power system stabiliser for multimachine power system using robust decentralised periodic output...

Power system stabiliser for multimachine powersystem using robust decentralised periodic outputfeedback

R. Gupta, B. Bandyopadhyay and A.M. Kulkarni

Abstract: Power system stabilisers (PSSs) are added to excitation systems to enhance the dampingduring low-frequency oscillations. The design of decentralised robust PSS for four machines withten buses using periodic output feedback is proposed. The nonlinear model of a multimachinepower system is linearised at different operating points and 16 linear state-space models areobtained. For each of these models an output injection gain is obtained using LQR technique.A decentralised robust periodic output feedback gain which realises these output injection gains isobtained using the linear matrix inequality approach. This method does not require states of thesystem for feedback and is easily implemented. The robust decentralised periodic output control isapplied to the nonlinear model of a multimachine system at different operating (equilibrium) pointsand gives encouraging results for the design of power system stabilisers.

1 Introduction

Power System Stabilisers are added to excitation systems toenhance the damping of an electric power system duringlow-frequency oscillation. Several methods are used for thedesign of PSSs. Tuning of supplementary excitation controlsfor stabilising system modes of oscillations has been thesubject of much research during the past 35 years [1].

Several methods are used for the design of PSSs. Twobasic tuning techniques have been successfully utilised withpower system stabiliser applications: the phase compen-sation method and the root-locus method. Phase compen-sation consists of adjusting the stabiliser to compensate forthe phase lags through the generator, excitation system andpower system such that the stabiliser path provides torquechanges which are in phase with the speed changes [2–8].This is the most straightforward approach, easily understoodand implemented in the field, and the most widely used.A number of sequential and simultaneous approaches forthe tuning of the these parameters has been reported in theliterature [9]. Synthesis by root locus involves shifting theeigenvalues associated with the power system modes ofoscillation by adjusting the stabiliser pole locations in thes-plane [10]. This approach gives additional insight intoperformance by working directly with the closed-loopcharacteristics of the systems, as opposed to the open-loopnature of the phase compensation technique, but is morecomplicated to apply in the case of multivariable systems,particularly in the field. A few attempts have been made fordesigning power system stabilisers for multimachine power

systems using multivariable control theory [11–16]. It isknown that, for multimachine systems, eigenvalue assign-ment is often too involved and complex and may not providesatisfactory results if the techniques mentioned are used [17].

Recently, modern control methods have been used byseveral researchers to take advantage of optimal controltechniques. These methods utilise a multivariable state-space representation of a multimachine power system modeland calculate a gain matrix which when applied as a statefeedback control will minimise a prescribed objectivefunction [18–21]. In practice, not all of the states areavailable for measurement. In this case, the optimal controllaw requires the design of a state observer. This increasesthe implementation cost and reduces the reliability of thecontrol system. Another disadvantage of the observer-basedcontrol system is that even slight variations of the modelparameters from their nominal values may result insignificant degradation of the closed-loop performance.Hence it is desirable to go for an output feedback design.Dynamic output feedback is one of the techniques availablefor designing a controller based on output feedback, whichessentially leads to a higher-order feedback system [22].

The static output feedback problem is one of the mostinvestigated problems in control theory. Complete poleassignment and guaranteed closed-loop stability is still notobtained by using static output feedback [23]. Anotherapproach to the pole placement problem is to consider thepotential of time-varying periodic output feedback. It wasshown by Chammas and Leondes [24] that a controllableand observable plant was discrete-time pole assignable byperiodically time-varying piecewise-constant output feed-back. Since the feedback gains are piecewise constant, theirmethod could be easily implemented and indicated a newpossibility. Such a control law can stabilise a much largerclass of systems than static output feedback [25–29]. In theperiodic output feedback technique the gain matrix isgenerally full [26]. This results in the control input of eachmachine being a function of the outputs of all machines.Centralised periodic output feedback PSSs requires signaltransmission between the generating units. This requirementin itself no longer constitutes a problem from the practical

q IEE, 2005

IEE Proceedings online no. 20041127

doi: 10.1049/ip-cta:20041127

R. Gupta and B. Bandyopadhyay are with Interdisciplinary Progamme inSystems and Control Engineering, Indian Institute of Technology, Powai,Mumbai-400076, India

