IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-88, NO. 6, JUNE 1969

Economic Dispatch of Generation via

Valve-Point LoadingLESTER H. FINK, SENIOR MEMBER, IEEE, HARRY G. KWATNY, AND JOHN P. McDONALD

Abstract-The valve-point loading logic described in this paper isintended to meet at any time in the most economical fashion a gen-eration commitment. This objective is approached by insuring that asgreat a portion of the load as practicable will be carried by unitsloaded to valve points, that the remainder of the load will be carriedby units reserved for regulation, and that in both categories theassignments will be made to those units which can provide therequisite capacity at the lowest cost.

INTRODUCTION

ECONOMIC dispatch involves the implementation of somesolution to the problem of apportioning a given total load

between a plurality of generating sources in such a way as tominimize the costs of generation and transmission. The classicalsolution, which says that costs are minimal when the incrementalcosts of generation at the several sources are equal, is based onmonotonically increasing incremental cost curves. This assump-tion is not everywhere valid because of the throttling losses in-curred when steam chest valves are barely open. These losses in-troduce negative slopes into the incremental cost curves. Thebasic desirability of and justification for operating at the localminima of the curves, referred to as valve-point loading, hasbeen discussed for many years [1], [2].

It is the intent of this paper to outline the logic behind and theconceptual development of a central control algorithm for Phila-delphia Electric Company's automatically controlled generation,which has been developed for implementing valve-point loading.However, before proceeding, it will be helpful to describe brieflythe operational relationship of Philadelphia Electric Company tothe Pennsylvania-New Jersey-Maryland Interconnection (PJM)of which it is a member. Unlike other interconnections, PJM fromits inception has sought to achieve maximum common economyby operating as a single control area with free-flowing internalties and a common running cost [3]. Actual interchange of energyamong members is recorded and accounted for after-the-fact onan hourly basis. Since PJM thus operates on a common com-posite incremental cost curve, the basic assumption on which thepresent logic is based, and which is logically prior to a choicebetween conventional and valve-point loading, is that the basiccommitment of Philadelphia Electric Company, as well as ofother members of PJM, to the Interconnection is to provide apreviously, implicitly agreed upon amount of generation (mega-watts) at any given incremental cost, to which is added responsi-bility for respondinig to a given fraction of the PJM area require-ment signal (AR) up to a given limit. Within the confines of this

Paper 69 TP 112-PWR, recommended and approved by thePower System Engineering Committee of the IEEE Power Group forpresentation at the IEEE Winter Power Meeting, New York, N.Y.,January 26-31, 1969. Manuscript submitted September 12, 1968;made available for printing December 13, 1968.

L. H. Fink and J. P. McDonald are with the Philadelphia Elec-tric Company, Philadelphia, Pa.H. G. Kwatny is with Drexel Institute of Technology, Philadel-

phia, Pa.

commitment, it is Philadelphia Electric Company's responsi-bility to provide this amount of generation in the most economicfashion. In other words, if through some oversight or error thereshould be a discrepancy between the individual incremental costcurves making up the composite curve being used by the Inter-connection office and the actual amount of generation which canbe provided at a given cost by any one company, the governingresponsibility is to provide the amount of generation implicit inthe cost curve that has been supplied to the Interconnection. Thispriority is a necessary, though not sufficient, condition for satis-factory Interconnection operation. It is the obligation of the mem-ber companies to keep the Interconnection office continually in-formed of the prevailing relationship between available genera-tion and incremental cost in order to avoid any conflict such asnoted above.Given this commitment, thermodynamic considerations in-

dicate that a given load can be supplied most economically byoperating as many units as possible at valve points. It is obvious,however, that valve-point loading can supply generation only indiscrete blocks and is inherently unsuited for matching generationto a continuously varying load. This means that a certain amountof capacity must be withheld from valve-point operation in orderthat it might be operated in a continuously varying manner so asto supply the difference between the valve-point loaded capacityand the continuously varying load which must be satisfied. Thisconsideration reveals the basic nature of regulating capacity andat least implicitly indicates the considerations that must be metin establishing the magnitude and responsiveness of regulatingcapacity.One very basic requirement for successful implementation of a

