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876 IEEE Transactionson Power Systems, Vol.10, NO. 2, May 1995

EIGEN ANALYSIS OF SERIES COMPENSATIONSCHEMES REDUCING THE POTENTIAL OF

SUBSYNCHRONOUS RESONANCE

M.R. Iravan i Abdel-Aty Edris ,Electrical Systems Division

Electric Power Resea rch InstituteDepartment of E lectrical and

Computer EngineeringUniversity of-Toronto

Toronto, OntarioCanadaM5S 1A 4

Abstract: Reference[ 11 describes a new c oncept for m itigation ofthe phenomenon of subsyn chron ous resonance (SSR ) based onasymm etrical series capacitor compensation atSSR frequencies. Thestudies reported in[ 11 are based on a digital time-domain simulationtechnique. Th is paper provides a quantitative evaluation of theconcept using a novel eigen analysis approach. The eigen analysisapproach represents the mathematical models of power systemcomponents in the three-phase basis, and can evaluate the impacts ofasymmetry and imbalance on the system dynamics in thesubsynchronous frequency range. The study results restate technicalfeasibility of the proposed SSR c ountermeasure.

This paper opens the avenue fo r examination of active power filtertopologies to introduce artificial asymmetry at SSR frequencies tocounteract torsional oscillations.Keywords: SSR; Eigen Analysis; Se ries Capacitor; Phase-Imbalance.

1. INTRODUCTION

In machine-network dynamics, the components of the generator air-gap torque co uple the electrical and mechanical sys tems at interactionfrequencies. Concep tually, machine-network interactions can becounteracted if the coupling at the interaction frequenciesis preventedor attenuated. Reference [I] introducesa method for counteractingSSR in se ries capacitor compensated systems, based on thedecoupling concept. The decoupling is achieved by introducing anew series capacitor compensation scheme which behavesas a set ofthree-phase asymmetrical capacitors at subsynchronous frequencies.Thus, attenuates subsynchronous frequency components of thegenerator air-gap torque. Hereinafter, this SSR countermeasureapproach is referred toas ASCC (Asymmetrical Series CapacitorCompensation). Reference[2] further elaborates on the proposedSSR countermeasure technique.

Another method to realize ASCC is described in[3]. This method isbased on setting voltage protection levels of conventional, three-phase, series capacitors at unequal values. This approach is intendedto reduce peak shaft transient torques during large-signaldisturbances, and is not effective for mitigation of small-signaltorsional oscillations. Th e studies reported in111, [2] and [3] arebased on digital time-domain simulation studies, using theElectromagnetic Transients Program (EMT P).

Conceptually, an ASCC scheme can also be realized by augmentingconventional, symmetrical series capacitor schemes with varioustypes of active powerfilters topologies, e.g. voltage sou rce inverters.Howev er, investigation of such app roaches requires an analytical toolfor evaluation of system dynamics during asymmetrical andorunbalanced operating conditions. This paper briefly introduces an

94 SM 557-9 PWRSby the IEEE Power System Engineering Committee of theIEEE Power Engineering Society for presentation atthe IEEE/PES 1994 Summer Meeting, San Fra ncisco, CA,

July 24 - 28 , 1994. Manuscript submitted December 30,1993; made available for printing May 3, 1994.

A paper recommended and approved

3412 Hillview Ave.Palo Alto, CA94304 U.S.A.

eigen analysis tool which models power system components based onthree-phase representation, and determines system eigen structureduring asymmetrical andor unbalanced small-signal dynamics.

The primary objective of this paper is to provide quantitativeevaluations of the ASCC schem es of[ l ] based on the use of theaforementioned eigen analysis softwa re tool. Technical feasibilityofthe ASCC schemes is tested on the first IEEE Benchmark Model.The eigen analysis results are verified by digital time-domainsimulation of the study system, using the EMTP.

2. EIGEN ANALYSIS TOOL

Existing eigen analysis software tools model three-phase electricpower sys tems based on the assumption of symmetrical and balancedoperating conditions. If only low-frequency (0.1- 2 Hz), small-signa l dynam ics are of conc ern, e.g. inter-area oscillations, networkdynamicsare neglected and only machine and controllers dynamicsare considered[5,6]. If high-frequency (5-180 Hz), small-signaldynamicsare of interest, e.g. SSR, network dynamic model andfurther details of machine and controller models are required[7].

