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Multi-stage fuzzy load frequency control using PSO

H. Shayeghi a,*, A. Jalili b, H.A. Shayanfar c

a Technical Engineering Department, University of Mohaghegh Ardabili, Ardabil, Iranb Islamic Azad University, Ardabil Branch, Ardabil, Iranc Center of Excellence for Power Automation and Operation, Electrical Engineering Department, Iran University of Science and Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 15 September 2007Accepted 18 May 2008Available online 3 July 2008

Keywords:LFCMSFRestructured power systemPower system controlPSO

a b s t r a c t

In this paper, a particle swarm optimization (PSO) based multi-stage fuzzy (PSOMSF) controller is pro-posed for solution of the load frequency control (LFC) problem in a restructured power system that oper-ate under deregulation based on the bilateral policy scheme. In this strategy the control is tuned on linefrom the knowledge base and fuzzy inference, which request fewer sources and has two rule base sets. Inthe proposed method, for achieving the desired level of robust performance, exact tuning of membershipfunctions is very important. Thus, to reduce the design effort and find a better fuzzy system control,membership functions are designed automatically by PSO algorithm, that has a strong ability to findthe most optimistic results. The motivation for using the PSO technique is to reduce fuzzy system effortand take large parametric uncertainties into account. This newly developed control strategy combines theadvantage of PSO and fuzzy system control techniques and leads to a flexible controller with simple stric-ture that is easy to implement. The proposed PSO based MSF (PSOMSF) controller is tested on a three-arearestructured power system under different operating conditions and contract variations. The results ofthe proposed PSOMSF controller are compared with genetic algorithm based multi-stage fuzzy (GAMSF)control through some performance indices to illustrate its robust performance for a wide range of systemparameters and load changes.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Global analysis of the power system markets shows that the fre-quency control is one of the most profitable ancillary services atthese systems. This service is related to the short-term balance ofenergy and frequency of the power systems. The most commonmethods used to accomplish frequency control are generator gov-ernor response (primary frequency regulation) and load frequencycontrol (LFC). The goal of LFC is to reestablish primary frequencyregulation capacity, return the frequency to its nominal valueand minimize unscheduled tie-line power flows between neigh-boring control areas. From the mechanisms used to manage theprovision this service in ancillary markets, the bilateral contractsor competitive offers stand out [1].

During the past decade, several proposed LFC scenarios havebeen attempted to adapt traditional LFC schemes to the changeof environment in the power systems under deregulation [2–4].In a power system, each control area contains different kinds ofuncertainties and various disturbances due to increased complex-ity, system modeling errors and changing power system structure.As a result, a fixed controller based on classical theory is not cer-tainly suitable for the LFC problem. It is desirable that a flexible

controller be developed. Efforts have been made to design load fre-quency controllers with better performance to cope with parame-ter changes, using various adaptive neural networks and robustmethods [5–10]. The proposed methods show good dynamical re-sponses, but robustness in the presence of model dynamical uncer-tainties and system nonlinearities were not considered. Also, someof them suggest complex state feedback or high order dynamicalcontrollers, which are not practical for industry practices. Recently,some authors proposed fuzzy PID methods to improve perfor-mance of the LFC problem [11–13]. It should be pointed out thatthey require a three-dimensional rule base. This problem makesthe design process is more difficult. To overcome this drawback,in author’s pervious papers [14,15] a improved control strategybased on fuzzy theory and GA technique have been proposed.The resulting structure is a multi-stage fuzzy (MSF) controllerusing two-dimensional inference engines (rule base) to performreasonably the task of a three-dimensional controller. The pro-posed method requires fewer resources to operate and its role inthe system response is more apparent, i.e. it is easier to understandthe effect of a two-dimensional controller than a three-dimen-sional one [14]. In order for a fuzzy rule based control system toperform well, the fuzzy sets must be carefully designed. A majorproblem plaguing the effective use of this method is the difficultyof accurately constructing the membership functions. Because, it isa computationally expensive combinatorial optimization and also

0196-8904/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.enconman.2008.05.015

* Corresponding author. Tel.: +98 451 5517374; fax: +98 451 5512904.E-mail address: hshayeghi@gmail.com (H. Shayeghi).