valve-point loading regime that has not been mentioned so faris that the amount of regulating capacity necessary to take up thedifference between block-loaded generation and actual load is afunction, among other things, of the amount of time required topick up or drop one or more blocks of generation. Furthermore,some minimum amount of time, say 10 minutes, must be providedduring which a unit remains at a given valve position without areversal of load, else that unit will not be valve-point loaded, butwill in effect have been treated as though it were regulating.These two considerations mean that successful operation ofvalve-point loading requires some amount of foreknowledge ofgeneration dispatch and therefore of load trend. We believe thatfailure to recognize this has been the basic weakness of pastattempts at valve-point loading by other utilities [4]. The presentprogram is predicated upon the use of a predictive filter de-signed to satisfy this requirement. The filter will be applied to theincremental cost signal received from the Interconnection, andthe predicted cost signal will then be converted to the correspond-ing megawatts. (It should be noted that if megawatt data areavailable, the predictor can be applied directly to megawatts.)The basic use of the filter output will be to classify the prevailingload trend as being certainly increasing, certainly decreasing, oressentially steady state. The judgment based on this classificationthen will determine the specific sequence of logical steps which

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, JUNE 1969

will be used in arriving at a set of dispatch signals. Moreover, anextremely useful by-product of the availability of a near termload forecast will be a forewarning to operators of actions whichthey must take in preparing units to permit operation in higheror lower load ranges. If the operator can accomplish these changesin time (15-20 minutes), it will provide smooth transfer from oneload range to another.A necesssary characteristic of statistical prediction is that it is

provided only within certain confidence limits. The amount ofregulating capacity R necessary for successful operation of thetype of program under discussion is thus a function, not only ofPhiladelphia Electric Company's commitment to PJM (1 per-cent of system capacity), but also of the width of the confidenceinterval in the prediction. For example, if the confidence intervalof the prediction is + 5 MW, the currently required regulatingcapacity should include 5MW to cover uncertainty in prediction.Thus, R is a dynamic quantity which constantly reflects systemconditions. Sufficient regulating capacity must be available toassure that the difference between block-loaded generation andactual load can be met within the duration of the forecast.

In order to put this algorithm in the context of the overall dis-patch logic, it may be noted that the area requirement signal isdispatched on a 2-second cycle, and that an economic dispatchcalculation is performed on a 12-second cycle. It is the intent ofthe Philadelphia Electric Company to recognize realistic limita-tions on the rate-of-response of individual units and to temperthe magnitude of the required change per dispatch period accord-ing to the approach of certain critical parameters to set limits.(Critical parameters will include, for instance, throttle pressure,superheat, reheat, transition, and impulse chamber tempera-tures, oxygen, and drum level.) Accordingly, changes in assignedgeneration following from the 12-second periodic economic dis-patch calculation will be stored and dispatched incrementally at2-second intervals along with the AR dispatch, with the sizeof the increments being determined by the current (dynamicallydetermined) allowable rates of response. The predictive filtercalculation will be performed at one-minute intervals in placeof every fifth economic dispatch calculation.

PARAMETERS OF CONTROLLED SYSTEM

As stated in the Introduction, one of the basic assumptions isthat the Philadelphia Electric Company's commitment to theInterconnection is to provide an agreed upon amount of mega-watts at a given incremental cost. It is important for economythat discrepancy between the composite incremental cost curve

supplied to the Interconnection and actual generation availableat a given cost be minimized. This is most important when a

change in available generation is made and would require allcompanies to update the Interconnection cost information when-ever changes occur.