Production grade eigen analysis so ftware packages ca nnot be used toinvestigate torsional damping effect of the ASCC schemes describedin [1,2]. The reason is that an ASCC scheme exploits asymmetricalcharacteristic of a system a s the fundamental concept to counteractSSR. And the eigen analysistools are not capab le to formulateasymmetricalhnbalanced three-phase systems for the investigation ofsmall-signal dynamics.

To overcome this barrier, a novel eigen analysis software tool isunder development w hich can model three-phase electrical systemsunder asymmetrical andlor unbalanced conditions, in the high-frequency (5-180 Hz) range. Th e package uses the"S matrixmethod" for calculation of system eigen structure [8]. Detaileddescription of the software is intended for a separate paper.Nevertheless, mathematical model of a set of three-phase,

asymmetrical capacitors are given in the following paragraphs tooutline the general formulation approach.

Figure 1showsa set of three-phase, unequal capacitorsC a. Cb andCC .Equation(1) describes dynamics of the capacitors.

where:

[vabcl= [va Vb %IT

Icabc] = diag[ca cc]

Equation (1) is transformed in a d-q-o reference frame [9] andrearrangedas Eqn. (2).

Fig. 1

1

'b-la+IC- c v c & -c

A set of unequal three-phase capacitors0885-8950/95/$04.000 1994IEEE

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877

where:

[TI s the 3 x 3 ransformation matrix[9].

Small-signal dynamics of the capacitor set is obtained from thelinearization ofEqn. (2) about an op erating point, as given by Eqn.(3).

( d j 4 p V d q o I = (1/3~ a ~ b C c X A ] [ A i d q o ]

+ ( 2 / 3 ~ b c C~ l [ i d 4 a - o ) ~ e ) (3 )

- [ Q { i & o - o p ( A w )

Elements of matrices [A],[B], and [Q] are given in Appendix A.Equation (3) describes small-signal (linearized) dynamics of anasymmetrical set of three-phase, lumped capacitors in a d-q-oreference frame. Equation (3) canbe further generalized if mutualcapacitor coupling also exists. Similar equations for three-phaselumped inductors (resistors) with and without mutual coupling canalso be developed.

3. TEST SYSTEM3.1 System Description

The first IEEE benchmark model forSSR studies is usedas he testsystem [4]. Figure2 shows a schem atic diagram of the test system.The reasons for selecting he first benchmark modelas the test systemare:

The system can exhibit three highly un stable torsional mod esin afairly narrow range of se ries capacito r compen sation levels.

Depending on the level of series capacitor compensation,th esystem can e xperience multiple torsional oscillations.

Fig. 2 A schem atic diagram of thetestsystem

Therefore, the studies can demonstrate the damping effect of anASCC scheme on highly unstableSSR modes and multi-modeoscillations. TheEMTP s used to simulate he test system and verifythe eigen an alysis results.

3.2 Component Model for Eigen Analysis

Figure 3(a) shows the original series compensation schem e of the testsystem withoutSSR countermeasure. Figure 3(b) shows the seriesresonant ASCC sch eme which is composed of the three-phase seriescapacitors (C) and series resonant circuits in two phase s[11. Figure3(c) shows the parallel resonant ASCC scheme whichis composed ofthe three-phase series capacitors(csl and cs2 in each phase) and twoparallel resonant circuits in phases 'a' and'c' [ 11.

It is assumed that the transmission line sections T1 andT2, tep-uptransformer, and generator are symm etrical three-phase components.Thus, without theSSR countermeasure schemes of Figs. 3(b) and3(c) , the test system is symmetrical and balanced. When either seriesor parallel resonant ASCCis in service, the three phases form anasymmetrical three-phase system. Although the line sections,transformer, and generator are symmetrical, their mathematical

L,

Fig. 3 (a) Symm etrical series compensation scheme

(b) Series resonant ASCC(c) P a d e l resonant ASCC

models for eigen analysis must permit unbalanced operation andexistence of 'zero-sequence' components.