A.M. Kulkarni is with Department of Electrical Engineering, IndianInstitute of Technology, Powai, Mumbai-400076, India

Paper first received 5th August 2003 and in revised form 18th August 2004

IEE Proc.-Control Theory Appl., Vol. 152, No. 1, January 2005 3

and technical viewpoints due to the rapid advancement inoptical fibre communication and their adoption by powerutilities. However, if a completely decentralised PSS can befound so that no significant deterioration in systemperformance is experienced compared with centralisedperiodic output feedback schemes, such a scheme wouldbe more advantageous in terms of practicability andreliability. In such schemes, not only is the cost ofimplementation drastically reduced but also the risk ofloss of stability due to signal transmission failure isminimised [30]. Also, due to the geographically distributednature of the power system, the decentralised controlscheme may be more feasible than the centralised controlscheme. In a decentralised power system stabiliser, thecontrol input for each machine is a function of the output ofthat machine only. This can be achieved by designing adecentralised PSS using the periodic output feedbacktechnique in which the gain matrix should have all off-diagonal terms zero or very small compared with thediagonal terms. In this technique a decentralised PSS for allmachines can be designed using one algorithm. Thus thedecentralised stabiliser design problem can be translatedinto a problem of diagonal gain matrix design for amultimachine power system. However, in a decentralisedperiodic output feedback PSS, a proper synchronisationsignal is required to be sent to all machines. Thisrequirement is comparatively less involved than transmit-ting measured signals as in a purely centralised controller.

We propose the design of a decentralised robust powersystem stabiliser for a multimachine system using periodicoutput feedback.

2 Periodic output feedback

2.1 Review

The problem of pole assignment by piecewise constantoutput feedback was studied by Chammas and Leondes [24]for linear time-invariant systems with infrequent obser-vation. They showed that, by use of periodically time-varying piecewise constant output feedback gain, the polesof the discrete-time control system could be assignedarbitrarily (within the natural restriction that they be locatedsymmetrically with respect to real axis) [27, 29].

Consider a discrete time-invariant system with samplinginterval t

xðk þ 1Þ ¼ FtxðkÞ þ GtuðkÞ ð1Þ

yðkÞ ¼ CxðkÞ ð2Þwhere x 2 Rn; u 2 Rm; y 2 Rp and Ft;Gt and C are constantmatrices of appropriate dimensions. The following controllaw is applied to this system. The output is measured at thetime instant t ¼ kt; k ¼ 0; 1; . . . : We consider constant holdfunction because they are more suitable for implementation.An output sampling interval is divided into N subintervals oflength D ¼ t=N; and the hold function is assumed constanton these subintervals. Thus the control law becomes

uðtÞ ¼ KlyðktÞ ð3Þ

ktþ lD � t � ktþ ðl þ 1ÞD; KlþN ¼ Kl ð4Þfor l ¼ 0; 1; . . . ;N � 1: Applying periodic output feedbackin (4), i.e. KyðktÞ is substituted for uðktÞ; the closed-loopsystem becomes

xðktþ tÞ ¼ ðFN þ GKCÞxðktÞ ð5Þ

The problem has now taken the form of static outputfeedback problem. Equation (5) suggests that an outputinjection matrix G be found such that

rðFN þ GCÞ< 1 ð6Þwhere rðÞ denotes the spectral radius. By observability onecan choose an output injection gain G to achieve any desiredself-conjugate set of eigenvalues for the closed-loop matrixðFN þ GCÞ; and from N � n it follows that one can find aperiodic output feedback gain which realises the outputinjection gain G by solving

GK ¼ G ð7Þfor K. The controller obtained from this equation will givedesired the behaviour, but might require excessive controlaction. To reduce this effect we relax the condition that Kexactly satisfy the linear equation and include a constrainton the gain K. Thus we arrive at the following inequations:

kKk< r1; kGK � Gk< r2 ð8ÞUsing the Schur complement it is straightforward to bringthese conditions in the form of linear matrix inequalities(LMI) [28] as

�r21I K

KT �I

" #< 0

�r22I ðGK � GÞ

ðGK � GÞT �I

" #< 0 ð9Þ

In this form the LMI Toolbox of Matlab can be used forsynthesis [31].

