Normal form analysis of interactions among multipleSVC controllers in power systems

Z.Y. Zou, Q.Y. Jiang, Y.J. Cao and H.F. Wang

Abstract: The static var compensator (SVC) has been widely employed in power systems toprovide reactive power and maintain busbar voltages. In the paper, a study case of negativeinteractions among multiple SVC controllers is reported, and an analytical approach based onnormal form theory is proposed for the analysis of nonlinear interactions among the multiple SVCcontrollers in power systems. Also, a nonlinear interaction index is developed to investigate theinteractions among multiple SVC controllers. The proposed approach has been applied to a typicalIEEE 4-machine 11-busbar system. The simulation results validate the proposed approach.

1 Introduction

For several years, the static var compensator (SVC) hasbeen employed to provide high-speed reactive compensationand to maintain the busbar voltages in power systems [1].The SVC is basically a shunt-connected static var generator/load whose output is adjusted to exchange capacitive orinductive current, so as to maintain or control specificpower system variables; typically, the controlled variable isthe SVC busbar voltage. However, with the growing use ofthe SVC, the co-ordination design problem of multiple SVCcontrollers in joint operation must be considered in practicalpower systems.

In recent years, normal forms theory has receivedconsiderable interest as a mean of dynamic analysis ofpower-system nonlinear behaviour [2–8]. With this techni-que, it is possible to obtain the simplest form of a set ofnonlinear differential equations and, hence, to identify andstudy the nature of system oscillations using conventionallinear analysis methodologies. Recently, the normal formstheory has been used to determine the interacting modes ofoscillation as well as to identify the nature and character-istics of the inter-area mode phenomenon. In [2, 3], thesignificance of second-order contributions to the normalforms has been investigated. Normal forms theory isemployed to detect the onset of the inter-area modephenomenon using several analytical indices. Furtherdevelopments in this methodology show that control modesmay have a significant interaction with electromechanicalmodes, specifically inter-area modes [2, 4, 5, 6].

In this paper, an analytical approach based on normalforms theory is proposed, for the study of nonlinearinteractions among multiple SVC devices installed in powersystems, which could provide some guidance to co-ordination design of multiple SVC controllers. Moreover,

a nonlinear interaction index is developed to identify andquantify the interactions of these SVC controllers. Accord-ing to this index, we can obtain a better understanding ofthe complex nonlinear dynamic phenomenon about thepower system installed with multiple SVC devices. Theproposed method is general and may be extended to includeother FACTS devices. A typical IEEE test system in [9] isemployed to illustrate the proposed methodology. Finally,detailed nonlinear time-domain simulations are conductedto check the validity of the analysis.

2 Mathematical model of power system with SVC

The power system consists of the dynamic models ofsynchronous machines and SVC interacting through thetransmission network, as shown in Fig. 1.

2.1 Generator dynamic equationsThe differential equations describing the dynamic behaviourof the synchronous generator and the excitation system aregiven by the following.

transmissionsystem

SVCm SVC2 SVC1

G0 G2 G1

Fig. 1 Configuration of power system with multiple SVCs

Z.Y. Zou, Q.Y. Jiang and Y.J. Cao are with the College of ElectricalEngineering, Zhejiang University, Hangzhou, Zhejiang 310027, China

H.F. Wang is with the Department of Electrical and Electronics Engineering,University of Bath, BA2, 7AY, UK

E-mail: yijiacao@zju.edu.cn

r IEE, 2005

IEE Proceedings online no. 20049009

doi:10.1049/ip-gtd:20049009Originally published online: 21st April 2005

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 4, July 2005 469

Rotor swing equations:

_dk ¼ o0ðok � 1Þ ¼ f1k

_ok ¼1

2HkðPmk � ðE0dk

Idk þ E0qkIqk Þ � Dkðok � 1ÞÞ ¼ f2k

ð1ÞInternal voltage equations:

_E0qk¼ 1

T 0d0k

ðEfdk � ðxdk � x0dkÞIdk � E0qk

Þ ¼ f3k

_E0dk¼ 1

T 0q0k

ð�E0dkþ ðxqk � x0qk

ÞIqk Þ ¼ f4k

ð2Þ

Static excitation system:

_Efdk ¼1

TAð�Efdk þ KAðVtrefk � VtÞÞ ¼ f5k

Vt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2dt þ v2qt

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðE0dkþ x0qIqk Þ

2 þ ðE0qk� x0d Idk Þ

2q

ð3ÞThe state variables and parameters in these equations havethe usual physical meaning as in [9].