The eigen a nalysis tool describedin Section2 represents the systemcomponentsas follows. Series inductanceof each transm ission lineis modelled as a set of three-phase mutually-coupled, lumpedinductances including the effect of flux linkage of neu tral conductor[lo]. Line resistancesare modelled as a set of three-phase balancedresistances (without mutual coupling) connected inseries with thecoupled inductances. Shunt capacitance of each line sectionisrepresented by two sets of symm etrical three-pha se capac itanceslocated atboth ends.

The step-up transformer s represented bya set of lumped, mutually-coupled inductan ces which also permit neutral connection to groundWinding resistances are assumed tobe equal in the three phases andare ocatedin series with the set of mutually coupled inductances.

Electrical system of the generator is represented in the Park's d-q-oframe based on the assu mption ofthree identical and symm etricallyspaced stator windings, and two windingson each rotor axis.Mechan ical system of the generator is modelledas six rigid masses inwhich adjacent masses are connected by shaft spring constants.Viscous and material damping of the mechanical system areneglected.

Series compensating capac itors, i.e. capac itors c in F igs. 3(a) and3(b) and capacitorscSl and cs2 of Fig. 3(c),are modelledas sets ofthree-phase equal capacitors. Correspond ing elements of series andparallel resonant circuits of Figs. 3(b) and 3 (c) are represented byunequal h - p h a s eelements.

4. STUDY RESULTS

The system operating point for the reported studies corns pond to

0.90 per unit power transfer fromthe

generator, at0.96 powerfactor

and the rated terminal voltage, unless otherw ise stated. Dynam ics ofthe gen erator excitation and governor systemsare not includedin thesystem model.

4.1 Case 1

Plot (a) of Fig. 4 shows real part( Q ) of the eigenvalue of the firsttorsional mode (f=15.71 Hz) when series compensation levelis

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EIGENVALUEREALPART 0

(Us)

50 60 70 80 90

COMPBNSATION LEVEL.(%)

Fig. 4 Real part of the eigenvalu eof the first mode(a) No countermeasurein service (Eigen Analysis)(b) Series resonant ASCC in s ervice (Eigen Analysis)(c) Serie s resonant ASCCin service(EMTP)

varied from 40% to 100%. Plot (a) shows that withoutSS Rcountermeasure,t~ of the first mode can possess large positivevalues and consequently rapidly growing sh aft oscillations.

Plot (b) ofFig. 4 show s of the first mode when the series resona ntASCC of Fig. 3(b ) is in service. Parameters of the series resonantcircuit arec/c, = 0.58 and c/cc = 0.93. Plot (b) clearly shows thatthe series resonant ASCC can effective ly ncreas e the damping ofthefirst mode. Plot (c) ofFig. 4 also shows theQ of the first modecalculated from the EM TP simulation results when the series resonantASCC is in service. Close agreement between plots (b) and (c)verifies accuracy ofthe eigen ana lysis results.

The procedures to calculate the real part of an eigenvaluecorresponding o a torsional modeare as follows.

A sh aft torsional response corresponding to the m ode of interestis obtained from the EMTP studies within the time window of0.0s to 3.0s.

Al l the positive encounter ed peak values within the time windoware determined.

0.8 -

DEVIATIONO

-0.8 1 I I I I I 10 0.5 1 1.5 2 2.5 3

= s)

TORQUEDEVIATION 0

(PU)

-0.8

I,0.5 1 1.5 2 2.5 3

= )Fig. 5 First torsima1 mode of shaft segment LPB-G at70%

compensation level(a) No countermeasure in serv ice(b) Ser ie s mn a n t ASC Cin service

A least square curve fitting technique is used to match a sing leexponential function of the form( A exp(0 r ) ) to the peak values.

CY identifies the rateof growthor decay of the torsional respon seand is approx imately the same as the real part of the eigenvaluecorrespondingto the torsional mode.

Figure 5 shows torsional response of the shaft section LPB-G w ithand without the series resonant ASCC in service. The oscillatio nsareexcitedas a result of a sudden chang e in the series compensation levelfrom 73% to 70%. Comparison of Figs. 5(a) and 5(b) clearly show sthe effectiveness ofthe countermeasure scheme for mitigation of theSSR phenomenon. Decrement factors of the oscillations llu stratednFigs. 5(a) and 5(b) closely agree with the corresponding eigenanalysis results ofFig. 4.

