a novel approach to conventional power system stabilizer design using tabu search

A novel approach to conventional power system stabilizer design usingtabu search

M.A. Abido1,*

Electrical Engineering Department, King Fahd University of Petroleum and Minerals, KFUPM Box 183, Dhahran 31261, Saudi Arabia

Received 18 June 1998; received in revised form 29 October 1998; accepted 17 December 1998

Abstract

A tabu search (TS) based power system stabilizer (PSS) is presented in this article. The proposed approach uses the TS algorithm to searchfor the optimal settings of conventional lead-lag power system stabilizer (CPSS) parameters. Incorporation of the TS algorithm in the PSSdesign significantly reduces the time consumed in the design process. One of the main advantages of the proposed approach is the fact that theTS algorithm leads to the optimal solution regardless of the initial guess. The performance of the proposed PSS under different disturbancesand loading conditions is investigated for single machine infinite bus and multimachine power systems. The eigenvalue analysis andsimulation results show the effectiveness of the proposed PSSs to damp out the local and the interarea modes of oscillations and workeffectively over a wide range of loading conditions and system parameter variations.q 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Conventional PSS; Tabu search; Dynamic stability; Low frequency oscillations

1. Introduction

Power systems are experiencing low frequency oscilla-tions due to disturbances. The oscillations may sustain andgrow to cause system separation if no adequate damping isavailable. To enhance system damping, the generators areequipped with power system stabilizers (PSSs) that providesupplementary feedback stabilizing signals in the excitationsystems. These stabilizers extend the power system stabilitylimit by enhancing the system damping of low frequencyoscillations associated with the electromechanical modes[1,2].

DeMello and Concordia in 1969 [3] presented theconcepts of synchronous machine stability as affected byexcitation control. They established an understanding ofthe stabilizing requirements for static excitation systems.In recent years, several approaches based on the moderncontrol theory have been applied to PSS design problem.These include optimal control, adaptive control, variablestructure control, and intelligent control [4–7].

Despite the potential of modern control techniques withdifferent structures, power system utilities still prefer aconventional lead-lag power system stabilizer (CPSS) struc-ture [8,9]. The reasons behind that might be the ease of on-

line tuning and the lack of assurance of the stability relatedto some adaptive or variable structure techniques. Kundur etal. [10] have presented a comprehensive analysis of theeffects of the different CPSS parameters on the overalldynamic performance of the power system. It is shownthat the appropriate selection of CPSS parameters resultsin satisfactory performance during system upsets.

Many different techniques has been reported in the litera-ture pertaining to optimum location and coordinated designproblems of CPSSs. Generally, most of these techniques arebased on phase compensation and eigenvalue assignment[8–18]. Different techniques of sequential design of PSSsare presented [11–13] to damp out one of the electromecha-nical modes at a time. Generally, the dynamic interactioneffects among various modes of the machines are found tohave significant influence on the stabilizer settings. There-fore, considering the application of stabilizer to onemachine at a time may not finally lead to an overall optimalchoice of PSS parameters. Moreover, the stabilizersdesigned to damp one mode can produce adverse effectsin other modes. In addition, the optimal sequence of designis a very involved question. The sequential design of PSSs isavoided in [14–18] where various methods for simultaneoustuning of PSSs in multimachine power systems areproposed. Unfortunately, the proposed techniques are itera-tive and require heavy computation burden due to systemreduction procedure. This gives rise to time consumingcomputer codes. In addition, the initialization step of these

Electrical Power and Energy Systems 21 (1999) 443–454

0142-0615/99/$ - see front matterq 1999 Elsevier Science Ltd. All rights reserved.PII: S0142-0615(99)00004-6

* Tel.: 1966-3-8604379; fax:1966-3-8603535E-mail address:[email protected] (M.A. Abido)

1 On Leave from EE Department, Menoufia University, Egypt.

www.elsevier.com/locate/ijepes

algorithms is crucial and affects the final dynamic responseof the controlled system. Hence, different designs assigningthe same set of eigenvalues were simply obtained by usingdifferent initializations. Therefore, a final selection criterionis required to avoid long runs of validation tests on thenonlinear model.

Recently, H∞ based techniques and sequential loopclosure method [19,20] have been applied to PSS designproblem. However, the importance and difficulties in theselection of weighting functions ofH∞ optimization

problem have been reported. In addition, the order of theH∞ based stabilizer is as high as that of the plant. This givesrise to complex structure of such stabilizers and reducestheir applicability. Although the sequential loop closuremethod is well suited for on-line tuning, there is no analy-tical tool to decide the optimal sequence of the loop closure.

