A FUZZY LOGIC BASED DISCRETE MODE POWER SYSTEM

STABILIZER

M. Chetty

ABSTRACT

A Fuzzy Power System Stabiliser (FPSS) design using discrete inputs ispresented. The FPSS basically utilises only one measurable plant signal asinput, namely, the generator speed. The speed signal is discretised resultingin three inputs to the FPSS. A simplified power system represented by asingle generator transmitting power to the infinite bus is considered forinvestigations. The system is simulated in simulink while the FPSS isimplemented using Fuzzy Logic Toolbox. There are six rules for the fuzzifica-tion and two rules for defuzzification. To provide satisfactory behaviour inthe entire region of operation of the generator, additional signals namely,generator’s active and reactive power are used as inputs to FPSS. Simulationstudies show the superior performance of the proposed FPSS compared to anoptimally designed Conventional Power System Stabiliser.

KeyWords: Controller design, power system stabiliser, fuzzy logic, stability.

Manuscript received November 30, 2000; revised April 13,2001; accepted September 26, 2001.

M. Chetty is with Computational Intelligence Research Group,Gippsland School of Computing and Information Technology,Monash University, Churchill, VIC- 3842, AUSTRALIA.

327 Asian Journal of Control, Vol. 4, No. 3, pp. 327-332, September 2002

–Brief Paper–

I. INTRODUCTION

Design of Power System Stabiliser (PSS) is an im-portant issue from the viewpoint of power system stabilitysince it damps out local plant modes and inter-area modesof oscillations without compromising the stability of othermodes. A conventional PSS comprises of a wash outcircuit and a cascade of two phase lead networks. Todesign such a PSS, many strategies [1-6] have been pro-posed in the past that included techniques such as rootlocus, Bode plots, eigenvalue assignment, multivariablefrequency domain techniques, optimal controller design,etc. Numerous applications of new techniques [7-11]based on expert system, neural network and rule basedfuzzy logic for PSS designs have also been reported.The methodology of the FPSS appears promising since thepower system is quite complex for analysis and also sincethe methodology enables to realise a controller in lessdevelopment time, at lower development cost and withbetter performance.

In this paper, the proposed FPSS design is imple-mented on a simplified power system model representedby a single generator transmitting power to the infinite

bus. MATLAB’s simulink and fuzzy logic toolbox havebeen used for system simulation. Generator speed is usedas an input to the FPSS which is then descretised resultingin three discrete inputs. The rule base consists of six rulesfor the fuzzification and two rules for defuzzification. Toprovide a satisfactory behaviour of the stabiliser due tovariations in plant parameters (due to changes in theoperating point of the generator), additional signals ofgenerator’s active and reactive power are used as inputs toFPSS and gain settings for the three FPSS inputs are madeadaptive. It is shown through computer simulation, that astabiliser designed by the proposed method provides asuperior performance compared to an optimally designedCPSS reported earlier [2] on the same system with identi-cal operating conditions.

II. THE SYSTEM STUDIED

The system to be investigated comprises of a syn-chronous generator connected to an infinite bus through along transmission line as shown in Fig. 1(a). A third ordermodel represents the generator. The synchronous genera-tor under consideration is considered to be equipped withan IEEE Type-1 excitation system. The exciter modelneglects saturation and the voltage limits of amplifieroutput. The prime mover torque is considered constant.

The block diagram representation of the power sys-tem is shown in Fig. 1(b). The symbols and notations arethe same as those given in [2], [12]. The relations in the

Asian Journal of Control, Vol. 4, No. 3, September 2002 328

block diagram apply to a two-axis machine representationwith a field circuit in the direct axis and without damperwindings. The interaction between the speed and voltagecontrol equations of the machine is expressed in terms ofsix constants K1, ……, K6. These constants with theexception of K3 (which is only a function of impedance)are dependent upon the actual real and reactive powerloading as well as the excitation levels in the machine [12].These constants are evaluated using the relations given inAppendix-B. The dynamic power system model in statespace form follows.