It is assumed that the total (gross) system desired generationGt in megawatts has been computed and includes 1) the net mega-watts corresponding to the Interconnection cost signal for Phila-delphia Electric Company, and 2) the station auxiliary powerrequirements for the Philadelphia Electric Company system. Atany given time the actual system generation is Ga. The economicdispatch routine, with knowledge of the near term load trend,will make adjustments to Ga which will bring it to Gt and con-

tinually satisfy regulating requirements.The actual system generation Ga is provided by base-loaded

(manual) units Gai, hydro units Gah, and units under automaticcontrol Gas

Ga = Gaa + Gam+ Gah.

The actual automatically controlled generation Gaa is com-posed of two parts, 1) block-load (valve-point) generation Ba, and2) generation provided by regulating units Naa which will becalled actual trim generation.

Gaa = Ba + Naa.It is by changing Ba, Naa, and Gah that the dispatch routine

automatically adjusts Ga. To make these decisions the routinemust have a detailed knowledge of the cost and size of availablegeneration.

Block-Load IncrementsThe amount of additional generation obtained by opening a

valve from fully shut to fully opened position is called a block-load increment and is designated AB. At any given time all unitson automatic control will have turbines with a number of valvesfully opened. The current block-load assigned generation is

B = 1AB.

The sum is over all fully opened valves of all automatic units.It must be noted that B = Ba only when all valve-point assign-ments have been met; i.e., no unit is in the process of changingfrom one valve to another.The fully opened position of the next valve on the ith turbine

will provide ABi MW of generation. This quantity is the block-load increment of the next available block on the ith turbine.The quantities ABi are maintained in a "block-load availabilitylist." Also included is the cost associated with the block-loadincrement. The incremental cost for the next available block-loadincrement on the ith turbine is designated Xi and is equal to theaverage incremental cost over the range of that valve, since thetotal cost for the additional AB1 MW is the integral under theincremental cost curve between the appropriate valve points.The average cost is this integral divided by ABi.

Regulating Increments

The dispatch routine will specify a number of units to operatebetween valve points. The amount of generation provided by eachsuch unit will be determined on the basis of the area requirementsignal, differences between B and Ba, and regulation commitment(which reflects the necessary rate of following AR). The totalactual megawatts generated in this manner is Naa. All unitsoperating in such a fashion are designated "regulating units."Regulation status is assigned or removed by the dispatch routineas needed to maintain regulation requirements. Regulating unitsare selected on an economic basis from a list of units availablefor this purpose. In addition to the economic constraint, it is pos-sible by program control to include other constraints.Loading a turbine between valve points represents a block of

generation nearly equal to the block-load increment AB for thatvalve. When in regulating status that valve may or may not bepermitted to approach the fully closed position, and therefore, inregulating status AN < AB. AN is referred to as a block of"regulating capacity." With AN < AB, the difference may bereferred to as the regulating minimum AN which is carried to avoidexcessive throttling losses. Since a unit in regulating status should,on the average, be at the midpoint of the valve range, the totalcost for this number of megawatts may be taken as the integralunder the first half of the incremental cost curve for that valverange. The average cost is this integral divided by the megawattscorresponding to the midpoint and on the ith turbine is definedas X2.

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FINK et al.: ECONOMIC DISPATCH VIA VALVE-POINT LOADING

z0

4

crw

z

w

0I--

TRIM GENERATIONAVAILABLE N R

ACTUAL GENERATIONOF ALL AUTO ACTUAL TRIM R

UNITS Gaa GENERATION Naa I

ACTUAL BLOCK-LOADEDGENERATION Bo

TIME

Fig. 1. Parameters of controlled system, graphical interpretation.

z

0

w

zw

0D

0)

0

4 -iE+TRIM GENERATION

AVAILABLE N R

DESIRED GENERATION

ON ALL AUTO ASI

UNITS Gd NIED TRIM R

dGENERATION NoE

ASSIGNED BLOCK-LOADEDGENERATION B

(ALSO HY,DRO Gah)

TIME

Fig. 2. Parameters of dispatch algorithm, graphical interpretation.

