Subsynchronous resonance of a powerstation with two identical generating

unitsG.D. Jennings, B.Sc.Eng., Grad.S.A.I.E.E., Prof. R.G. Harley, M.Sc.Eng.,

Ph.D., Mem.I.E.E.E., C.Eng., M.I.E.E., M.S.A.I.E.E., and D.C. Levy, M.Sc.Eng.,M.S.A.I.E.E., C.Eng., M.I.E.E.

Indexing terms: Generators, Stability

Abstract: The paper investigates the effect of adding a second 1072 MVA turbogenerator to an existing one in apower station feeding a series capacitor compensated transmission line. In carrying out the investigation, thenew two-generator system is modelled as one equivalent generator with increased capacity, and also as a multi-machine system with two separate generators. A comparison of these models shows that, for this type of stabil-ity study and this transmission system configuration, there is no advantage gained in modelling the twogenerators separately rather than as one equivalent generator. The results further show that conditions can existwhere the system is stable with one generator but unstable with two, and vice versa.

List of principal symbols

d, q = d, g-axis of a machine rotating reference frameD, Q = D, Q-axis of the synchronously rotating reference

frame[D] = shaft damping matrix[G] = rotational voltage inductance matrixH = inertia constant in secondsi = vector of axis currents[/] = identity matrix[J] = shaft inertia matrix[X] = shaft stiffness matrix[L] = machine inductance matrixN = compensation levelp = derivative operator d/dtPt = generator terminal power[A] = machine resistance matrixs( = ith natural frequencyT = forcing torque vectorTe = electric to rqueTm = shaft mechanical torquev = vector of axis voltagesVb = infinite bus voltageVt = generator terminal voltageH>,- = ith mode shape vectorXc = transmission line capacitive reactance0 = null vectorq> = flux linkageA = small change operator)H = ith eigenvalue8 = ro tor angle vectorco0 = system frequency in rad/s (electrical)co = angular velocity of rotor in rad/s (electrical)

1 Introduction

A large power station sited in the Western Cape area ofSouth Africa will transmit power to the national grid via a1400 km transmission line. The present line is rated at400 kV and is equipped with series capacitor com-pensation. The possibility of subsynchronous resonance(SSR) occurring in this system has been investigated andreported elsewhere [1]; this first paper has been followed

Paper 4275C (P10, P9), received 23rd May 1985

The authors are with the Department of Electrical Engineering, University of Natal,King George V Avenue, Durban, 4001, South Africa

by many others dealing with topics such as system model-ling, multi-machine SSR and possible SSR counter-measures [2-11].

All previously reported investigations were based onlyon the first 1072 MVA generator at this power station;however, a second identical generator is currently beingcommissioned.

Conventional representation in a stability analysis of apower station containing more than one generator usuallyentails 'lumping' all the generators into one equivalentgenerator with a rating equal to the sum of the ratings ofthe individual generators [12]. This is a valid simplifica-tion if the stability study is concerned only with the tran-sient oscillations of energy between the electrical networkand the rotating inertias of the generators. However, anSSR investigation also includes an analysis of the oscil-lation of energy between the various individual inertias ofa generator shaft interacting with the electrical system.Thus the shaft's natural frequencies and the torsional mod-elling of the shaft become important. If two generators areconsidered to be a single equivalent generator of increasedrating, then the station is transformed from two coupledidentical resonant systems to a single resonant system, anda certain amount of detail about the station's mechanicalresonance characteristics is lost. The combined resonantproperties of two coupled resonant systems are differentfrom those of one of the systems individually. The effect ofthis on the SSR behaviour of the Western Cape system hasnot yet been published.

This paper investigates two aspects of this problem. Itfirst evaluates whether, for stability purposes, the two gen-erators have to be analysed as a true multi-machinesystem, or whether they can be replaced by one 'equivalentgenerator' of twice the rating. The results show that theloss in detail in the shaft torsional characteristics incurredby considering a single equivalent generator of increasedinertia instead of the real two-generator system, does notseriously affect the SSR stability studies of this particularpower system; this finding does, however, depend on theimpedances between these generators as well as betweenthem and the infinite bus.

This paper then carries on to reconsider this station'spotential SSR problem at high levels of series com-pensation, with particular emphasis on the effects ofadding the second generator. It compares the results withthose obtained previously by the authors when only one

IEE PROCEEDINGS, Vol. 133, Pt. C, No. I, JANUARY 1986 33

generator was in operation, and shows that the previousresults were optimistic.

2 Theory

To investigate the non-linear power systems of Figs. 1 and2 it is necessary to describe each system element by a

transformerinfinitebus

VtP, station bus

Fig. 1 Schematic diagram of the two-generator system

Fig. 2 Schematic diagram of the equivalent generator system and theone-generator system

mathematical model. The well known [13] two-axis equa-tions describing the generator, transmission system andmechanical system of Fig. 2 are summarised below for thesake of completeness.

