Determination of stabiliser settings inmultimachine power systems

V.N. Degtyarev and B.J. Cory

Indexing terms: Power systems and plant, Stabilisers

Abstract: The paper considers a novel approach to simultaneous calculation of the adjustable parameters of allthe stabilisers in a multimachine power system to obtain the best dynamic performance. The parameter opti-misation problem is based on an eigenvalue assignment formulation, and an algorithm enabling the needed shiftof dominant system modes is presented. The proposed algorithm is used to set gains of two stabilisers in athree-machine low damped power system.

1 Introduction

Improving the steady-state stability of large interconnectedpower systems has received a great deal of attention inrecent years. Automatic excitation control of synchronousgenerators is one of the most effective ways to solve thisproblem. However, the use of high-gain and short-time-constant voltage regulators has particularly aggravated theproblems of steady-state instability. To overcome theseproblems, power system stabilisers (PSS) [1] have beeninstalled and are widely used in power systems of theUSSR and USA [1, 2]. Recently, the first stabilisers havebeen installed by the North of Scotland Hydro-ElectricBoard [3]. Methods and algorithms to co-ordinate theactions of these stabilisers are of great importance in theproper design and overall stability of a power system.

In the USSR the D-separation technique in its numer-ous modifications has been used to choose settings of sta-bilisers [1, 4, 5]. Such procedures are rather tedious andtime-consuming and often do not lead to the required sol-ution [6] in a multimachine system where several stabili-sers are installed. Another drawback of this method is thatit takes into account only complex eigenvalues, thus losinginformation about possible dominant real values.

Several papers [3, 6-9] have demonstrated the useful-ness of eigensystems analysis techniques in linearisedstudies of power systems and in particular for the siting ofstabilisers. The authors either use different optimisation

approaches with sensitivities of dominant eigenvaluesobtained by different means [3, 7] or sophisticated opti-misation methods to overcome problems of nonlinear costfunctions [6, 8].

Recent papers [10, 11] have presented similar methodsby which stabiliser settings can be determined in a systemwhere all generating units are equipped with stabilisers,which, to some extent, is always possible [6]. Applying themethods on a system only partly equipped with stabiliserscan, however, easily lead to a reduction of damping of theweakly controlled modes, and even to unstable situations.Also, the methods generally cannot guarantee improve-ment of stability margins in the presence of dominant realeigenvalues.

This paper presents and illustrates a new method fortuning stabilisers in multimachine power systems. Themethod unites eigenvalue analysis with the basis of the D-separation technique. It provides a shift of both real andcomplex dominant eigenvalues, mainly associated withmechanical motion, and excludes the appearance ofunstable modes. Designs of most stabilisers used in powersystems have two parameters responsible for the stabilisingeffect [6], so the method is devised and described here toadjust only two parameters of each stabiliser; however, itcan be expanded to a larger number. The method is illus-trated by application to the 3-machine system of Fig. 1and the two parameters to be optimised are Ka and Kb inthe PSS of Fig. 2.

163.0(6.7)

18 kV163

230kV-163 76.4

"(6.7)

(D1.025/9.3°

(9.2)-75.9

(-0.8)86.6

(-8.4)

(-10.7)

• load C

-24.123OkV

24.2,-85? 85,(-24.3)

(7J1.026

/3.7°

-84.3

I -40.7

(50.0)125

load A40.9

(8)1.016

/o.r

(3.0)60.8

(-18.0)

H1.3)H8.7)

(22.9) 230 kV

-59.45

-30.55

30.7

Paper 4790C (P9), first received 29th October 1985 and in revised form 11th June1986

Dr. Cory is, and Dr. Degtyarev was formerly, with the Department of ElectricalEngineering, Imperial College, Exhibition Road, London SW7 2BT, UnitedKingdom. Dr. Degtyarev is now with the Kisheniev Polytechnic, Moldavia, USSR

-71.6

71.6

16.5kV-- 0

85.0(-10.9)

(9)1.032/2.0°

(-1146)

(3)1.025A7°

U.013(-16.54) I ^

90.0 (30.0)t

loadB

(1.0)1.026

e23.9)

(27.0)

07,v_y(27

1.04/0.0°

71.6(27)

Fig. 1 Multimachine power system

MW flows(MVar) flowsVoltage p.u. /angle

308 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

Fig. 2 Simplified transfer functions of different AVR + PSS branches

2 Proposed algorithm

We consider a system described by a set of linear, first-order differential and algebraic equations, e.g. a powersystem subjected to small disturbances. In the s-domainthis system can be described [1] by the set of equations

