Electric Power Systems Research 60 (2001) 77–83

A fuzzy logic-based self tuning power system stabilizer optimizedwith a genetic algorithm

J. Lu, M.H. Nehrir *, D.A. PierreElectrical and Computer Engineering Department, Montana State Uni�ersity, 610 Cobleigh Hall, Bozeman, MT 59717-0378, USA

Received 6 October 2000; accepted 27 August 2001

Abstract

This paper presents an approach for designing power system stabilizers (PSS) with a fuzzy logic based parameter tuner. In theinitial design step, Prony analysis is used to identify linear models for the synchronous generator at a large number of operatingpoints, consisting of various power outputs and machine terminal voltages. Next, optimal parameter settings for a conventionalPSS are generated using the linearized models. From the operating point settings, a selection of fuzzy rules is used to tune thestabilizer parameters online according to real-time measurements. The membership functions of the fuzzy parameter tuner areoptimized using a genetic algorithm (GA). Simulation studies show that the proposed stabilizer performs well over a wide rangeof operating conditions and provides better dynamic performance than a fixed parameter PSS. © 2001 Elsevier Science B.V. Allrights reserved.

Keywords: Power system stabilizer; Fuzzy logic; Prony analysis; Genetic algorithms

www.elsevier.com/locate/epsr

1. Introduction

Power systems are complex non-linear systems andoften exhibit low frequency electro-mechanical oscilla-tions due to insufficient damping caused by adverseoperating conditions. Power system stabilizers (PSS) arewidely used to suppress these oscillations and enhancethe overall stability of power systems. Conventional(fixed parameter) stabilizers, consisting of cascade con-nected lead–lag compensators derived from a linearmodel representing the generator at a certain operatingpoint have long been used to damp the oscillations.Various methods have been suggested for both obtain-ing the linear models and designing stabilizers. How-ever, the linear control strategies often do not providesatisfactory results over a wide range of operatingconditions. To overcome this drawback, several othercategories of stabilizers have been proposed in theliterature. In the past decade, there has been noticeableresearch on the application of intelligent fuzzy logic-based PSS [1–5] and its optimization to adapt tochanging system operating conditions [6,7].

In this paper, a fuzzy logic-based control strategy toimplement a self-tuning PSS is introduced, where thePSS parameters are adjusted by a fuzzy parametertuner according to on-line measurements. The underly-ing idea is as follows: since we can design a linearstabilizer, which has satisfactory performance at a par-ticular operating point, we can do this at many differ-ent operating points, it is then possible to develop asynthetic scheme incorporating these individual stabiliz-ers which can respond to various operating conditionswith excellent performance. This is done by tuning theparameters of the stabilizer on-line according to theknowledge of the individual stabilizers.

Unlike conventional fuzzy logic applications, whererules are generated based on operator’s experience orgeneral knowledge of the system in a heuristic way, inthis application an optimization technique is used forselecting both individual rules and membership func-tions. Therefore, a sound knowledge base can be guar-anteed. The implementation includes three main parts,plant identification (i.e. obtaining linear models); indi-vidual stabilizer design; and membership function opti-mization. Identification involves obtaining linearmodels for the generator over a wide range of operating

* Corresponding author. Tel.: +1-406-994-4980; fax: +1-406-994-5958.

0378-7796/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0378-7796(01)00170-5

J. Lu et al. / Electric Power Systems Research 60 (2001) 77–8378

points. Prony-based analysis [8] is used to obtain suchlinear models. Among the operating points some, dis-tributed uniformly over a grid, are selected as ‘pivotpoints’ to design conventional stabilizers on, and theothers are used to optimize the membership functions.If the generator happens to operate at a pivot point, thePSS parameters are tuned exactly to the values ac-quired. Otherwise, the stabilizer parameter settings areobtained using knowledge of neighboring pivot pointsand fuzzy logic operations. The performance of such astabilizer at a non-pivot point depends on the member-ship functions, which determine the relative weights ofthe neighboring pivot points’ influence on the currentoperating point. To optimize the shape of these mem-bership functions, an objective function for the overallperformance of the fuzzy PSS over these non-pivotpoints is extremized with respect to the shape of themembership functions using a genetic algorithm (GA).