If the ith turbine is available for regulation but is not currentlyin regulating status, this turbine along with the regulating capac-

ity of the next block ANa is recorded in a "regulation availabilitylist." If the ith turbine is currently in regulating status, it does notappear in the regulation availability list. However, this turbinealong with the block-load increment ABi does appear in the block-load availability list. Should the computer decide to utilize thisblock-load increment, the next valve range is added to the

regulation availability list. The regulating cap)acity lost inassigning this block-load increment is reassigned to a unit on theregulation availability list if required.The "trim generation available" N is defined as the sum of

the regulating capacities of all uniits in regulating status, i.e.,

N = 2; N (sum over all units supplying regulating capacity).

This also includes any hydrogenieration that is to be consideredas regulating capacity.The computer insures that there is always a mililnmuI of i R

MW assigned regulating capacity that could absorb ehanges ofR MW in systemn generation.Fig. 1 shows a graphical interpretation of some of the pre-

viously defined quantities. This represents the condition whereAR = 0 and B = Ba.

DISPATCH ALGORITIIM

As area requirement and incremental cost force PhiladelphiaElectric Company's generation up or down, its generation capac-ity is added or subtracted in such a manner as to maintain theminimum (4R) MW of regulation capacity. Therefore, thesatisfaction of regulating requirements forms the basis for allchanges in system automatic generation.

Durinig each dispatch cycle, there are four things to be con-sidered in allocating system automatic generation:

1) the prevailing load trend2) generation assigned and actual generation3) total regulating capacity requirement 2R4) cost of available generation.

As was mentioned in the Introduction, some amount of fore-knowledge of load trend is essential to successful operation of avalve-point loading program. This dispatch algorithm will use apredictive filter to classify the prevailing load trend as being cer-tainly increasing, certainly decreasing, or essentially steady state.The predictive filter will be the subject of a, future paper.A decision to change generation can not be made on just the

difference between the desired generation Gt and the actual sys-tem generation Ga, because during a previous dispatch cycle adecision to load a particular valve block may have been made,and that block has not been completely loaded. Therefore, it isnecessary to consider assigned generation rather than actualgenerationi for block-loaded units. Ba and Naa have previouslybeen defined, respectively, as the actual gen-eration from valve-point loaded uniits and actual trim generation. To incorporatethis in the logic, two new terms must now be defined.

Gd = desired generation on units on automatic control, includ-ing hydro units

Na = assigned trim generation.

Gd is found by subtracting the manual generation Gamn from thesystem desired generation Gt. B is equal to the assigned valve-point generation. The quantity N is defined as the sum of all ANfor regulating units. The assigned trim generation is foutnd by

Na - Gd - (1 - 7) Gah- B - N.

This assumes that Sy pu of the hydrogeneration is available asregulating capacity. The difference between B and Ba is a veryimportant quantity in monitoring system performance. Specif-ically, the rate of which Ba approaches B is the combined rateof change of those units changing valve assignmelnts. Generally,it reflects the rate at which the actual generation is approachingthe desired generation and, therefore, reflects the company'sability to respond to load changes.

Since any difference between B and Ba is carried by the reg-ulating units, it is quite possible that sluggish response rates cancause the entire regulatinig capacity to be used to compensate forthis deficiency. With this possibility the quanitity B - Bal shouldbe compared to R and the load dispatcher alerted any time |B-Ba| is greater than 1/2 R.

In order to combine conveniently the above conisiderationswith the regulation capacity commitment, two new quantitiesare defined. The quantity E+ is defined as the regulating capacityabove required and is found by

E+ = (N - Na) - R.

The quantity E- is defined as the excess regulating capacity be-low required and is found by

E- = Na - R.

These quantities can be interpreted graphically as shown inFig. 2.

It is necessary that certain incremental cost data be maintainedin tables or lists along with information on current block sizes andcurrent allowable rates of change. The block-load incremental

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, JUNE 1969

cost for the next available block-load increment on the ith tur-bine is designated Xi and is recorded in a block-load availabilitylist along with the block size AB1. The regulating incremental costfor the next available block of regulating capacity is designatedXi and is recorded in the regulation availablity list along with theblock size ANt.