Energy Conversion and Management 49 (2008) 2570–2580

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extraction of an appropriate set of membership function from theexpert may be tedious, time consuming and process specific. Thus,to reduce fuzzy system effort cost, in our pervious paper [15] a GAtechnique based on the hill climbing method have been proposed.Although, GA seems to be good methods to solve optimizationproblems, when applied to problems consisting of more numberof local optima, the solution from GA are just near global optimumareas. Also, it takes long simulation time to obtain the solution.Moreover, when the number of parameter is more, optimizationproblem is complex and coding chromosomes with more gens forincreasing algorithm accuracy is caused GA convergent speed willbecome very slow, so that convergent accuracy may be influencedby the slow convergent speed. Here, to overcome these drawbacks,a PSO based MSF (PSOMSF) controller is proposed. In this study,PSO technique is used for tuning membership functions of MSFcontroller. This method is proposed to improve optimization syn-thesis such that the global optima are guaranteed and the speedof algorithms convergence is extremely improved, too. PSO algo-rithm can be used to solve many of the same kinds of problemsas GA and does not suffer from of GA’s difficulties [16–18]. PSOis a novel population based metaheuristic, which utilize the swarmintelligence generated by the cooperation and competition be-tween the particle in a swarm and has emerged as a useful toolfor engineering optimization. It has also been found to be robustin solving problems featuring nonlinearing, non-differentiabilityand high dimensionality. The proposed PSOMSF controller is testedon a three-area restructured power system under different operat-ing conditions in comparison with the GAMSF [15] controllerthrough some performance indices. Results evaluation show thatthe proposed method achieves good robust performance for widerange of system parameters and load changes in the presence ofsystem nonlinearities and is superior to the other controllers.

2. Generalized LFC model

In the deregulated power systems, the vertically integrated util-ity no longer exists. However, the common LFC objectives, i.e.restoring the frequency and the net interchanges to their desiredvalues for each control area, still remain. The deregulated powersystem consists of GENCOs, TRANSCOs and DISCOs with an openaccess policy. In the new structure, GENCOs may or may not partic-ipate in the LFC task and DISCOs have the liberty to contract withany available GENCOs in their own or other areas. Thus, variouscombinations of possible contracted scenarios between DISCOsand GENCOs are possible. All the transactions have to be clearedby the independent system operator (ISO) or other responsibleorganizations. In this new environment, it is desirable that a newmodel for LFC scheme be developed to account for the effects ofpossible load following contracts on system dynamics.

Based on the idea presented in [19], the concept of an ‘ aug-mented generation participation matrix’ (AGPM) to express the pos-sible contracts following is presented here. The AGPM shows theparticipation factor of a GENCO in the load following contract witha DISCO. The rows and columns of AGPM matrix equal the totalnumber of GENCOs and DISCOs in the overall power system,respectively. Consider the number of GENCOs and DISCOs in areai be ni and mi in a large scale power system with N control areas.The structure of AGPM is given by

AGPM ¼AGPM11 � � � AGPM1N

..

. . .. ..

.

AGPMN1 � � � AGPMNN

2664

3775 ð1Þ

where,

Fig. 1. The generalized LFC scheme in the restructured system.

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AGPMij ¼

gpf ðsiþ1Þðzjþ1Þ � � � gpf ðsiþ1ÞðzjþmjÞ

..

. . .. ..

.

gpf ðsiþniÞðzjþ1Þ � � � gpf ðsiþniÞðzjþmjÞ

26664

37775

For i, j = 1, . . .,N and

si ¼Xi�1

k¼1

ni; zj ¼Xj�1

k¼1

mj and s1 ¼ z1 ¼ 0

In the above, gpfij refers to ‘generation participation factor’ andshows the participation factor of GENCO i in total load followingrequirement of DISCO j based on the contract. Sum of all entriesin each column of AGPM is unity. The diagonal sub-matrices ofAGPM correspond to local demands and off-diagonal sub-matricescorrespond to demands of DISCOs in one area on GENCOs in an-other area. Block diagram of the generalized LFC scheme in arestructured system is shown in Fig. 1. The nomenclature used isgiven in Appendix A. Dashed lines show interfaces between areasand the demand signals based on the possible contracts. Thesenew information signals are absent in the traditional LFC scheme.As there are many GENCOs in each area, ACE signal has to be dis-tributed among them due to their ACE participation factor in theLFC task and

