Electric Power Systems Research 60 (2001) 115–121

A fuzzy logic-based adaptive power system stabilizer formulti-machine systems

J. Lu, M.H. Nahrir *, D.A. PierreDepartment of Electrical and Computer Engineering, Montana State Uni�ersity, 610 Cobleigh Hall, Bozeman, MT 59717-0378, USA

Received 23 February 2001; accepted 12 September 2001

Abstract

This paper presents an approach for designing fuzzy logic based adaptive power system stabilizers (PSS) for multi-machinepower systems. The approach is based on the traditional frequency domain method. In addition, a fuzzy signal synthesizer isintroduced to achieve adaptiveness. In this approach, two linear stablizers are designed to accommodate two extreme loadingcases. A fuzzy logic mechanism is used to generate one single control signal by properly combining the outputs of the linearstablizers. The fuzzy controller is optimized using a least squares error criterion. Simulation studies of a one-machine infinite-busand two multi-machine systems show that the proposed stabilizer provides satisfactory damping against low frequency oscillationsunder different operating conditions. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Power system stabilizer; Fuzzy logic; Self-tuning

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1. Introduction

Power systems are usually large nonlinear systems,which are often subject to low frequency electro-me-chanical oscillations when working under some adverseloading conditions. Power system stabilizers (PSS) areoften used as an effective and economic means ofdamping such oscillations. Many approaches are avail-able for PSS design, most of which are based either onclassical control methods [1–3] or on intelligent controlstrategies [4–7].

A frequency domain method, based on classical con-trol theory, is widely viewed as a simple and robustmethodology. In this approach, transfer functions areidentified for a part of the excitation-generation system,namely, transfer functions between the voltage refer-ence point and the electric torque. From these transferfunctions, the frequency domain responses of the sys-tem are obtained. To contribute sufficient damping tothe modes of interest in the system without causingadverse effects on other modes, the stabilizer should

have a characteristic that is close to the inverses of theidentified transfer functions. Therefore, a linear com-pensator is manually designed to approximate the de-sired compensation characteristic. Conventional PSSdesigned with this philosophy work well, especially atthe operating point for which they are designed. How-ever, an adaptive stabilizer can perform better if it iscapable of tracking and responding to the changingoperating condition by providing a better match of thedesired stabilizer characteristics, which may be affectedby the operating condition.

In our proposed approach, a Takagi–Sugeno typefuzzy model [8] is used to develop the PSS. Firstly, twolinear stabilizers are manually designed to accommo-date two extreme loading cases, i.e. a heavy and a lightcondition. At any intermediate operating point, theoutput of the proposed stabilizer is formed from aweighted combination of the outputs of the two stabi-lizers. A least squares error criterion is used to deter-mine weighting coefficients for the stabilizercharacteristic to approach an ideal one. The twoweighting coefficients reveal how close a particularoperating point is to each extreme point. Assigninglinguistic terms to the extreme points and consideringthe weighting coefficients as membership grades for the

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

E-mail address: hnehrir@ece.montana.edu (M.H. Nahrir).

0378-7796/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 8 -7796 (01 )00169 -9

J. Lu et al. / Electric Power Systems Research 60 (2001) 115–121116

operating point, fuzzy logic is used to describe therelationship between intermediate points and the twoextreme points. By applying this procedure for a seriesof pre-chosen operating points and using interpolation,two membership curves are obtained. When used in realtime, the two stabilizers (for extreme cases) work simul-taneously and independently providing two stabilizingsignals. Fuzzy reasoning is then applied to determinehow the two stabilizing signals are to be mixed togenerate one output signal for the operating conditionat that time. As a result, a higher-ordered adaptivestabilizer is achieved.

It should be noted that the proposed stabilizer, whichis designed based on the frequency domain method, isfairly insensitive to changes in the external networkstructures; it is the changes in the system operatingconditions that the fuzzy controller introduced in thispaper aims to compensate.

A detailed description of the proposed design proce-dure is given in Section 2. In Section 3, simulationresults are given for a single-machine-infinite-bus sys-tem, a four-machine 13-bus system and a 16-machine68-bus system to demonstrate the effectiveness of theproposed method. Conclusions are drawn in Section 4.