2.2 Multimodel synthesis

For multimodel representation of a plant it is necessary todesign controller that will robustly stabilise the multimodelsystem. Multimodel representation of plants can arise inseveral ways. When a nonlinear system has to be stabilisedat different operating points, linear models are sought to beobtained at those operating points. Even for parametricuncertain linear systems, different linear models can beobtained for extreme points of the parameters. These modelsare used for stabilisation of the uncertain system.

Consider a family of plant S ¼ fAi;Bi;Cig; defined by

_xx ¼ Aix þ Biu ð10Þ

y ¼ Cix; i ¼ 1; . . . ;M ð11ÞBy sampling at the rate of 1=D we get a family of discretesystems fFi;Gi;Cig. We can find output injection gainsGi such that ðFN

i þ GiCiÞ has required set of poles. Nowconsider the augmented system defined as follows:

~FF ¼

F1 0 � � � 0

0 F2 � � � 0

..

. ... . .

. ...

0 0 � � � FM

26664

37775; ~GG ¼

G1

G2

..

.

GM

26664

37775; ~GG ¼

G1

G2

..

.

GM

26664

37775

ð12ÞThe linear equation

~FF ~NN�1 ~GG � � � ~GG K0

..

.

K~NN�1

264

375 ¼ ~GG ð13Þ

has a solution if ð ~FF; ~GGÞ is controllable with controllabilityindex ~nn; and ~NN � ~nn=(number of inputs or outputs). It hasbeen shown in [27], that the controllability of individualmodel generically implies the controllability of the

IEE Proc.-Control Theory Appl., Vol. 152, No. 1, January 20054

augmented system. Thus, for an exact solution, theminimum value of ~NN would be 80 (because ~NN � controll-ability index of the augmented system=number of inputs oroutputs) and that would result 80 gain matrices of size 4 4each. But, from the practical view point, computation ofgains in this way may not be useful because large number ofgain changes may make the implementation task difficult.Also, so computed gains may turned out to be large whichmay magnify the noise. To reduce this effect of gain, werelax the condition such that (13) satisfies exactly and alsoinclude a constraint on the gain. This can be explained in thefollowing.

Consider the four-machine and ten-bus system at onecondition leading to one 20th-order model. An outputinjection gain matrix can be obtained using DLQR theory.If a periodic output feedback gain has to be obtained for thissystem only, then minimum five gain matrices ðK0;K1;K2;K3;K4Þ each of 4 4 dimension are needed which willexactly realise the output injection gain computed for thissystem. Equation (7) has a solution if N � controllabilityindex of this system=number of inputs or outputs. For onesystem controllability index is 20. So a minimum five gainsequences are needed.

As we are dealing with robust stabilisation, we have tofind a K which will satisfy GiK ¼ Gi ði ¼ 1; . . . ; 16Þ for allthese equations. However, if we restrict our gain sequenceto five, there would not exist a common K which wouldsatisfy GiK ¼ Gi for i ¼ 1; . . . ; 16. That’s why thiscondition is relaxed and a robust K has been sought to beobtained by satisfying the equation GiK ¼ Gi approxi-mately. If the gain sequences are taken 80, there will be aunique solution for K and if it is more than 80, a nonuniquesolution may exist. Hence an approximate solution has beenobtained using an LMI approach and a five-gain sequencehas been used. The smaller the gain sequence, the easier willbe the implementation task. In this problem N ¼ 5 givesgood results.

Thus we consider the following inequalities:

kKk< r1

kGiK � Gik< r2i; i ¼ 1; . . . ;M ð14Þ

where r2 small means that the periodic output feedbackcontroller with gain K is a good approximation of theoriginal design output injection controller and r1 smallmeans low noise sensitivity. This can be formulated in theframework of linear matrix inequalities as in the case of thefast output sampling feedback technique [28]

�r21I K

KT �I

� �< 0

�r22iI ðGiK � GiÞ

ðGiK � GiÞT �I

� �< 0 ð15Þ

In this form the LMI Toolbox of Matlab [31] can be used forsynthesis. The robust periodic output feedback controllerobtained by this method requires only constant gains andhence is easier to implement.