2.2 Representation of SVCSVC is a shunt compensation component. The blockdiagram of the SVCmodel is shown in Fig. 2. In this model,SVC is basically represented by a variable reactance withmaximum inductive and capacitive limits to control theSVC busbar voltage. For the ith SVC, a total reactanceBSVC is assumed and the following differential equationholds:

_BSVCi ¼ ðKSVCiðVSVCrefi � VSVCiÞ � BSVCiÞ=TSVCi ¼ f6k ð4Þwhere KSVC and TSVC are the gain and time constant of theSVC. The model is completed by the algebraic equationexpressing the reactive power injected at the SVC node:

Q ¼ �BSVCV 2SVC ð5Þ

2.3 Network representationWithout loss of generality, we assume that in the n-machinepower system, m SVC devices are installed. Assume, for thesake of simplicity, that the internal generator nodes areordered in the form {1,y,n}; followed by load nodes whereSVCs are installed {n+1,y, n+m} and finally, the loadbusbars without voltage support {n+m+1,y,N}. Elim-inating the terminal generator busbars and load busbarswithout SVCs, the nodal balance equation is expressed innetwork D–Q co-ordinates as

Ig

..

.

0

264

375 ¼

Ygg � � � Ygl

..

.� � � ..

.

Ylg � � � Yll

264

375

Vg

..

.

Vl

264

375þ 0

..

.

jBSVCV l

24

35 ð6Þ

where Ygg, Ygl, Y lg and Y ll are submatrices of the reducedadmittance matrix, and

Ig ¼ ½ ID1þ jIQ1

ID2þ jIQ2

� � � IDn þ jIQn �T

Vg ¼ ½E0D1þ jE0Q1

E0D2þ jE0Q2

� � � E0Dnþ jE0Qn

�T

V l ¼ ½ VDnþ1 þ jVQnþ1 VDnþ2 þ jVQnþ2 � � � VDnþm þ jVQnþm �T

BSVC ¼ diag½BSVC1BSVC2

� � � BSVCm �

2.4 Co-ordinate transformationTo express network currents in machine (d–q) co-ordinates,the complex transformation is introduced as follows:

Ig ¼ IDQ ¼ TIdq;Vg ¼ VDQ ¼ TEdq ð7Þwhere

T ¼ diag½ ejd1 ejd2 � � � ejdn �and

Edq ¼ ½E0d1 þ jE0q1 � � � E0dn þ jE0qn �T

Substituting (7) into (6), we have

Idq ¼ T�1YggTEdq þ T�1YglVl ð8Þand

0 ¼ YlgTEdq þ YllV l þ jBSVC V l ð9ÞSolving for the load terminal voltages V1 results in

Vl ¼ �ðYll þ jBSVCÞ�1YlgTEdq ð10Þand

Idq ¼T�1YggTEdq�T�1Ygl

ðYll þ jBSVCÞ�1

YlgTEdq

ð11Þ

From (11), the current of the kth generator with oneSVC installed at busbar p can then be expressed on d–qaxes as

Idk þ jIqk ¼Xn

m¼1Y kme�jðdk�dmÞðE0dm

þjE0qmÞ

� ð Y kp

Y pp þ jBSVCÞXn

m¼1Y pme�jðdk�dmÞ

ðE0dmþ jE0qm

Þ

ð12Þ

2.5 Mathematical model of power systemLet X¼ [d, x, Eq

0, Ed0, Efd, BSVC]

T as the state vector.Substituting (12) into (1)–(4) yields the nonlinear systemrepresentation:

_X ¼ FðXÞ ¼ ½f 1; f 2; f 3; f 4; f 5; f 6�T ð13Þ

where F(x) represents a smooth n-dimensional vector field.Based on this mathematical model, a second-ordermathematical model of a power system obtained by theperturbation theory is used to analyse the dynamicinteraction among multiple SVC controllers.