Pnij¼1aji ¼ 1. We can write [20]:

di ¼ DPLoc;j þ DPdi;DPLoc;j ¼Xmi

j¼1

DPLj�i; DPdi ¼Xmi

j¼1

DPULj�i ð2Þ

gi ¼XN

j¼1 and j 6¼i

TijDfj ð3Þ

fi ¼ DPtie;i;sch

XN

k¼1 and k 6¼i

DPtie;ik;sch ð4Þ

DPtie;ik;sch ¼Xnl

j¼1

Xmk

i¼1

apf ðslþjÞðzkþiÞDPLðzkþiÞ�k

�Xnk

i¼1

Xmi

j¼1

apf ðskþiÞðzlþjÞDPLðzlþjÞ�l ð5Þ

DPtie;i�error ¼ DPtie;i�actual � fi ð6Þ

qi ¼ ½q1i � � � qki � � � qnii �; qki ¼XN

j¼1

Xmj

t¼1

gpf ðsiþkÞðzjþtÞDPLt�j

" #ð7Þ

DPm;k�i ¼ qki þ apfkiDPdi; k ¼ 1;2; . . . ;ni ð8Þ

DPm,ki is the desired total power generation of a GENCO k in area iand must track the demand of the DISCOs in contract with it inthe steady state.

A three-area power system, shown in Fig. 2 is considered as atest system to illustrate the effectiveness of the proposed controlstrategy. It is assumed that each control area includes two GENCOsand DISCOs. The power system parameters are given in Tables 1and 2.

3. PSO based MSF controller scheme

Fuzzy logic control is one of the most successful areas in theapplication of fuzzy theory and is excellent alternatives to the con-ventional control methodology when the processes are too com-plex for analysis by conventional mathematical techniques[25,26]. Because of the complexity and multi-variable conditionsof power systems, conventional control methods may not give sat-isfactory solutions. On the other hand, their robustness and reli-ability make fuzzy controllers useful for solving a wide range ofcontrol problems in power systems. In this paper, particle swarmoptimization based multi-stage fuzzy controller is proposed forsolution of the LFC problem. The motivation of using the proposedPSOMSF controller is to take large parametric uncertainties, systemnonlinearities and minimize of area load disturbances into account.As MSF controller is elaborately explained in author’s pervious pa-per [14], here we points out only the salient features of it briefly.This control strategy combines fuzzy PD controller and integralcontroller with a fuzzy switch. The fuzzy PD stage is employed topenalize fast change and large overshoots in the control inputdue to corresponding practical constraints. The integral stage isalso used in order to get disturbance rejection and zero steadystate error.

It should be noted that the exact tuning of membership func-tions in MSF control strategy is very important to achieve the de-sired level of system robust performance. Because it is acomputationally expensive combinatorial optimization problemand also extraction of an appropriate set of membership functionfrom the expert may be tedious, time consuming and process spe-cific. In order to overcome this drawback and reduce fuzzy systemeffort and cost, PSO algorithm is being used to optimal tune ofmembership functions in the proposed MSF controller. Particleswarm optimization (PSO) algorithm, which is tailored for optimiz-ing difficult numerical functions and based on metaphor of humansocial interaction, is capable of mimicking the ability of humansocieties to process knowledge [20]. It has roots in two main com-ponent methodologies: artificial life (such as bird flocking, fishschooling and swarming); and evolutionary computation.Although, the PSO algorithm is initially developed as a tool formodeling social behavior, it has been applied in different areas.

Table 1Control area parameters

Parameter Area 1 Area 2 Area 3

KP (Hz/pu) 120 72 91TP (s) 20 14.3 10.6B (pu/Hz) 0.8675 0.785 0.87Tij (pu/Hz) T12 = T13 = T23 = 0.545

Table 2GENCOs parameter

MVAbase (1000 MW)parameter

GENCOs (k in area i)

1-1 2-1 1-2 2-2 1-3 2-3

Rate (MW) 1000 800 1100 900 1000 1020TT (s) 0.36 0.42 0.44 0.4 0.36 0. 4TH (s) 0.06 0.07 0.06 0.08 0.07 0.08R (Hz/pu) 2.4 3.3 2.5 2.4 2.4 3.3a 0.5 0.5 0.5 0.5 0.5 0.5

D13

G13 G23

D21

G11 G21G12 G22

D1 2

Area 1 Area 3

Area 2

Gij : GENCOi-j

Dij : DISCOi-j

D11

D2 2

D23

Fig. 2. A three-area restructured power system.