2.3 Decentralised robust periodic outputfeedback

In periodic output feedback for a multimachine system, thegain matrix is generally full [26]. This results in the controlinput of each machine being a function of the outputs of allmachines. Decentralised robust periodic output feedbackcontrol can be achieved by making the off-diagonalelements of K0;K1; � � � ; K ~NN�1 matrices zero. So thestructure of Ki ði ¼ 0; � � � ;N � 1Þ matrix is assumed as

Ki ¼

ki11 0 0 0

0 ki22 0 0

0 0 ki33 0

0 0 0 ki44

2664

3775; i ¼ 0; � � � ;N � 1 ð16Þ

With this structure of Ki; the problem can be formulated inthe framework of linear matrix inequalities using (14) and(15) and the desired matrices obtained. Now it is evidentthat the control input of each machine is a function of theoutput of that machine only and this makes the powersystem stabiliser design using the periodic output feedbacktechnique a decentralised one. In a decentralised PSS, toactivate the proposed controller at same instant, a propersynchronisation signal is required to be sent to all machines.

3 Case study

The nonlinear differential equations governing the beha-viour of a power system can be linearised about a particularoperating point to obtain a linear model that represents thesmall-signal oscillatory response of a power system.Variations in the operating conditions of the system resultin variations in the parameters of the small-signal model.A given range of variation in the operating conditions of aparticular system thus generates a set of linear models eachcorresponding to one particular operating condition. Sinceat any given instant the actual plant could correspond to anymodel in this set, a robust controller would have to importadequate damping to each one of this entire set of linearmodels.

3.1 Linearisation of multimachine system

A simple model of a power system is considered. A fieldwinding and one damper winding on the q-axis [32, 33]is assumed for synchronous machines. The loads arerepresented by constant impedances and mechanicalpower is assumed to be constant. A single time-constantmodel of the AVR and static exciter are used.

A four-machine, ten-bus power system [33] is consideredfor designing a periodic output feedback controller using theMatlab LMI Toolbox. The single-line diagram of the systemis shown in Fig. 1. The machine data, line data and load flowdata are given in the appendix (Section 6.1) [33]. Sixteenlinearised plants corresponding to 16 operating conditionswith variations in generator power, network and loaddistribution (60 to 100% variations from peak loads) areobtained as given in the Appendix. The multimachine powersystem is modelled using the Simulink Toolbox of Matlaband the linear state-space models are obtained for the same.Discrete models are obtained for sampling time t ¼ 0:1 sec.PSS is activated at all generators. For each PSS, generatorslip is taken as the feedback signal.

Fig. 1 Single line diagram of four-machine and ten-bus system


3.2 Computation of robust decentralisedperiodic output feedback controller gains

Using the method discussed in Section 3 stabilising outputinjection gain matrices Gið20 4Þ are obtained usingDLQR theory [30, 34, 35].

Using the LMI approach, (14) and (15) are solved usingdifferent values of r to find the robust decentralised gainmatrix K. The robust decentralised periodic output feedbackgain matrix K ð20 4Þ is obtained as given in the Appendix(Section 6.2).

The closed-loop responses with this robust decentralisedgain K for all the linearised models are satisfactory and ableto stabilise the outputs. The eigenvalues of ðFN þ GKCÞ arefound to be within the unit circle.

3.3 Simulation with nonlinear model

A Simulink-based block diagram including all the nonlinearblocks is generated [33]. The slip signal with robustdecentralised gain K and a limiter is added to Vref signal.The output of the PSS is limited to prevent the PSS acting tocounter the action of AVR. Different operating points are

taken in different plants. The fault location is considered atone or two buses in different plants. The disturbanceconsidered is a self-clearing fault at different buses clearedafter 0.1 s. The limits of PSS output are taken as �0:1. In thisdecentralised PSS, to activate the proposed controller atsame instant, a synchronisation signal is required to be sentto all machines at 0.1 s.