3 Analytical approach based on normal form

3.1 Normal form of vector fieldConsider a nonlinear system described by the differentialequation (13). Let XSEP be a stable equilibrium point (SEP)and XSEP satisfies that F(XSEP)¼ 0.

We expand (13) as a Taylor series about the stableequilibrium point XSEP and obtain (14), using again X and

+

−

VSVCref

VSVC

KSVC

Bmax

Bmax

BSVC1+sTsvc

Fig. 2 SVC model

470 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 4, July 2005

xi as the state variables,

_xi ¼ AiX þ1

2XT H iX þ higher order terms ð14Þ

where Ai is the ith row of Jacobian A which is equal to

½@F=@X �XSEP, H i ¼ ½@2Fi=@xj@xk�XSEP

.

Denote by J the (complex) Jordan form of A, and by Uthe matrix of the right eigenvalues of A, then thetransformation X¼UY yields, for linear and second-orderterms of (14), the equivalent system

_yj ¼ ljyj þ YT CY ¼ ljyj þXN

k¼1

XN

l¼1Cj

klykyl ð15Þ

where

C j ¼ 1

2

XN

p¼1V T

jp UT HpU� �

¼ Cjkl

� �and V denotes the matrix of associated left eigenvectors. Ifthe second-order nonresonance condition holds (namely,lj+lkali), then the normal form transformation of (15) isdefined by

Y ¼ Z þ h2ðZÞ ð16Þwhere

h2jðZÞ ¼XN

k¼1

XN

l¼1h2j

klzkzl;

h2jkl ¼ Cj

kl=lk þ ll � lj

In Z co-ordinates, the system (16) takes on the form

_zj ¼ ljzj ð17ÞEquations (14)–(17) allow us to obtain explicit second-ordersolutions for the system in the different co-ordinate systems:

zjðtÞ ¼ zj0eljt ð18Þ

yjðtÞ ¼ zj0eljt þXN

k¼1

XN

l¼1h2j

klzk0zl0eðlkþllÞt ð19Þ

xiðtÞ ¼XN

j¼1uijzj0eljt

þXN

j¼1uij½XN

k¼1

XN

l¼1h2j

klzk0zl0eðlkþllÞt� ð20Þ

where the initial conditions in Z co-ordinates are obtainedby the solution of the nonlinear optimisation problem:

min f ðZ0Þ ¼ Z0 þ h2ðZ0Þ � Y0 ð21ÞComparison of (18) and (19) leads to the definition of thenonlinear interaction index, Index, for mode j as follows [2, 7].

IndexðjÞ ¼ jmaxk;lðh2jklzk0zl0Þ=zj0j ð22Þ

where maxk;lðh2jklzk0zl0Þ is the complex form where maxk;l

jh2jklzk0zl0j occurs.The Index is computed to assess the degree of

nonlinearity at mode j and provides a normalised measureof the relative size of the nonlinearity at the initial value.Moreover, the Index will be used in the following Section toanalyse the interactions among the multiple SVCs.

3.2 Normal form analysis proceduresComputation of the nonlinear Index and analysis ofnonlinear interactions among different SVC controllerscan be carried out as in the following procedure:

(a) For a given disturbance, determine the postdisturbancestable equilibrium point XSEP. Move the origin to thepostdisturbance SEP via mathematical transformation.

(b) Expand the mathematical model of the power systemwith multiple SVCs in (13) as a Taylor series about the SEPand retain up to second-order terms.

(c) Derive the representation of the second-order Taylor-series expansion in the Jordan variables Y by X¼UY.Determine the initial conditions in the Jordan and normalform variables from the relations Y0¼U�1X0.

(d) Compute the Jordan form coefficients Cjkl and the

normal form coefficients h2jkl.

(e) The nonlinear algebraic (21) is solved to obtain the initialconditions of normal form variables. Numerically, this isone of the key steps in applying the normal forms techniqueto the power-system problem.