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Moreover, it has been recognized as a computational intelligencetechnique intimately related to evolutionary algorithms. Detailsof the original PSO algorithm can be found in Kennedy et al. [21–23]. PSO is a populated search method for optimization of contin-uous nonlinear functions resembling the movement of organismsin a bird flock or fish school. Its key concept is that potential solu-tions are flown through hyperspace and are accelerated towardsbetter or more optimum solutions. Its paradigm can be imple-mented in simple form of computer codes and is computationallyinexpensive in terms of both memory requirements and speed. Itlies somewhere between evolutionary programming and geneticalgorithms. As in evolutionary computation paradigms, the con-cept of fitness is employed and candidate solutions to the problemare termed particles or sometimes individuals, each of which ad-justs its flying based on the flying experiences of both itself andits companions. It keeps track of its coordinates in hyperspacewhich are associated with its previous best fitness solution, andalso of its counterpart corresponding to the overall best value ac-quired thus far by any other particle in the population. Vectorsare taken as presentation of particles since most optimizationproblems are convenient for such variable presentations. In fact,the fundamental principles of swarm intelligence are adaptability,diverse response, proximity, quality, and stability. It is adaptivecorresponding to the change of the best group value. The allocationof responses between the individual and group values ensures adiversity of response. The higher dimensional space calculationsof the PSO concept are performed over a series of time steps. Thepopulation is responding to the quality factors of the previous bestindividual values and the previous best group values. The principleof stability is adhered to since the population changes its state ifand only if the best group value changes [24]. Fig. 3 shows thestructure of the proposed PSOMSF controller for solution of theLFC problem. In this structure, input values are converted totruth-value vectors and applied to their respective rule base. Theoutput truth-value vectors are not defuzzified to crisp value aswith a single stage fuzzy logic controller but are passed onto thenext stage as a truth value vector input. The darkened lines inFig. 3 indicate truth value vectors.

In this effort, all membership functions are defined as triangularpartitions with seven segments from �1 to 1. Zero (ZO) is the cen-ter membership function which is centered at zero. The partitionsare also symmetric about the ZO membership function as shown inFig. 4. The remaining parts of the partition are negative big (NB),negative medium (NM), negative small (NS), positive small (PS),positive medium (PM) and positive big (PB).

There are two rule bases used in the MSF controller. The first iscalled the PD rule bases as it operates on truth vectors form the er-ror (e) and change in error (De) inputs. A typical PD rule base for

the fuzzy logic controller is given in Table 3. This rule base re-sponds to a negative input from either error (e) or change in error(De) with a negative value thus driving the system to ward thecommanded value. Table 4 shows a PID switch rule base. This rulebase is designed to pass through the PD input if the PD input is notin zero fuzzy set. If the PD input is in the zero fuzzy set, then thePID switch rule base passes the integral error values ð

ReÞ. This rule

base operates as the behavior switch, giving control to PD feedbackwhen the system is in motion and reverting to integral feedback toremove steady state error when the system is no longer moving.The operation used to determine the consequence value at theintersection of two input fuzzy value is given as

ci;j ¼ Pða�i bjÞ; i; j ¼ 1;2; . . . Nm ð9Þ

where, ai is the membership value of ith fuzzy set for a given e in-put, and bj is likewise for a De input. The operator used to determinethe membership value of the kth consequence set is

Ck ¼X

Ci;j; i; j ¼ 1;2; . . . Nm ð10Þ

The defuzzification uses the weighted average method where Ck isthe peak point of the kth output fuzzy membership function.

d ¼X

C�kck=X

Ckk ¼ 1; . . . ;N ðsets in output pointÞ: ð11Þ

3.1. Membership functions tuning strategy by PSO

In the proposed PSOMSF controller, we must tune the linguistichedge combinations which are difficult to be contributed accordingto human experience and knowledge. To acquire an optimal com-bination, this paper employs PSO technique [27] to improve opti-mization synthesis and find the global optimum value of fitnessfunction. PSO is a population based evolutionary search algorithmscharacterized as conceptually simple, easy to impalement andcomputationally efficient. As it is reported in [28], this optimiza-tion technique can be used to solve many of the same kinds ofproblems as GA, and does not suffer from some of GAs difficulties.PSO has also been found to be robust in solving problem featuring

Fuzzyfy D

Fuzzyfy P PID Switch Rule Base

Fuzzyfy I

Defuzzify

PSO Technique

Ki/s

Kds

HF Controller

PD Rule Base

Nominal Model of Area i

di i i

ACEiyref

PSOMSF Controller

ρζ

Fig. 3. Structure of the proposed PSOMSF control strategy.

-1 -b -a 0 a b 1

NB NM NS ZO PS PM PB

Fig. 4. Symmetric fuzzy partition.