2.1 Synchronous generatorSubject to the usual assumptions [13] pertaining to thetwo-axis theory, the mathematical model of a synchronousgenerator, expressed in terms of currents and a rotor refer-ence frame, takes the following state-space form:

where co is the rotor speed. The vector of axis windingcurrents is given by

i = PL ir, U, i i, i, ,~]T (2)• Lld» lfd > ' M ' lq' lkql> lkq2J K^)

and that of voltages byv = lvd,vfd,O,vq,O,OY (3)

In the above model, the synchronous generator is rep-resented by six coils with one damper circuit on the d-axisand two damper circuits on the g-axis. The axis currentsfrom eqn. 1 are used to calculate the electrical torque Te asfollows:

Te = -ir[G]iey0/2 (4)

2.2 Transmission systemThe transmission system is assumed to consist of a lumpedseries combination of resistance, inductance and capac-itance in each of the stator phases.

The two-axis voltage vector at the generator terminalsis given by

vm = vbus — vc — vL — vR (S\dqz dqz dqz dqz 'dqz v**/

where the terms on the right hand side of eqn. 5 are theinfinite bus voltage and the voltage drops across thecapacitance, inductance and resistance as given by eqns. 6,7 and 8, respectively:

vLd = Lpid + wLiq and vL

q = Lpiq - coLid

v% = Rid and uf = RL

(7)

(8)

2.3 Mechanical systemThe mechanical system of a single generating unit isassumed to consist of four turbine stages, a generator andan exciter [1], all represented in Fig. 3 as six inertias inter-connected by five torsional springs (the connecting shafts),

infinitebus

J i6 , J 2 6 2 J363 J464 J5 65 J 6 6 6

K12 (———i K23r—n K^r—n K « 5 r f ± i K56HP LP1 LP2 LP3 GEN EXC

Fig. 3 Mechanical model of a single turbogenerator

and which can be described by a set of second order differ-ential equations of the form

0 = [J]p2d + [_D-]p8 + [K]5 + T (9)

where [J] is a diagonal matrix of inertias, [D] is a diago-nal matrix of damping coefficients, [K] is a symmetricmatrix of shaft stiffnesses, J is a forcing torque vector(acting on each inertia) and S is an angular position vector.

2.4 Two-generator systemThe two generators, coupled through the transformers asshown in Fig. 1, are analysed using multi-machine tech-niques, the theory for which has been explained elsewhere[14]. The generators and network are represented insteady ABC variables for the initial load-flow calculationand the entire system is represented in d, q variables foreigenvalue and transient response calculations.

In a multi-machine system the d, q reference frames ofindividual generators rotate at different speeds, and thestator variables of all generators are therefore transformedinto a common synchronously rotating D, Q referenceframe. In the common reference frame each generator isrepresented by a model which consists of a resistance,inductance and a voltage source which does not neglectstator transformer-voltage and speed-voltage terms.

The state-space formulation of the total system can besummarised in the following form [14]:

The state vector of the entire system is a composite vectorconsisting of the following components:

(11)

and

u = [ybDQ,bDg,bQg, vfd,FJ

Pv<d = iJC - covi and pvi = L/C + covcd (6)

(12)

where the subscript '£' denotes currents in non-machineinductors and the subscript '#' denotes quantities associ-ated with the generators, while [̂ >dr] and [<p9r] are thecomposite vectors made up of the q>dr and q>qT vectors ofeach generator, [/J are the mechanical states of the gener-ators (angles and speeds), vbDQ contains the £>, Q-axis com-ponents of the infinite busbar voltage, bDg and bQg are thevectors of all the D, Q components of the generator volt-ages bg, vfd is the vector of generator field voltages and Ftis the vector of prime mover input powers.

34 IEE PROCEEDINGS, Vol. 133, Pt. C, No. J, JANUARY 1986

For the purpose of the eigenvalue calculations, the gen-erator equations, transmission system equations and themechanical equations in single machine and multi-machineform are linearised about some steady operating point assummarised in Appendix 7.2.

2.5 Single per-unit equivalent generatorConventionally, a number of generators can be replacedby a single equivalent generator with a rating equal to thesum of the ratings of the individual generators. Thus, thetwo generators of Fig. 1 can be simplified to the equivalentgenerator of Fig. 2 with a rating of 2144 MVA, i.e. twicethat of a single generator. The base values for one of thephysical generators (1072 MVA) and those for the equiva-lent generator (2144 MVA) are given in Appendix 7.1.

The per-unit values of the electrical parameters in theequivalent system of Fig. 2 are given by the per-unit valuesin the two-generator system of Fig. 1 multiplied by theratio of equivalent-generator base power to single gener-ator base power, that is by a factor of two. However, as thegenerator and transformer in Fig. 2 represent the parallelcombination of generators and transformers in Fig. 1, theirparameters in physical units are half those of a single gen-erator and transformer in Fig. 1. Thus in per-unit, the gen-erator and transformer of Fig. 2 have the same electricalparameters as those in Fig. 1.