(f) The nonlinear interaction Index developed is derived by(22) and explicit time-domain closed-form solutions alsocan be obtained by (20).

4 Case studies

4.1 Test system installed with two SVCsLet us consider a two-area, four-machine test system, asshown in Fig. 3. The transient models for the generatorsand the first-order simplified model for the excitationsystems are used. Two SVCs are separately located atbusbar 3 and busbar 13. This test system is speciallydesigned to study control interactions between two SVCs.The system data and the controller parameters of two SVCsare given in Appendix 8.

In the following analysis, the variation of two electricalparameters, i.e. the electrical coupling and the short-circuitcapacity (SCC), are required to investigate the controlinteractions between two SVCs. The electrical coupling is

predominantly determined by the impedance linking thehigh-voltage busbars (busbar 3 and busbar 13). And, in thisconfiguration, both SVCs’ high-voltage busbars present areactance connecting them to generators. Changing thisvalue, the short-circuit capacity measured at these busbarsmay be easily modified.

Regarding the electrical distance between the SVCdevices, two different situations will be analysed, i.e. theuncoupled case and the coupled case. For each one of theseconfigurations, two situations, i.e. the high SCC and the lowSCC, will also be considered to check the influence on theinteraction. The line parameters of the uncoupled case andthe coupled case and the values of the high SCC and the

1 1020

2

3 13

GEN 1

101 120110 11

GEN 3

GEN 412

SVCSVCGEN 2

Fig. 3 Single-line diagram of the test system

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 4, July 2005 471

low SCC at busbar 3 and busbar 13 in each configurationare listed in Table 1.

4.2 Nonlinear normal form analysisIn this study case, there are one inter-area mode, i.e. mode4/5, and two local modes, i.e. modes 9/10 and 11/12, whichare shown in Table 2. Moreover, these inter-area and localmodes are seen as the critical electromechanical modesconcerned.

Emphasis in the present analysis is placed on the studyand identification of interaction between two SVCs. In thefollowing analysis, the interactions in two cases as describedin the preceding text will be investigated by the normal formmethod.

4.2.1 Case 1 (uncoupled case): Using the proce-dure developed in Subsection 3.2, the initial conditions ofnormal form variables and interaction Index are computed.Here, the nonlinear index developed is used to quantifythese interaction phenomena and assess the extent of thenonlinearity at critical mode j. Table 3 provides a list of theinteraction index data, in this case at the critical modesdescribed here. As shown, comparing the values of thenonlinear interaction index in three operation situations, i.e.normal operation without SVC, two SVCs in jointoperation with the high SCC and two SVCs in jointoperation with the low SCC, the variation of the interactionIndex in the critical modes is small and can be neglected.These results show that, when two SVCs maintain a weakconnection between their commutations busbars, with hightransfer impedance (uncoupled condition), no controlinteractions may be expected. And an important conclusionis that, in a large network, with several embedded SVCdevices, the control design in each station can be donewithout considering the other station, if the transferimpedance between two SVCs is high.

4.2.2 Case 2 (coupled case): Table 4 providesanother list of the interaction index data at the critical

modes described, when two SVCs are in closely coupledcondition. From the results in Table 4, it can be observedthat, comparing the values of the interaction index in thenormal operation without SVC, the changes of theinteraction Index at three critical modes with both SVCsin joint operation are not significant, when the SCC of theSVC-supporting busbars is high. However, when two SVCsare operating jointly with the low SCC of the SVC-supporting busbars, the values of the nonlinear interactionIndex become obviously large and reveal a strong controlinteraction between both SVC devices. Hence, we canconclude, only under the coupled and low SCC operationconditions, two SVC controllers experience a stronginteraction and great care should be exercised during thedesign stage of these controllers, to guarantee that they willoperate properly.

In fact, when multiple SVCs are installed at differentbusbars in the power system, the nonlinear factor plays animportant role in the system behaviour. Thus, it is thestrong nonlinearity in the power system that causessignificant changes in the computed indices, even if the

SCC for the busbars has a small change, which can be seenfrom the data of Table 1.

