a tabu search based approach to power system stability enhancement via excitation and static phase...

Electric Power Systems Research 52 (1999) 133–143

A tabu search based approach to power system stabilityenhancement via excitation and static phase shifter control

M.A. Abido *, Y.L. Abdel-MagidElectrical Engineering Department, King Fahd Uni6ersity of Petroleum and Minerals, Box 183, Dhahran 31261, Saudi Arabia

Received 15 July 1998; accepted 3 December 1998

Abstract

Power system stability enhancement through control of excitation and static phase shifter (SPS) has been investigated. Thedesign problem of excitation and SPS controllers is formulated as an optimization problem. An eigenvalue-based objectivefunction to increase the system damping is proposed. Then, tabu search (TS) algorithm is proposed to search for optimalcontroller parameters. Different control schemes have been proposed and tested on a weakly connected power system withdifferent disturbances, loading conditions, and parameter variations. It was observed that although excitation control enhances thepower system stability, the SPS controller provides most of the damping and improves the voltage profile of the system. Thenonlinear simulation results show the effectiveness and robustness of the proposed control schemes over a wide range of loadingconditions and system parameter variations. © 1999 Elsevier Science S.A. All rights reserved.

Keywords: Power system stabilizer; Static phase shifter; Tabu search algorithm

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1. Introduction

Since the 1960s, low frequency oscillations have beenobserved when large power systems are interconnectedby relatively weak tie lines. These oscillations maysustain and grow to cause system separation if noadequate damping of electromechanical modes is avail-able [1,2].

In the early literature, DeMello and Concordia in1969 [3] presented the concepts of synchronous machinestability as affected by a lead-lag compensator, usuallycalled power system stabilizer (PSS), for damping ma-chine oscillations. Then, power system engineers haveshowed great interest and made significant contribu-tions in PSS design [4–9]. Recently, several approachesbased on modern control theory have been applied toPSS design problem. These include optimal control,adaptive control, variable structure control, and intelli-gent control [10–14].

Although PSSs provide supplementary feedback sta-bilizing signals in the excitation systems and extend thepower system stability limit by enhancing the system

damping of low frequency oscillations associated withthe electromechanical modes, they suffer a drawback ofbeing liable to cause a great variations in the voltageprofile and they may even result in leading power factoroperation under severe disturbance conditions [15,16].

The recent advances in power electronics have led tothe development of the flexible alternating currenttransmission systems (FACTS). FACTS are designed toovercome the limitations of the present mechanicallycontrolled power systems and enhance power systemstability by using reliable and high-speed electronicdevices. One of the promising FACTS devices is thestatic phase shifter (SPS). An SPS is a device thatchanges the relative phase angle between systemvoltages. Therefore, the real power flow can be regu-lated to mitigate the low frequency oscillations andenhance power system transient stability. The effective-ness of the SPS in improving power system stability hasbeen investigated in several studies with promising re-sults [17–25]. This provides an alternative choice to theconventional PSS in suppressing power systemoscillations.

Recently, Jiang et al. [22] proposed an SPS controltechnique based on nonlinear variable structure controltheory. In their control scheme the phase shift angle is

* Tel.: +966-3-8605235/4379; fax: +966-3-8603535.E-mail address: [email protected] (M.A. Abido)

0378-7796/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved.PII: S 0 3 7 8 -7796 (99 )00013 -9

M.A. Abido, Y.L. Abdel-Magid / Electric Power Systems Research 52 (1999) 133–143134

determined as a nonlinear function of rotor angle andspeed. However, in real life power system with a largenumber of generators, the rotor angle of a single gener-ator measured with respect to the system reference willnot be very meaningful. Tan and Wang [23] proposeda direct feedback linearization technique to linearizeand decouple the power system model to design theexcitation and SPS controllers. Despite the potential ofmodern control techniques with different structures,power system utilities still prefer a conventional lead–lag controller structure [4–6]. The reasons behind thatmight be the ease of on-line tuning and the lack ofassurance of the stability related to some adaptive orvariable structure techniques. It is shown that the ap-propriate selection of conventional lead–lag stabilizerparameters results in effective damping to low fre-quency oscillations [4,6].