2. Proposed stabilizer

2.1. The structure of the stablizer

The proposed fuzzy logic PSS consists of two con-ventional linear stabilizers and a fuzzy logic-based sig-nal synthesizer. These two stabilizers are designed forextreme (heavy and light) loading conditions, andtherefore they generate stabilizing signals working bestunder those extreme conditions. It is intuitive to assumethat a proper combination of them would work best forthe conditions between these extreme ones. How faraway the current condition is from each extreme casedetermines how much the respective stabilizing signal isweighed in the output of the synthesizer. The synthe-sizer combines the two individual signals in such a waythat the signal fits the loading condition optimally. Fig.1 illustrates the structure of the stablizer.

The linear stabilizers could be typical second orhigher order filters, depending on the characteristics ofthe system. Second-order filters are usually adequate,since the signal synthesizer may act as a refiner tofine-tune the control signal.

The fuzzy logic synthesizer accepts one variable indi-cating the generator loading condition, and generatesone output. For that input variable two linguistic termsused to represent the two extreme cases, and accord-ingly there are two membership curves. There are tworules in the following form:

Rule (i ): IF the loading condition is Ei THEN theoutput signal is Si.

Here Ei denotes one of the extreme cases, and Si

denotes the corresponding stabilizing signal.We adopt a Takagi–Sugeno fuzzy model [8], so Si is

crisp and the defuzzification is simply a weighted sum-mation that is expressed as follows:

S=�i

wi · Si (1)

where wi is the truth value (weight) of the ith rule,which is obtained by comparing the input variableagainst the membership function curve. Based on Eq.(1), the overall transfer function of the proposed PSS isthen:

H(s)=K �i

wi · Hi(s) (2)

where Hi(s) is the transfer function of the ith linearstabilizer (called basis functions hereafter), and K is areal gain factor.

From Eq. (2) we can see that the performance of theproposed PSS depends on the linear stabilizer Hi(s) andthe membership functions.

2.2. Con�entional PSS design

The conventional frequency domain PSS design isused [2]. Under the operating points of interest, thedynamic equations are linearized to form the standardstate space model, with the rotor angle dynamics prop-erly removed. From this model, the transfer functionfor the excitation-generation part is obtained:

G(s)=�Te(s)�Vs(s)

(3)

where Te is the electric torque and Vs is the input to thevoltage regulator. The frequency response characteris-tics obtained from G( j2�ƒ) show the phase lag causedby the excitation-generation sub-system. Ideally, the

Fig. 1. The overall structure of the proposed stablizer.

J. Lu et al. / Electric Power Systems Research 60 (2001) 115–121 117

transfer function of a compensator H(s) aimed tocompletely compensate for the phase lag of G(s), is justthe inverse of G(s). However, in most cases it is physi-cally impractical to have 1/G(s) as the transfer functionof the compensator, and therefore it has to be designedmanually. We assume that the transfer function takesthe form of a second or higher order filter and tune itstime constants so that its leading phase characteristiccancels out the phase lag of G(s) over the frequencyrange of interest (0–3 Hz in this study). The magnitudegain of H(s) is of less concern, and therefore, a nominalgain is assumed.

2.3. Fuzzy logic signal synthesizer design andoptimization

As discussed in Section 2.1, membership functionshave a significant impact on the behavior of the pro-posed PSS, and optimal performance can be achievedonly with optimized membership functions. In our de-sign, an optimization algorithm is used to determinemembership curves point by point with a small stepsize. Interpolation is used to obtain the membershipvalues for intermediate points.