In the following subsection, detailed time-domainsimulations are performed to validate the correctness andvalidity of the conclusion obtained.

4.3 Time-domain simulationAs seen in Fig. 3, two SVCs are separately installed inbusbar 3 and busbar 13. The function of SVC installed is tomaintain the busbar voltage. The SVC controllers weredesigned individually and we assumed that satisfactorycontrol performance of the two SVC controllers had beenobtained.

Figures 4 and 5 present the results of the system dynamicresponse performance after the application of a +1.5%voltage reference step of the SVC, when two SVCs are inuncoupled operating condition. The electrical distancebetween two SVCs is then modified to make two SVCs inthe coupled condition. From the simulation results shown

Table 1: Adopted electrical parameters in uncoupled caseand coupled case

Electrical Parameters Uncoupledcase

Coupledcase

Impedance linking busbars 3and 13

0.011+0.11j 0.0011+0.011j

High SCC (MVA) at busbar 3 3416.42 5259.79

Low SCC (MVA) at busbar 3 2606.30 4819.31

High SCC (MVA) at busbar 13 3263.09 5222.72

Low SCC (MVA) at busbar 13 2521.74 4702.01

Table 2: Eigenvalues of test system

Mode Eigenvalue Frequency Mode type Source

4/5 �0.312573.8433j 0.6117 inter-area GEN1,2,3,4

9/10 �1.345777.0778j 1.1265 local GEN3,4

11/12 �0.829277.4845j 1.1912 local GEN1,2

Table 3: Nonlinear Index when two SVCs are in uncoupledcase

Modes Normal Operation(without SVC)

High SCC (twoSVCs)

Low SCC (twoSVCs)

Index Index Index

4/5 1.9314 1.9389 3.9276

9/10 2.1792 1.9539 2.3685

11/12 1.3538 1.5442 5.1551

Table 4: Nonlinear Index when two SVCs are in coupledcase

Modes Normal Operation(without SVC)

High SCC (twoSVCs)

Low SCC (twoSVCs)

Index Index Index

4/5 2.7621 5.7041 32.183

9/10 3.3726 4.4070 47.491

11/12 1.6272 2.1935 21.821

472 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 4, July 2005

in Figs. 6–9, we can conclude that, when the SCC of theSVC-supporting busbar is high, there is practically nocontrol interaction. However, the controller performance oftwo SVCs becomes deteriorated under the coupled and lowSCC condition.

From nonlinear analysis and time-domain simulation, wecan conclude that, under high SCR operating conditions, itdoes not matter if SVC devices are coupled or not, as thereis practically no control interaction. This is because the

0 1 2 3 4 5 6 7 8 9 10

time, s

0.960

0.958

0.956

0.954

0.952

0.950

0.948

0.946

0.944

0.942

busb

ar v

olta

ge o

f SV

C1,

p.u

.

Fig. 4 Control performance of SVC1 in case 1

0 1 2 3 4 5 6 7 8 9 10

0.915

0.920

0.925

0.930

time, s

busb

ar v

olta

ge o

f SV

C2,

p.u

.

Fig. 5 Control performance of SVC2 in case 1

0 1 2 3 4 5 6 7 8 9 100.985

0.990

0.995

1.000

1.005

time, s

busb

ar v

olta

ge o

f SV

C1,

p.u

.

Fig. 6 Control performance of SVC1 in coupled and high SCRcondition

0 1 2 3 4 5 6 7 8 9 100.982

0.984

0.986

0.988

0.990

0.992

0.994

0.996

0.998

time, s

busb

ar v

olta

ge o

f SV

C2,

p.u

.

Fig. 7 Control performance of SVC2 in coupled and high SCRcondition

0 1 2 3 4 5 6 7 8 9 100.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

time, s

busb

ar v

olta

ge o

f SV

C1,

p.u

.

Fig. 8 Control performance of SVC1 in coupled and low SCRcondition

0 1 2 3 4 5 6 7 8 9 100.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

time, s

busb

ar v

olta

ge o

f SV

C2,

p.u

.

