Stability enhancement of a multimachine power system using fuzzy logic based power system stabilizer tuned through genetic algorithm

Stability enhancement of a multimachine power system using fuzzy logicbased power system stabilizer tuned through genetic algorithm

P. Lakshmi* , M. Abdullah Khan

School of Electrical and Electronics Engineering, College of Engineering, Anna University, Chennai—600 025, India

Received 27 July 1998; received in revised form 8 October 1998; accepted 11 May 1999

Abstract

The paper presents a procedure for application of Fuzzy logic Power System Stabilizer (FPSS) for stability enhancement of a multimachinepower system. The location of Power System Stabilizer (PSS) is determined using critical modes and participation factors for differentoperating conditions. A simple configuration is chosen for FPSS with a speed deviation of synchronous generator and its derivative as inputsignals. The parameters of the stabilizers are tuned through simulation of non-linear model of power system under chosen disturbance usingCrowding Genetic Algorithm (CGA) to minimize a Root Mean Square Deviation (RMSD) index of the concerned state variable whichmaximizes the damping of the critical mode. Results obtained from a two area, five generator power system indicate the improvedperformance of FPSS over conventional PSS.q 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Fuzzy logic power system stabilizer; Crowding genetic algorithm; Root mean square deviation

1. Introduction

Modern power systems are highly complex and non-linear. Their operating conditions can vary over a widerange. The increasing size of generating units, the loadingof the transmission lines and the operation of high speedexcitation systems nearer to their operating limit are themain causes affecting small signal stability of powersystems. Dynamic load changes and action of controllerscreate small oscillations in power systems, which preventfull exploitation of available generating capacity.

Conventional Power System Stabilizers (CPSS) are usedto damp out small oscillations and they are designed basedon a linearized model, which is valid around a particularoperating point [1]. The structure and the parameters ofCPSS are determined to provide optimal performance atthis operating point. The most widely used CPSS is thelead-lag compensator. The gain settings of this compensatorare fixed by tuning at some specific operating condition toprovide optimal performance. Parameters of the powersystem stabilizer need to be retuned to get the desiredperformance as the power system configuration changeswith time. Self-tuning PSS [2–3] provides better dynamicperformance over a wide range of operating conditions. It isdifficult to realize a self-tuning PSS, as it requires parameter

identification, state observation and feedback gain compu-tation, which are very time consuming.

The rule-based control [4] and the fuzzy logic control [5–9] are simple, alternate control schemes which have beenproposed to overcome the above mentioned problems. Outof these schemes, fuzzy logic control appears to possessmany advantages such as lesser computation time androbustness. Fuzzy logic technique has been found to be abetter substitute for conventional control techniques, whichare based on highly complex mathematical models.

Design of Fuzzy logic based Power System Stabilizer(FPSS) is an active area of research and satisfactory resultshave been obtained [5–9]. Although, fuzzy logic controlintroduces a good tool to deal with complex, non-linearand ill-defined systems, it still has a drawback that there isno systematic procedure for tuning the parameters of theFPSS. The parameters are tuned by trial and error method[10], which is time consuming. Therefore, there is a need foran effective method for tuning the parameters of the FPSS soas to maximize the damping of power system oscillations.Genetic Algorithm (GA) is a global search technique basedon the operations observed in natural selection and genetics[11]. GA has been recently applied to hydro generatorgovernor tuning [12], load flow problems [13] and economicdispatch problems [14]. GA can be used irrespective of thecomplexity of the performance index.

This paper presents a method for stability enhancement ofa multimachine power system using FPSSs whose

Electrical Power and Energy Systems 22 (2000) 137–145

0142-0615/00/$ - see front matterq 1999 Elsevier Science Ltd. All rights reserved.PII: S0142-0615(99)00043-5

www.elsevier.com/locate/ijepes

* Corresponding author. Tel.:191-044-4903144; fax:191-044-2350091.

parameters are tuned through GA. This method comprises ofthree stages: (i) identification of critical modes and locationof FPSS; (ii) selection of suitable configuration of FPSS;and (iii) tuning of crucial parameters of FPSS to maximizethe damping of the critical swing modes, both local andinterarea modes of the system being studied. The perfor-mance of the stabilizer has been tested for various distur-bances using a sample power system with five machines,eight buses and two areas. From the simulation results, theFPSS is found to be effective for both the local and interareamodes of oscillations.