It is convenient to define a number of system quantities whichare also of interest. The quantities X* and X* represent the highestincremental cost of all currently employed block-load incrementsand regulating block increments, respectively. The quantities Sand S* represent the sum of the two lowest cost block-load incre-ments currently available and the sum of the two highest costblock-load increments currently loaded, respectively.With all these definitions and conditions, it is possible now to

construct a logic diagram of the dispatch algorithm.

DISPATCH LOGIC

In the development of an automatic economic dispatch system,the actual dispatch logic is the heart of the system, but is only asmall portion of the total system logic. Since this paper dealsonly with the dispatch logic, certain functions will be assumedpreviously completed.

1) The necessary inputs from each station have been read,scaled, and checked.

2) The current status of each machine is checked against pre-vious dispatches to determine if it is responding to load requestsBa -e B and to update allowable rates of change.

3) The Philadelphia Electric Company's running cost for thisdispatch cycle and the megawatts Gt corresponding to this costhave been computed.

4) All unit information has been updated in the stored tables,and the block costs have been adjusted for transmission lossesbased on the system actual generation.

5) The predictive filter has classified the load trend as beingcertainly increasing, certainly decreasing, or essentially steadystate.

6) Decisions relative to hydrogeneration have already beenmade.

7) The list of available units has been checked to determineif, because of a recently removed constraint, a unit is operatingfar above or below system incremental cost. Appropriate actionhas been initiated to bring this unit to the right economic level.

8) The following calculations have been made and adjustedfor any decisions made in 6) or 7) above:

B = 2 AB

N = 2 AN

N= 2.N

Na = G- Gam- (1 - y) Gah - B - N

E+ = N - Na- R

E- = Na - R.

9) The quantities X* and X* have been determined, and thetables of unit costs have been ordered by ascending costs.

Fig. 3 is a composite diagram of the dispatch logic for all threeload trends. Each load-trend condition will be explained sep-arately, and that portion of Fig. 3 associated with that situationwill be shown in subsequent diagrams.The dispatch logic for each load trend can be divided into three

subgroups. The first subgroup contains those logic decisions con-cerned with changes in block-load increments-changes in B.

UPWARD TREND DOWNWARD TREND STEADY STATE TRENDSET ALL TREND FSE-T ALL TREND SET ALL TRENDSWITCHES TO SWITCHES TO SWITCHES TOl ~ ~ ~~~23.. >

I1 $. L ~~~~~~~~~~~SAND S*I|CALCULATE| |ASSIGN A ES N ESWNW T BE ,;E+ AND E-l

JAXaXTREND NO \YES X NO ASSIGN ABE UNIT.

1> YESX

NEW :S CALCULATE3,* N

YES NO TRENDSIYES NEWS+

NEI YEYS S |

I ~~~~~UNIT I E9S <

N

Fig. 3. Dispatch logic showing all three load-trend conditions.

The next subgroup is concerned with changes in regulating blockincrements to insure that regulation requirements are satisfied-changes in N. The last subgroup involves exchanging regulatingblock increments such that the regulation requirements are al-ways satisfied as economically as possible.

Upward Load TrendAn explanation of how a series of logic decisions functions is

best shown with the aid of simple examples. This approach willbe used with respect to each of the subgroups of the upward load-trend dispatch logic.The function of the first subgroup (blocks Ul through U6 in

Fig. 4) is to assign the cheapest block-load increments availableto be loaded when the parameter E- exceeds the size of thatblock.

