23)a

(5)c (iv)-(d) (17)d (24)b

(6)a (11)a (18)c (25)a

(7)d (12)a (19)c (26)a

(27)d (31)b (35)d (39)a

(28)d (32)a (36)c (40)b

(29)a (33)d (37)a (41)c

(30)a (34)b (38)d

Chapter12

(1)d (22)b (43)b (64)b

(2)a (23)c (44)d (65)b

(3)d (24)a (45)a (66)d

(4)b (25)a (46)d (67)a

(5)a (26)b (47)c (68)b

(6)b (27)b (48)a (69)c

(7)b (28)c (49)b (70)a

(8)a (29)a (50)c (71)c

(9)b (30)b (51)c (72)d

(10)b (31)a (52)d (73)c

(11)c (32)b (53)b (74)a

(12)d (33)c (54)a (75)b

(13)a (34)c (55)b (76)d

(14)c (35)c (56)c (77)c

(15)d (36)b (57)b (78)a

(16)b (37)a (58)b (79)d

(17)b (38)b (59)d (80)a

(18)c (39)c (60)d (81)b

(19)d (40)a (61)c (82)b

(20)a (41)b (62)e

(21)a (42)a (63)d

Chapter13

(1)d (6)c (11)b (16)c

(2)a (7)a (12)d (17)d

(3)d (8)d (13)d (18)a

(4)a (9)c (14)a (19)b

(5)d (10)a (15)d (20)a

Acknowledgments

Weexpressourgratitudetothosewhohavehelpedusinmanywaysinmakingthisbookareality:

Toourcollegeadministrationforprovidingusaconduciveatmosphereandextendingimmensesupporttocarryoutthiswork.Toourdepartmentcolleagues,especiallythosewho,withtheirvaluableinputsbywayofsuggestions,content-relatedguidanceanddiscussions,helpedustosuccessfullycompletethisbook.Toourstudentsfortheirservices.Toallthosewhohaveeitherdirectlyorindirectlyhelpedusinbringingoureffortstofruition.Toourfamiliesfortheirconstantandfavorableencouragementateverystepoftheway.

Wearegratefultoourpublishers,PearsonEducation,India,formakingthisbookareality.WespecificallythankSojanJose,JenniferSargunar,M.E.Sethurajan,andThomasMathewRajeshfortheireditorialinputsandforthesuccessfulcompletionoftheproject.

S.SivanagarajuG.Sreenivasan

Copyright©2009DorlingKindersley(India)Pvt.Ltd.

LicenseesofPearsonEducationinSouthAsia.

NopartofthiseBookmaybeusedorreproducedinanymannerwhatsoeverwithoutthepublisher’spriorwrittenconsent.

ThiseBookmayormaynotincludeallassetsthatwerepartoftheprintversion.ThepublisherreservestherighttoremoveanymaterialpresentinthiseBookatanytime,asdeemednecessary.

ISBN9788131726624

ePubISBN9789332503380

HeadOffice:A-8(A),Sector62,KnowledgeBoulevard,7thFloor,NOIDA201309,India.

RegisteredOffice:11LocalShoppingCentre,PanchsheelPark,NewDelhi110017,India.

Forinstructor’sresources,clickhttp://www.pearsoned.co.in/SSivanagaraju/

http://www.pearsoned.co.in/SSivanagaraju/

Power System Operation and Control - GitHub

Documents

Transcript of Power System Operation and Control - GitHub