The relevant concepts and techniques used in thispaper are introduced in Section 2. The stabilizer designis discussed in Section 3. Simulation results are pre-sented in Section 4 to demonstrate the effectiveness ofthe proposed stabilizer.

2. Fuzzy logic and genetic algorithm

In this section, a brief overview of fuzzy logic andGAs is given. Detailed information on these subjects isreadily available in subject-related textbooks, e.g. [9,10].The application of a GA to optimize fuzzy logic-basedcontroller performance is not new, e.g. [11]. In thisstudy, fuzzy logic is used for online reasoning and theGA is used for optimization of the membership func-tions in the off line design phase, as described inSection 3.

Fuzzy logic involves three procedures, fuzzification;inference; and defuzzification. Fuzzification convertscrisp input values into fuzzy linguistic terms with theircorresponding memberships. The inference procedure isreasoning using fuzzy linguistic rules, which are basedon some knowledge acquired by experience or otherknowledge extraction means. Defuzzification convertsthe results of the reasoning procedure back into crispvalues. In cases where the output of fuzzy reasoning iscrisp, defuzzification is unnecessary. While rules play anessential role in the inference procedure, membershipfunctions are also important in obtaining a properoutput from the fuzzy controller. Unlike rules whichare in linguistic form, membership functions can oftenbe defined by mathematical functions, which meansthat it is possible to use a numerical technique tooptimize them to improve the performance of the fuzzycontroller. In particular, triangular and trapezoidalmembership functions can be characterized by severalnumbers, i.e. their corner points. A stochastic optimiza-

tion algorithm such as a GA can then be used tooptimize the membership functions and improve theperformance of the fuzzy logic controller.

GAs are stochastic optimization algorithms whichhave proved to be effective in various applications. Agiven GA emulates the process of evolution and naturalselection, which is based on an idea that the forcedriving species to evolve can be imitated in an artificialcontext. A typical GA maintains a population of solu-tions and implements a ‘survival of the fittest’ strategyin the search for better solutions. It has been shown tobe capable of finding global optima in complex prob-lems by exploring virtually all regions of the state spaceand exploiting promising areas through mutation,crossover and selection operations applied to individu-als in the populations. In this study a GA is used toobtain proper corner points for a set of trapezoidalmembership functions.

3. The proposed stabilizer

3.1. The structure of the stabilizer

The proposed fuzzy logic stabilizer consists of aconventional power system stabilizer (CPSS) and afuzzy logic-based parameter tuner. Using the knowl-edge from a rule base, prepared off line, the fuzzy tuneradjusts the parameters of the CPSS according to thereal-time operating information. Fig. 1 illustrates thestructure of the stabilizer. All the rules for the fuzzytuner and the shape of the membership functions areobtained through a GA based optimization method, aswill be explained in the next three subsections.

The transfer function of the CPSS used is as follows:

H(s)=KsTw

1+sTw

1+sT1

1+sT2

1+sT3

1+sT4

(1)

where K is the PSS gain, and T1–T4 are the stabilizertime constants, all of which are determined by the fuzzyparameter tuner. Tw is the time constant of a washoutfilter which is set to 5 s in this study.

Fig. 1. The structure of the proposed stabilizer.