Given a loading condition, the desired characteristicof the PSS, i.e. HD(s), is the inverse of G(s). Supposethe basis functions Hi(s) have already been obtained asexplained in the preceding subsection, and the corre-sponding weights for that particular loading conditionare to be determined to form a suitable stabilizer. Thegoal of our design is to obtain the weights that makeH(s) as close as possible to the desired transfer functionHD(s). The performance of the stabilizer is then evalu-ated in terms of squared errors. To achieve this, wedefine a series of radian frequencies, at which the erroris evaluated:

�k=2k�fmax

n, k=1, …, n (4)

where fmax is the maximum frequency of interest (3 Hzin this study), and n is the number of evaluating points.Here we simply assume the gain factor K to be one,since it can be contained in w1 and w2 and therefore istrivial. For the transfer function H(s)=w1H1(s)+w2H2(s), we define the error function below and try tominimize it:

f(w1, w2)= �n

k=1

�H( j�k)−HD( j�k)�2

= �n

k=1

�w1H1( j�k)+w2H2( j�k)−HD( j�k)�2

(5)

For notational briefness, we define Ri,k and Ii,k as thereal and imaginary parts of Hi( j�k), respectively:

Hi( j�k)=Ri,k+ jIi,k, i=1,2 (6)

Similarly, we define RD,k and ID,k as the real andimaginary parts of HD( j�k):

HD( j�k)=RD,k+ jID,k (7)

Next, several aggregation terms are defined:

A1= �n

k=1

(Ri,k2 +I i,k

2 ), i=1, 2 (8)

A3= �n

k=1

(RD,k2 +ID,k

2 ) (9)

A4= �n

k=1

(R1,k · R2,k+I1,k · I2,k) (10)

A5= �n

k=1

(R1,k · RD,k+I1,k · ID,k) (11)

and

A6= �n

k=1

(R2,k · RD,k+I2,k · ID,k) (12)

With these terms, the expression for f(w1, w2) of Eq.(5) is reduced to:

f(w1, w2)

=A1w12+A2w2

2+A3+2A4w1w2−2A5w1−2A6w2

(13)

Now, it is clear that we have a problem of quadraticminimization at hand, and the optimal values of w1 andw2 are obtained through an algebraic equation:

�f�w1

=�f

�w2

=0 (14)

which is reduced to:

�A1 A4

A4 A2

n�w1

w2

n=�A5

A6

n(15)

Suppose the generator in consideration operateswithin its power limits (Pmin–Pmax), and we apply theprocedure to obtain the weights w1 and w2, for differentloading conditions between Pmin and Pmax as follows:

Pl=1m

(Pmax,−Pmin)+Pmin, l=0, 1 …, m (16)

where there are (m+1) operating points.As a result, a series of wi,1, i=1, 2 and l=0–m, are

obtained. In cases where we want w1,0 and w2,m (the firstand the last weights) to be unity, a normalizationprocedure is applied; that is, H1(s) and H2(s) are multi-plied by w1,0 and w2,m, respectively. Then, all the wi,l

and w2,l are divided by w1,0 and w2,m respectively. Now,we have a series of normalized wi,1 and Hi(s). Withsome interpolation involved, the coefficients wi,l formthe membership function curves, shown in Fig. 2, whichare used to determine the truth value of the ith rule.

J. Lu et al. / Electric Power Systems Research 60 (2001) 115–121118

Fig. 2. Membership function curves.

Fig. 3. Membership function curves.

To achieve best performance, the gain of H(s) (K givenin (Eq. (2))) should be adjusted properly. Simulation oreigenvalue analysis can be used for this purpose. Theformer is used in this study.

When used on-line, the signal synthesizer simplychecks the steady-state value of the active power (ob-tained by measuring the mechanical power or measuringthe electric power and removing its transient componentsusing a low pass filter) against the membership curves todecide the appropriate weights, and then blends them asillustrated in Fig. 1.

3. Simulation results

To test the performance of the proposed stabilizer,simulation studies were performed on three systems: aone-machine-infinite-bus system, a 2-area-4-machine-13-bus system and a 16-machine-68-bus system [9].Three-phase short-circuits were applied on the systems underdifferent operating conditions. Faults occurred at theends of transmission lines and then were cleared after0.05 s.

3.1. Simulation results-a one-machine-infinite-bussystem

A generator was connected to an infinite bus througha transformer and two parallel lines. Machine data isgiven in Appendix A. For this system, Pmax is 9.9 p.u.,and we select P1�{1, 2, 3, 4, 5, 6, 7, 8, 9, 9.9}.