A(s)X = 0

where X is a vector of variables

(1)

= (Xl,X2,...,Xr)

and

A(s) =

al 1 \ 2 aln + blns

bdlsakl + bdl

ak+ 1 1 + bfc+fc 2 + fa 2

ae 2 + be 2 S

<*e+l 2

k+ln

*e nbens

'ml

*m+l

Here rows 1, . . . , k represent differential equations of gen-erators; rows k + 1 , . . . , e represent differential equationsof an AVR and PSS; coefficients au in rows e + 1, . . . , nare coefficients of algebraic equations of regulators(e + I, ..., m) and the network (m + 1, . . . , n). Some of thecoefficients a{j, btj in rows k + 1, . . . , e are equal to thestabilising parameters Kai, Kbi which have to be tuned,while others are untuned parameters or their com-binations. The set of eqns. 1 may be reduced to differentialequations only, but variables connected with the tunedparameters Ktj must not be discarded. The proposed algo-rithm includes the following steps:

Step 1: After transformation of eqn. set 1 with initialvalues of the tuned parameters to state-space (differential)

form, we have

where Br, Ar are first-order differential matrices of coeffi-cients atj, b{j. System eigenvalues can now be obtained,and among them z dominant ones are to be selected. Thistransformation is run with the values of stabiliser param-eters Kai, Kbi equal to initial values, based on practicalexperience. If there is lack of the latter they can beassumed equal to zero. The number of dominant eigen-values is usually equal to twice the number of generators ifthey are complex, or less if there are no infinite buses andsome of the mechanical motion eigenvalues are real. It isvery rare that dominant eigenvalues are not mainly associ-ated with mechanical motion. If such eigenvalues appear,this implies improper tuning of AVRs or PSSs with oscil-lations of greater than 2 Hz. However, this puts norestrictions on the described method.

Step 2: Among the dominant eigenvalues we choose Mpivot ones, the real parts of which are furthest to the rightin the complex plane. We now define a new value for thereal part of each pivot eigenvalue, which must be less thanor equal to the next left most chosen dominant real parteigenvalue in the complex plane. Thus we obtain M newvalues of pivot modes as

X[ = a,- + (as\ - a,) ± ja>i = <xlst ±jcOi i = 1, . . . , M

where a*, is a new negative real part of the chosen eigen-values.

Step 3: This step is similar to the procedure of the D-separation technique [1], and requires reduction of eqn. 1through a Gaussian elimination technique to one of thefollowing forms, s being equal to X\:

i = l , . . . , L (3a)

D2(A'i) = KaiBu + Cu = 0 i = I, ..., L (3b)

where Z n , Z2i, Z, are complex values, B2i,Bl{, Clt,C21 are real quantities, and L is the number of stabilisersin the power system.

Eqn. 1 is transformed to the form of eqn. 3a in the case

IEE PROCEEDINGS, Vol. 133, Pi. C, No. 6, SEPTEMBER 1986 309

where a pivot eigenvalue is complex. Transformation tothe forms of eqns. 3b or c is needed only when k\ are realwith no imaginary parts. The symbol' in the description ofthe tuned parameters Kai, Kbi is used to underline thisdifference.

Step 4: By solving eqns. 3, e.g. eqn. 3a being separated forreal and imaginary parts, a set of new parameter values forKai, Kbi is generated, each of which, in the case of acomplex pivot eigenvalue a pair of Kai, Kbi, can shift apivot eigenvalue to the prespecified position X\. This newset will provide

K1 =

K aLI + d

(4)

where Ktj represents pairs of parameters K'aij and K'bij

corresponding to complex eigenvalues; Kaij Kbij corre-spond to real pivot eigenvalues, if there are any; / is thenumber of complex pivot eigenvalues; index i deals withthe number of the stabiliser in the power system; index jreflects the number of its pivot eigenvalue. So the numberof rows in eqn. 4 is equal to L, while the number ofcolumns depends on the chosen number of pivot eigen-values, a pair of conjugate values being considered as one.

Step 5: Increments of parameters are now computed as

AK1 =KX -K° (5)

where K° is the K1 values with parameters adopted duringprevious steps. In the matrix AK1 all elements which

Step 7: New values of Kai, Kbi are substituted in eqn. 1and the whole procedure is repeated.

Having repeated these steps several times with the samevalue a^, a new value of a stability margin a^,, the realpart of the right-most eigenvalue, is obtained. However, ingeneral a^ # a*,, because it is impossible to control Zdominant modes by 2L parameters, when Z > 2L. Yet,after completing these steps, an improved stability marginis obtained. If &„ does not provide an adequate or desiredstability margin, step 8 must be run.