∆ω= 12H

(K1 ∆δ – K2 ∆Eq′ ) + 1

2H∆Tm

∆δ = 2πf ∆ω

∆Eq′ = 1

Tdo, K3

( – ∆Eq, – K3 K4 ∆δ + K3 ∆Efd)

∆Efd = 1TE

( – KE ∆Efd + ∆VR)

∆VE = – ∆VE

TF

+KF

TFTE

( – KE ∆Efd + ∆VR)

∆VR = 1TA

( – ∆VR – KA K6 ∆Eq, – KA K5 ∆δ

– KA ∆VE + KA UPSS)

The set of equations can be arranged as

x = Ax + γ ∆Tm

where the state vector

x = [∆ω, ∆δ, ∆Eq, , ∆Efd, ∆VE, ∆VR]t

A is the plant matrix and γ is the disturbance matrix. ∆Tm.

III. DESIGN OF FUZZY PSS

The FPSS design mainly consists of choosing appro-priate inputs followed by designing the fuzzifier, rulebase, inference mechanism and the defuzzifier. In thispaper, the proposed FPSS utilizes the speed deviation, ∆ωof the synchronous generator as the input. Three separatesignals, generated from ∆ω using time delay blocks, arethen fed as inputs in the FPSS. The Simulink representa-tion of the complete FPSS model is shown in Fig. 2.

3.1 The FPSS inputs

The first input U1 to the FPSS comes directly from∆ω, i.e.

Generator InfiniteBus

(a) Single machine infinite bus power system

∆Tm

∆Em ∆VR

∆VE

∆Eq,

K1

K2 K4

K6

K5

Σ

Σ Σ Σ

∆ω

∆ω

∆δ

PSS

UPSS

12Hs

2πfs

1KE + sTE

KA1 + sTE

sKF1 + sTF

K31 + T, K3do

IEEE Type-1 excitation System

+

+

+ +–

––

–

–

–

(b) Model of the power system

Fig. 1. The power system under consideration.

Fig. 2. Block diagram of complete FPSS.

KoutGain

Fuzzy Logic ContrbllerMux

Mux

Sum

U3KT

U2(K-1)T

U1(K-1)T

U1KT

U2KT

Unit Delay

G2

G3

Gain

Gain

output of FPSS

Saturation

Saturation

Sum

SaturationGain

G1

Unit Delay

-K-

Gain

Speed deviation

input

1—z

1—z

–+

–

+

329 M. Chetty: A Fuzzy Logic Based Discrete Mode Power System Stabilizer

U1 = ∆ω

As the speed signal is discretised, the speed signal for theKth sample is

U1KT = ∆ωKT

The second and third inputs U2 and U3 of the FPSS areobtained by using a ‘sample and hold circuit’ with asampling period, T.

The second input to the FPSS at any sampling instantKT is given as

U2KT = U1KT – U1(K – 1)T

The sample U1(K – 1)T is one sampling period before thesample U1KT input.

The third input to FPSS at the end of the Kth sampleis U3KT and is derived from two consecutive samples of U2as

U3KT = U2KT – U2(K – 1)T

The second input U2 and the third input U3 definedin such manner can be viewed [13] as respectively the 2nd

level difference and the 3rd level difference of input U1.The FPSS takes all three inputs separately and am-

plifies them respectively via the gains G1, G2 and G3. Sincethe speed deviation under steady operation is zero, thenominal values of the inputs to FPSS are set equal to zero.The crisp values of inputs to FPSS are scaled between therange [–1, 1] because each input membership functions isdesigned to accept inputs within this range. An identicalrange is adopted for the output membership functions.Choosing a positive and negative range to the input (aswell as output) membership functions allows the FPSS toinject either positive or negative stabilising signals into theexcitation system. Thus, accelerating or deceleratingtorque, as necessary, can be applied to the generator rotor.The next step is to determine the number and shape of themembership functions.

3.2 Input and output membership functions

Numerous membership functions were investigatedfor this particular design of FPSS. The input (antecedent)membership functions that proved most promising werethe arctangent functions describing the gaussian curveand these were therefore chosen.

For the inputs, two membership functions, namely,Pin (positive) and Nin (negative), one for each of the threeinputs is chosen. For example, the Pin1 is the positive inputmembership function of the first input while Nin2 is thenegative input membership function of the second input.Hence, a total of six membership functions are used for thefuzzification.