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nonlinearing, non-differentiability and high-dimensionality. It is asthe search method to improve the speed of convergence and findthe global optimum value of fitness function. In this work, thePSO module works offline. PSO searches the optimal linguistichedge combination according to the controlled plants. Accordingto Fig. 4 for exact tuning of used membership functions in the pro-posed method we must find the optimal value for a and b param-eters, where 0 < a < b < 1.

This new approach features many advantages; it is simple, fastand can be coded in few lines. Also, its storage requirement is min-imal. Moreover, this approach is advantageous over evolutionaryand genetic algorithms in many ways. First, PSO has memory. Thatis, every particle remembers its best solution (local best) as well asthe group best solution (global best). Another advantage of PSO isthat the initial population of the PSO is maintained, and so there isno need for applying operators to the population, a process that istime and memory-storage-consuming. In addition, PSO is based on‘‘constructive cooperation” between particles, in contrast with the

genetic algorithms, which are based on ‘‘ the survival of the fittest”.PSO starts with a population of random solutions ‘‘particles” in aD-dimension space. The ith particle is represented by Xi =(xi1,xi2, . . . ,xiD). Each particle keeps track of its coordinates inhyperspace, which are associated with the fittest solution it hasachieved so far. The value of the fitness for particle i (pbest) is alsostored as Pi = (pi1,pi2, . . . ,piD). The global version of PSO keeps trackof the overall best value (gbest), and its location, obtained thus farby any particle in the population. PSO consists of, at each step,changing the velocity of each particle toward its pbest and gbestaccording to Eq. (12). The velocity of particle i is represented asVi = (vi1,vi2, . . . ,viD). Acceleration is weighted by a random term,with separate random numbers being generated for accelerationtoward pbest and gbest. The position of the ith particle is then up-dated according to Eq. (13) [27].

vid ¼ w� vid þ c1 � randðÞ � ðPid � xidÞ þ c2 � randðÞ � ðPgd � xidÞð12Þ

xid ¼ xid þ vid ð13Þ

where, pid = pbest and pgd = gbest. Several modifications have beenproposed in the literature to improve PSO algorithm speed and con-vergence toward the global minimum. One modification is tointroduce a local-oriented paradigm (lbest) with different neighbor-hoods. It is concluded that gbest version performs best in terms ofmedian number of iterations to converge. However, lbest versionwith neighborhoods of two is most resistant to local minima. PSOalgorithm is further improved via using a time decreasing inertiaweight, which leads to a reduction in the number of iterations[27]. Fig. 5 shows the flowchart of the proposed PSO algorithm. Itshould be noted that choice properly fitness function is veryimportant in synthesis procedure. Because different fitness func-tions promote different PSO behaviors, which generate fitness valueproviding a performance measure of the problem considered [14].For our optimization problem, the flowing fitness function isintroduced:

f ðITAEÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP3i¼1ITAEi

q3

; ITAEi ¼Z t

0tjACEijdt ð14Þ

Here, PSO procedure is applied to exact tune of membershipfunctions of the proposed MSF controller for the solution of LFCproblem. Results of membership function set values are listed inTable 5 and Fig. 6 shows the minimum fitness value evolving pro-

Table 3PD rule base

De

NB NM NS ZO PS PM PB

e NB NB NB NB NB NM NS ZONM NB NB NB NM NS ZO PSNS NB NB NM NS ZO PS PMZO NB NM NS ZO PS PM PBPS NM NS ZO PS PM PB PBPM NS ZO PS PM PB PB PBPB ZO PS PM PB PB PB PB

Table 4PID switch rule base

PD values

NB NM NS ZO PS PM PBRe NB NB NM NS NB PS PM PB

NM NB NM NS NM PS PM PBNS NB NM NS NS PS PM PBZO NB NM NS ZO PS PM PBPS NB M NS PS PS PM PBPM NB NM NS PM PS PM PBPB NB NM NS PB PS PM PB

Evaluate the fitness of each particle

Optimal value of member ship functions hedge

Satisfyingstoppingcriterion

Update pbest and gbest

End

Start

Select parameters of PSO: N, C1, C2, C and w

Generate the randomly positions and velocities of particles

Initialize, pbest with a copy of the position for particle, determine gbest

Update velocities and positions according to Eqs. (12) and (13)

No

Yes

Fig. 5. Flowchart of the proposed PSO technique for exact tuning of membership functions.

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cess. In order to acquire better performance, number of particle,particle size, number of iteration, C1, C2, C is chosen as 20, 8, 500,2, 2 and 1, respectively. Also, the inertia weight, w, is linearlydecreasing from 0.9 to 0.4.