Figure 6 shows real parts of the eigen values of the third and fourthtorsional modes when series compensation level is varied from 10%to 70%. Param eters of the series resonant ASCCare c/ca = 0.58 andc/cc = 0.93 which are the same values used to obtain plots (b) and(c) of Fig. 4. Figure 6 indicates that the SSR counterme asure alsocan effectively damp out the 3rd and 4th oscillatory modes in theentire range of se ries compensation.

4.2 Case 2

Figure 7 depic ts the real part of the eigenv alue of the first mode withand without parallel resonant ASCC in service. The operatingconditions are exactly the sam eas those of Fig. 4. The tuningfrequencies of phases 'a' and 'c' are selected at 60-15.71=44.29Hzand 60-25.53=34.47Hz. Resistive elementsRa and Rc are equal to

EIGENVALUE

(Us)REALPART 0 -

20 30 40 50 60

COMPENSATION LEVEL(%)

EIGENVALUE

(Us)REALPART 0 -

20 30 40 50 60


Fig. 6 Real pa rts of the eigenvalues of3rd and 4th modes(a)No countermeasurein service (Eigen Analysis)(b) Series resonant ASCC in service (Eigen Analysis)

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I f : \ I


(11s)

50 60 70 80 90


Fig. 7 Real part of the eigenvalue of thefirst mode(a) No countermeasurein service (Eigen A nalysis)(b) Parallel resonant ASCCin service (EigenAnalysis)(c) Parallel resonant ASCCin service(EMTF')

0.86R. Parallel resonant circuits of phases 'a' and 'c' are installed on30% of the net series capacitor.

Figure7 indicates that the parallel resonant scheme can effectively

increase the damping of the first mode and provide adequate dampingin the practical range of series compensation(40% o 70%). Closeagreem ent between the sim ulation and the eigen analysis results, i.e.plots (b) and (c) of Fig.7, verifies the accuracy of the eigenvaluestudies.

Figure8 shows time-domain simula tion results with and without theparallel resonant scheme in service. Decreme nt factors of thetorsional oscillation s ofFig. 8 closely agree with the correspondingresults ofFig. 7. The oscillations are excited as a result ofa suddenchange in the series compensation level from73% o 70%.

Figure 9 shows damping effect of the parallel resonan t ASCC on th ethird and fourth torsional modes when series compensation level isvaned from 10% to 70%. Figure 9 illustrates that th e parallelresonantASCC also can effectively mitigate both mo des in the wholerange of series compensation values.

0.8 7

TORQUED@VlATlONO

(PU)

-0.8 I I 1 i I I I0 0.5 1 1.5 2 2.5 3

0.8 TIME 6)

TORQUEDEVIATION0

(PU)

-0.8

0 0.5 I 1.5 2 2.5 3

-nhm s)Fig. 8 First torsion al mode of the shaft segme nt LPB-G at70%

compensation level(a) No countermeasure in service(b) Parallel resonant ASCCin service

Mode 3i (a ) i

0.1 ..........

EIGENVALUEREiALPART 0

20 30 40 50 60


0.1 .......... :..... ...............................................

' (a) i


-0.1 ;.A ................

420 30 40 50 60


Fig. 9 Real parts of the eigenvalues of 3rd and 4th modes(a) No countermeasure in se rvice(b) Parallel resonant ASCCin service

4.3 Case 3

Cases 1 and2 investigate damping effects of series and parallel

resonant schemes on the SSR phenomenon excited by sm all-signalperturbations. Case3 studies the effects of the two scheme s on peaktransient torsional stresse sas a result of large-signal disturbances.

Figure 10 illustrates the effect of the series and parallel resonantASCC's on the transient mechanical stresses of the shaft sectionLPB-G. Transient stresses are initiatedas a result of a three-p hase oground fault at the generator high voltage bus,Fig. 2. The fault iscleared after three cyclesby the line circuit breakers. Fault clearingisfollowed by a successful reclosure after another 15 cycles (3-15cycles disturbance). Priorto the fault, the g enerator delivers 0.5per-unit power at0.97 lagging power factor at series compensation levelof 70%. Parame ters of series and parallel resonant schemes are thosegivenfor Cases 1 and2 respectively.