In most recent articles [21,22], genetic algorithm (GA), asa heuristic search algorithm, has been applied to theproblem of PSS design. Various eigenvalue based objectivefunctions were optimized to select the optimal set of CPSS

M.A. Abido / Electrical Power and Energy Systems 21 (1999) 443–454444

Fig. 1. The computational flow chart of TS algorithm.

parameters. The results are promising and confirming thepotential of GA to search for the optimal settings of CPSSparameters. However, GA is characterized as memorylesssearch algorithm. Generally, using a flexible memory ofsearch history can significantly enhance the search process.

In the last few years, tabu search (TS) algorithm [23–27]appeared as another promising heuristic algorithm for hand-ling the combinatorial optimization problems. TS algorithmuses a flexible memory of search history to prevent cyclingand to avoid entrapment in local optima. It has been shownthat, under certain conditions, the TS algorithm can yieldglobal optimal solution with probability one [26]. In powersystems, TS was applied to a number of power system opti-mization problems with impressive successes [28–30].However, to the best of the author’s knowledge, the poten-tial of the TS algorithm to the PSS design problem has notyet been exploited.

In this article, a novel approach to the PSS designproblem is developed using the TS algorithm. The problemof CPSS design is formulated as optimization problem.Then, the TS algorithm is employed to solve this optimiza-tion problem with the aim of getting the optimal or nearoptimal settings of the PSS parameters. The proposed designapproach was applied to both a single machine infinite bussystem and a multimachine power system. The eigenvalueanalysis and simulation results were carried out to assess theeffectiveness of the proposed PSS to damp out the electro-mechanical modes of oscillations and enhance the dynamicstability of power systems.

2. Problem statement

2.1. PSS structure

To enhance the power system dynamic stability andextend the power system stability limit, synchronous

generators are equipped with PSSs. A widely used CPSSis considered in this study. It can be described as [1]

Ui � KcisTw

1 1 sTw

�1 1 sT1i��1 1 sT2�

�1 1 sT3i��1 1 sT4� Dvi ; �1�

whereTw is the washout time constant,Ui the PSS outputsignal at theith machine, andDv t the speed deviation of thismachine. The time constantsTw, T2 and T4 are usuallyprespecified. The stabilizer gainKci and time constantsT1i

andT3i remain to be determined.

2.2. Power system model

A power system can be modeled by a set of nonlineardifferential equations as:

X � f �X;U�; �2�whereX is the vector of the state variables andU is thevector of input variables. In this study,X ��d;v;E0q;Efd�T andU is the PSS output signals.

In the design of PSSs, the linearized incremental modelsaround an equilibrium point are usually employed [1–3].Therefore, the state equation of a power system withnmachines andm stabilizers can be written as:

D X � A DX 1 BU; �3�whereA is 4n × 4n matrix and equals2f/2X, while B is 4n ×mmatrix and equals2f/2U. BothA andB are evaluated at theequilibrium point.DX is 4n × 1 state vector whileU is m× 1input vector.

2.3. Identification of electromechanical modes and psslocations

The state equation of linearized model (3) of undrivensystem can be rewritten as:

D X � A DX:

M.A. Abido / Electrical Power and Energy Systems 21 (1999) 443–454 445

Fig. 2. Objective function variations of example 1.

Then the 4n eigenvalues of the system matrixA can bedetermined. Out of these eigenvalues, there aren 2 1 modesof oscillations related to machine inertias [31].

For the stabilizers to be effective, it is extremely impor-tant to identify the eigenvalues associated with electro-mechanical modes and the machines to which theseeigenvalues belong [18]. Participation factors method [32]and sensitivity of PSS effect (SPE) method [33] are used forthis purpose.

2.4. Objective function and PSS tuning

To increase the system damping to electromechanicalmodes, an objective functionJ defined below is proposed.

J � max {Re�li�; i [ set of electromechanical modes};

�5�wherel i is theith eigenvalue associated with electromecha-nical modes. This objective function is proposed to shiftthese eigenvalues to the left ofs-plane in order to improve

the damping factor and insure some degree of relative stabi-lity.

The problem constraints are the parameter bounds. There-fore, the design problem can be formulated as the followingoptimization problem.

minimize J �6�

subject to Kminci # Kci # Kmax

ci ; �7�

Tmin1i # T1i # Tmax

1i ; �8�

Tmin3i # T3i # Tmax

3i : �9�The minimum and maximum values of the stabilizer gain

are set as 0.1 and 100, respectively [2]. The minimum valuesof T1 andT3 are set slightly aboveT2 andT4 values, respec-tively, while their maximum values are set up to 20 times ofT2 and T4 values, respectively [2]. It is obvious that theproblem constraints are quite mild.