2. Approach adopted

The approach adopted in this work comprises the follow-ing three steps:

(i) Identification of the critical rotor swing modes (modeswith damping ratioz # 0:45).(ii) Identification of location of FPSS for the criticalmodes based on the participation factors.(iii) Optimal tuning of parameters like the width of thelabels (WL) of the linguistic variables and the gain (KF) ofFPSS through GA for effective damping of the criticalmode.

3. Identification of critical modes and location of FPSSfor multimachine power system

In a multimachine power system, stabilizers are located atmachines where the critical rotor swing modes are to beeffectively damped. Theoretically it is possible to have aFPSS at each machine. But in practice, there are only fewplaces where the FPSS would result in better damping of allrotor modes. Improper location of stabilizers would lead tohigh stabilizer gains and would result in severe deviation involtage profile under disturbance conditions. In some cases,

the stabilizer may even cause operation of generators atleading power factor. Siting the stabilizers at the best loca-tions is an important factor to obtain a stable closed loopsystem with well-damped rotor modes and small stabilizergains.

The concept of participation factor [15] is used to deter-mine the best location of the FPSS. The participation factorsare computed using the right and the left eigenvectors of thesystem matrix of the power system corresponding to thespecified operating condition. The right eigenvector givesthe mode shape by describing the activity of the state vari-ables when that particular mode is excited whereas the lefteigenvector gives the mode composition by describing theweighted combination of the state variables needed toconstruct that mode.

The system is linearized at an operating condition. Usingeigen structure analysis, the critical rotor swing modes(modes with damping ratio# 0.45) are determined. Thenature of the swing modes (whether local, interarea or intra-plant modes) are also identified. The critical modes arearranged in the increasing order of the damping factor.First the most critical rotor swing mode is taken and bycomparing the participation factors of the speed componentof various machines, the machine with the highest partici-pation factor is chosen as the best location for damping thatmode. In the same way, the best location for PSS for damp-ing other critical swing modes are obtained. This procedureis repeated for different operating conditions with signifi-cant changes in loading pattern and network configuration.FPSSs are to be located at all the best locations thus identi-fied.

4. Selection of suitable configuration of the FPSS

A simple configuration is chosen for the FPSS (Fig. 1).The shaft speed, bus frequency and real power arecommonly used input signals to the PSS. Among thesesignals, the shaft speed is very easy to measure and isfound to be quite effective. In this paper, speed deviationof the synchronous machine and its derivative are chosen asinputs to the FPSS. GainKF is used to scale the outputvariableVS of the PSS (Fig. 1). The inputs are normalizedusing their estimated peak values. The peak values are esti-mated from the simulation of the given power system with-out any PSS for different disturbances. Seven labels arechosen for the linguistic variables used to represent theinputs and the output. The labels are LP (Large Positive),MP (Medium Positive), SP (Small Positive), VS (Very

P. Lakshmi, M. Abdullah Khan / Electrical Power and Energy Systems 22 (2000) 137–145138

Fig. 1. Block diagram representation of FPSS.

Fig. 2. Membership function.

Small), SN (Small Negative), MN (Medium Negative) andLN (Large Negative).

Linear triangular membership function is used in thedesign of FPSS and the width of the membership functionWL (Fig. 2) for bothDv andD _v is chosen to be equal for allthe labels in order to keep the number of parameters to beoptimized a minimum. The Rule Table [6] giving acomplete set of rules which define the relation betweenthe inputs and the output of the FPSS is used in this work.The Rule Table is then transformed into a fuzzy relationmatrix [6]. The stabilizer output is determined using themin–max composition rules (Table 1).

5. Tuning of FPSS parameters

The steps of the proposed tuning algorithm are as follows.