For the equivalent generator's mechanical system to beequivalent to that of the two-generator system, its shaft'stotal inertia must be equal to the sum of the total inertiasof the two generators in order to have the same quantity ofstored rotational energy at synchronous speed. However,as the base inertia for the equivalent generator is twicethat for a single generator, the per-unit values of inertiawill be the same for both systems.

Similarly, to carry through the torsional characteristicsof the shaft to the equivalent generator, the physical valuesof shaft stiffnesses of the equivalent generator must betwice those of the single generator's shaft. As in the casewith the inertia, due to the doubling of the base value ofstiffness for the equivalent generator, the per-unit values ofshaft stiffness are the same in both cases.

The inertia constant H in seconds is the same for allgenerators, as both the stored kinetic energy at synchro-nous speed and the base power increase by the samefactor. The inertia J in eqn. 9 is given by 2H/co0.

The equivalent system is summarised in Table 1 wherethe elements in the table give the ratio of equivalent-generator system values to the two-generator systemvalues.

Table 1: Ratio of equivalent-generator system parameters totwo-generator system parameters

Generatormechanical

Generator electrical Transmissionand transformer line

p.u. physical p.u.1 2 1

physical p.u. physical0.5 2 1

and 12 are also calculated for two generators in operation,but the system is analysed as a multi-machine system. The

bUU

500

400

300

200

100

0

100

-i

-

E1

-

E2

/

\

R2

'MK5

see Fig.4B

MK^_«..MK3— MK2

_.. MK1

MK0

R1 ~M

-15 -10 -5real , s -1

Fig. 4A Eigenvalue loci for the equivalent generator system as Xc isvaried from 0.28 p.u. to 2.12 p.u. (2144 MVA base)

120 r

100

a 80co>o.§ 60

MK4

MK3

MK2

MK1

20-1.5 -1.0 - 0 5

c = 1.04p.u.(N =

-x =1.36p.u.(N=67.3°/.)

-xc=1.A3p.u.(N = 70.8%>)

_xc = V67p.u.(N=82.70/.)

= 2.02p.u.(N = 99.8°/.)

0 0.5 1 0real . s"1

1.5 2.0

Fig. 4B Expanded view of modes MK1 to MK4 for the equivalent gen-erator system as Xc is varied from 0.28 p.u. to 2.12 p.u. (2144 MVA base)

3 Results

The results in this section are calculated for three differentgenerator configurations. Figs. 6, 9 and 11 are calculatedfor only one generator as a base case which is the samestation representation that has been used in the previousstudies [1-11]. Figs. 4A and 4B, on the other hand, arecalculated for two generators in operation, but considerthem as a single equivalent generator so that the analysismay be carried out as a single machine study. Figs. 5, 10

system parameters are summarised in Appendix 7.1. Forall the results, the initial conditions assumed are: Vb =1.0 p.u., Vt = 1.1 p.u. and Pt = 0.6 p.u.

3.1 Eigen value - loci diagramsEigenvalue-loci were generated from repetitive calculationsof the eigenvalues of the linearised state-space equations.In the case of complex conjugate pairs, only those loci withpositive imaginary parts are drawn.

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986 35

3.1.1 Variation in series compensation: In this sectionthe system stability as a function of the series capacitorcompensation is investigated by calculating a locus ofeigenvalues for each generator system representation asthe compensation level is varied from about 14% (Xc =0.08 ohm) to about 105% {Xc = 0.57 ohm) where:

% compensation level = N =

100Xc/(line reactance XL) (13)

Equivalent generator against two separate generatorsFig. 4A shows the eigenvalue-loci in the direction of thearrows for the two-generator system represented as anequivalent-generator system. Table 2 contains a set ofeigenvalues from these loci calculated at a compensationlevel of 74% (Xc = 0.4 ohm). This is the same value of Xcthat was used in the first study on this system [1].

Table 2: Equivalent-generator-system eigenvalues calculatedfor a compensation level of 74% (Xc = 0.4 ohm)

Mode Real Imaginary

E1E2MK5MK4-]MK3<MK2jMKiJMK0MR1]R2>R3

Supersynch. fluxSubsynch. fluxSupersynch. mech.

.Subsynchronousmechanical

i

Hunting modeMagnetic stability

Dampercircuits

-11.2-10.4

-0.79-1.05-0.72-0.30-0.61-1.04-0.11-1.10

-10.8-17.6

±559.0±69.8

±582.0±110.0±99.2±77.3±42.9±5.10

0.00.00.00.0

As the compensation level is increased, the frequency ofthe supersynchronous electrical mode El increases, whilethat of the subsynchronous mode E2 decreases. Mechani-cal modes MK4 and MK5 are not affected by com-pensation level changes (as the generator mass is near anodal shaft point [1] for these modes), but the damping ofmodes MK1, MK2 and MK3 is influenced. The frequencyand damping of the hunting mode MKO increases withincreasing compensation due to the stronger electricalcoupling of the generator to the infinite bus.