The proposed approach employs the TS algorithm to


Fig. 3. System response of example 1 with fault test.

solve this optimization problem and search for optimal ornear optimal set of PSS parameters, {Kci ;T1i ;T3i ; i �1;2;…;m} :

3. Tabu search algorithm

TS is a higher level heuristic algorithm for solving combi-natorial optimization problems. It is an iterative improve-ment procedure that starts from any initial solution andattempts to determine a better solution. TS was proposedin its present form a few years ago by Glover [24–27]. It has

now become an established optimization approach that israpidly spreading to many new fields. Together with otherheuristic search algorithms such as GA, TS is singled out as‘‘extremely promising’’ for the future treatment of practicalapplications [24]. Generally, TS is characterized by its abil-ity to avoid entrapment in local optimal solution and preventcycling by using flexible memory of search history.

The basic elements of TS are briefly stated and defined asfollows:-

Current solution, xcurrent

It is a set of the optimized parameter values at any iteration.It plays a central role in generating the neighbor trial solutions.

MovesThey characterize the process of generating trial solutionsthat are related toxcurrent.

Set of candidate moves, N(xcurrent)It is a set of all possible moves or trial solutions,xtrials, in the


Table 1System eigenvalues of example 1

No PSS PSS [1] Proposed PSS

1 0.295^ j4.960 2 1.157^ j4.397 2 2.986^ j5.4362 10.393^ j3.284 2 4.602^ j7.408 2 2.988^ j5.655— 2 0.201, 2 18.677 2 0.204, 2 18.243

Fig. 4. System response of example 1 with parameter change test.

neighborhood ofxcurrent. In the case of continuous variableoptimization problems, this set is too large or even infiniteset. Therefore, one could operate with a subset,S(xcurrent)with a limited number of trial solutionsnt, of this set, i.e.S , N andxtrial [ S(xcurrent).

Tabu restrictionsThese are certain conditions imposed on moves that makesome of them forbidden. These forbidden moves are listedto a certain size and known as tabu. This list is called thetabu list. The reason behind classifying a certain move asforbidden is basically to prevent cycling and avoid returningto the local optimum just visited. The tabu list size plays agreat role in the search of high quality solutions. The way toidentify a good tabu list size, is simply watch for the occur-rence of cycling when the size is too small and the deteriora-tion in solution quality when the size is too large caused byforbidding too many moves. In some applications a simplechoice of the tabu list size in a range centered at seven seemsto be quite effective [25]. Generally, the tabu list size shouldgrow with the size of the given problem. In our implemen-tation, the size seven is found to be quite satisfactory.

Aspiration criterion (Level)It is a rule that override tabu restrictions, i.e. if a certainmove is forbidden by tabu restriction, the aspiration criter-ion, when satisfied, can make this move allowable. Different

forms of aspiration criteria are used in the literature [23–27]. The one considered here is to override the tabu status ofa move if this move yields a solution which has betterobjective function,J, than the one obtained earlier withthe same move. The importance of using aspiration criterionis to add some flexibility in the TS by directing it towardsthe attractive moves.

Stopping criteria These are the conditions under whichthe search process will terminate. In this study, the searchwill terminate if one of the following criteria is satisfied: (a)the number of iterations as the last change of the best solu-tion is greater than a prespecified number; or, (b) the numberof iterations reaches the maximum allowable number. In ourimplementation, the search will terminate if the best solu-tion does not change for more than 20 iterations or thenumber of iterations reaches 200.

The general algorithm of TS can be described in steps asfollows:

Step 1 Set the iteration counterk� 0 and randomly gener-ate an initial solutionxinitial. Set this solution as thecurrent solution as well as the best solution,xbest,i.e. xinitial � xcurrent� xbest.

Step 2 Randomly generate a set of trial solutionsxtrials inthe neighborhood of the current solution, i.e. create


Fig. 5. Single line diagram for New England system.

S(xcurrent). Sort the elements ofS based on theirobjective function values in ascending order asthe problem is a minimization one. Let us definexi

trial as theith trial solution in the sorted set, 1#i # nt. Here,xtrial

i represents the best trial solutionin S in terms of objective function value associatedwith it.

Step 3 Seti � 1. If J�xitrial� . J�xbest� go to Step 4, else set

xbest� xitrial and go to Step 4.

Step 4 Check the tabu status ofxitrial: If it is not in the tabu

list then put it in the tabu list, setxcurrent� xitrial; and

go to Step 7. If it is in tabu list go to Step 5.Step 5 Check the aspiration criterion ofxi

trial: If satisfiedthen override the tabu restrictions, update theaspiration level, setxcurrent� xi

trial; and go to Step7. If not seti � i 1 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 is satis-

fied then stop, else setk � k 1 1 and go back toStep 2.