The input membership functions Pin and Nin areshown in Fig. 3. The membership functions for each of thethree inputs are considered identical. This shape of themembership function produces an effect similar to a PIDcontroller and compensates any large differences comingfrom the 1st level difference, 2nd level difference or 3rd leveldifference. For example, if any of FPSS input is large (say,U1KT – U1(K – 1)T =>large), then the FPSS would output alarge compensating signal of appropriate polarity due tothe chosen shape of the membership functions. Thus, thetendency of the generator rotor would be to neither accel-erate nor decelerate (i.e. maintain constant value).

For the outputs, two membership functions, namely,Pout and Nout are used for the deffuzification processwhere the subscript indicates the output membershipfunction. The functions consist of two opposite slopedlines as shown in Fig. 4.

The reason for choosing a linear relationship isbecause the output membership function is usually a linearrepresentation of the input membership functions. In thedesign, this was accomplished using the trapezoidal mem-bership functions. Once the membership functions areformulated, these are used to design the rule base.

3.3 Rule base

The designed rule base of the FPSS consists of thefollowing types of rules:

1. If (Ui is Pini) then (outFLC is Pout) (1)2. If (Ui is Nini) then (outFLC is Nout) (1)

PinNin1

0.5

0–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

input variable

1

0.5

0–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

output variable “outFLC”

Nout Pout

Fig. 4. Output membership functions.

Fig. 3. Input membership functions.

Asian Journal of Control, Vol. 4, No. 3, September 2002 330

Where i = 1, 2, 3.

Thus, there are all together six rules comprising therule base. Each rule has been given equal importance andprovided with a weight of 1.

3.4 Inference mechanism and defuzzification

The input membership functions Pini (i = 1, 2, 3) andNini (i = 1, 2, 3) individually fuzzifies the three crisp inputsacting on it which then appear as inputs to Pout and Noutmembership functions. The individual contribution com-ing from each rule depends not only on the membershipfunctions but also on the type of operators used in theinference mechanism.

In the current design, the max function is used forboth, implication of the inputs as well as aggregation ofthe individual fuzzy outputs. The overall fuzzy output isthen defuzzified using centroid method to obtain a crispoutput corresponding to the center of the area under theoverall fuzzy output. This crisp value is between –1 and+1. The defuzzified output is then amplified by the gainKout (see Fig. 2) before it is injected into the excitationsystem of the synchronous generator. Figure 5 shows theoverall relationship between the inputs, rules, member-ship functions and illustrates the outputs illustrating theprocess of input fuzzification and output defuzzificationfor an operating condition corresponding to which U1=0.8, U2 = 0.2 and U3 = 0.

3.5 Gain settings and adaptation

At a given operating point ( e.g. Po = 0.8, Qo = 0.6)of the generator, numerous simulations are carried out byvariation of the gain settings G1, G2 and G3 and observingthe generator speed response. At this operating point, aspecific gain combination G1 = 250, G2 = 1000, G3 =1000, that resulted in a satisfactory generator response waschosen. Similarly, at other operating conditions, simula-tions are again carried out to obtain the corresponding gaincombinations. Based on these results, an adaptation mecha-nism for the gain settings G1, G2 and G3 is developed

which maintains a satisfactory performance for changes inthe plant (due to variations in the operating condition).A set of IF-THEN-ELSE rules constructed for this pur-pose selects the appropriate gain combination for aparticular operating condition. These rules are of thefollowing type

If (Pmin1 ≤ Po ≤ Pmax1) and (Qmin1 ≤ Qo ≤ Qmax1)

then G1 = G1o, G2 = G2o, G3 = G3o

Here, G1o, G2o, G3o are the gain settings correspond-ing to the operating condition (i.e. Po and Qo). Theminimum and maximum limits for Po(or Qo) are Pmin(orQmin1) and Pmax1(or Qmax1).

V. SIMULALTION RESULTS

The power system under consideration is essentiallythe same as that studied by earlier researchers [2,3]. Thesystem parameters and the operating conditions are thesame as given in [2]. Various studies have been performedto investigate the effect of the proposed FPSS. The resultsare compared with a Conventional Power System Stabiliser(CPSS).