Fig. 4B shows an expanded view of mechanical modes 1to 4. The system first becomes unstable at an Xc of about1.04 p.u. (0.278 ohm) when mode MK3 moves into theRHP. Mode MK2 moves into the RHP when Xc =1.36 p.u. (0.367 ohm) and mode MK1 moves into the RHPwhen Xc = 1.67 p.u. (0.448 ohm). Thus the maximum per-missible compensation level with the equivalent generatorbefore the first occurrence of instability is 51.5% (Xc =0.278 ohm).

Figs. 5A and 5B contain the loci calculated for the two-generator multi-machine representation of the system, andare similar to the equivalent generator loci of Fig. 4. Table3 contains a set of eigenvalues calculated for the two-generator system when the compensation level is 74%{Xc = 0.4 ohm).

A comparison of Figs. 5A and 5B with Figs. 4A and 4Band of the eigenvalues in Table 3 with those in Table 2shows the following:

(a) The modelling of the two generators separately doesnot produce a duplicate set of the six mechanical modesMKO to MK5 and the four rotor circuit modes Rl, R2, R3and M of a single generator system. Instead, for each gen-erator mode of Figs. 4A and 4B there now appears a pairof modes in Figs. 5A and 5B. These pairs are the so-called

common (or symmetrical) and anti- (or anti-symmetrical)modes associated with symmetrical resonant systems [15](see Appendix 7.3). For mechanical modes MK4 and

600r

500-

400 -

300-

200-

100-

-100

El

-

- E2

- <

MR 2 R2

MK5

•E3

see Fig 5B

— MK2

.... MK1

, MKO

Rl Tio - •••-•—

-15 -10 -5real . s"

Fig. 5A Eigenvalue loci for the two-generator system as Xc is variedfrom 0.14 p.u. to 1.06 p.u. {1072 MVA base)

120

100-

2 80-

aI 60oE

40 h

20-1.5 -1.0 - 0 . 5

MK4MK3-

MK2'.

MK1'

i

——— ** n

MK3

——^ , —MK2

MK1

j ( c = 0.52p.u.(N=51.5V.)

S ( c = 0.62p.u.(N = 61.27.)

,% =0.68p.u.(N=67.3V.)

^xc=0715p.u.(N=70.8*/.)

,xc=0835pu.(N=82.7°/.)

N<c=1.01 p.u. (N =998*/.)

0.5real . s "

1.0 1.5 2.0

Fig. 5B Expanded view of modes MK1 to MK4 for the two-generatorsystem as Xc is varied from 0.14 p.u. to 1.06 p.u. (1072 MVA base)

MK5, only one mode is seen as in each case the commonmode and anti-mode are almost coincident because thesemodes are not excitable from the electrical network andare thus not coupled. For the rotor circuit real modes (M,Rl, R2, R3), the anti-modes are initially more dampedthan the common modes. However, this is not obvious aseach common mode locus moves through its anti-modewhen the compensation level is increased, and at highcompensations the anti-modes are less damped than thecommon modes. For the mechanical modes (MK1 toMK3) the anti-mode (MKT to MK3') is, in each case,

36 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986

Table 3: Two-generator-system eigenvalues calculated for acompensation level of 74% (Xc = 0.4 ohm)

Mode

E1E2E3MK5MK4-\MK3 1MK2(M K VMKOMR1 1R2 >

R3 3

Supersynch. fluxSubsynch. fluxSynch, fluxSupersynch. mech.

Subsynchronousmechanical

Hunting modeMagnetic stability

Dampercircuits

Commonreal

-11.2-10.4

-2.25-0.79-1.05-0.72-0.30-0.61-1.04-0.11-1.10

-10.8-17.6

modeimaginary

±559.0±69.8

±314.0±582.0±110.0

±99.2±77.3±42.9

±5.100.00.00.00.0

Anti-modereal

-0.79-1.05-0.87-0.87-0.98-1.68-0.19-1.41

-12.8-19.1

imaginary

±582.0±110.0±101.0

±78.5±43.6

±7.360.00.00.00.0

slightly more damped and at a higher frequency than itscorresponding common mode. The hunting mode MKOstarts at a frequency and damping less than its anti-modeMKO', but at high compensation levels the frequency ofthe common mode is greater than that of the anti-mode.The difference between common mode and anti-modevalues depends on how tightly the generators are coupledto the infinite bus; the closer they are to the infinite bus,the smaller this difference becomes.

(b) The anti-mode loci of the two-generator system areonly slightly affected by a change in compensation level.However, the common mode loci are identical to the lociof the equivalent generator system in Figs. 4A and 4B.Thus, for the system in Fig. 1, the level of compensationpermitted before the onset of instability can be calculatedwith either a two-generator multi-machine representationor a single equivalent generator representation. This con-clusion is significant as the two-generator model requiresan additional eighteen differential equations, and so theuse of the equivalent generator model results in a largereduction of computer time.

All further results in this paper investigate the effect whichthe addition of the second generator has on the previousstability studies [1] done for only one generator.