The basis functions are obtained as follows:

H1(s)=2.30(0.316s+1)(0.254s+1)(0.8766s+1)

(s+1)(0.001s2+0.01s+1)(17)

H2(s)=0.36(0.0292s+1)(0.4965s+1)

(0.0163s+1)(0.001s2+0.01s+1)(18)

The respective membership functions for the abovebasis functions are obtained as shown in Fig. 3. In Fig.3, the membership curve w2 does not increase monoton-ically. This is because at light loading conditions a largemagnitude gain of H(s) is desired to minimize the errordefined in (Eq. (5)) and the membership functions aretuned to meet this gain requirement since the magnitudegains of the basis functions are fixed during membershipfunctions calculation.

The time constant of the washout filter was set at 10s, and the gain factor K was selected so that the DC gainof the stabilizer (excluding the washout filter) at theheaviest loading condition is 30. The overall transferfunction of the stabilizer is:

H(s)=K10s

10s+1[w1 · H1(s)+w2 · H2(s)] (19)

A conventional stabilizer Hc(s) was also designed forcomparison using the frequency domain method [2].

Hc(s)=3010s

10s+1(0.1s+1)2

(0.01s+1)2 (20)

With the proposed PSS applied, Figs. 4 and 5 show thegenerator speed deviation as a function of time, for a lightloading condition (P=2.5 p.u.) and a heavy loadingcondition (P=7.8 p.u.), respectively. For comparisonpurpose, the generator responses are also shown when noPSS is applied and when a conventional PSS is applied.

Fig. 4. Generator response under light loading (P=2.5 p.u.).

J. Lu et al. / Electric Power Systems Research 60 (2001) 115–121 119

Fig. 5. Generator response under heavy loading (P=7.8 p.u.).

Fig. 7. Membership function curves.

H(s)=K20s

20s+1[w1 · H1(s)+w2 · H2(s)] (23)

A conventional stabilizer Hc(s) was also designed forcomparison:

Hc(s)=5020s

20s+1(0.23s+1)2

(0.02s+1)2 (24)

Figs. 8 and 9 show the speed deviation of generatorc2 as a function of time, for a light loading condition(P=2.5 p.u.) and a heavy loading condition (P=6.5p.u.), respectively. For comparison, generator responsesare shown with the proposed fuzzy PSS, with a conven-tional PSS, and with no PSS applied. Note that underheavy loading condition (Fig. 9), the system is unstablewithout an acting stabilizer. It is clear from Fig. 8 thatunder light loading the proposed PSS is more effectivethan the conventional one in damping the generatoroscillations; it also performs at least as effectively as theconventional controller under heavy loading.

It is clear that the effectiveness of the proposed PSS issuperior to a conventional PSS under both loadingconditions.

3.2. Simulation results—a 4-machine-13-bus system

Fig. 6 shows a 4-machine, 13-bus system, where thegenerators are located in two distant areas [9]. A stabi-lizer for generator c2 is designed using the proposedmethod. For this generator, Pmax is 8.0 p.u., and weselect P1�{1, 2, 3, 4, 5, 6, 7, 8}.

The basis functions are as follows:

H1(s)=1.05(0.06s+1)(0.09s+1)(0.32s+1)

(0.02s+1)3 (21)

H2(s)=0.34(0.13s+1)(0.28s+1)

(0.015s+1)2 (22)

The respective membership functions for the abovebasis functions are obtained as shown in Fig. 7.

The time constant of the washout filter was set at 20s, and the gain factor K was selected so that the DCgain of the stabilizer (excluding the washout filter) atthe heaviest loading condition is 50. The transfer func-tion of the stabilizer is:

Fig. 8. Generator response under light loading (P=2.5 p.u.).Fig. 6. A Two-Area-Four-Bus System.

J. Lu et al. / Electric Power Systems Research 60 (2001) 115–121120

Fig. 9. Generator response under heavy loading (P=6.5 p.u.). Fig. 11. Generator response under light loading (P=3.5 p.u.).