In the last few years, tabu search (TS) algorithm[26–30] appeared as a promising heuristic algorithmfor handling the combinatorial optimization problems.TS algorithm uses a flexible memory of search his-tory to prevent cycling and to avoid entrapment inlocal optima. It has been shown that, under certainconditions, the TS algorithm can yield global optimalsolution with probability one [29]. In power systems,TS has been applied to a number of power systemoptimization problems with impressive successes [31–33]. However, the potential of TS algorithm to solvepower system control problems has not yet beenexploited.

In this paper, the potential of TS algorithm to exci-tation and SPS control problem is investigated.In addition, an assessment of the effects of the excita-tion and SPS control when applied independently andalso through coordinated application has been carriedout. The controller design problem is formulated asan optimization problem. Then, TS algorithm is em-ployed to solve this problem with the aim of gettingthe optimal or near optimal settings of the con-troller parameters. Different control schemes have beenproposed and tested on a weakly connected powersystem. Based on eigenvalue analysis and simulationresults, it was observed that the proposed controlschemes provide good damping of electromechanicalmodes of oscillations and enhance power systemstability.

2. Linearized power system model

In this study, a single machine infinite bus systemwith an SPS shown in Fig. 1 is considered. The genera-tor is connected to the infinite bus via a transmissionline. The line impedance is Z=R+ jX and the genera-tor has a local load of admittance YL=g+ jb. Thegenerator is represented by the third-order model com-prising of the electromechanical swing equation andthe generator internal voltage equation [1,2]. The swingequation is divided to the following equations

rd=vb(v−1) (1)

rv=(Pm−Pe−D(v−1))

M(2)

where, Pm and Pe are the input and output powers ofthe generator respectively; M and D are the inertiaconstant and damping coefficient, respectively; d and v

are the rotor angle and speed, respectively; r is thederivative operator d/dt. The output power of thegenerator can be expressed in terms of the d-axis andq-axis components of the armature current, i, andterminal voltage, 6, as

Pe=ndid+nqiq (3)

The internal voltage, Eq% , equation is

rEq%=(Efd− (xd−xd% )id−Eq% )

Tdo%(4)

Here, Efd is the field voltage; Tdo% is the open circuitfield time constant; xd and xd% are d-axis reactance andd-axis transient reactance of the generator, respectively.

The IEEE Type-ST1 excitation system shown in Fig.2 is considered in this study. It can be described as

rEfd=(KA(Vref−n+uPSS)−Efd)

TA

(5)

where, KA and TA are the gain and time constant ofthe excitation system respectively; Vref is the referencevoltage. As shown in Fig. 2, a conventional lead–lagPSS is installed at the feedback loop to generate astabilizing signal. In Eq. (5), the terminal voltage 6 canbe expressed as

n= (nd2+nq

2)1/2 (6)

and;

nd=xqiq (7)

nq=Eq%−xd% id (8)

where xq is the q-axis reactance of the generator.Fig. 3 illustrates the block diagram of an SPS. The

phase shift angle of the SPS, F, is expressed as

rF=(Ks(Fref−uSPS)−F)

Ts

(9)Fig. 1. Single machine infinite bus system with an SPS.

M.A. Abido, Y.L. Abdel-Magid / Electric Power Systems Research 52 (1999) 133–143 135

Fig. 2. IEEE type-ST1 excitation system with conventional lead–lag PSS.

Fig. 3. SPS with conventional lead–lag controller.

where, Fref is the reference angle; Ks and Ts are the gainand time constant of the SPS. As shown in Fig. 3, aconventional lead–lag controller is installed at the feed-back loop to generate the SPS stabilizing signal uSPS.The expressions of d-axis and q-axis components of thearmature current, id and iq, are given in Appendix A.