J. Lu et al. / Electric Power Systems Research 60 (2001) 77–83 79

Table 1Structure of the rule base

w̄= [w11, w12, …, w1n

w21, w22, …, w2n, …, wm1, wm2, …, wmn ]T (2)

p̄=P · w̄. (3)

Typical shapes of the membership functions for theinput variables are shown in Fig. 2. One characteristicof this type of membership functions is that for anycrisp input only one or two linguistic terms are in-volved, and in the latter case, the sum of the member-ships corresponding to the two terms is unity.Therefore, the shapes of the membership functions canbe completely described with a set of boundaries whichdetermine the interval of overlap of the two linguisticterms. For example, for the three linguistic terms(shown in Fig. 2), boundary points b1, b2, b3, and b4 areenough to specify the membership functions.

3.2. Indi�idual rules

The rule base consists of a group of individual rules,which are obtained based on the linear models of thesystem at various operating points. The ‘if’ part of eachrule has several fuzzy linguistic terms covering theranges of the input variables. For every rule, a crispoperating point (named pivot point in this paper) mustbe selected to obtain the crisp stabilizer parametervector p̄ ij*. A convenient way to achieve this is to choose‘center’ values within the range of linguistic terms forevery input variable and then designate all possiblecombinations of these center values (one for each inputvariable in every combination) as ‘pivot points’. Asshown in Fig. 2, a1, a2, and a3 might be chosen as centervalues of LT1, LT2 and LT3, respectively. The selectionof the ‘center’ values is heuristic and somewhat arbi-trary. However, the optimization of membership func-tions will compensate for this arbitrariness.

For example, suppose we use active power and termi-nal voltage as the operating parameters. Then, wedefine light, medium and heavy output for active poweras 0–0.5, 0.5–0.9, and 0.9–1.2 p.u., respectively. Also,we define low and high voltage as 0.8–1.05, and 1.05–1.3 p.u., respectively. Then, we may choose 0.3, 0.7,and 1.0 to be associated with the three power linguistic

Let vector p̄ denote the CPSS parameter set, i.e.p̄= [K T1 T2 T3 T4]T. The dimension of p̄ is reduced incase some time constants are identical, or if they aredesignated as constants.

Assuming there are only two real-time measuredoperating parameters as input to the fuzzy tuner, therules take the following form: rule (i, j ): if (input 1) isLT1i, and (input 2) is LT2j, then p̄ is set to p̄ ij*, wherei=1, …, m and j=1, …, n.

Here LT1i and LT2j are the linguistic terms for thetwo input variables to the fuzzy tuner, respectively. Thestructure of the rule base is shown in Table 1. Variablesi and j are the indices of the linguistic terms and m andn are the total number of linguistic terms for the inputvariables. p̄ ij* denotes, the optimal CPSS parametersetting corresponding to the particular operating condi-tion defined by the two input variables. The CPSSdesign methodology used will be explained in Section3.3. The input variables may be chosen from variousmeasurements such as voltages, active and reactivepowers. Proper selection of these variables will deter-mine the steady state condition or operating point ofthe generator uniquely.

The output of an individual rule is a crisp vector p̄ ij*instead of a fuzzy linguistic term which needs defuzzifi-cation. In turn, the output of the fuzzy tuner is theweighted sum of the outputs of all applicable rules,where the input variables together with their corre-sponding membership functions determine the weights,each of which shows truth value or ‘degree of appli-cability’ of one rule.

Next we define a matrix P to be used in the ‘then’part of the rule base, i.e. P= [p̄11* , p̄12* , …, p̄1n* ,p̄21* , p̄22* , …, p̄2n* , …, p̄m1* , p̄m2* , …, p̄mn* ]. Assume the cur-rent crisp input to the fuzzy tuner is (OP1, OP2). Thevalue OP1 is then evaluated to determine its degree ofmembership w1i in LT1i, for i=1, 2, …, m ; similarly,OP2 is then evaluated to determine its degree ofmembership w2j in LT2j, for j=1, 2, …, n. Weightswij are then assigned using wij=min(w1i, w2j). Each wij

is an element of a weight vector w̄ which yields thefuzzy tuner output when multiplied by P, as shownbelow: Fig. 2. The membership functions.