Figure 1O(a) demo nstrates that the transien t torque is dominate d bythe first torsional mode (15.71Hz). When compared with Fig.10(a ), Figs. 10(b ) and 1O(c) show tha t both series and pa rallelresonantschemes can reducepeak magn itudes of mechanicalstressesimposed on the shaft segment. The first few encou ntered peaktorques are not noticeably affected by the counterm easure schem es.Nevertheless, the damping effect is pronouncedafter M.7.s.

Overv oltage protection sc hemes of series capacitors are not includ edin the simulation model for the study results presented inFig. 10. Figure 11 shows the same investigation results when the models ofnonlinear ZnO protection schem es of series capacitor are included inthe simulation studies. The ove rvoltage protection le vel provided by

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-2 --3

I I I I

' I (C)I I I

0 0.5 1 1.5 2

TIMEW

-2 --3

I I I0 0.5 1 1.5 2

TIMEL9

(C)1 I I

iI

~

1 I II

ZnO elem ents is set at 2.15 per-unit, based on the capacitor peak ratedvoltage. In series and parallel resonant schemes of Figs. 3(b) and

3(c), capacitor segmentsc, ca. c,, csl and $2 are equipped with ZnOprotection scheme s. Figu re 11 indicates that bothseries and parallelresonant schemes noticeably reduce the peak mechanicalstresses as aresult of the 3-15 cycles disturbance, even when overvoltageprotection schemesare n service.

4.4 Discussions

The primary objectiveof this paper is to provide a quantitativeevaluation of theseries and parallel resonant schemesof Figs. 3(b)and 3(c). Thus, no particular attempt was made to optimize theparameters of the two schemes with respect to the damping oftorsional modes. The introduced eigen analy sis tool can be used tocalculate sensitivities of torsional modes with respect to theparameters of the two counterm easure schemes, and to optim ize thecorresponding parameter values [7].

The detuning machine-network approach investigated in this workcan also be realized based on theuse of active power topologies. Theactive power filters can be installed either in seriesor parallel withseries capacitors at the capacitor siteor even at remote locations, e.g.neutral end ofY onnected windingsof the system transformer. Th eintroduced eigen analysis technique provides a comprehensiveanalysis tool to investigate and design such power electronic circuits.

5. CONCLUSIONS

This paper provides a quantitative evaluation of machine-network

0 0.5 1 1.5 2

-(S)

2 I

LPB-G I

-1 I

-2i (b)

Torque

(PU) O I

Y

-3 i I I I 10 0.5 1 1.5 2

detuning conceptfo r preventiodmitigation of subsynchronousresonance. The required detuning is achieved by augmenting series

capacitors with eitherseries or parallel resonant circuits.A newlyintroduced eigen analysis approachis used fo r the studies. Three-phase representation of system components are usedfo rdetermination of the system eigen structure. Thus, impacts ofasymmetry and imbalance on the torsional dynamics are accuratelyrepresented. The eigen analys is results are verified by digital time-domain simulation, using the EMTP. The studies show that:

Both series and parallel resonant schemes can effectively enhancedamping of all subsynchronous oscillatory modesas a result ofsmall-signal perturbations.

Both series and parallel resonant circuits reduce peak transientstresses imposed on the turbine-generator shaft segments afterlarge-signal disturbances.

6. REFERENCES

[l ]

[2]

[3]

141

A. A. Edris, "Series Compensation Schemes Reducing thePotential of Subsynchronous Resonance", IEEE Trans.,Vol.

A.A. Edris, "Subsynchronous Resonance CountermeasureUsing Phase Imbalance", IEEE Trans., Vol. PWRS-8, No.4,November1993.M.R. Iravani, "A Method for Reducing Transient TorsionalStresses of Turbine-Generator Shaft Torques", IEEE Trans.,

IEEE SSR Working Group, "First Benchmark Model for

PWRS-5, NO. 1, pp. 219-226, 1990.

Vol. PWRS-7, NO. 1, pp. 28-36, 1991.