4. Simulation results

In the simulation study, the linear model of turbine DPVki/DPTki

in Fig. 1 is replaced by a nonlinear model of Fig. 7 (with ±0.05 lim-

it). This is to take GRC into account, i.e. the practical limit on therate of the change in the generating power of each GENCO. The re-sults in Refs. [20,26] indicated that GRC would influence the dy-namic responses of the system significantly and lead to largerovershot and longer settling time.

The proposed PSOMSF controller is applied for each control areaof the restructured power system as shown in Fig. 2. To illustraterobustness of the proposed control strategy against parametricuncertainties and contract variations, simulations are carried outfor four cases of possible contracts under various operating condi-tions and large load demands. Performance of the proposedPSOMSF controller is compared with the GAMSF [15] controller.

4.1. Case 1

In this scenario, DISCOs have the freedom to have a contractwith any GENCO in their or another areas. Consider that all the DIS-COs contract with the available GENCOs for power as per followingAGPM. All GENCOs participate in the LFC task.

AGPM ¼

0:25 0 0:25 0 0:5 0

0:5 0:25 0 0:25 0 0

0 0:5 0:25 0 0 0

0:25 0 0:5 0:75 0 0

0 0:25 0 0 0:5 0

0 0 0 0 0 1

26666666664

37777777775

It is assume that a large step load 0.1 pu MW is demanded byeach DISCOs in all areas. Power system responses with 25% de-crease in system parameters are shown in Fig. 8. Using the pro-posed method, the frequency deviation of the all areas arequickly driven back to zero and has small settling time. Also thetie-line power flow properly converges to the specified value, ofEq. (5), in the steady state case (Fig. 8), i.e.; DPtie21,sch = 0 andDPtie23,sch = +0.025 pu MW.

4.2. Case 2

In this case, It may happen that a DISCO violates a contract bydemanding more power than that specified in the contract. Thisexcess power must be reflected as a local load of the area but not

Table 5Optimal values of parameters a and b

Membership function ACEi DACEiR

ACE Output

PSO a 0.0563 0.0288 0.0362 0.1232Result b 0.1384 0.4014 0.4091 0.8016

-

d

-d

1TTki

1s

ΔPT kikiΔPV

Fig. 7. Nonlinear turbine model with GRC.

0 10 20 30-1

0.2

ΔF1

0 10 20 30-1

0

0.8

ΔF2

0 10 20 30-1

-0.4

0

ΔF3

0 10 20 30-0.04

0

0.06

ΔPti

e 1-2

0 10 20 30

-0.02

0

0.04

0.1

ΔPtie

2-3

0

Time (Sec) Time (Sec)

Fig. 8. Deviation of frequency and tie-lines power flows; solid (PSOMSF) and dashed (GAMSF).

0 50 100 150 200 250 0 350 400 450 5000.75

0.8

0.85

0.9

0.95

Iteration

Fitn

ess

Val

uw

Fig. 6. Minimum fitness value convergence.

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as the contracted demand and taken up by GENCOs in the samearea. Consider case 1 again with modification that DISCO of areas1, 2 and 3 demands 0.07, 0.04 and 0.02 pu MW of excess power,respectively. The total local load in areas is computed as

DPLoc;1 ¼ 0:1þ 0:1þ 0:07 ¼ 0:27;DPLoc;2 ¼ 0:1þ 0:1þ 0:04 ¼ 0:24;DPLoc;3 ¼ 0:1þ 0:1þ 0:01 ¼ 0:21 pu MW

Power system responses in this scenario with 25% increase insystem parameters are shown in Fig. 9. Using the proposed meth-od, the frequency deviation of all areas and tie-line power flows arequickly driven back to zero and have small settling time. Also, thetie-line power flows properly converges to the specified value,Eq. (5), in the steady state, i.e.: DPtie21,sch = 0 and DPtie31,sch =+ 0.025 pu MW.

4.3. Case 3

Consider case 2 again. Assume, in addition to the specified con-tracted loead demands 0.1 pu MW, a bounded random step loadchange as a large uncontracted demand as shown in Fig. 10 ap-pears in each control area, where

�0:05 6 DPdi 6 þ0:05 pu MW

The propose of this scenario is to test the robustness of the pro-posed controller against uncertainties and random large load dis-turbances. The power system responses in this case with 25%decrease in system parameters are shown in Figs. 11–13. The sim-ulation results demonstrate that the proposed control strategytrack the load fluctuations and meet robustness for a wide rangeof load disturbances and possible contract scenarios under plantparameter changes.