5.1 Conventional PSS

The CPSS model is given by the transfer function

CPSS(s) = Kc

sTw

1 + sTw

1 + sT1

1 + sT2

1 + sT3

1 + sT4

where Tw is the wash out time constant. T1-T4 are PSS timeconstants and Kc is the PSS gain. For the system underconsideration, the tuned parameters of the CPSS by ISEtechnique for system under consideration reported in [2]are chosen. These parameters are Kc = 47.0, Tw = 10, T1 =T3 = 0.35sec, T2 = T4 = 0.05 sec.

5.2 Selection of sampling time, T

The complete power system model with FPSS issimulated using MATLAB’s Simulink toolbox whichuses ode5 solver. For obtaining generator time responses,the ode5 solver time step, t is chosen as t = 0.005 sec. Thesampling time of the unit delay blocks (which generatesinputs U2 and U3) has to be equal to or multiple of the stepsize, t. Figure 6 shows the effect of variation of samplingtime, T, of the delay blocks. It can be observed from thefigure that the sampling time T = 0.005 sec provides asatisfactory behaviour. Any increase in the sampling timefrom this value results in deterioration of system behaviour.Hence, this value of sampling time is chosen and keptfixed for further investigations.

U1 = 0.8 U2 = 0.2 U3 = 0 outFLC = 0.135

1

2

3

4

5

6

1 1 1–1

1–1

Fig. 5. Inference and defuzzification.

331 M. Chetty: A Fuzzy Logic Based Discrete Mode Power System Stabilizer

5.3 Performance of the FPSS

The nominal operating point of the generator isconsidered at Po = 0.8 and Qo = 0.6. In power systemstability studies, the PSS is incorporated to improve thegenerator speed deviation response by reducing the set-tling time and the overshoot. This augments the dynamicstability of the power system. In the investigations re-ported here, a 0.1 pu step change in the turbine torque isconsidered as the disturbance and the influence of PSS onthe generator speed responses due to disturbance in tur-bine torque are examined. Figure 7 shows the three speeddeviation responses, namely, (i) Open loop, (ii) WithCPSS and (iii) With FPSS. The open loop system withoutPSS is highly oscillatory. From the figure, it can be seenthat the FPSS has increased the power system stability byreducing the settling time and overshoot of the speeddeviation response when compared with the CPSS.

The effect on the generator speed deviation due to

20% variations of active power Po and the reactive powerQo from the nominal point is shown respectively in Fig. 8and Fig. 9. The responses are observed to be satisfactory.

VI. CONCLUSIONS

A FPSS design utilising discrete inputs derived fromgenerator speed deviation is presented. Structure anddesign of the proposed stabiliser are highlighed. Theactive and reactive power inputs to the FPSS enables anadaptation of the gain settings for the discretised speedsignal inputs. The proposed FPSS comprises of only sixrules. An improved damping and system dynamics isobserved by the inclusion of the FPSS for varying operat-ing conditions of generator. The effect of FPSS due tovariation of active and reactive power is also presented Acomparison between the simulation responses due to theproposed FPSS and an optimally designed CPSS showsthe superior performance of the FPSS over a wide range ofoperating conditions.

ACKNOWLEDGEMENT

The author wishes to thank Mr. N. Trajkoski for the

8

6

4

2

0

–2

–40 0.5 1 1.5 2 2.5 3

T = 0.01T = 0.005T = 0.02

× 10–3

Time (sec)

spee

d de

viat

ion,

rad

/sec

Fig. 6. Effect of sampling time, T.

Open-LoopCPSSFPSS

21.5

10.5

0–0.5

–1–1.5

–2

× 10–3

0 1 2 3Time (sec)

spee

d de

viat

ion,

rad

/sec

Fig. 7. Speed responses to a 10% step change in turbine torque.

8

6

4

2

0

–2

–40 0.5 1 1.5 2 2.5 3 3.5

Po = 0.64Po = 0.8Po = 0.96

× 10–4

Time (sec)

spee

d de

viat

ion,

rad

/sec

Fig. 8. Effect of variation of active power, Po.

Qo = 0.48Qo = 0.6Qo = 0.72

76543210

–1–2–3

× 10–4

0 0.5 1 1.5 2 2.5 3 3.5Time (sec)

spee

d de

viat

ion,

rad

/sec

Fig. 9. Effect of variation in reactive power, Qo.