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Computer Simulation of Subsynchronous Resonance", IEEETrans., Vol. PAS-96, No.5, pp. 1565-1572, 1977.P. Kundur, G.J. Rogers, D.Y. Wang, L. Wang, M.G. Lauby,"A Comprehensive Computer Program Package for Sm allSignal Stability Analysis of Power Systems", IEEE Trans.,

N. Martins, L.T.G. L ima, "Eigenvalue and Frequency DomainAnalysis of Small-Signal Electromechanical StabilityProblems", IEEE PES Special Publication 90TH0292-3-PWR,

M.R. Iravani, "A Software Tool for Coordination ofControllers in Power Systems", IEEE Trans., Vol. PW R SJ .

Vol. PWRS-5,NO .5, pp. 1076-1083, 1991.

pp. 17-33, 1989.

No. 1, pp. 119-125, 1990.N. Uchida, T. Nagao, "A New Eigen-Analysis Method ofSteady-State Stabilitv Studies for Large Power Svstems:SMatri; Method" , IEEE Trans., Vol. PWYRS-3,No. i, p. 706-714, 1988.P.C. Krause, "A nalysis of Electric Machinery", McGraw H ill,New York, 1 986, pp. 133-160.V. Del Toro, "Electric Power Systems", Prentice Hall,Toronto, 1992, pp. 382-388.

APPENDIX A

Element of matrix [A],Eqns. (3):

A11 = K1+2K2sin28,

A12 = A21 = -K3 + 2K3 sin28, - K2 sin28,A13 =A31 = - 2 K ~ c o s 8 , - 2 K 3 ~ i n ~ ,

A22 = (K1+ 2K2)- K2 sin28, + K3 sine,A23 = A32 = -2 K2 sin8, + 2K3 cose,A33 = K4K1= 2A+ B /2+ C f2 , K2= ( -2A+ B + C ) /2

K3=-(&/2)(B-C), K4 = A + B + C

A = Cbc, , B = ceca, c = a c b

Elements of mam x[QJ, Eqn. (3):

q12 = -q21 = 1, rest of elementsare zero

Elements of matrix[B], Eqn. (3):

R1= K2 sin(28,)+K3 cos(28,)B12 = &L1= K3 sin(28,)- K2 c0s(2B0)

h3 = h1 = K2 sine, - K3 cose,&e!= -K2 sin(28,) + K3 cos(28,)+, = &32= K2 sine, +K3c ose,4 3 = o

M.R. Iravani (M '85) received his B.Sc. d egree in electrica lengineering in 1976 from Tehran Polytechnique University andstarted his career as a consulting engineer. He received his MSc .and Ph.D. degrees also in electrical engineering from the Universityof Manitoba, Canada in 1981, and 1985 respectively. Presently heisa professor at the University of Toronto. His research interestsinclude power electronics, power systems dynamics, and control.

Abdel-Aty Edris was born in Cairo, Egypt in 1945. He receivedhisB.Sc. from Cairo University in 1967, the MS. from Ain-ShamsUniversity, Egypt in 1973, the Ph.D. from Chalmers UniversityofTechnology, Sweden in 1979.Dr. Edris joined ASEA (now ABB) in Vasteras, Sweden in 1981.From 1982 to 1986 he was involved in power systems analysis ofHVDC and reactive power compensation projects.From 1986 to1990 he worked with developm ent projects introducing new conceptsimproving power systems performance. From 1 990 to 1992 he waswith Transmission and Relaying Centerof ABB's Advanced SystemsTechnology in Pittsburgh. In 199 2, Dr.Edris joined Electric PowerResearch Institute (EPRI)as a Manager of Flexible AC TransmissionSystems (FACTS).Dr. Edris is a member ofth e IEEE SSR WG, anda member oftheCIGREWG for Power System s Dynam ic Analysis.

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Discussion

0. P. Malik (The University of Calgary, Calgary, Alberta,Canada):

It is an interesting approach to use eigen analysis for thestudy of subsync hronous reson ance. The d iscussion would like

to obtain some clarification from the authors on the followingpoints.