In the design of PSS and SPS damping controller, thelinearized incremental model around an equilibriumpoint is usually employed [1–3]. Linearizing the expres-sions of id and iq, see Appendix A, and substituting intothe linear form of Eqs. (1)–(9) yield the followinglinearized power system model

rX=AX+BU (10)

Here, the state vector X is [Dd, Dv, DEq% , DEfd ]T and thecontrol vector U is [uPSS, DF]T.

3. Proposed control schemes

3.1. PSS and SPS controller structure

A widely used conventional lead–lag controller struc-ture for both excitation and SPS as shown in Figs. 2and 3 is considered in this study. In this structure, the

washout time constant Tw, the exponent p, and the timeconstants T2PSS and T2SPS are usually pre-specified. Thecontroller gains, KPSS and KSPS, and time constants,T1PSS and T1SPS, remain to be determined. The bestlocation of the SPS is at the generator terminal as itgives the greatest change of electrical distance betweenthe generator and the disturbance point [23]. Hence, thespeed deviation, Dv, is available and used as the inputsignal to both PSS and SPS controller. This makes theproposed controllers easy for on-line implementation.

3.2. Proposed control schemes

To investigate the ability of PSS and SPS controllerto damp out the low frequency oscillations associatedwith the electromechanical modes, three different con-trol schemes are proposed as follows.� Scheme (a) where PSS only is considered. In this

case, the tuning parameters are KPSS and T1PSS.� Scheme (b) where SPS controller only is considered.

In this case, the tuning parameters are KSPS andT1SPS.

� Scheme (c) where coordinated design of both PSSand SPS controller is considered. In this case, thetuning parameters are KPSS, T1PSS, KSPS, and T1SPS.


3.3. PSS and SPS controller design

As far as the electromechanical modes are con-cerned, the first step in the design process is to identifythe eigenvalues of the system matrix A associated withthese modes. Participation factors method [34] is usedfor this purpose. An eigenvalue-based objective func-tion J defined below is proposed.

J=min{z %s of the electromechanical modes} (11)

where z is the damping ratio. This objective function isproposed to improve the time domain control systemspecifications such as damping factor, overshoots, andsettling time.

The problem constraints are the optimized parame-ter bounds. Therefore, the design problem can be for-mulated as the following optimization problem.

Maximize J (12)

Subject to

KPSSmin5KPSS5KPSS

max (13)

T1PSSmin 5T1PSS5T1PSS

max (14)

KSPSmin5KSPS5KSPS

max (15)

T1SPSmin 5T1SPS5T1SPS

max (16)

The minimum and maximum values of the controllergains are set as 0.1 and 100, respectively [2]. Theminimum value of T1PSS is set slightly above the valueof T2PSS [2]. The maximum value of both T1PSS andT1SPS is set to 1.0 s.

The proposed approach employs TS algorithm tosolve this optimization problem and search for optimalor near optimal set of the optimized parameters.

4. Tabu search algorithm

TS is a higher level heuristic algorithm for solvingcombinatorial optimization problems. It is an iterativeimprovement procedure that starts from any initialsolution and attempts to determine a better solution.TS was proposed in its present form a few years agoby Glover [27–30]. It has now become an establishedoptimization approach that is rapidly spreading tomany new fields. Together with other heuristic searchalgorithms, TS has been singled out as ‘extremelypromising’ for the future treatment of practical appli-cations [24]. Generally, TS is characterized by its abil-ity to avoid etrapment in local optimal solution andprevent cycling by using flexible memory of searchhistory.

The basic elements of TS are briefly stated anddefined as follows:

� Current solution, xcurrent,: it is a set of the optimizedparameter values at any iteration. It plays a centralrole in generating the neighbor trial solutions.

� Moves: they characterize the process of generatingtrial solutions that are related to xcurrent.

� Set of candidate moves, N(xcurrent),: it is the set of allpossible moves or trial solutions, xtrials, in the neigh-borhood of xcurrent. In case of continuous variableoptimization problems, this set is too large or eveninfinite set. Therefore, one could operate with asubset, S(xcurrent) with a limited number of trialsolutions nt, of this set, i.e. S¦N andxtrial�S(xcurrent).