Fig. 9 Control performance of SVC2 in coupled and low SCRcondition

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 4, July 2005 473

main linking variable among SVC controllers is the ACvoltage. Hence, in stiff AC systems, no interactions shouldbe expected, regardless of the electrical distance among thedevices. On the other hand, under the coupled and low SCRconditions, a strong interaction phenomenon among multi-ple SVC controllers really does exist and the co-ordinationdesign of the controllers must be properly considered insuch cases.

5 Conclusion

In this paper, a systematic analytical technique based onnormal forms theory is proposed for the analysis ofnonlinear interactions among multiple SVCs installed inpower systems. The nonlinear interaction index calculatedby normal forms method is used to indicate the interactionextent. The developed nonlinear analysis approach, basednormal forms, leads to a better understanding of the systemnonlinear behaviour and can be employed to locate anddesign controllers, and can also be extended to accom-modate more complex multimachine power system andother FACTS devices. The results obtained by the normalform analysis show a good agreement with detailednonlinear time-domain simulation of the study system.

6 Acknowledgment

This work is supported by an Overseas Young ScholarsCollaboration Fund (number 50228007) and an InnovativeGroup Fund of NSFC and partly by EPSRC in the UK.

7 References

1 Mathur, R.M., and Varma, R.: ‘Thyristor-based FACTS controllersfor electrical transmission systems’ (IEEE Press, 2002)

2 Thapar, J., Vittal, V., Kliemann, W., and Fouad, A.A.: ‘Application ofthe normal form of vector fields to predict interarea separation in powersystems’, IEEE Trans. Power Syst., 1997, 12, (2), pp. 844–850

3 Starret, S.K., and Fouad, A.A.: ‘Nonlinear measures of mode-machineparticipation’, IEEE Trans. Power Syst., 1998, 13, (2), pp. 389–394

4 Vittal, V., Bhatia, N., and Fouad, A.A.: ‘Analysis of the inter-areamode phenomenon in power systems following large disturbances’,IEEE Trans. Power Syst., 1991, 6, (4), pp. 1515–1521

5 Ling, C.M., Vittal, V., Kliemann, W., and Fouad, A.A.: ‘Investigationof modal interaction and its effects on control performance in stressedpower systems using normal forms of vector fields’, IEEE Trans. PowerSyst, 1996, 11, (2), pp. 781–787

6 Barocio, E., and Messina, A.R.: ‘Normal form analysis of stressedpower systems: incorporation of SVC models’, Int. J. Electr. PowerEnergy Syst., 2003, 25 (1), pp. 79–90

7 Jang, G., Vittal, V., and Kliemann, W.: ‘Effect of nonlinear modalinteraction on control performance: use of normal forms technique incontrol design part I: general theory and procedure’, IEEE Trans.Power Syst., 1998, 13, (2), pp. 401–407

8 Jang, G., Vittal, V., and Kliemann, W.: ‘Effect of nonlinear modalinteraction on control performance: use of normal forms technique incontrol design part II: case studies’, IEEE Trans. Power Syst., 1998, 13,(2), pp. 408–413

9 Kundur, P.: ‘Power system stability and control’ (EPRI Power SystemEngineering Series, McGraw-Hill, New York, 1994)

8 Appendix: Parameters of the four-machine testsystem

Generator:

Generator base MVA¼ 900MVA;

H1¼H2¼H3¼H4¼ 6. 5s, D1¼D2¼D3¼D4¼ 0.0, Xd1

¼Xd2¼Xd3¼Xd4¼ 1.8, Xq1¼Xq2¼Xq3¼Xq4¼ 1.7, X 0d1¼ X 0d2 ¼ X 0d3 ¼ X 0d4 ¼ 0:25;X 0q1 ¼ X 0q2 ¼ X 0q3 ¼ X 0q4 ¼0:25;

Exciter: KA1¼KA2¼KA3¼KA4¼ 200, TA1¼TA2¼TA3¼TA4¼ 0.01s

The controller parameters of two SVCs:

The rating of SVC¼7100Mvar;

Ksvc1¼ 100, Tsvc1¼ 0.05; Ksvc2¼ 60, Tsvc2¼ 0.02.)

474 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 4, July 2005