For further illustration, the computational flow chart ofthe general TS algorithm is shown in Fig. 1.

4. Example 1: single machine system

4.1. PSS design

To evaluate the effectiveness of the proposed approach to

PSSs design, a single machine infinite bus system is consid-ered first. The detailed system model and parameters aregiven in [1]. For comparison with the CPSS reported in[1], Tw and T2 are set to be 5 and 0.1 s, respectively andone lead-lag block is considered. Therefore the optimizedparameters areKc andT1.

TS algorithm has been applied to search for optimalsettings of the optimized parameters. The number of trialsolutions is selected to be 15. The final values of the opti-mized parameters areKc � 18.247 andT1 � 0.2697 s. Theconvergence rate of the objective functionJ with thenumber of iterations is shown in Fig. 2.

4.2. Simulation results

For comparison purposes, the system operating point isspecified asP� 1.0 pu,Q� 0.015 pu, andV� 1.05 pu [1].The performance of the proposed PSS under different distur-bances is compared to that of [1] with a transfer functiongiven by

U � 7:0915 s

1 1 5 s

� �1 1 0:685 s1 1 0:1 s

� �Dv: �10�


Fig. 6. Objective function variations of example 2.

Table 2The optimal values of the proposed PSS parameters for example 2

PSS Location Kc T1 T3

G5 95.18 0.099 0.187G7 47.35 0.335 0.101G9 94.17 0.247 0.254

Table 3System eigenvalues without PSSs of example 2

Point#1 Point#2 Point#3

2 0.150^ j9.735 2 0.047^ j9.954 2 0.169^ j9.7072 0.097^ j9.733 2 0.127^ j9.986 2 0.028^ j9.6672 0.160^ j9.392 2 0.187^ j9.622 2 0.269^ j9.3452 0.106^ j8.256 2 0.165^ j8.247 2 0.198^ j8.1252 0.160^ j8.070 2 0.099^ j8.070 2 0.090^ j8.1162 0.105^ j7.400 2 0.024^ j7.225 2 0.050^ j7.1692 0.121^ j6.766 2 0.120^ j6.560 2 0.030^ j6.4612 0.049^ j6.237 1 0.071^ j6.083a 1 0.187^ j5.829a

2 0.034^ j4.334 2 0.024^ j4.223 1 0.083^ j3.990a

a Unstable modes.

The system eigenvalues with and without PSS are givenin Table 1. It is shown that the proposed PSS enhancesgreatly the damping of electromechanical mode ofoscillation.

Two different disturbances are applied to assess the

effectiveness of the proposed PSS to enhance systemdamping. These disturbances are three phase fault andparameter change tests. It is worth mentioning that thenonlinear system model is used to carry out thesimulations.


Fig. 7. System responses of example 2 with the operating point#1.

4.2.1. Fault testThe behavior of the proposed PSS under transient condi-

tions was verified by applying a three phase fault at theinfinite bus whent � 1 s. The fault duration was 0.1 s.The system response is shown in Fig. 3. It can be seen

that the proposed PSS damps out the low frequency oscilla-tions very quickly and improves the system settling time.

4.2.2. Parameter change testTo verify the robustness of the proposed PSS, the system


Fig. 8. System responses of example 2 with the operating point#3.

inertia was reduced by 20% of its nominal value while a20% pulse in the input torque has been applied fromt �1.0 s tot � 6.0 s. The system response is shown in Fig. 4.It can be seen that the first swing in the torque angle issignificantly suppressed. While the maximum overshoot is19% with the PSS given in [1], its value is reduced to 11%with the proposed PSS. This extends the stability marginand improves the disturbance tolerance ability of thesystem.

5. Example 2: multimachine power system

5.1. Test system and optimum PSS locations

To assess the effectiveness of the proposed PSS toimprove the stability of multimachine power systems, theten-machine 39-bus New England power system shown inFig. 5 is considered. Each machine is represented by a fourth

order nonlinear model. Generator G1 is an equivalent powersource representing parts of the US–Canadian inter-connection system. Details of the system data are given in[7]. The participation factor method [32] and the SPEmethod [33] were used to identify the optimum locationsof PSSs. The results of both the methods indicate that thegenerators G5, G7, and G9 are the optimum locations forinstalling PSSs.