Step 1:The damping of the most critical mode is maxi-mized by tuning the parametersWL andKF of the FPSSconnected to the corresponding machine (with no otherFPSSs included in the system) by optimizing a fitnessfunction using Crowding Genetic Algorithm (CGA)through a full simulation of a chosen disturbance and achosen operating condition which excite the criticalmode.Step 2:Step 1 is repeated for each one of the other criticalmodes one at a time, and the parameters of FPSS of theconcerned machine are tuned (with no other FPSSsincluded in the system) for the corresponding mode.Step 3:After tuning the FPSS parameters separately forall the critical modes, simulation is carried out again forthe most critical mode using the respective disturbanceand operating condition with all the tuned FPSSs includedin the system. If the damping of the most critical mode isacceptable, go to Step 4. Otherwise, the FPSS parametersof that mode are retuned using CGA for the correspond-ing disturbance and operating condition keeping the otherFPSSs parameters the same.Step 4:Repeat Step 3 for each one of the other criticalmodes.

5.1. Crowding genetic algorithm

The parameters of the FPSS are tuned using CrowdingGenetic Algorithm (CGA) [14]. GA operates on a popula-tion of current approximations initially drawn at random,from which improvement is sought. GA starts with agroup of binary strings called a population. Each binarystring is called a chromosome. A binary string/chromosomeconsists of many substrings. The number of binarysubstrings is equal to the total number of parameters ofthe PSS. Each binary substring represents the feasiblevalue for each parameter of interest.

If the length of the binary substring is fixed as S and thenumber of parameters of the PSS is NP, then the length ofthe string is given by Sp NP. All the binary strings in the

initial population are generated randomly. A ‘fitness’ valuederived from the objective function of the problem isassigned to each member of the population. The memberwith higher value of fitness will have more opportunities topass on genetically important information to successivegenerations.

In the standard genetic algorithm [16], the entire popula-tion is replaced by offsprings using crossover and mutationproperties. In Crowding Genetic Algorithm [14], twoparents are chosen randomly. Using crossover and mutation,two offsprings are produced. Comparing the fitness func-tions of both the parents and children, the best two stringswill go for the next generation. This is repeated for the otherstrings. Uniform crossover technique [18] is used in thispaper, where the convergence speed is faster than onepoint and two point crossover.

The fitness function used in this work is a Root MeanSquared Deviation (RMSD) index evaluated for a state vari-ablex�t� in which the respective critical mode is dominant.

RMSD index�XNT

k�1

�x�k�2 xss�2NT

( )1=2

�1�

subject to the following constraints

W Lmin , WLi , W Lmax; i � 1;2;…n

K Fmin , KFi , K Fmax; i � 1;2;…n

wheren is the number of stabilizers,x�k� the response atsampling time t � kDT; xss the steady state responsecomputed at the end of simulation period by taking theaverage of the last two peaks of the response, NT the totalnumber of samples upto the final time of simulation,WLi thewidth of the label for theith stabilizer;i � 1; 2;…n; KFi thegain of the FPSS;i � 1; 2;…n; WLmin andWLmax the lowerand upper bounds forWL, KFmin and KFmax the lower andupper bounds forKF.

6. Sample system study and results

The machine model used in the simulation is a ‘two axismodel’ [17] with state variablesd , v , E 0q and E 0d: IEEEType 1S [17] fast exciter model has been used for the

P. Lakshmi, M. Abdullah Khan / Electrical Power and Energy Systems 22 (2000) 137–145 139

Table 1Rule Table for FPSS output

D _v

Dv LN MN SN VS SP MP LP

LP VS SP MP LP LP LP LPMP SN VS SP MP MP LP LPSP MN SN VS SP SP MP LPVS MN MN SN VS SP MP MPSN LN MN SN SN VS SP MPMN LN LN MN MN SN VS SPLN LN LN LN LN MN SN VS

excitation system in all the units. Constant admittance typeloads and steam turbines with electro-hydraulic governorsare used in the simulation studies.

Fig. 3 shows a five machine, seven bus system adoptedfrom Ref. [9], with a weakened tie. This system is chosen toinvestigate the validity of the proposed method of designingthe FPSS. The system has two power generating areas A1and A2. Generators 1, 2 and 4 are of larger capacity than thegenerators 3 and 5. Area A1 consists of generators 2, 3 and 5whereas generators 1 and 4 form area A2. These two areasare connected through a weak tie-line between buses 6 and7. Appendix A gives the parameters for all the generatingunits, transmission lines, loads, exciters and governors.

6.1. Location of the FPSS

An eigen structure analysis of the sample system for thefollowing operating conditions are made:

(a) Full load operating condition.(b) Loading same as case (a) but with one of the two tielines between buses 6 and 7 removed.(c) Case (a) with loading increased by 25%.