Step 8: Choose a new value of a% > a*, and run steps 1-7.If now a new value of stability margin aj, is better thanbefore, one can proceed to selection of a new value of ast.This steps helps to avoid difficulties of passing throughlocal extrema which are inevitable with a cost functionsuch as the stability margin [6].

3 Multimachine system example

The three-machine power system previously studied inReference 12 and shown in Fig. 1 is considered in thispaper. Generator and system data are given in Tables 1and 2. Linearised system equations are presented in theAppendix.

Table 1: Generator data

Generator

xdX'd

T'dc

, p.u., p.u.. pu.

, s

1

0.1460.06080.09698.9623.64

2

0.89580.11980.89586.06.4

3

1.31250.18131.31255.893.01

Table 2

Branch

RXB/2

: System data.

1-4

00.057*0

4-6

0.0175 0.092

0.079

p.u. values

6-9

0.0390.1700.179

3-9

00.05860

9-8

0.01190.10080.1045

8-7

0.00850.07200.0745

7-2

00.06250

7-5

0.0320.1610.153

5 ^

0.0100.0850.088

exceed maximum values of parameters Ka, Kb are put tozero. If all the elements of AK1 are equal to zero, a^ mustbe reduced and the previous steps repeated.

Step 6: The elements in rows of eqn. 5 differ in sign andvalue. For the purposes of guiding the choice of the incre-mental change to parameters Ka, Kb, we introduce matrixS, whose columns are

where S'aij = sign AK'aij, S^,-= sign AK'bij, SaiJ = signAKaij, and Sbij = sign AKbij.

In matrix AX1 we can choose co-ordinates i,j of anelement with minimum value, given by one of the equal-ities

= Ka = Kbij

If in row i, corresponding nonzero values of S'aij, Saij orS'bij> Sbij have different values, the final increment isadopted as AKlj with the other being put to zero. If in rowi corresponding nonzero values of S'aij, SaiJ or S'bij, S'bij

are equal, the final increments of parameters for the stabili-ser i are to be calculated as an average of increments inrow i, previously multiplied by vector K}A given by

In this system PSSs are to be applied to generators G2and G3. The chosen type of PSS is similar to those whichare widely employed in USSR power systems. Simplifiedtransfer functions of different AVR + PSS branches arepresented in Fig. 2.

Channels for voltage deviation are not responsible forthe stabilising effect, so parameters KOu, Klu were taken tobe constant in the calculations at KOu = — 50 and Klu =— 4.3. The transfer function of the excitation system ofgenerator Gl differs from that shown in Fig. 2 by theabsence of frequency deviation and frequency derivativechannels.

During step 1, among the system eigenvalues presentedin Table 3 and calculated without any PSSs, we choose thedominant as

/>! = -2.374 ±;11.37 k2 = -0.36 ±;9.712

/ 3 = -0.09

Thus the system is characterised by a slow damped aperio-dic movement (i3) and relatively slow periodic motion (A2).To test the proposed algorithm it was required to find PSSsettings corresponding to the maximum possible value ofthe stability margin.

The maximum PSS parameter Ka was set to ± 7 andparameter Kb was set to ± 3 .

310 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

Table 3:

numbervalue

numbervalue

numbervalue

Eigenvalues of system without PSS

-2.374 ±y11.37

^11 ^12-132 -92.19

A20 / l 2 1

-5.893 -24.23

A2

-0.36±y9.712

-82.74 -84.54

A22 A 2 3

- 2 0 . 7 8 - 3 7 . 7 3

-30.09

-65.101

A2A

-37.92

-417.86

±/33.01

A25

-38.01

-538.46 -6589.50

-6.905 ±/31.1

^26- 1 7 . 7 1

A21

- 1 7 . 8 6

-?590.1 -590.5

-97.492±y25.11

^28-38.46

^29-38.46

-9130.9

-1.608

^30-17.86

-131.8

-2.205

During the second step we obtain an isolated real pivoteigenvalue of / = —0.09 and choose l\ = —0.36, k\t =— 0.36, which is equal to the real part of the next-leftdominant eigenvalue A2. Solving eqns. 3b and c we obtainmatrix K1 as

00 0

1.322 -

1.633-3.641-4.49J

In this initial stage we have only one pivot eigenvalue andmatrix AK1 is equal to Kl.

In matrix Ki etefiients of the fourth column exceed themaximum values of parameters, so they were set to zero.Matrix S was obtained as

TO 0 1 01

and finally, after step 7 of the algorithm, we obtain

Ka! = 1.322 Ka2 = Kbl = Kb2

= 0 at the first iteration

Convergence of the algorithm is demonstrated in Table 4and in Fig. 3, where some routine steps are not shown. Theprocedure was stopped after 11 iterations, although it waspossible to obtain a slightly greater value of the maximumstability margin after several more iterations.