Simulation results of different generators with faults areshown in Figs. 2 and 3without and with controller. As shownin the plots, the proposed controller is able to damp out theoscillations in 6 to 8 s after clearing the fault. Figures 4 and 5show, in the case of a synchronisation delay of 0.1 s. betweeneach PSS, they are still effective although performance is notas good as those shown in Figs. 2 and 3. Note that slips arecomputed relative to the centre of inertia which is defined as

Scoi ¼Pn

i¼1 HiSiPni¼1Hi

� Si ð17Þ

Fig. 2 Open- and closed-loop responses with fault using robustdecentralised periodic output feedback controller

a Generator 1; model 1, fault at bus 1b Generator 2; model 2, fault at bus 2

Fig. 3 Open- and closed-loop responses with fault using robustdecentralised periodic output feedback controller


Fig. 4 Open- and closed-loop responses with fault andsynchronising delay using robust decentralised periodic outputfeedback controller


Fig. 5 Open- and closed-loop responses with fault andsynchronising delay using robust decentralised periodic outputfeedback controller



4 Conclusions

A design scheme of a robust power system stabiliserfor a multimachine (four generators) power system usingdecentralised periodic output feedback has been developed.The slip signal is taken as output and periodic outputfeedback is applied at an appropriate sampling rate. It isfound that the robust decentralised controller provides gooddamping enhancement for various operating conditions.

5 References

1 Larsen, E.V., and Swann, D.A.: ‘Applying power system stabilizerspart-I: general concepts’, IEEE Trans. Power Appar. Syst., 1981,PAS-100, (6), pp. 3017–3024

2 Larsen, E.V., and Swann, D.A.: ‘Applying power system stabilizerspart-II: performance objective and tuning concepts’, IEEE Trans.Power Appar. Syst., 1981, PAS-100, (6), pp. 3025–3033

3 Schlief, F.R., and White, J.H.: ‘Damping for the North-West-Southwesttie-line oscillations - A: Analog simulator study’, IEEE Trans. PowerAppar. Syst., 1966, PAS-85, pp. 1239–1246

4 Schlief, F.R., and Angell, R.R.: ‘Damping of system oscillations with ahydrogenerating unit’, IEEE Trans. Power Appar. Syst., 1967, PAS-86,pp. 438–442

5 Concordia, C., and De mello, F.P.: ‘Concepts of synchronous machinestability as affected by excitation control’, IEEE Trans. Power Appar.Syst., 1969, PAS-88, pp. 316–329

6 Schlief, F.R., Hunkins, H.D., Hattan, E.E., and Gish, W.B.: ‘Control ofrotating exciters for power system damping: pilot applicationsand experience’, IEEE Trans. Power Appar. Syst., 1969, PAS-88,pp. 1259–1266

7 Warchol, E.J., Schlief, F.R., Gish, W.B., and Church, J.R.: ‘Alignmentand modeling of Hanford excitation control for system damping’, IEEETrans. Power Appar. Syst., 1971, PAS-90, pp. 714–724

8 Gerhart, A.D., Hillesland, T., Jr., Luini, J.F., and Rockfield, M.L.:‘Power system stabilizer field testing and digital simulation’, IEEETrans. Power Appar. Syst., 1971, PAS-90, pp. 2095–2100

9 Fleming, R.J., Mohan, M.A., and Parvatisam, K.: ‘Selection ofparameters of stabilizers in multimachine power systems’, IEEETrans. Power Appar. Syst., 1981, PAS-100, pp. 2329–2333

10 Bollinger, K.E., Laha, A., Hamilton, R., and Harras, T.: ‘Powerstabilizer design using root locus methods’, IEEE Trans. Power Appar.Syst., 1975, PAS-94, pp. 1484–1488

11 Aladeen, M., and Trinh, H.: ‘Design of a distributed power systemstabilizer’, J. Electr. Electron. Eng. Aust., 2002, 22, (1), pp. 1–8

12 Aladeen, M., and Lin, L.: ‘A new reduced order multimachine powersystem stabilizer design’, Electr. Power Syst. Res., 1999, 52, pp. 97–114

13 Nanda, J., Kothari, M., Bhattacharya, M., Aladeen, M., and Kalam, A.:‘Tuning of power system stabilizers multimachine systems using ISEtechnique’, Electr. Power Syst. Res., 1998, 46, pp. 119–131

14 Crusca, F., and Aladeen, M.: ‘Mixed H-infinity controller designfor power systems’. IEEE Conf. on System Engineering, Kobe, Japan,15–17 September 1992, pp. 266–269