Two-generator system against one-generator systemThe effect of adding a second generator can be seen bycomparing Figs. 6A and 6B with Figs. 4A and 4B as Figs.6A and 6B show the loci calculated for the system withonly one generator. Table 4 contains a set of eigenvaluesfor the one-generator system calculated with a com-pensation level of 74% (Xc = 0.4 ohm).

Table 4: One-generator-system eigenvalues calculated for acompensation level of 74% (Xc = 0.4 ohm)

Mode Real Imaginary

E1E2MK5MK4>jMK3(MK2MK1JMKOMR1 )

IR2 >R3 3

Supersynch. fluxSubsynch. fluxSupersynch. mech.

Subsynchronousmechanical

Hunting modeMagnetic stability

Dampercircuits

-9.85-9.56-0.79-1.05+0.25-0.12-0.76-1.24-0.17-1.18

-11.4-18.1

±540.0±88.7

±582.0±110.0

±98.2±78.9±43.2

±5.980.00.00.00.0

The main differences between the loci of Figs. 4A and4B and Figs. 6A and 6B can be explained by noticing thatthe effect of adding a second generator, and hence a doub-

ling of the power transferred between the power stationand the bus, is the same as a per-unit equivalent generator

600 r

500

400

300

200

100

-100

MK5

El

E2

R2

see

MKA

MKO

R1~

Fig

- "

M

. 6B

" M K 3MK2

-15 -10 -5 0 5real , s'1

Fig. 6A Eigenvalue loci for the one-generator system as Xc is variedfrom 0.14 p.u. to 1.06 p.u. (1072 MVA base)

120

100

. 8 0 -

60-

40-

20-2

MK4

MK3

MK2

MK1

xc=O58p.u.(N = 5690/.)

V x c = O.76p.u.(N = 75.30/.)

yx =0.75p.u.(N = 74.3°/.)

X x c =O.89p.u . (N = 88.10/.)

xc=0.96p.u. (N=951°/o )

-1 1real

Fig. 6B Expanded view of modes MK1 to MK4 for the one-generatorsystem as Xc is varied from 0.14 p.u. to 1.06 p.u. (1072 MVA base)

transferring the same power across a transmission line oftwice the distance. Hence, as the transmission line resist-ance and reactance have doubled in per-unit, whereas thegenerator and transformer parameters are the same asbefore, there is an increase in the per-unit values of R, XL

and Xc (between the internal voltage of the equivalent gen-erator and the infinite bus) of 97% in R, 69% in XL and100% in Xc. This results in the following:

{a) For a low value of compensation, there is an effectivehigher value of reactance between the equivalent gener-ator's internal voltage and the infinite bus. Thus, as thegenerator is not modelled with a voltage regulator, the

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986 37

locus at M in Fig. 4, which is associated with the magneticstability of the generator [13], starts in the RHP. Further-more, this increased reactance means that there is a weakerelectrical coupling between the equivalent generator andthe infinite bus, and this results in a decrease in thedamping and frequency of the hunting mode MKO as canbe seen in Table 2.

(b) The damping of the electrical modes El and E2 hasincreased with the addition of the second generator. This isdue to the fact that the real parts of the eigenvalues for anRLC circuit are given by —R/2L and the per-unit resist-ance has increased by more than the per-unit inductance.The effect of this is that mode E2 does not interact asstrongly with the equivalent generator mechanical modes,and this is demonstrated by the fact that MK1, MK2 andMK3 do not move as far into the RHP.

(c) For any compensation level, the frequency of modeEl has increased and that of mode E2 has decreased. Thisis due to the fact that the imaginary parts of the eigen-values of an RLC circuit can be approximated to l/y/LCfor R2/4l3 <̂ 1/LC and, as the increase in L is less than thedecrease in C (increase in Xc), the resonant frequency ofthe RLC circuit between the equivalent generator internalvoltage and the bus voltage is higher. The consequence ofthis is that the subsynchronous mode E2 interacts with themechanical modes at a lower value of Xc. Thus at highcompensation levels, the hunting mode MKO starts tointeract with the subsynchronous mode E2 and changesdirection towards the RHP.

(d) Mechanical mode MK3 is stable for two generatorsbut unstable for one generator at a compensation level of74%. This is a result of the combined effect of (b) and (c).

The effects of (b) and (c) above on the stability of themechanical modes are of an opposite sense; however, theircombined effect can be determined by looking at the realparts of the mechanical modes in Figs. 4B and 6B. Thisinformation for the one-generator system is contained inFig. 7, which shows the real parts of modes MK1 to MK3

2.0

1.5 -

7 10(A

g 0.5

0

-0.5

-1 0

Fig. 8 shows the real parts of the active mechanicalmodes for the equivalent generator (dotted curves MK12,

stable" (safe)

r/A

IK3

\ MK2V\MK1/

stable /(unsafe)//\ B\ /c

1 1 I

50 60 90 10070 80compensation,°/o

Fig. 7 Real part of mechanical modes MK1 to MK3 for the one-generator system

as a function of the compensation level. The system isstable at those compensations at which all three curvesMK1, MK2 and MK3 are negative; that is at a com-pensation less than A or between B and C. However,whereas the area before A is a safe stable area, the onebetween B and C is an unsafe stable area as a decrease inXc (or increase in XL) due to the switching out of a branchof a parallel transmission line can cause the system tomove to an operating point with an unstable mechanicalmode. Hence this system should only be operated at acompensation level less than A.