� Tabu restrictions: these are certain conditions im-posed on moves that make some of them forbidden.These forbidden moves are listed to a certain sizeand known as tabu. This list is called the tabu list.The reason behind classifying a certain move asforbidden is basically to prevent cycling and avoidreturning to the local optimum just visited. The tabulist size plays a great role in the search of highquality solutions. The way to identify a good tabulist size, is simply watch for the occurrence of cyclingwhen the size is too small and the deterioration insolution quality when the size is too large caused byforbidding too many moves. In some applications asimple choice of the tabu list size in a range centeredat 7 seems to be quite effective [29]. Generally, thetabu list size should grow with the size of the givenproblem.

� Aspiration criterion (level): it is a rule that overridetabu restrictions, i.e. if a certain move is forbiddenby tabu restriction, the aspiration criterion, whensatisfied, can make this move allowable. Differentforms of aspiration criteria are used in the literature[26–30]. The one considered here is to override thetabu status of a move if this move yields a solutionwhich has better objective function, J, than the oneobtained earlier with the same move. The impor-tance of using aspiration criterion is to add someflexibility in the TS by directing it towards theattractive moves.

� Stopping criteria: these are the conditions underwhich the search process will terminate. In thisstudy, the search will terminate if one of the follow-ing criteria is satisfied: (a) the number of iterationssince the last change of the best solution is greaterthan a pre-specified number; or, (b) the number ofiterations reaches the maximum allowable number.The general algorithm of TS can be described in steps

as follows:Step 1: set the iteration counter k=0 and randomlygenerate an initial solution xinitial. Set this solution asthe current solution as well as the best solution, xbest,i.e. xinitial=xcurrent=xbest.


Table 1The optimal settings of the controller parameters of the proposed schemes

Prop. Scheme (c)Prop. Scheme (a) Prop. Scheme (b)

KSPS T1SPSKPSS KPSST1PSS T1PSS KSPS T1SPS

0.27218.015 95.376 0.065 18.160 0.117 65.965 0.118

Fig. 4. Objective function variations of the proposed schemes.

Step 2: randomly generate a set of trial solutions xtrialsin the neighborhood of the current solution, i.e. createS(xcurrent). Sort the elements of S based on theirobjective function values in descending order as theoptimization problem is a maximization one. Let usdefine x trial

i as the i th trial solution in the sorted set,15 i5nt. Here, x trial

1 represents the best trial solutionin S in terms of objective function value associ-atedwith it.Step 3: set i=1. If J(x trial

i )\J(xbest) go to step 4, elseset xbest=x trial

i and go to step 4.Step 4: check the tabu status of x trial

i . If it is not in thetabu list then put it in the tabu list, set xcurrent=x trial

i ,and go to step 7. If it is in tabu list go to step 5.Step 5: check the aspiration criterion of x trial

i . If sat-isfied then override the tabu restrictions, update theaspiration level, set xcurrent=x trial

i , and go to step 7.If not set i= i+1 and go to step 6.Step 6: if i\nt go to step 7, else go back to step 4.Step 7: check the stopping criteria. If one of them issatisfied then stop, else set k=k+1 and go back tostep 2.

5. Tabu search for controller design

TS algorithm has been applied to search for optimalsettings of the optimized parameters. In our implementa-

tion, the tabu list size 7 and number of trial solutions 15are found to be quite satisfactory. In addition, the searchwill terminate if the best solution does not change formore than 20 iterations or the number of iterationsreaches 200. The final settings of the optimized parame-ters for the proposed schemes are given in Table 1. Theconvergence rate of the objective function J with thenumber of iterations is shown in Fig. 4.