J. Lu et al. / Electric Power Systems Research 60 (2001) 77–8380

terms and 0.9 and 1.2 to be associated with the twovoltage terms, respectively. The pivot points will be(0.3, 0.9), (0.3, 1.2), (0.7, 0.9), (0.7, 1.2), (1.0, 0.9) and(1.0, 1.2). We then design individual linear stabilizersfor these pivot points to obtain p̄ ij*.

3.3. CPSS design methodology

As explained in the preceding two subsections, everyrule actually contains a CPSS for a particular operatingcondition. Therefore, a CPSS design methodology isrequired. Several different methods exist for designingCPSS. In this study Prony analysis is used to obtainlinear transfer functions for the non-linear system atdifferent operating points [8]. Then, the dominatingpoles of the transfer function are chosen and an opti-mization algorithm, described in the next subsection, isused to maximize the damping ratio of these controllingmodes with respect to the stabilizer parameters. Detailsfollow.

Suppose the transfer function of the linearized systemis G(s) and that of the stabilizer to be optimized is H(s)as given in Eq. (1). The stabilizer parameters K, T1, T2,T3, and T4 are to be chosen. Noticing that the feedbackof the stabilizer into the exciter is positive rather thannegative, as shown in Fig. 1, the closed loop transferfunction is:

G(s)1−G(s)H(s)

=�i

ai+ jbi

s− (ci+ jdi)(4)

In (Eq. (4)) ai, bi, ci, and di are real numbers and j isthe unity imaginary number. For a term in the summa-tion that has bi=di=0, the time domain counterpart isaie

ci t. For a complex pole pair in the summation,the time domain expression becomes �ai

2+bi2 ·

eci t · cos(dit+�i), which is obtained by performing theinverse Laplace transformation on the correspondingpartial fraction terms of the poles. The term�ai

2+bi2

reflects the magnitude of the mode and should be takeninto consideration when choosing the dominatingmodes. This will prevent some trivial modes with verysmall magnitudes from being weighted excessively inthe performance index. After eliminating these modesand the modes lying far left in the s-plane, the remain-ing ones are the dominating modes to be taken intoaccount in CPSS design. The objective function used,which is to be maximized, is the smallest one among alldamping ratios of the controlling modes. An equivalentdescription is to minimize the largest damping ratioangle �, as illustrated in Fig. 3.

3.4. Optimization of the membership functions

As discussed in Section 3.1, the membership func-tions can be represented by a set of boundary pointsshown in Fig. 2. Hence, it is possible to use a numerical

Fig. 3. Optimization criterion.

optimization technique to optimize the performance ofthe fuzzy tuner with respect to the membership func-tions. Every optimization method requires an objectivefunction, which is to be extremized. Therefore, a keyissue is to formulate the performance index into anexplicit expression in terms of the membership func-tions, which are actually a set of real numbers. Aproperly chosen performance index expression shouldreflect the performance of the fuzzy tuner accurately.Once the performance index is formulated the member-ship functions can be obtained using an appropriateoptimization method.

In Section 3.3, we developed an evaluation functionto assess the performance of a stabilizer parametersetting at a particular operating point. We use thatevaluation function to construct a single objective func-tion for the fuzzy parameter tuner over a full range ofoperating points. For evaluation purpose, we choose aselection of evenly distributed operating points (evalu-ating points) at which the performance is evaluated. Atthese evaluating points, Prony analysis is again appliedto obtain linear models of the non-linear system. Anexample of a combined view of the pivot points andevaluating points, together with the membership func-tions for two input variables P1 and P2, is shown in Fig.4. We evaluate the performance of a set of membershipfunctions by, (1) obtaining stabilizer parameter settingsover the evaluating points by using the membershipfunctions and the already available stabilizer parametersettings for pivot points; (2) obtaining the performanceindices for individual evaluating points using the linear(Prony-based) models obtained at these points; and (3)adding the indices obtained in step (2) to obtain a singleindex. The above steps can be formulated in the follow-ing form:

PI(MF)= �p�EP

minq�CP(p, H(p, MF))

�(q) (5)

where, the abbreviations are, PI, performance index;MF, membership functions; EP, the set of all theevaluating points; H(p, MF), the resulting CPSS trans-

J. Lu et al. / Electric Power Systems Research 60 (2001) 77–83 81

fer function based on operating point and membershipfunctions; CP(p, H), the set of controlling modes at anoperating point p, with a given transfer function (H)for the CPSS; �(q), the damping ratio for an oscillationmode q.