MK3,

-1.060 70 80

compensation ,'U90 100

Fig. 8 Real part of mechanical modes MK1 to MK3 for the one-generator system (solid curves) and the equivalent generator system (dashedcurves)

MK22 and MK32) superimposed on those (MKll5 MK2t

and MK3i taken from Fig. 7) for one generator. It can beseen from Fig. 8 that, although the two-generator system isstable at 74% compensation, as predicted in Table 3, thisis an unsafe stable operating point. The system with twogenerators has two unsafe stable areas, between E and Fand between G and H, and a safe stable area below D. Fig.8 shows that in the regions EF and GH, the system isstable with two generators but not with one generator;similarly in the regions DA and BC, one generator is stablebut not two. Fig. 8 therefore shows that for certain valuesof compensation, the system can be put into sub-synchronous resonance by merely synchronising or de-synchronising the second generator. Especially significantis the fact that part of the safe stable area DA for onegenerator is an unstable area for two generators.

It is thus seen from Fig. 8 that the effect of (c) abovepredominates and instability occurs at a lower com-pensation level with two generators. The compensationlevels at which the modes in Fig. 8 go unstable for onegenerator and two generators are compared in Table 5.

Table 5: Compensation Xc in ohms giving instability ofmechanical modes for one and two generators

Mechanical modeMK1 MK2 MK3

One unitTwo units

0.516 0.403 0.3090.448 0.367 0.278

The stability characteristics of the system with one andtwo generators are markedly different as the compensationlevel is varied and so the second generator must be con-sidered in a stability study.

3.1.2 Variation in the power level: Figs. 9 and 10 showthe eigenvalue-loci as the generator terminal power Pt isincreased from 0.05 p.u. to 0.95 p.u. while the com-pensation is fixed at 74% {Xc = 0.4 ohm). Fig. 9 is basedon only one generator present, but Fig. 10 is based on twogenerators represented by the .multi-machine model. Onlythe subsynchronous modes are shown as the super-synchronous modes are unaffected by a change in thepower level. The loci for the equivalent generator modelare not shown as they are identical to the common modeloci of the two-generator model shown in Fig. 10.

38 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986

The loci for the one-generator system in Fig. 9 showlittle movement in the mechanical modes MK1 to MK3, as

120

100

80(0

I 601-

>• 40oc'§» 20_E

0

-20-20

-

R3

E2 /

I

• . .

/ MK4

I

MK1

MKO

sr

MK3

MK2

-15 -10 -2real . s"

-1

Fig. 9 Eigenvalue loci for the one-generator system as P, is varied from0.05 p.u. to 0.95 p.u. with the compensation level fixed at 74%

100-

80-

60-

5 20 -

-20

; - jII1?3' R3 R2' R2 II

IMKO'

R"?

MK3'" -MK3

M~K2' M~K2

MK1' MK1

MKO

" R1 M 7 ^0— -

-20 -15 -10 -3real . i

-2 -1

Fig. 10 Eigenvalue loci for the two-generator system as P,for each gen-erator is varied from 0.05 p.u. to 0.95 p.u. with the compensation level fixedat 74%

well as the electrical mode E2. The three rotor modes Rl,R2 and R3 and the mechanical mode MK4 indicate amore rapid decay rate with increasing power level. Themagnetic stability (locus M) and the hunting mode MKOtend to a less stable condition as the load angle increases.Fig. 10 shows the common mode and anti-mode loci forthe two-generator case, and it is seen that an increase inthe power level causes the anti-mode loci to move in thesame direction as the common mode loci. However, thedecrease or increase in damping of the anti-modes is, inmost cases, greater than that of the common modes. As inthe case with one generator, the active subsynchronousmechanical modes are hardly affected by the power levelvariations so that it is unlikely that the system will bemoved from a subsynchronously stable to an unstable (orvice versa) operating point by an adjustment in the powerlevel.

3.2 Transient behaviourTo evaluate the transient effects of the second generator,the differential equations of Section 2 which describe thedifferent generator systems, are numerically integrated togenerate transient response curves when the system is sub-jected to a disturbance. The disturbance considered is atemporary symmetrical 10% reduction in the infinite busvoltage for a duration of 200 ms. The compensation levelchosen is 74% (Xc = 0.4 ohm), and the system eigenvaluesgiven in Tables 3 and 4 govern the dynamic behaviour.

The response is calculated for the one-generator system(the results for which appear in Fig. 11) and for the two-generator system (the results for which appear in Fig. 12).In each case the variables shown are the generator current,load angle, electric torque and shaft torque on either sideof the generator as well as the fast Fourier transform ofcertain variables.