6. Simulation results

To assess the effectiveness and robustness of theproposed control schemes, three different loading condi-tions given in Table 2 were considered. Moreover,different disturbances and system parameter variationswere applied. For comparison purposes, Tw, T2PSS, andT2SPS are set to be 5, 0.1, and 0.1 s, respectively and one

Table 2Loading conditions

F (Deg.)6 (pu)Q (pu)P (pu)Loading

0.01.05Nominal 1.0 0.0151.05 0.0Leading PF 0.70 −0.20

0.40 1.05Heavy 5.01.10


Table 3System eigenvalues with and without control

PSS [1]No control Prop. scheme (a) Prop. Scheme (c)Prop. scheme (b)

−1.15794.397a −2.85095.309a+0.29594.960a −3.16193.727a −5.03594.243a

−10.39393.284 −4.60297.408 −3.11795.796 −6.49497.795 −6.69795.060– −13.36191.791−0.209, −18.676−0.204, −18.257−0.201, −18.677– – −0.200, −10.000−12.201–

–0.211–– ––

a Eigenvalues associated with the electromechanical mode.

Fig. 5. d response to the fault test with nominal loading.

lead–lag block is considered i.e. p=1. Theperformance of the proposed control schemes iscompared to that of PSS given in Ref. [1] with atransfer function given by

U=7.091� 5s

1+5s��1+0.685s

1+0.1s�Dv (17)

It is worth mentioning that all the time domainsimulations were carried out using the nonlinear powersystem model. The system data is given in Appendix B.

6.1. Nominal loading

At this loading condition, the system eigenvalueswith and without the proposed control schemes aregiven in Table 3. It is shown that the open loop systemis unstable and the PSS [1] stabilizes the system with adamping ratio z of 0.254. The corresponding dampingratios of the proposed schemes (a), (b), and (c) are0.473, 0.647, and 0.765, respectively. It is quite clearthat the proposed control schemes enhance greatly thedamping of electromechanical mode of oscillation.However, better damping characteristics can beachieved with proper coordinated design of PSS andSPS controller as shown in proposed scheme (c).

Two different disturbances were applied to assess theeffectiveness of the proposed control schemes toenhance system damping. These disturbances are asfollows.

6.1.1. Fault testThe behavior of the proposed control schemes under

transient conditions was verified by applying a threephase fault at the infinite bus at t=1 s. The faultduration was 0.1 s. The system response is shown inFigs. 5 and 6. It can be seen that the first swing in thetorque angle is significantly suppressed with theproposed schemes (b) and (c). This means that SPSoutperforms PSS in damping of low frequencyoscillations and increasing of stability margin. Theresults also show that the proposed control schemesimprove greatly the system settling time and voltageprofile.

The effectiveness of the proposed control schemes toincrease the critical clearing time (CCT) of the fault isalso investigated. With PSS [1], it is found that CCT is0.136 s. With the proposed scheme (a), CCT is slightlyimproved to be 0.141 s. However, the significantimprovement has been achieved with applying SPS inthe proposed schemes (b) and (c). With these schemes,


the CCT’s are found to be 0.173 and 0.186 s, respec-tively.

6.1.2. Parameter 6ariation testTo verify the robustness of the proposed schemes to

system parameter variation, the system inertia has beenreduced by 20% of its nominal value while a 20% pulseincrease in the input torque has been applied fromt=1.0 s to t=6.0 s. The system response is shownin Figs. 7 and 8. The simulation results show thatthe system damping is improved with the proposedschemes.

6.2. Leading PF loading

It may become necessary to operate the generator ata leading power factor. In this case, the stability margin

is reduced and it becomes important to test the proposedcontrol schemes under this difficult situation. A threephase fault test has been applied at the infinite bus for0.1 s. The system response is shown in Figs. 9 and 10.It is shown that the system damping characteristics aresignificantly enhanced with the proposed controlschemes (b) and (c).

6.3. Hea6y loading

6.3.1. Fault testA three phase fault disturbance at the infinite bus for

0.05 s was applied. The results are shown in Figs. 11 and12. It can be seen that the proposed schemes (b) and (c)suppress the first swing in torque angle and extend thesystem stability limit. In addition, the voltage profile isgreatly improved with these control schemes.

Fig. 6. Heat-power feasible region for the co-gen. unit 3.

Fig. 7. d response to the parameter variation test with nominal loading.


Fig. 8. Terminal voltage response to the parameter variation test with nominal loading.