15 Crusca, F., and Aladeen, M.: ‘Mutivariable frequency-domain tech-niques for the systematic design of stabilizers for large scale powersystems’, IEEE Trans. Power Syst., 1991, 6, (3), pp. 1133–1139

16 Anderson, J.H., Baulch, C., Aladeen, M., and Crusca, F.: ‘A comparisonof mutivariable time- and frequency-domain design methods for powersystem stabilizers including links with reported field trials’, IEEETrans. Power Syst., 1987, PWRS-6, (3), pp. 189–196

17 Yu, Y.N., and Li, Q.H.: ‘Pole placement power system stabilizersdesign of an unstable nine-machine system’, IEEE Trans. Power Syst.,1990, PWRS-5, pp. 353–358

18 Saleh, F.A., and Mahmoud, M.S.: ‘Design of power system stabilizerusing reduced-order models’, Electr. Power Syst. Res., 1995, 33, pp.219–226

19 Yu, Y.N., and Siggers, C.: ‘Stabilization and optimal control signalfor a power system’, IEEE Trans. Power Appar. Syst., 1971, PAS-90,pp. 1469–1481

20 Kumar, B., and Richards, E.F.: ‘An optimal control law by eigenvalueassignment for improved dynamic stability in power systems’, IEEETrans. Power Appar. Syst., 1982, PAS-101, pp. 1570–1577

21 Yu, Y.N., Vongsuriya, V.K., and Vedman, L.N.: ‘Application of anoptimal control theory to a power system’, IEEE Trans. Power Appar.Syst., 1970, PAS-89, pp. 55–62

22 Shrikant Rao, P., and Sen, I.: ‘Robust pole placememt stabilizer designusing linear matrix inequalities’, IEEE Trans. Power Syst., 2000, 15,(1), pp. 3035–3046

23 Syrmos, V.L., Abdallah, C., Dorato, P., and Grigoriadis, K.: ‘Staticoutput feedback: a survey’, Automatica, 1996, 33, (6), pp. 577–590

24 Chammas, A.B., and Leondes, C.T.: ‘Pole assignment by piecewiseconstant output feedback’, Int. J. Control, 1979, 29, pp. 31–38

25 Huang, T.L., Chen, S.C., Hyvang, T.Y., and Yang, W.T.: ‘Powersystem output feedback stabilizer design via optimal subeigenstructureassignment’, Electr. Power Syst. Res., 1991, 21, pp. 107–114

26 Gupta, R., Bandyopadhyay, B., and Kulkarni, A.M.: ‘Design ofdecentralised power system stabilizer for multimachine power systemusing periodic output feedback technique’. Proc. Int. Conf. on Quality,Reliabilty and Control, ICQRC2001, Mumbai, India, December 2001,pp. C45-1-8

27 Werner, H., and Furuta, K.: ‘Simultaneous stabilization based on outputmeasurement’, Kybernetika, 1995, 31, pp. 395–411

28 Werner, H.: ‘Multimodel robust control by fast output sampling– anLMI approach’. Proc. American Control Conf., Philadelphia, PA, USA,June 1998, pp. 3719–3723

29 Patre, B.M., Bandyopadhyay, B., and Werner, H.: ‘Control of discretetwo-time scale system by using piecewise constant periodic outputfeedback’, Syst. Sci., 1997, 23, pp. 23–37

30 Aldeen, M., and Crusca, F.: ‘Multimachine power system stabiliserdesign based on new LQR approach’, IEE Proc. Gener., Trans. Distrib.,1995, 142, (6), pp. 395–411

31 Gahenet, P., Nemirovski, A., Laub, A.J., and Chilali, M.: ‘LMI Toolboxfor Matlab’, The Math Works Inc., Natick, MA, 1995

32 Kundur, P.: ‘Power system stability and control’ (McGraw-Hill,New York, 1993)

33 Padiyar, K.R.: ‘Power system dynamics stability and control’ (Wiley,1999)

34 Narasimhamurti, N.: ‘Using Matlab control system Toolbox: studentwork book’ (University of Michigan, Dearbon)

35 Bruneer, U.: ‘Review of classical and modern control theoryusing Matlab: student edition, part 1’ (Prentice Hall, USA, September2002)

6 Appendix

6.1 Machine and network data

See Tables 1–4.