Fig. 5 depicts a condition where at some previous instant oftime the load Gd and generation assignments B and N were suchthat E+ and E- were zero. Since that time Gd has increased anamount X, but no changes have occurred in B or N. Since alldispatch decisions relative to block-load increments are made on

consideration of E-, the question is does the parameter E- carrythe correct information. It should be noted that if there is no

discrepancy between the composite incremental cost curve usedby the Interconnection and the actual generation available at agiven cost, the increase X will correspond exactly to the mega-watts available from the cheapest available block-load inerement.However, because of the incremental cost being weighted bytransmission loss factors and the smoothing involved in calculat-ing the Interconnection incremental cost curve, it is unlikely thatno discrepancy will exist. In this real-world situation the com-puter will supply the required megawatts as determinied from theincremental cost at the minimum cost to the Philadelphia Elec-tric Company.

Reference to Fig. 5 shows that since the regulating requirement=tR is always taken symmetrically with respect to Gd, and B hasnot changed, the change X is reflected in Na. This is saying thatunless some change is made in B, the regulating units will supplyX MW more. If this occurs, E+ will equal -X, and E- will equalX.Now blocks Ul through U6 in Fig. 4 will make a decision based

on the value of E- as to what changes if any should be made in B.Since it has been assumed that X is equal in magaitude to the

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FINK et at.: ECONOMIC DISPATCH VIA VALVE-POINT LOADING

Fig. 4. Dispatch logic, upward load trend. Fig. 6. Dispatch logic, downward load trend.

R

q ^ T ~Gd

E+I i

R

RNa

BEFORE CHANGE AFTER CHANGEIN Gd IN Gd

Fig. 5. Change in parameters of dispatch algorithmwith increase in Gd.

cheapest ABi available, the decision will be to load that incre-ment. In Fig. 5 this will increase B to the lower limit on R (aftercondition) and, on recalculating, E+ and E- will both be againequal to zero.Had the situation been that X was greater than ABE, E- would

still be positive, and the next cheapest block would also havebeen checked for possible loading. If the difference betweenX andthe first block was not equal to or greater than the second blocksize, this difference would be assigned to the regulating units.The function of the second subgroup (blocks U7 through Ull),

is to add regulating capacity to insure that the regulating require-ments +R are met. Again referring to Fig. 5, it is assumed that Xrepresents an amount of megawatts that could not be suppliedexactly by a block-load increment. By picking up this load onthe regulating units, a situation where we have excess lowerregulation and a deficit in upper regulation has been created.Reference to blocks U7 through Ull in Fig. 4 shows that thelogic will iterate through all available regulating blocks, assign-ing them by increasing cost to be loaded until E+ becomes equalto or greater than zero or there are no more blocks available.At the input to the last subgroup (blocks U12 through U16),

E- is less than the next cheapest available block-load increment,and the regulating requirement is satisfied. It is the function of thelast subgroup to determine if the regulating requirement could be

R

Gd

la I

E+

I

t -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Nat, R

A 4-

II

BEFORE CHANGE _ AFTER CHANGEIN Gd IN Gd

Fig. 7. Change in parameters of dispatch algorithmwith decrease in Gd.

satisfied more economically. It does this using two criteria. First,the cost of the available regulating block must be lower than thehighest cost block currently supplying regulation. Secondly therelative megawatt size of the two blocks must be such as to keepE+ positive. If an available regulating block satisfies both criteria,the high-cost block will be operated as a block-loaded unit at theupper valve and the next regulating block increment will bemade available, and the lower-cost regulating block will beassigned to regulation.

Downward Load Trend

Fig. 6 shows the dispatch logic associated with a downward loadtrend. This logic can be divided into similar subgroupings as wasdone for the upward trend. Again decisions on block-load incre-ments are based on E-, and decisions on regulating incrementsare based on E+.

In Fig. 7 a decrease in Gd of YMW is depicted, and the changesin Na, E+, and E- are shown. Na has decreased by Y and as be-fore reflects the condition where the regulating units absorb thechange. E- becomes equal to - Y, and E+ becomes equal to + Y.

In the first subgroup (blocks Dl through D5 of Fig. 6),the highest cost block-load increment loaded will be dropped ifE- is less than zero. This process will continue until E- becomesequal to zero, or positive, or all units have been checked.