Fig. 9. d response to the fault test test with leading PF loading..

Fig. 10. Terminal voltage response to the fault test with leading PF loading.


Fig. 11. d response to the fault test with heavy loading.

Fig. 12. Terminal voltage response to the fault test with heavy loading.

Fig. 13. d response to the Vref pulse test with heavy loading.


Fig. 14. Terminal voltage response to the Vref pulse test with heavy loading.

6.3.2. Vref pulse testA 5% pulse decrease in reference voltage disturbance

was also applied. The disturbance duration is 5 s. Thesimulation results are shown in Figs. 13 and 14. It isshown that the system is unstable with the conventionalPSS [1]. It is quite clear that the proposed control schemesprovide good damping characteristics to low frequencyoscillations.

7. Conclusions

In this study, power system stability enhancementthrough excitation and SPS control has been investi-gated. The controller design problem is formulated as anoptimization problem. Then, TS algorithm is proposedto search for optimal settings of controller parameters.Three different control schemes have been proposed andapplied to a weakly connected power system. The pro-posed schemes were tested under different disturbances,loading conditions, and system parameter variations.The simulation results show that1. the potential of TS algorithm to solve the problem of

PSS and SPS controller design;2. the SPS controller outperforms the PSS in damping

of low frequency oscillations and improves the voltageprofile;

3. better damping characteristics can be obtained bycoordinated control of excitation and SPS;

4. the effectiveness and robustness of the proposedcontrol schemes over a wide range of loading condi-tions and system parameter variations.

Acknowledgements

The author would like to acknowledge the support of

King Fahd University of Petroleum and Minerals, SaudiArabia. The author would also like to acknowledgeMenoufia University, Shebin El-Kom, Egypt.

Appendix A

Referring to Fig. 1, the voltage 6%=6Ú−F and its dand q components can be written as

6d%=Eq% sin F−xd% id sin F+xqiq cos F (A.1)

6d%=Eq% cos F−xd% id cos F+xqiq sin F (A.2)

The load current iL=6%YL and the line current iline= i−iL. The infinite bus voltage 6b=6%− ilineZ. The compo-nents of 6b can be written as

6b sin d=c16d%−c26q%−Rid+Xiq (A.3)

6b cos d=c26d%−c16q%−Xid+Riq (A.4)

where, c1=1+Rg−Xb ; and c2=Rb+Xg. Substitutingfrom Eqs. (A.1) and (A.2) into Eqs. (A.3) and (A.4), thefollowing two equations can be obtained

c5id+c6iq=6b sin d+c3Eq% (A.5)

c7id+c8iq=6b cos d−c4Eq% (A.6)

where, c3=c2 cos F−c1 sin F; c4=c1 cos F+c2 sin F;c5=xd% c3−R ; c6=xqc4+X ; c7= −xd% c4−X ; and c8=xqc3−R. Solving Eqs. (A.5) and (A.6) simultaneously, idand iq expressions can be obtained. Linearizing Eqs. (A.5)and (A.6), Did and Diq can be expressed in terms of, Dd,DEq% , and DF.

Appendix B

The system data are as follows.M=9.26 s; Tdo=7.76;D=0.0; xd=0.973; xd%=0.19; xq=0.55; R= −0.034;


X=0.997; g=0.249; b=0.262; KA=50; TA=0.05;Ks=1.0; Ts=0.05; �F�510°; �uPSS�50.2 pu; �Efd �57.3pu.

All resistances and reactances are in pu and timeconstants are in seconds.

With the nominal loading condition given in Table2, the system matrices are

A=ÃÃ

Ã

Æ

È

0.0−0.0588−0.090095.5320

3770.00.00.0

0.0−0.1303−0.1957−815.93

0.00.0

0.1289−20.00

ÃÃ

Ã

Ç

É

, and

B=ÃÃ

Ã

Æ

È

0.00.00.0

1000

0.00.07750.0185105.66

ÃÃ

Ã

Ç

É

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a tabu search based approach to power system stability enhancement via excitation and static phase...

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