The results reveal that there are four oscillatory rotorswing modes for each operating condition. The rotormodes of the system are shown in Table 2(a)–(c) for theabove operating conditions. The mode shapes are found


Fig. 3. Two area power system.

Table 2Rotor modes of the system for the three operating conditions (a)–(c)

Mode no. Rotor mode Frequency (Hz) Damping factorz Nature of the mode

Operating condition (a)1 2 0.02601 j 4.5555 0.73 0.01 Local mode2 2 1.08931 j 7.9126 1.26 0.14 Local mode3 2 1.14731 j 1.8903 0.30 0.52 Interarea mode4 2 2.83631 j 3.8716 0.62 0.60 Local modeOperating condition (b)1 2 0.18831 j 4.7684 0.76 0.04 Local Mode2 2 1.13321 j 8.0383 1.28 0.14 Local Mode3 2 1.10501 j 2.1595 0.34 0.46 Interarea mode4 2 2.63701 j 4.0269 0.64 0.55 Local ModeOperating condition (c)1 0.21641 j 4.5166 0.72 2 0.05 Local Mode2 2 1.17781 j 7.4154 1.18 0.16 Local Mode3 2 1.04851 j 2.5313 0.40 0.38 Interarea Mode4 2 3.23861 j 4.0119 0.64 0.63 Local Mode

from the normalized right eigenvector components corre-sponding to rotor speeds of all the machines. Using thisconcept of mode shapes, it is inferred that mode 1 is alocal mode in area A1 and this is the most crictical modeunder the operating condition (a). In this mode machine 3 isswinging against machines 2 and 5. The second mode forthe same operating condition is a local mode of area A2 andmachine 1 is swinging with machine 4. Mode 3 is an inter-area mode, with machines of area A1 swinging againstmachines of area A2. The last mode is a local mode ofarea A1 and machine 3 is swinging against machines 2and 5. In the same way, mode shapes are identified for theother operating conditions.

The participation factors for the corresponding conditionsare shown in Table 3(a)–(c) respectively. With the assump-tion that a damping ratio less than or equal to 0.45 is critical,two critical modes namely mode 1 and mode 2 are observed(Table 2(a)). Comparing the participation factors of thespeed component of all the machines of these two criticalmodes, the machines 3 and 1 are having the highest parti-cipation factor for modes 1 and 2, respectively.

The modes 1 and 2 of operating condition (b) areobserved as critical modes and machines 3 and 1 are havingthe highest participation factors for the respective modes.Operating condition (c) has three critical modes namelymodes 1, 2 and 3 and machines 3, 1 and 2 are respectivelyidentified for these modes. Hence PSSs are to be located atmachines 1, 2 and 3.

6.2. Tuning of the FPSS

Simulation of the non-linear model of the power system iscarried out for three different operating conditions withdifferent disturbances (Case 1, Case 2 and Case 3) as

explained below using the configuration of FPSS and tuningprocedure mentioned in Sections 4 and 5.

In order to facilitate comparison with ConventionalPower System Stabilizer (CPSS), simulation was alsocarried out replacing FPSS by CPSS. The transfer functionof the CPSS used is as follows

G�s� � KCSTW

�1 1 STW��1 1 ST1�2�1 1 ST2�2

�2�

TW was chosen as 10 s. The time constantsT1 andT2 werechosen as 0.3012 and 0.0536 s, respectively, using thedesign procedure given in Ref. [17]. The CPSSs werelocated on machines 1, 2 and 3. GainKC was tuned usingthe CGA.

Case 1:A small disturbance namely change in voltagesettingVref of the exciter system of the machine 3 was simu-lated on operating condition 6.1(a). FPSS parameters ofmachine 3 were tuned for the relative rotor angled30

(defined asd3 2 d0) where d0 is the system referencedefined as

d0 �

XNM

i�1

Midi

XNM

i�1

Mi

where NM is the total number of machines in the system andMi is the moment of inertia of theith machine.

Case 2:A small disturbance namely change in voltagesettingVref of the exciter system of the machine 1 is simu-lated on operating condition 6.1(b) and FPSS parameters ofmachine were tuned for the relative rotor angled10.