3.3. Simulation results—a 16-machine-68-bus system

This system has 16 generators, seven of which actu-ally represent external systems [9]. A stabilizer is de-signed for one generator (c9) using the proposedmethod. For this generator, Pmax is 10.0 p.u., and weselect P1�{l, 2, 3, 4, 5, 6, 7, 8, 9, 10}

The basis functions are as follows:

H1(s)=0.1860.0169s2+0.26s+10.0004s2+0.04s+1

(25)

H2(s)=0.0350.04s2+0.4s+1

0.000784s2+0.056s+1(26)

The respective membership functions for the abovebasis functions are given in Fig. 10.

The time constant of the washout filter was 10 s, andthe gain factor K was set so that the DC gain of thestabilizer (excluding the washout filter) at the heaviestloading condition is 100. The transfer function of thestabilizer has the same form as in (Eq. (19)).

The conventional stabilizer Hc(s) designed for theabove generator is:

Hc(s)=10010s

10s+1(0.18s+1)2

(0.024s+1)2 (27)

With the proposed PSS applied, Figs. 11 and 12 showthe speed deviation of generator c9 as a function oftime, for a light loading condition (P=3.5 p.u.) and aheavy loading condition (P=7.5 p.u.), respectively.The generator responses are also shown when a conven-tional PSS is applied and when no PSS is applied.Again, under heavy loading condition (Fig. 12), thesystem is unstable without an acting stabilizer. In thiscase superiority may not be so much as in the previouseases, but we are still able to observe someimprovement.

Fig. 12. Generator response under heavy loading (P=7.5 p.u.).Fig. 10. Membership function curves.

J. Lu et al. / Electric Power Systems Research 60 (2001) 115–121 121

4. Conclusions

The design of a fuzzy logic-based adaptive powersystem stabilizer for multi-machine power systems waspresented in this paper. The stabilizer uses a Takagi–Sugeno type fuzzy logic signal synthesizer to blend thestabilizing signals obtained from pre-designed linearstabilizers based on the system operating condition. Thelinear stabilizers are designed using the classical fre-quency domain method. The rules of the fuzzy synthe-sizer are simple and the membership functions areobtained using a quadratic optimization algorithm.Simulation results show that the proposed fuzzy adap-tive stabilizers can effectively enhance the damping oflow frequency oscillations and perform better thanconventional stabilizers.

Acknowledgements

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

Appendix A

The data for the Single Machine System

Generator parameters:armature resistance Ra=0 p.u.;leakage reactance X1=0.015 p.u.;d-axis open-circuit transient T �do=5.0 s;

time constantd-axis open-circuit subtransient T �do=0.031 s;

time constantd-axis synchronous reactance Xd=0.2 p.u.;d-axis transient reactance X �d=0.025 p.u.;d-axis subtransient reactance X�d=0.02 p.u.;q-axis open-circuit time T �qo=0.66 s;

constantT�qo=0.061 s;q-axis open-circuit subtransient

time constantq-axis synchronous reactance Xq=0.19 p.u.;

X �q=0.042 p.u.;q-axis transient reactanceX�q=0.02 p.u.;q-axis subtransient reactanceH=2.8756 s;inertia constantD=0.damping coefficient

Exciter parameters (IEEE TypeST-3 Model)

TR=0.0 s;input filter time constantKA=7.04;voltage regulator gainTA=0.4 s;voltage regulator time constantTB=6.67 s;voltage regulator time constantTC=1.0 s;voltage regulator time constant

maximum voltage regulator VRmax=7.57 p.u.;output

minimum voltage regulator VRmin=0 p.u.;output

maximum internal signal VImax=0.2 p.u.;minimum internal signal VImin=−0.2 p.u.;

KJ=200;first stage regulator gainpotential circuit gain coefficient KP=4.365;

�P=20°;potential circuit phase anglecurrent circuit gain coefficient K1=24.2;

XL=0.091 p.u.;potential source reactanceKC=1.096;rectifier loading factorEfdmax=6.53 p.u.;maximum field voltageKG=1;inner loop feedback constant

maximum inner loop voltage VGmax=6.53 p.u.feedback

Transformer reactance 0.02 p.u.;Transmission line reactance 0.04 p.u.The AVR is set to maintain the

terminal voltage at 1.05 p.u.;1.08 p.u.Voltage of infinite bus

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