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, JUNE 1969

The second subgroup (blocks D6 through D1I) checks to seethat if E+ is positive and larger than the size of the highest costregulating block loaded, that block will be dropped from regula-tion and valve-point loaded.The third subgroup (blocks DlI through D15) is identical to

blocks U12 through U16 in Fig. 4 and performs the same func-tioil except that the unit coming off regulation will be block-loaded at the lower valve point.

Steady-State Load Trend

The dispatch logic for the steady-state condition shown in Fig.8 is basically a combination of the downward and upward load-trend logic except that changes in block-load assignments aremade after a more conservative check of load change.

Block-load changes are made on the basis of E-. When positive,it indicates excess lower regulating capacity which can only bereduced by increasing B. When negative, it indicates a deficit inregulating capacity which requires a reduction in B.When the predictive filter has classified the current load trend

as steady state, there is no indication as to which direction theload will move when it does begin to change. In the dispatch logicfor an upward load trend, the cheapest block-load increment waspicked up when E- exceeded that block size. In the dispatch logicfor a downward trend, the highest cost block-load incerement wasunloaded when E- exceeded the block size. In the steady-statedispatch logic, the criterion is that E- must exceed two block-loadincrements. The parameters S and S* are defined, respectively,as the sum of the two cheapest block-load increments availableand the sum of the two highest cost block-load increments loaded.The parameter E- is compared to S* in block 8843 when it is nega-tive, and it is compared to S in block SS4 wheni it is positive.The rest of the logic is similar to the other dispatch logic

diagrams and will not be discussed here. However, it should benoted that when regulating blocks are exchanged, the unit comingoff regulation is block-loaded at the lower valve point.

Assignment of RegulationOnce the amount of generation that must be assigned to reg-

ulating units Na has been determined, it is necessary to establishthe proportion that will be assigned to each of the regulatingunits. The quantity -yGah is carried by hydro. The remainder Na'= Na - YGah is assigned to regulating steam units. The criteriathat are observed in calculating this assignmiient are 1) that theassignment to each unit should be directly p)roportional to itsregulating capacity AN and inversely proportional to its regulat-ing cost X, both relative to the other regulating units, and 2) thatthe total regulating requirement be exactly satisfied without 3)violating the upper or lower regulating limit oni any unit. Ac-cordingly, for each dispatch period, a set of regulating assignmentfactors pi is calculated according to the logic showii in Fig. 9.

A ANiXi= N

satisfies the first criterion, where A is the average regulating costgiven by

- 2AN1ixA= N

anid the kj assures satisfaction of the constraints established by thesecond and third criteria, namely

2pi =1, i AN 'i Na.NaI

SS17

Fig. 8. Dispatch logic, steady-st ate load.

Fig. 9. Dispatch logic, assignment of regulation.

for all i. The kj are tentatively chosen according to the secondcriterion, checked, and, if necessary, corrected to avoid violationof the third criterion.

NOMENCLAITURE

B total assigned block-load generationBa actual block-load (valve-point) generationAB block-load incrementE+ regulating capacity above required amountE- excess regulating capacity below required aimiouniitGa total actual system generationGaa actual generatioil provided by steam uinits utnder auto-

matic controlGah actual generation provided by hydro unitsGam actual generation provided by base-load (manual) UlnitsGd desired generation on units on automatic controlGt total system desired generationk normalizing factor for regulating assignmentN trim generation available, equaling the sum of reg-

ulating capacities in regulating statusNa assigned trim generationNa' trim generation assigned to regulating steam unitsNaa actual trim generation (provided by regulating uiniits)

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FINK et al.: ECONOMIC DISPATCH VIA VALVE-POINT LOADTNG

AN block of regulating capacityAN regulating rninimum of a single regulating unitN total system regulating minimumR assigned regulating capacity requirement; calculation

involves consideration of 1) 1 percent of system capacityat some rate and 2) confidence interval of prediction

Xi block-load incremental cost of the next available blockon the ith turbine

Xi incremental cost for a block of regulating capacity oIn theith turbine

X* highest incremental cost of all currently employed block-load increments

X* highest incremental cost of all currently employed reg-ulating blocks

S sum of the two lowest cost block-load increments cur-rently available

S* sum of the two highest cost block-load increments cur-rently loaded

a regulating capability of given unit relative to requiredregulating capacity

f regulating assignment factor, nonnormalized-y per unit of hydrogeneration available for regulationAp

average regulating costregulating assignment factor, normalized and con-strained.