Case 3:To excite the interarea mode, a three-phase to


Table 3Participation factors for the operating conditions (a)–(c)

Rotor modes

Machine no. Mode 1 Mode 2 Mode 3 Mode 4

Operating condition (a)Dv1 0.1406 0.5327 0.1533 0.0503Dv2 0.3118 0.0263 0.4741 0.2266Dv3 0.4206 0.1245 0.1331 0.6506Dv4 0.0976 0.3186 0.2285 0.1533Dv5 0.2637 0.1882 0.3554 0.2802Operating condition (b)Dv1 0.1582 0.5391 0.1526 0.0524Dv2 0.3287 0.0316 0.5179 0.2274Dv3 0.4130 0.1402 0.1284 0.6415Dv4 0.1084 0.2455 0.2361 0.1704Dv5 0.2767 0.2371 0.3043 0.2576Operating condition (c)Dv1 0.0031 0.0176 0.0009 0.0013Dv2 0.0062 0.0007 0.0073 0.0033Dv3 0.0095 0.0049 0.0004 0.0122Dv4 0.0015 0.0116 0.0045 0.0018Dv5 0.0054 0.009 0.0056 0.0053

Fig. 4. RMSD index for the Case 1.

Table 4RMSD index for Case 1

Type of PSS Relative rotor angled30

Without PSS 0.710CPSS 0.695FPSS 0.682

ground fault at bus 6 on one of the two tie lines betweenbuses 6 and 7 was simulated on operating condition 6.1(c).The fault was cleared in four cycles by disconnecting thefaulted transmission line. After sixteen cycles, the faultedtransmission line was reclosed successfully. FPSS para-meters of machine 2 were tuned for the tie-line (MW)flow of line 6–7.

The following GA parameters were used in the search forboth CPSS and FPSS:

Population size� 20:String length� 8 bits:Crossover probability� 0:8:Mutation probability� 0:001:Convergence tolerance� 0:001:Maximum number of generations� 300:WLmax for WL � 0:7:WLmin for WL � 0:1:KFmax for KF � 5:KFmin for KF � 0:001:KCmax for KC � 40:KCmin for KC � 0:01:

High crossover probability and low mutation probabilityhave been chosen [14,18] in view of the small population size.

6.3. Results of tuning

The optimal parameters obtained for FPSS of machine 3

are WL � 0:451 pu andKF � 0:52: Fig. 4 shows that thevariation of the RMSD index with number of generationsfor Case 1 using the CGA. The total number of generationstaken is 177.

FPSS parameters of machine 1 is tuned for the relativerotor angle d10 for Case 2 and the optimal parametersobtained for FPSS of machine 1 areWL � 0:324 pu andKF � 0:588: The total number of generations taken is 216.FPSS parameters of machine 2 is tuned for the tie-line(MW) flow of line 6–7. The optimal parameters obtainedfor FPSS of machine 2 areWL � 0:409 pu andKF � 0:723and the total number of generations taken is 243.

The values ofKC1 � 13:03; KC2 � 12:47 and KC3 �14:89 were obtained for the Cases 1, 2 and 3, respectively,for CPSS. Total number of generations for the three casesare 146, 189 and 176.

Simulation was carried out for Case 1 with FPSSs locatedat machines 1, 2 and 3 and with their parameters set to theoptimal values determined using CGA. The time responseof the relative rotor angled30 without stabilizer, with CPSSand with FPSS are given in Fig. 5. The figure shows that thedamping of the mode is adequate.

A comparison of the time responses with CPSSs andFPSSs can be made from Fig. 5 as well as from Table 4which shows that the performance of FPSS is superior tothat of CPSS.

Simulation was carried out for Case 2 with all the opti-mally tuned FPSSs located at machines 1, 2 and 3. The


Fig. 5. System response for Case 1.

Fig. 6. System response for Case 2.

system response of the relative rotor angled10 (Fig. 6)shows that the damping is inadequate for both FPSS andCPSS. RMSD index values for the rotor angled10 are shownin Table 5.

The FPSS parameters of machine 1 is retuned keeping theother parameters same. The retuned parameters are:WL �0:489 pu andKF � 0:663 of FPSS andKC1 � 15:97 forCPSS. The time response obtained through simulationwith the retuned parameters is shown in Fig. 7. The oscilla-tions are damped effectively compared to Fig. 6. The RMSDindex values for FPSS and CPSS is shown in Table 5.