4 Conclusions

A new co-ordinated synthesis method of applying PSS wasdeveloped by combining eigenvalue analysis and the bestof the D-separation technique. This method has the follow-ing features:

(a) It is simultaneously able to select the generators towhich the PSS can be effectively applied and to chooseadequate settings of the PSS for these generators.

(b) It is effective not only for local modes, but also forlow-frequency or real interarea modes.

(c) It avoids instability of roots mainly associated withexcitation systems. Its effectiveness has been shown by anumerical example with a 3-machine system.

12

+ cx

11 987 65 Z 3

>39 8 7 6 5 3A / 2

-2.A -2.0 -1.6 -1.2 -0 .8 -0M 0

Fig. 3 Movement of dominant eigenvalues through optimisation pro-cedure

1EE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986

5 Acknowledgments

The authors are grateful to their respective institutions forresearch facilities, and to the British Council for financialsupport for Dr. V. Degtyarev.

6 References

1 VENIKOV, V.A.: 'Transient processes in electrical power systems'(English translation, Mir Publishers, Moscow, 1980)

2 FARMER, R.G., and AGRAWAL, B.L.: 'State-of-art technique forpower system stabilizer tuning', IEEE Trans., 1983, PAS-102, pp.699-707

3 RAMSAY, B., and SULLEY, J.L.: 'Eigenvalue analysis of the effectson dynamic stability of plant in the North of Scotland Hydro-ElectricBoard network'. Proceedings of 8th PSCC, Helsinki, 19-24 Aug. 1984,pp. 990-996 (Butterworths)

4 ZHENENKO, G.N., and FOUAD, A.A.: 'Steady-state stabilityanalysis with frequency methods: optimization of excitation systemparameters', IEEE Trans., 1984, PAS-103, pp. 715-721

5 GERTSENBERG, G.R., SOVALOV, S.A., PEREL'MAN, I.F., andYUSIN, V.M.: 'Using high-power excitation regulators on thermalpower stations', Elektricheskie Stantsii, Jan. 1977, no. 1, pp. 35-39 (inRussian)

6 GRUZDEV, I.A., TRUSPEKOVA, G.Kh., and USTINOV, S.M.:'Simultaneous coordination of voltage regulators settings on the baseof numerical search', Elektrichestvo, Mar. 1984, no. 3, pp. 51-53 (inRussian)

7 LEFEBVRE, S.: 'Tuning of stabilizers in multimachine powersystems', IEEE Trans., 1983, PAS-102, pp. 290-299

8 DOI, A., and ABE, S.: 'Coordinated synthesis of power system stabili-zers in multimachine power systems', ibid., 1984, PAS-103, pp. 1473-1479

9 GOOI, H.B., HILL, F.F., MOBARAK, M.A., THORNE, D.H., andLEE, T.H.: 'Coordinated multi-machine stabilizer setting withouteigenvalue drift', ibid., 1981, PAS-100, pp. 3879-3887

10 LIM, CM., and ELANGOVAN, S.: 'Design of stabilisers in multi-machine power systems', IEE Proc. C, Gen., Trans. & Distrib., 1985,132,(3), pp. 146-153

11 SIVAKUMAR, S., SHARAF, A.M., and HAMED, H.G.: 'Coordi-nated tuning of power system stabilizers in multimachine powersystems', Electr. Power Syst. Res., 1985, 8, pp. 275-284

12 ANDERSON, P.M., and FOUAD, A.A.: 'Power system control andstability' (Iowa State University Press, 1977)

7 Appendix

Linearised equations of the 3-machine system are given as:

23.64SAO)! + 0.6642A£Q1 + 1.067A/4! + lAo^ = 0

O.OO265SA<512 - Aco1 - Aco2 = 0

6.4SA«2 + U9AIq2 + 0.9087A£g2 + 1A«2 = 0

0.002655^! 3 - Acot - Aco3 = 0

3-OlSAcWi + 1.4027 AIq3 + 0.6044A£^3 + lAco3 = 0

0.7634SA/rf1 + 8.96SA£^1 + AEq^ - A£r l = 0

4.656SA/<22 + 6SAEq2 + AEq2 - A£r2 = 0

6.6628S/d3 + 5.89£<j3 - £ r 3 = 0

1.457A£Q1 - 0.3382A£g2 - 0.2032A£^3

+ 0.959A<512 + 0.5705A<513 - AIqt = 0

0.6308A£Q1 + 0.821A£g2 + 0.0632A£q3

-0.197A<512 + 0.1331A(513 - A/g2

311

= 0

IEE PROCEEDINGS, Vol . 133, Pt. C , No. 6, SEPTEMBER 1986

0.4447 A£Q1 + 0.0404A£g2 + 0.042A£g3

+ 0.1908A<512 - 0.3501 A<513 - AIq3 = 0

1.7198A£Q1 + 0.5357A£q2 + 0.4066A£g3

+ 0.6053A<512 + 0.2851A<513 - AIdY = 0

0.05898A£Q1 - 0.8385A£q2 + 0.0948A£q3

+ 0.1611 Ad12 - 0.0887A<513 - AId2 = 0

0.0991A£G1 - 0.1066A£g2 - 0.6219A£g3

-0.0722A<512 + 0.5456A<513 - AId3 = 0

0.0966A/*/! + 0.006AIq1 + 0.9971A£Q1 - AU1 = 0

0.5427A/d2 + 0.7114A/g2 + 0.6059A£g2 - AU2 = 0

0.9182A/d3 + l.O158A/g3 + 0.6996A£<?3 - AL/3 = 0

x - A£Q1 + 0.0852A/d1 = 0

-SAcoU2 + Aw2/0.00265 - 0.5295SA/g2

+ 0.7748SA£g2 + 0.694SA/rf2 = 0

-SAcoU3 + Aw3/0.00265 - 0.8958SA/g3

+ 0.7551SA£g3 + 0.991SA/rf3 = 0

Abstracts of papers published in other Parts of the IEE PROCEEDINGSThe following papers of interest to readers of /££ Proceedings Part C, Generation, Transmission & Distribution haveappeared in other Parts of the /££ Proceedings:

Coppers for electrical purposesV.A. CALLCUT

/££ Proc. A, 1986,133, (4), pp. 174-201The development of electrical machines was facilitated bythe availability of high conductivity copper conductors,and the nonferrous manufacturers are still meeting therequirements of the electrical and electronics industry.Having shown that copper is still in plentiful supply, thereview describes over 80 standardised wrought coppersand copper alloys and 37 casting materials. Compositionsare given together with basic mechanical and physicalproperties in the annealed, worked or heat-treated condi-tions as appropriate. Machining and jointing techniquesare described and the suitability of the materials for enduse requirements is discussed. Sources of further informa-tion are given.

Model of the formation of a dry band on an NaCl-pollutedinsulationC. TEXIER and B. KOUADRI

/££ Proc. A, 1986,133, (5), pp. 285-290A model of the appearance of a dry band on a flat insula-tor subjected to saline fog pollution has been experimen-tally and theoretically investigated with the followingsimple hypotheses: (a) the surface temperature of the insu-lator is uniform, (b) the mass of evaporated water is pro-portional to the temperature difference between theconducting pollution layer and the ambient atmosphere,(c) the product of electrical resistance into the residualmass of pollution remains constant at constant tem-perature, (d) a dry stripe appears when, after evaporation,the residual mass of pollution is lower than a critical

threshold. After a theoretical evaluation of the differentterms of the energy-balance equation, the temporal evolu-tion of the current, of the mass of pollution and of thesurface temperature, was numerically computed togetherwith the delay for appearance of a dry band, the energyrequired and the average dissipated power. Qualitativeand quantitative agreement was found between the theo-retical model and experimental data with a0.23 x 0.035 m2 plane insulator. A sheet of blotting paperwetted by a NaCl solution simulated the pollution. Thistheoretical model predicts the time formation of a dryband under conditions where formation is known to occur,not whether or not formation will occur.

An energy strategy for the UKA. GOODLAD, R. EDEN, G. LEACH, A. HENNEY andA.E. VICARS-MILES

/££ Proc. A, 1986,133, (5), pp. 311-318The papers are taken from the colloquium on 'An energystrategy for the UK' organised by the Engineering andSociety Executive Committee (M2) of the IEE's Manage-ment and Design Division. Five papers are presented. Thekeynote speech on the UK Government's approach toenergy policy by the Under Secretary of State for Energy iscomplemented by a review of the issues which an energypolicy should address. An alternative 'Low energy strategyfor the UK', published six years ago, is assessed in terms ofachievements in energy efficiency to date. The structure ofthe electricity supply industry is critically appraised and aproposal is madet to reshape it. Finally, a view of theenergy scene from the oil industry's position is putforward. The discussion sessions are summarised.

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 6, SEPTEMBER 1986 313