5.2. PSS design

In this example, the optimized parameters areKci, T1i andT3i, i � 1,2, and 3.Tw, T2, andT4 are set to be 10, 0.05, and0.05 s, respectively. TS algorithm is applied to search foroptimal settings of these parameters. The number of trialsolutions is selected to be 30. The final values of the opti-mized parameters are given in Table 2. The convergencerate of the objective functionJ with the number of iterationsis shown in Fig. 6.


Fig. 9. Objective function values of example 1 with different initializations.

Fig. 10. Stabilizer gainKc values of example 1 with different initializations.

5.3. Simulation results

To demonstrate the effectiveness of the proposed PSSsover a wide range of loading conditions, three differentoperating conditions are considered in this study. The elec-tromechanical modes without PSSs are given in Table 3. Itis clear that these modes are poorly damped and some ofthem are unstable. The electromechanical modes with theproposed PSSs are given in Table 4. It is obvious that thesystem damping to the electromechanical modes is greatlyimproved.

For further illustration, a six-cycle three-phase faultdisturbance at bus 29 at the end of line 26–29 is consideredfor the time simulations. The performance of the proposedPSSs is compared to that of [7]. The speed deviations of G5,G7, and G9 are shown in Figs. 7 and 8 with the operatingpoints 1 and 3, respectively. It is clear that the systemperformance with the proposed PSSs is much better andthe oscillations are damped out much faster. This illustratesthe superiority of the proposed PSS design approach to getan optimal or near optimal set of PSS parameters. In addi-tion, the proposed PSSs are quite efficient to damp out thelocal modes as well as the interarea modes of oscillations.

6. Discussion

Some comments on the proposed approach are now in thefollowing order:

1. Unlike the methods of [14–18], the proposed TS basedapproach does not rely on the initial solution. Startinganywhere in the search space, TS algorithm ensures theconvergence to the global minima. In this study, differentinitial solutions are considered by changing the speed ofthe random number generator that generates the initialsolution. The convergence of the objective function andthe optimized parameters of example 1 are shown inFigs. 9–11. The results shown in Fig. 8 emphasize thatthe proposed approach finally leads to the optimal solu-tion regardless of the initial one.

2. Based on the aforementioned conclusion, the proposedapproach can be used to improve the solution quality ofother methods described in [8–18].

3. Exact assigning of the dominant electromechanical modecan be carried out using the proposed approach by insert-ing another stopping criterion during the search process.This stopping criterion will terminate the search if thereal part of the dominant mode is less than or equal acertain chosen negative number.

7. Conclusions

In this study, the TS algorithm is proposed to the PSSdesign problem. The proposed design approach employs theTS to search for optimal settings of the CPSS parameters.The proposed design approach is applied to a singlemachine infinite bus system and a multimachine powersystem with different disturbances, parameter variations,and loading conditions. The main features of the proposedapproach can be summarized as:

1. The proposed PSSs are of decentralized nature as only


Table 4System eigenvalues with the proposed PSSs of example 2

Point#1 Point#2 Point#3

2 0.589^ j13.77 2 0.553^ j13.85 2 0.559^ j13.952 0.150^ j9.735 2 0.126^ j9.987 2 0.167^ j9.7072 0.212^ j9.060 2 0.276^ j9.061 2 0.308^ j8.9342 0.160^ j8.246 2 0.177^ j8.254 2 0.180^ j8.1142 0.164^ j8.077 2 0.200^ j8.130 2 0.202^ j8.1272 0.223^ j7.521 2 0.294^ j7.511 2 0.282^ j7.3832 0.438^ j5.541 2 0.374^ j5.868 2 0.443^ j5.4262 2.100^ j2.801 2 2.082^ j2.505 2 1.943^ j1.9552 1.182^ j1.655 2 0.799^ j1.880 2 0.702^ j2.108

Fig. 11. Time constantT1 values of example 1 with different initializations.

local measurements are employed as the stabilizer inputs.This makes the proposed PSS easy to tune and install.

2. All PSSs are designed simultaneously taking into consid-eration the interaction among them.

3. The solution quality of the proposed approach is inde-pendent of the initialization step.

4. As eigenvector calculations and sensitivity analysis arenot required to evaluate the proposed objective function,heavy computations of the design process are avoided.

5. The eigenvalue analysis reveals the effectiveness of theproposed TS based PSS to damp out local as well asinterarea modes of oscillations.

6. The simulation results show that the proposed TS basedPSSs can work properly over a wide range of loadingconditions and system parameter variations.

Acknowledgements

The author would like to acknowledge the support ofKing Fahd University of Petroleum and Minerals, SaudiArabia and the Menoufia University, Shebin El-Kom,Egypt.

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a novel approach to conventional power system stabilizer design using tabu search

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