ACKNOWLEDGMENT

The advice and assistance of P. M. Davidson, Station EconomyDivision, throughout the development of this algorithm is grate-fully acknowledged.

REFERENCESII] G. L. Decker and A. D. Brooks, "Valve point loading of tur-

bines," AIEE Trans. (Power Apparatus and Systems), vol. 77,pp. 481-486, August 1958.

[21 A. P. Hayward, "Economic scheduling of generation by valvepoints," AIEE Trans. (Power Apparatus and Systems), vol. 80,pp. 963-965, 1961 (February 1962 sec.).

t3] L. T. Anstine, J. H. Henderson, F. A. Kramer, F. H. Light, G.A. Pall, F. M. Reed, and H. G. Stewart, "Application of digitalcomputer techniques to the economic scheduling of marginalgenerating units for the Pennsylvania-New Jersey-Marylandinterconnection," IEEE Trans. Power Apparatus and Systems,vol. 83, pp. 316-320, April 1964.

14] R. J. Ringlee and D. D. Williams, "Economic system operationconsidering valve throttling losses: IL-Distribution of systemloads by the method of dynamic programming," AIEE Trans.(Power Apparatus and Systems), vol. 81, pp. 615-622, 1962(February 1963 sec.).

Discussion

W. 0. Stadlin (Leeds anid Northrup Company, North Wales, Pa.):An important point that the authors have observed in applying adispatch computer is that theoretical optimization should not be at-tempted at the price of excessive regulation of generation amongunits. Availability of separate control-valve servos oinsome new steamturbine electrical governor designs presents the possibility of valveoperation in any desired relationship, instead of in a fixed sequence,thereby further enhancing the computer program.

Additional comments are desired to the following questions:1) Theoretically, exact valve-point loading could show that some

generators should be unloaded to a lower valve point even during alnincreasing load trend. Is this possibilitv inieluded in the auithors' dis-patch algorithm?

2) From a control viewpoint, how well can the actual valve posi-tion be correlated with the megawatt output of a machine?

3) What are the expected savings to be realized from valve-pointloading versus the conventional approach of fitting a monotonic func-tion to the generator characteristics?

Manutscript received February 7, 1969.

Lester H. Fink, Harry G. Kwatny, and John P. McDonald: Weappreciate MIr. Stadlin's interest and thank him for his commelits.The answer to his first question is implicit in 7) in the Section Dis-patch Logic. Removal of constraints on a unit which had been forced,for some reason exterior to the dispatch logic, to operate far aboveor below the system running cost would, at this point in the logic,result in that unit's being moved to the proper economic level regard-less of the load trend. This is the only such provision.We cannot answer the second question until we have concluded ouir

current tests on valve-position measurement techniques. Our intenitis to monitor the actual mechanical position of the valves, and eachtime a uinit is operated at any full-open valve position, the table show-ing megawatt versus valve position will be updated to the cuirrentcorrect relationship for use in future calculations.

Finally, with regard to the third question, Light and Gille [5] esti-mated that for our steam system, which then totaled 3256 MW, sav-ings bv valve-point loading should approximate $60 000 annually.We have no more recent estimate. In [4] savings of 0.1 percent to 0.2percent are projected.

REFERENCES[5] F. H. Light and J. A. Gille, "Economic operation at valve

cracking points," presented at the AIEE Winter General Meet-ing, New York, N.Y., January 28-February 2, 1962.

Manuscript received March 21, 1969.

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