3.2.1 One-generator system: The eigenvalues in Table 4for the one-generator system indicated that mode MK3would be unstable, and this is illustrated in Fig. 11 wherethe transient curves contain a growing oscillatory com-ponent. The fast Fourier transforms (FFTs) of the shafttorques in Fig. 11 contain the mechanical modes at 6.9,12.5, 15.6 and 92.6 Hz. These compare well with the fre-quencies predicted by the eigenvalues in Table 4 andshown in Table 6. Mode MK4 at 17.5 Hz is not clearlyseen due to its non-excitability from the electrical network.The frequency components at 25, 28 and 31 Hz are due tothe mutual coupling between the armature and field cir-cuits [1].

The FFT of the generator current contains the fre-quency components 34, 37, 43, 49, 50, 51, 57, 63 and66 Hz. These frequency components are the complemen-tary mechanical frequencies and appear at 50 ± fm, wherefm are the natural frequencies of the multi-inertia shaft [1].

3.2.2 Two-generator system: In Fig. 12 the results aredisplayed for one of the generators only, as both gener-ators are operating under identical conditions. The eigen-values of Table 3 indicated that the system would bestable, and this is seen to be true in the decaying oscil-lations of the displayed transient curves. However, this isnot a safe operating point as a decrease in Xc (or increasein XL) due to the switching out of a branch of a paralleltransmission line could cause the system to move to anoperating point with an unstable subsynchronous mecha-nical mode. Furthermore, if one of the two generators wasswitched out, then the resulting one-generator systemwould experience subsynchronous instability as seen inFig. 11.

The FFTs of the current and shaft torques in Fig. 12show similar frequency components to those in Fig. 11. Acomparison of the shaft torque frequencies in Fig. 12 withthose predicted from the eigenvalues of Table 3, andshown in Table 6, identifies them as belonging to thecommon modes and not to the anti-modes.

Table 6: Mechanical frequencies for one and two generators

Mode No Two generators One generator

common modefrequency, Hz

anti-mode modefrequency, Hz frequency, Hz

012345

0.86.8

12.315.817.592.6

1.26.9

12.516.117.592.6

1.06.9

12.515.617.592.6

The common modes are excited in this case and not theanti-modes, as the fault considered is remote from thestation, and the two generators act in unison. However, ifthe anti-mode frequencies were to be excited as well, theywould be difficult to distinguish from the common modefrequencies (or the one-generator system frequencies) asthey are in such close proximity. Thus, although the addi-tion of the second generator alters the damping of some ofthe system modes (and hence their stability), it does nothave much affect on the frequency of these modes. This

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986 39

2.0

25 50 75

frequency , Hz

torque LP3-GEN

-160.025 50 75

frequency , Hz

means that whether there is one generator operating ortwo generators with either the common or anti-modes

generator current54

21 53

0)"5 52co

51

50I

0.750

0.700

^ 0.650

3 0.600o~

2 0.550

0.500

0.450

0.030

0.020

0.010

0

2 -0 .010

-0.020

-0.030

station feeding a series capacitor compensated transmis-sion line. It has also compared the effect of modelling thegenerator load angle

0.5 1.0time. s

1.5 2.0

generator E torque

0.5 1.0

time i s

1.5 2.0

torque GEN-EXC

0.5 1.0time, s

1.5 2.0

FFT torque GEN-EXCCD

CD•o

§• -110.0

50 75frequency, Hz

100

Fig. 11 Transient response curves for the one-generator system after a temporary 10% drop in the infinite bus voltage at a compensation level of 74%

excited, the frequency spectrum of the mechanical oscil-lations will be practically the same. This could be signifi-cant when, for example, considering the use of shuntreactor controllers to damp SSR oscillations.

The results in this section have considered two identicalgenerators operating under identical conditions, and donot take into account variations in parameters between thegenerators or differences in the electrical load on each gen-erator.

4 Conclusion

This paper has investigated the effect of adding a second1072 MVA turbogenerator to the first one in a power

new two-generator system as a single equivalent generator,or as a multi-machine system with two separate gener-ators. The following conclusions can be drawn from theresults presented:

(a) The addition of the second generator effectivelyresults in a change in the interconnecting tie-line charac-teristics, and so previous stability studies for the one-generator case are no longer valid, but are ratheroptimistic. Instability will occur at a lower value of com-pensation than was calculated for the one-generatorsystem. In fact an Xc of about 10% lower than previouscalculations will have to be used to maintain a stableMK3; this value does, however, depend on the model of

40 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986

the transmission system, including shunt loads, and is cur- generators as two separate generators rather than as anrently the subject of a further investigation,

generator current

0.80

0.70

0.60

050

0.40

2.0

25 50 75frequency , Hz

torque LP3-GEN

100

1.0 1.5time . s

2.0

25 50 75frequency , Hz

equivalent single generator. This may not necessarily be

generator load angle

Q>

O>Co

oo

65

64

63

62

.0

.0

.0

.0

- J n /W\/w»/ \/i / v '

- v \[\ J\l\ATV

0.020

0.010

0

-0.010

-0.020

1.0 1.5time, s

2.0

2.0

torque GEN-EXC

0.5 1.0t ime , s

-150.025 50 75

frequency , Hz100

Fig. 12 Transient response curves for the two-generator system after a temporary 10% drop in the infinite bus voltage at a compensation level of 74%

(b) There are also certain operating conditions when astable operating point with one generator may be unstablewhen a second generator is added (and vice versa). Hencethe system can be put into an unstable subsynchronouscondition by merely synchronising or desynchronising thesecond generator.