With the performance index being defined, then anumerical optimization technique can be applied tooptimize the membership functions for the input vari-ables. Here we used a GA for this purpose [9]. Thissearching algorithm is chosen because the functionmapping relationship from the membership function tothe overall performance index is irregular and demandsa robust algorithm to achieve the optimum. A floatingpoint version of the algorithm is used, as solutions inthis problem are not discrete. In the first step, an initialsolution population having a pre-set size is generatedrandomly. Then, the selection procedure is applied,using the GA crossover operators, to choose ‘good’individual solutions, which will be parents to producethe next ‘generation’. Mutation operators are also ap-plied to alter individual solutions with the hope toobtain better solutions by chance. This selection andreproduction procedure continues until certain termina-tion criteria are met. In this study, the population sizeis set to 30, the algorithm stops after 50 generations,and various genetic operators are used [12].

4. Simulation results

The performance of the proposed fuzzy PSS wasevaluated in simulation studies of a one-machine infi-nite-bus system. A (fourth order) transient model wasused for the generator, which accounted for non-linear-ities, and a standard IEEE ST-1 type model was usedfor the excitation/automatic voltage regulator system,making the over all generator-excitation system modelof seventh order. Generator and excitation system

Fig. 5. Machine angular speed vs. time, light loading; (P=0.35,X0=0.25, Xpf=0.5 p.u.).

parameters used are given in Appendix A. Variousmeasurements can be used as input signals to the fuzzytuner; in this study we used the steady state value ofactive power (P), which can be obtained by filteringinstantaneous electric power, and the reactance (X)from the generator terminal to the infinite bus.

In a multimachine system, the procedure described inthis paper can be used to design PSS for those genera-tors which could be modeled as connected to a largesystem through a transmission line.

System identification (Prony analysis) was performedto obtain linear models of the system under variousoperating conditions with active power evenly rangingfrom 0.2 to 1.0 p.u. and reactance from 0.2 to 0.7 p.u.,both in steps of 0.1 p.u. We used five linguistic termsfor active power and three for reactance. For activepower, the center crisp values which represent the fivelinguistic terms are 0.2, 0.4, 0.6, 0.8, and 1.0 p.u., andfor reactance, they are 0.2, 0.4, and 0.6 p.u. Then,individual stabilizers are designed for the pivot points.After that, the membership functions are optimized asexplained in subsections 3.3 and 3.4.

A three-phase short-circuit fault was applied at theremote end of the transmission line and cleared after0.1 s. Simulation results were obtained under variousoperating conditions to evaluate the performance androbustness of the stabilizer. The post-fault line reac-tance may or may not change, depending on whetherthe line is tripped or reclosed successfully. For evalua-tion purposes, the performance of the system with theproposed fuzzy stabilizer was compared with that whena conventional stabilizer was used and when there wasno stabilizer.

Figs. 5–7 show the machine angular speed as afunction of time for three loading conditions, light,

Fig. 4. Combined view of pivot and evaluating points and member-ship functions.

J. Lu et al. / Electric Power Systems Research 60 (2001) 77–8382

Fig. 6. Machine angular speed vs. time, medium loading; (P=0.6,X0=0.25, Xpf=0.5 p.u.).

half of the real axis of the s-plane when attempt isbeing made to maximize the damping ratio.