Asian Journal of Control, Vol. 4, No. 3, September 2002 332

assistance in simulating the system in Simulink and hishelp in compiling some of the results.

APPENDIX A

The nominal parameters of the system under inves-tigation are as follows. All data is in pu except the inertiaconstant, H and the time constants are given in seconds.

Operating conditions

Po = 0.8, Qo = 0.6, f = 50 Hz, Vto = 1.0

Generator

H = 5.0s, Tdo′ = 6.0s, xd = 1.6, xd

′ = 0.32, xq = 1.55

IEEE Type-I excitation system

KA = 50.0, TA = 0.05s, KE = –0.05, TE = 0.05s,

KF = 0.05, TF = 0.5s

Transmission line

xe = 0.4

APPENDIX B

The constants K1, ……, K6 are evaluated using therelations given below. The subscript o means steady statevalue. The steady state values of d-q axis voltage andcurrent components for the single machine infinite bussystem are calculated using the phasor diagram relationsas

K1 =

xq – xd′

xe + xd′ IqoV∞ sin δ0 +

EqoV∞xe + xq

cos δo

K2 =V∞

xe + xd′ sin δo

K3 =xd

′ + xexd + xe

K4 =xd – xd

′

xe + xd′ V∞ sin δo

K5 =xq

xe + xq

Vdo

Vto

V∞ cos δo –xd

′

xe + xd′

Vqo

Vto

V∞ sin δo

K6 =xe

xe + xd′

Vqo

Vto

REFERENCES

1. Larsen, E.V. and D.A. Swann, “Applying Power Sys-tem Stabilisers: Part I, Part II and Part III,” IEEETrans. Power Apparatus Syst., Vol. PAS-100,pp.3017-3046 (1981).

2. Kothari, M.L., J. Nanda and K. Bhattacharya, “Dis-crete Mode Power System Stabilisers,” IEE Proc. Pt.C, Vol. 140, No. 6, pp. 523-531 (1993).

3. Aldeen, M. and M. Chetty, “A Dynamic Ouput Feed-back Power System Stabiliser,” Proc. The Contr.’95 Conf., Vol. 2, pp. 575-579, Melbourne (1995).

4. Samarasinghe, V.G.D.C. and N.C. Pahalawaththa,“Design of Universal Variable-Structure Controllerfor Dynamic Stabilization of Power Systems,” IEEProc. Gener., Transm. Distrib., Vol. 141, No. 4, pp.363-368(1994).

5. Anderson, P.M. and A.A. Fouad, Power System Con-trol and Stability, IEEE Press, New York (1994).

6. Ahmed, S.S., L. Chen and A. Petroianu, “Design ofSuboptimal H∞ Excitation Controllers,” IEEE Trans.Power Syst., Vol. 11, No.1, pp. 312-318 (1996).

7. Ohtsuka, K., T. Taniguchi, T. Sato, S. Yokokawa andY. Ueki, “A H∞ Optimal Theory-Based GeneratorControl System,” IEEE Trans. Energy Conver., Vol.7, No. 1 (1992).

8. El-Metwally, K.A. and O.P. Malik, “A Fuzzy LogicPower System Stabiliser,” IEE Proc. Gener. Transm.Distrib., Vol. 145, No.3, pp. 277-281 (1995).

9. Lie, T.T and A.M. Sharaf, “An Adaptive Fuzzy LogicPower System Stabiliser,” Electr. Power Syst. Res.,Vol. 38, pp. 75-81 (1996).

10. Hsu, Y.Y. and C.R. Chen, “Tuning of Power SystemStabiliser Using Artificial Neural Network,” IEEETrans. Energy Conver., Vol.6, No. 4, pp. 612-619(1991).

11. Hiyama, T., “Application of Neural Network to RealTime Tuning of Digital Type PSS,” Proc. Int. PowerEng. Conf. (IPEC), Singapore, pp.392-397 (1993).

12. Mello, De F.P. and C. Concordia, “Concepts ofSynchronous Machine Stability as Affected byExcitation,” IEEE Trans. Power Apparatus Syst., Vol.PAS-88, pp. 316-327 (1969).

13. Terano, T., K. Asai and M. Sugeno, Applied FuzzySystems, Academic Presss Inc., pp.86-93 (1994).