0 10 20 30-1.4

0.5

0

ΔF1

0 10 20 30-1.5

0

1

ΔF2

0 10 20 30-1.5

0

0.5

ΔF3

0 10 20 30-0.08

0

0.04ΔP

tie1-

2

0 10 20 30-0.07

0

0.09

ΔPtie

2-3

Fig. 9. Deviation of frequency and tile lines power flows; solid (PSOMSF) and dashed (GAMSF).

0 40 80 120 160 200

0.16

0.2

0.24

0 40 80 120 160 200

0.16

0.2

0.24

0 40 80 120 160 200

0.16

0.2

0.24

ΔPd

1ΔP

d2

ΔPd

3

Fig. 10. Random disturbance load patterns in areas 1, 2 and 3.

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To demonstrate performance robustness of the proposed meth-od, the integral of the time multiplied absolute value of the error

(ITAE) and figure of demerit (FD) based on the system performancecharacteristics are being used as

0 40 80 120 160 200

-1.2

-0.6

0

0.6ΔF

1

0 40 80 120 160 200-1.5

0

1.5

ΔF2

0 40 80 120 160 200-1.4

-0.6

0

0.4

ΔF3

Fig. 11. Deviation of frequency, solid (PSOMSF) and dashed (GAMSF).

20 60 100 140-0.1

0

0.15

ΔF1

20 40 60 80 100 120 140

0

0.15

ΔF2

20 40 60 80 100 120 140-0.12

0

0.08

ΔF3

Fig. 12. Scales of deviation of frequency, solid (PSOMSF) and dashed (GAMSF).

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ITAE ¼Z 10

0tðjACE1ðtÞj þ jACE2ðtÞj þ jACE3ðtÞjÞdt ð15Þ

FD ¼ ðOS� 7Þ2 þ ðUS� 4Þ2 þ ðTs� 0:1Þ2 ð16Þ

where, overshoot (OS), undershoot (US) and settling time (for 3%band of the total load demand in area 1) of frequency deviationarea 1 is considered for evaluation of the FD. Numerical results ofperformance robustness for cases 1 and 2 are listed in Tables 6and 7, whereas the system parameters are varied from �25% to25% of the nominal values. Figs. 14 and 15 shows the maximum var-iation percent of ITAE and FD from theirs values in the nominalconditions.

Remark 4.1. Examination of Figs. 14 and 15 reveals that theproposed control strategy achieves good robust performanceagainst parametric uncertainties and is superior to othercontrollers.

Remark 4.2. The worst case, as seen from Tables 6 and 7, occurswhen the all system parameters have been had +25% increasingfrom their nominal values.

Remark 4.3. we have considered different cases for LFC control ofa three-area power system. The simulation results show that incomparison with GAMSF controller, the system performance is sig-nificantly improved by the PSOMSF controller designed in thispaper for a wide range of load disturbances and possible contract

0 40 80 120 160 200-0.08

0

0.08ΔP

tie 1

-2

0 40 80 120 160 200

0

0.06

0.12

ΔPtie

2-3

Fig. 13. Deviation tile lines power flows; solid (PSOMSF), dashed (GAMSF).

Fig. 14. Values of performance indices under cases 1: (a) ITAE (b) FD.

Table 6Value of ITAE

Test no. PPC (%) Scenario 1 Scenario 2

PSOMSF GAMSF PSOMSF GAMSF

1 �25 764.9 866.0 1526.6 2030.22 �20 787.0 899.0 1561.2 2076.63 �15 803.4 942.2 1593.4 2289.44 �10 824.6 981.8 1616.0 2518.15 �5 869.4 1046.9 1641.2 2755.6

0 (Nominal) 0 918.7 1128.9 1733.6 2989.96 +5 983.1 1269.6 1846.9 3263.37 +10 1059.1 1469.8 2002.1 3439.58 +15 1128.6 1631.1 2150.3 3520.19 +20 1187.5 1738.2 2281.1 3645.4

10 +25 1232.0 1920.3 2406.5 3678.2

Table 7Value of FD

Test no. PPC (%) Scenario 1 Scenario 2

PSOMSF GAMSF PSOMSF GAMSF

1 �25 295.7 584.9 1155.7 2219.32 �20 271.3 551.7 991.5 2103.33 �15 212.84 534.3 831.4 2191.94 �10 203.7 514.0 686.4 2278.55 �5 201.9 511.7 580.1 2313.7

0 (Nominal) 0 202.2 541.1 557.5 2260.66 +5 204.3 614.4 554.8 23156.07 +10 206.2 757.6 558.5 2183.18 +15 207.6 869.7 552.3 1859.89 +20 209.1 874.9 545.3 1806.7

10 +25 209.2 1010.1 558.2 1991.1

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scenarios under different plant parameter changes even in thepresence of GRC.