Simulation was carried out for Case 3 with optimal para-meters of FPSSs of 2 and 3 and retuned parameters of FPSS1. The time response of the real power flow in tie-line 6–7(Fig. 8) shows that the oscillations are damped effectivelywith FPSS as compared to CPSS. RMSD index evaluatedfor the MW flow in tie-line 6–7 is shown in Table 6.

Again, the adequacy of damping was checked for Case 1

through simulation using the retuned parameters formachine 1 and the response was found to be the same asgiven in Fig. 5. From the results (both the time response andthe RMSD index values) of the three cases studied, it isinferred that the performance of FPSS is better than theCPSS.

7. Conclusions

The paper gives a general design procedure for locatingfuzzy logic based power system stabilizer in a multimachinepower system and the parameters of all the stabilizers aretuned through genetic algorithm. It is easy to implement andthe RMSD index values are comparatively low for all thedisturbances. Simulations of the response of the proposedstabilizer to various disturbances, changes in networkconfiguration and loading pattern have demonstrated theeffectiveness of the FPSS. It is also seen that the FPSS


Fig. 7. System response for Case 2 with retuned parameters.

Fig. 8. MW flow in tie-line (6–7) for Case 3.


Type of PSS MW flow in tie Line 6–7

Without PSS 120.701CPSS 88.1296FPSS 80.9706


Type of PSS Relative rotor angled10 with optimalparameters

Relative rotorangled10 withretuned parameters

Without PSS 6.3861 6.3861CPSS 4.0202 3.0952FPSS 3.7821 2.3470

can damp both local and interarea modes of oscillationseffectively.

Appendix A

Differential equations of synchronous machines

_d � v 2 1

Tj _v � Tm 2 Te 2 Dv

T 0qo_E 0d � 2E 0d 2 �xq 2 x 0q�Iq

T 0qo_E 0d � 2E 0fd 2 E

Te � E 0dId 1 E 0qIq 2 �x 0q 2 x 0d�IdIq

E � E 0q 2 �xd 2 x 0d�Id


Machine no. 1 2 3 4 5

Parameters of the machinesxd (pu) 0.1026 0.1026 1.0260 0.1026 1.0260xq (pu) 0.0658 0.0658 0.6580 0.0658 0.6580x0d (pu) 0.0339 0.0339 0.3390 0.0339 0.0339x0q (pu) 0.0405 0.0405 0.4050 0.0405 0.4050T 0do(s) 5.67 5.67 5.67 5.67 5.67T 0qo(s) 0.9 0.9 0.9 0.9 0.9H (s) 80.0 80.0 10.0 80.0 10.0Parameters of the excitersKA 190.0 190.0 190.0 190.0 190.0TA (s) 10.0 10.0 10.0 10.0 10.0KF 0.08 0.08 0.08 0.08 0.08TF (s) 1.0 1.0 1.0 1.0 1.0TR (s) 0.04 0.04 0.04 0.04 0.04Parameters of the governorsTS (s) 0.2 0.2 0.2 0.2 0.2TB (s) 0.42 0.42 0.42 0.42 0.42TA (s) 0.1 0.1 0.1 0.1 0.1TC (s) 0.03 0.03 0.03 0.03 0.03

Transmission lines data in pu

S.no. Line no. R X B/2

1 1–7 0.00435 0.01067 0.015362 2–6 0.00468 0.04680 0.004043 3–6 0.01002 0.03122 0.032044 3–6 0.01002 0.03122 0.032045 5–6 0.00711 0.02331 0.027326 6–7 0.06048 0.19178 0.105727 7–8 0.01724 0.04153 0.060148 4–8 0.00524 0.01184 0.01756

Loads in puL1 � 7.52 j 5.0L2 � 8.52 j 5.0L3 � 7.02 j 4.5

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Machine no. 1 2 3 4 5

P (pu) 5.1076 8.5835 0.8055 8.567 0.8501Q (pu) 6.8019 4.3836 0.4353 4.6686 0.2264V (pu) 1.0750 1.050 1.025 1.075 1.025

Note: All the parameter values are based on system base of 100 MVA

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Stability enhancement of a multimachine power system using fuzzy logic based power system stabilizer tuned through genetic algorithm