Transfonnation to the d-q-o frame of reference isextremely useful for the study of symmetrical and balan cedconditions. For the study of asymmetrical cond itionsas nthis paper, d-q-o transformation osses its advantages. Th eauthors appropriately start with abc frame of reference ineqn.(l), b ut then transform it to d-q-o frame of reference.Even after linearization of eqn.(2) for small signalanalysis, eqn(3 ) still contain s differential terms, i.e. tim efunctions. Eigen analysis being a frequency domaintechnique, the authors d o not explain how they go fromeqn. (3 ) to eigen-analysis.

Following from point (i), i t is claimed at a numb er ofplaces in the paper, eg. Abstract, Intro ductio n, Discussion,Conclusion, that a new eigen-analysis approach isintrodu ced. This claim is certainly invalid, as the neweigen analysis tool purported to have been introduced hasnot been introduced in this paper nor any reference isprovided where this tool is described. It is stated to beunder development and "intended for a sep arate paper".That paper should have been published first beforethispaper.

Will the authors confirm that:(a) the generator model used hastwo amper windings on

the q-axis and one damper winding on the d-axis?(b ) for th e con ditions in Fig.5, the system with no

counter measures is s table for a series compensationlevel of 73% but unstable for70%?

Power systems are non-linear, and their ch aracteristics andparameters depend upon the operating conditions.Furthe r, the effect of governo r and excitation controller isneglected in this study. This discusser is interested inhearing the authors ' com ments on whether the inclusionofnon-linearities and the e ffect of th e abo ve controllers willaffect the results and co nclusions of this paper.

Manuscript received August22 , 1994.

M.R. Irav ani and A.A. Edris: We would like tothank Professor Malik for his interest in our paper

and his comm ents.The followingis our response toProfesso r Malik's comments.

(i) The main idea behind transforming the systemequations in a d-q-o reference frameis toconv ert the machine time-variantODEs into aset of time-invariant ODEs, whichis necessary

for eigen analysis. Th e transformationperforms the task whether the networkissymmetrical and balanced or not.If thenetwork is symmetrical and balanced, thetransformation also has another advantage,i.e.it reduces the three abc frame state variables(e.g. Avo, Avb and AV,) t o t wo s t a t e

variables (e.g. dVd and Avq). However, w ecannot exploit this property for our system.With respectto Eqn. (3),it should be noted tha tterms A 8 and d(A8) /d t are present in thetransformed equation whetherthe network issymm etrical and balanced or not. How ever,this is not an impediment for eigen analysis,since rotor position and speedare two of thegenerator state variables, and they appearinvector Ax when all the system equations areorganizedin the general form of

A x = A A x ( Al )0

(ii) Th e objective of the paperis to dem onstrate thatthe conventional eigen analysis approach canalso be extended fo r those conditions where thesystem is asymmetrical and unbalanced, e.g.th e SSR countermeasures discussed in thepaper or power electronic devices which canartificially introduce asymmetry in the system.Justifying the need, the eigen a nalysis toolisdeveloped and a comprehensive report will bepublished in the near future.

(iii) (a) Th e generator model used has one dam perwinding and one field winding on the d-axis, and two damper windings on theq-axis.

(b) With out countermeasure, at both 73% and70% compe nsation levels,th e first modecan become unstableif the systems issubjected to a small-signal perturbation.Initially, compensation levelis adjusted a t73%. Since simulation starts fromasteady-state condition the first mo deis notexcited and not observable in the timeresponse. A sudden 3% change in thecompensation levelis introducedto excitcdthe f i rs t mode and demonstrate i tsinstabilityas shown in Fig. 5(a). Figure5(b) indicates that the unstable oscillationsare mitigated by the countermeasure.

(iv) A s long as small-signal torsional oscillation sare of concern, the eigen analysis resultsarevalid, evenif the initial steady-state opera tingpoint of the systemis associated with n onlinearoperating regions of transformers and thegenerator. If the systemis subjected to large-

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signal disturbance, e.g. faults, the eigenanalysis results are not necessarily valid.Section 4.3 of the paper elaborates on thistopic.

The reason fo r neglecting the impacts ofgovernor and excitations systems on the

torsional oscillations is their relatively slowresponses. Practically, instability of a torsional

mode is detected and mitigated within up to2seconds after the modehas been excited.During thistime frame, neither the excitationsystem (due tothe field time-constant) nor thegovernor system can react to the torsionaloscillations.

Manuscript received October 27 , 1994,

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