5. Conclusions

A new PSO based multi-stage fuzzy controller is proposed forsolution of the LFC problem in a restructured power system in thispaper. This control strategy was chosen because of increasing thecomplexity and changing structure of power systems. This newlydeveloped control strategy combines advantage a fuzzy PD andintegral controllers with a fuzzy switch to achieve the desired levelof robust performance, such as frequency regulation, tracking ofload demand and disturbance attenuation under load fluctuationfor a wide range of the plant parameters changes and system non-linearities. In order to reduce design effort and find better fuzzysystem control, a PSO based algorithm has been used to tune mem-bership functions automatically. The PSO algorithm proposed inthis paper is easy to implement without additional computationalcomplexity. Thereby experiments this algorithm gives quite prom-ising results. The ability to jump out the local optima, the conver-gence precision and speed are remarkably enhanced and thus thehigh precision and efficiency are achieved.

The effectiveness of the proposed method is tested on a three-area restructured power system for a wide range of load demandsand disturbances under different operating conditions. Comparedwith GAs in term of ITAE and FD, the PSO demonstrates its superi-ority in computational complexity, success rate and solution qual-ity and following conclusions can be drawn about the proposedmethod.

1. It is characterized as a simple heuristic of well balanced mech-anism with flexibility to enhance and due to non-model base, itcan be used to control a wide range of complex and nonlinearsystems.

2. It does not depend on the nature of the function it minimizesand is insensitive to the initial searching point, there by ensur-ing quality solution for different trial runs.

3. It is effective and ensures robust performance for a wide rangeoperating conditions under plant parameter changes and sys-tem nonlinearities. Also, it dose not require an accurate modelof LFC problem, has simple structure and easy to implement.Moreover, the design process is less demanding than that ofother controllers.

4. The system performance characteristics in terms of ‘ITAE’ and‘FD’ indices reveal that this control strategy is a promising con-trol scheme for the AGC problem in the deregulated powersystems.

Appendix A. Nomenclature

F area frequencyPtie net tie-line power flow turbine power

PT turbine powerPV governor valve positionPC governor set pointACE area control errorapf ACE participation factorD deviation from nominal valueKP subsystem equivalent gainTP subsystem equivalent time constantTT turbine time constantTH governor time constantR droop characteristicB frequency biasTij tie line synchronizing coefficient between areas i and jPd area load disturbancePLj�i contracted demand of Disco j in area iPULj�i un-contracted demand of Disco j in area iPm,j�i power generation of GENCO j in area iPLoc total local demandg area interfacef scheduled power tie-line power flow deviation (DPtie,sch.)

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Hossein Shayeghi received the B.S. and M.S. degrees inElectrical and Control Engineering in 1996 and 1998,respectively. He received his Ph.D. degree in ElectricalEngineering from Iran University of Science and Tech-nology, Tehran, Iran in 2006. Currently, he is an Assis-tance Professor in Technical Eng. Department ofUniversity of Mohaghegh Ardabili, Ardabil, Iran. Hisresearch interests are in the Application of Robust Con-trol, Artificial Intelligence to Power System ControlDesign, Operation, Planning and Power System Restruc-turing. He is a member of Iranian Association of Electricaland Electronic Engineers and IEEE.

Aref Jalili received the B.S. and M.S. degrees in ElectricalEngineering from Azad University, Ardabil and SouthTehran Branches, Iran in 2003 and 2005, respectively.Currently, he is a Ph.D. student in Electrical Engineeringat Science and Technology Research Branch of the AzadUniversity, Tehran, Iran. His areas of interest in researchare Application of Fuzzy Logic and Genetic Algorithm toPower System Control and Restructuring.

Heidarali Shayanfar received the B.S. and M.S. degreesin Electrical Engineering in 1973 and 1979, respectively.He received his Ph.D. degree in Electrical Engineeringfrom Michigan State University, U.S.A., in 1981. Cur-rently, he is a Full Professor in Electrical EngineeringDepartment of Iran University of Science and Technol-ogy, Tehran, Iran. His research interests are in theApplication of Artificial Intelligence to Power SystemControl Design, Dynamic Load Modeling, Power SystemObservability Studies and Voltage Collapse. He is amember of Iranian Association of Electrical and Elec-tronic Engineers and IEEE.

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