5. Conclusions

A fuzzy logic-based adaptive PSS was presented inthis paper. The stabilizer uses a fuzzy logic inferencemechanism to tune the parameters of the stabilizeraccording to online measurements. The fuzzy rules aredesigned based on linear models of the generator ac-quired with Prony analysis under various operatingpoints. The membership functions are obtained using aGA to maximize the damping ratio over a wide rangeof operating conditions. Simulation results show thatthe proposed fuzzy adaptive stabilizer can effectivelyenhance the damping of low frequency oscillations.

Acknowledgements

This work was supported by the National ScienceFoundation Grant ECS-9616631 and by Montana StateUniversity Engineering Experiment Station.

Appendix A

Generator parameters: T �do, T �qo=6.0 s, 0.535 s; Xd,X �d=0.8958, 0.1198 p.u.; Xq, X �q=0.8645, 0.1969 p.u.;Ra (armature resistance)=0, H=6.4 s.

Exciter parameters: TF=0.35 s, TE=0.314 s, TA=0.2 s; KA=20, KE=1, KF=0.063.

A voltage of 1.05 p.u was assumed for the infinitebus.

References

[1] Y. Hsu, C. Cheng, Design of fuzzy power system stabilisers formultimachine power systems, IEE Proceedings-C 137 (3) (1990).

[2] P. Hoang, K. Tomosivoc, Design and analysis of an adaptivefuzzy power system stabilizer, IEEE Trans-EC Ll (2) (1996).

[3] T. Hiyama, Application of rule-based stabilising controller toelectrical power system, IEE Proceedings-C 136 (3) (1989).

[4] T. Hiyama, Rule-based stabilizer for multi-machine power sys-tem, IEEE Trans-PWRS 5 (2) (1990).

[5] N. Hosseinzadeh, A. Kalam, A rule-based fuzzy power systemstabilizer tuned by a neural network, IEEE Trans-EC 14 (4)(1999).

[6] Y. Abdel-Magid, M. Abido, S. Al-Baiyat, A. Mantawy, Simulta-neous stabilization of multimachine power systems via geneticalgorithms, IEEE Trans-PWRS 14 (4) 1999.

[7] Y. Hong, W. Wu, A new approach using optimization for tuningparameters of power system stabilizers, IEEE Trans-EC 14 (3)1999.

[8] D. Trudnowski, J. Smith, T. Short, D. Pierre, An application ofprony methods in PSS design for multimachine systems, IEEETrans-PWRS 6 (1) 1991.

medium, and heavy, respectively. In these figures, Pdenotes active power and X0 and Xpf are the transmis-sion line pre-fault and post-fault reactances. All theunits unspecified are per unit. It is clear from thesefigures that in each case the damping of the electrome-chanical oscillations is enhanced with the proposed PSSas compared with using a conventional PSS or no PSS.In some cases, the improvement in the machine speed,when calculated in percentage, may be small, but thereduction in the magnitude of active power oscillations,measured in MW, can be significant.

It was noticed that in some cases, although theoscillations damped out quickly, the DC componentdecayed at a lower speed. This situation can be allevi-ated by taking into consideration the poles on the left

Fig. 7. Machine angular speed vs. time, heavy loading; P=0.95,X0=Xpf=0.6 p.u.

J. Lu et al. / Electric Power Systems Research 60 (2001) 77–83 83

[9] D. Goldberg, Genetic Algorithm in Search, Optimization,and Machine Learning, Addison-Wesley, Reading, MA,1989.

[10] C. de Silva, Intelligent Control—Fuzzy Logic Applications,CRC Press, Boca Raton, 1995.

[11] J. Wen, S. Chang, O.P. Malek, A synchronous generator fuzzyexcitation controller optimally designed with genetic algorithm,IEEE Trans-PWRS 13 (3) (1998).

[12] Z. Michalewicz, Genetic Algorithm+Data Structure=Evolu-tion Programs, Springer-Verlag, Berlin, 1994.