6.2 Robust decentralised periodic outputfeedback gains

K0 ¼

�0:5662 0 0 0

0 �0:5662 0 0

0 0 �0:1409 0

0 0 0 �0:1409

2664

3775;

K1 ¼

2:0876 0 0 0

0 2:0876 0 0

0 0 0:6453 0

0 0 0 0:6453

2664

3775;

K2 ¼

�3:0503 0 0 0

0 �3:0503 0 0

0 0 �1:1563 0

0 0 0 �1:1563

2664

3775;

K3 ¼

2:2099 0 0 0

0 2:2099 0 0

0 0 1:0343 0

0 0 0 1:0343

2664

3775;

K4 ¼

�0:6248 0 0 0

0 �0:6248 0 0

0 0 �0:3387 0

0 0 0 �0:3387

2664

3775:

Table 1: Machine data: four-machine and ten-bus system

Ra xd x 0d xq x 0

q H T 0do T 0

qo Xt D KE TE

Gen (pu) (pu) (pu) (pu) (pu) (pu) (pu) (pu) (pu) (pu) (pu) (pu)

1 0.00028 0.2 0.033 0.19 0.061 54.0 8.0 0.4 0.022 0.0 200 0.05

2 0.00028 0.2 0.033 0.19 0.061 54.0 8.0 0.4 0.022 0.0 200 0.05

3 0.00028 0.2 0.033 0.19 0.061 63.0 8.0 0.4 0.022 0.0 200 0.05

4 0.00028 0.2 0.033 0.19 0.061 63.0 8.0 0.4 0.022 0.0 200 0.05


Table 2: Line data for four-machine and ten-bus system

Series

resistance

Series

reactance

Shunt

reactance

From bus To bus ðRsÞ pu ðXsÞ pu ðBcÞ pu

1 6 0.01 0.012 0.0

2 5 0.01 0.012 0.0

9 10 0.022 0.22 0.33

9 10 0.022 0.22 0.33

9 10 0.022 0.22 0.33

9 5 0.002 0.02 0.03

9 5 0.002 0.02 0.03

3 8 0.001 0.012 0.0

4 7 0.001 0.012 0.0

10 7 0.002 0.02 0.03

10 7 0.002 0.02 0.03

6 5 0.005 0.05 0.075

6 5 0.005 0.05 0.075

8 7 0.005 0.05 0.075

8 7 0.005 0.05 0.075

Table 3: Load flow data for four-machine and ten-bussystem

Bus V � PG QG PL QL Bl

(pu) (deg.) (pu) (pu) (pu) (pu) (pu)

1 1.03 8.2154 7.0 1.3386 0.0 0.0 0.0

2 1.01 21.5040 7.0 1.5920 0.0 0.0 0.0

3 1.03 0.0 7.2122 1.4466 0.0 0.0 0.0

4 1.01 210.2051 7.0 1.8083 0.0 0.0 0.0

5 1.0108 3.6615 0.0 0.0 0.0 0.0 0.0

6 0.9875 26.2433 0.0 0.0 0.0 0.0 0.0

7 1.0095 24.6997 0.0 0.0 0.0 0.0 0.0

8 0.9850 214.9443 0.0 0.0 0.0 0.0 0.0

9 0.9761 214.4194 0.0 0.0 11.59 2.12 3.0

10 0.9716 223.2977 0.0 0.0 15.75 2.88 4.0

Table 4: Model parameter variations for four-machine and ten-bus system

Active power Load

Model ðPGÞ ðPLÞ Bus structure

1 [7 7 7.2 7] [9.59, 10.75] as per line data

2 [7 7 7.2 7] [9.59, 11.25] one line 9-10 disconnected

3 [7.2 7.1 7.0 6.9] [10.09, 11.25] as per line data

4 [7.2 7.1 7.0 6.9] [10.09, 11.75] one line 10-7 disconnected

5 [7 7 7.2 7] [10.59, 11.75] as per line data

6 [7.1 6.9 7.5 6.5] [10.59, 12.25] as per line data


8 [5 8 6.2 8] [11.09, 12.75] as per line data





13 [7 7 7.2 7] [11.59, 14.75] as per line data


15 [7.1 6.9 7.5 6.5] [11.59, 15.75] as per line data



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