(c) The modelling of two generators as a multi-machinesystem introduces anti-resonant modes which represent anout-of-phase resonance between the two generators. Forthe purpose of maximum permissible capacitor com-pensation calculations, these anti-modes appear to have noeffect, and no advantage is gained by modelling the two

true for other combinations of transformer and tie-lineswitching configurations, but is forming the basis of afurther investigation.

(d) The addition of the second generator has some effecton the damping of some of the system common modes buthas little effect on the frequency of these modes. For a faultremote to the station, the common modes are excited;however, if the anti-modes are excited they are sufficientlyclose in frequency to the common modes (and one-generator system modes) for the excitation in the frequencydomain to appear as if it were due to the common modesor due to one generator operating alone. Thus if any SSR

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986 41

countermeasure is successful in damping oscillations withonly one generator, it would also be successful in thedamping of any common mode or anti-mode oscillationswith two generators in operation.

5 Acknowledgments

The authors acknowledge the assistance of R.C.S. Peplowand H.L. Nattrass in the Digital Processes Laboratory ofthe Department of Electronic Engineering, University ofNatal. They are also grateful for financial support receivedfrom the CSIR and the University of Natal.

6 References

1 LIMEBEER, D.J.N., HARLEY, R.G., and SCHUCK, S.M.: 'Sub-synchronous resonance of the Koeberg turbogenerators and of alaboratory micro-alternator system', Trans. S. Afr. Inst. Electr. Eng.,1979, 70, (11), pp. 278-279

2 HARLEY, R.G., JENNINGS, G.D., LAHOUD, M.A., andHADINGHAM, M.F.: 'Level of mathematical model required topredict the response of a turbogenerator system with and withoutseries capacitors', Trans. S. Afr. Inst. Electr. Eng., 1984, 75, (3), pp. 28-39

3 HARLEY, R.G, JENNINGS, G.D., and LEVY, D.C.: 'Modalanalysis representation of a multi-inertia turbogenerator', Electr.Power Syst. Res., 1985, 8, (1), pp. 53-58

4 LIMEBEER, D.J.N., HARLEY, R.G., and LAHOUD, M.A.: 'Sup-pressing subsynchronous resonance with static filters', IEE Proc. C,Gener./Trans. & Distrib., 1981, 128, (1), pp. 33-44

5 LAHOUD, M.A., and HARLEY, R.G.: 'An optimal controller for thesuppression of subsynchronous resonance', Electr. Power Syst. Res.,1983, 6, (3), pp. 203-216

6 LIMEBEER, D.J.N., HARLEY, R.G., and LAHOUD, M.A.: 'Thesuppression of subsynchronous resonance with the aid of an auxiliaryexcitation control signal', Trans. S. Afr. Inst. Electr. Eng., 1983, 74, (8),pp. 198-209

7 LIMEBEER, D.J.N, HARLEY, R.G., and ERASMUS, S.J.: 'Aninvestigation of subsynchronous resonance by Fourier Transform-ation', Electr. Power Syst. Res., 1979, 2, (2), pp. 133-143

8 LAHOUD, M.A., and HARLEY, R.G.: 'Multi-machine sub-synchronous resonance: Part I — Line Representation', Trans. S. Afr.Inst. Electr. Eng., 1984, 75, (2), pp. 3-13

9 LAHOUD, M.A., HARLEY, R.G., and LEVY, D.C.: 'Multi-machinesubsynchronous resonance: Part II — Load Representation', Trans.S. Afr. Inst. Electr. Eng., 1984, 75, (2), pp. 14-27

10 HARLEY, R.G., and BALD A, J.C.: 'Subsynchronous resonancedamping by specially controlling a parallel HVDC link', IEE Proc. C,Gener. Trans. & Distrib., 1985, 132, (3), pp. 154-160

11 LAHOUD, M.A., and HARLEY, R.G.: 'Theoretical study of a shuntreactor SSR stabilizer for a nuclear powered generator', Electr. PowerSyst. Res., 1985, 8, (3), pp. 261-274

12 KIMBARK, E.W.: 'Power system stability' (John Wiley and Sons,New York, 1967), 1

13 ADKINS, B., and HARLEY, R.G.: 'The general theory of alternatingcurrent machines: Applications to practical problems' (Chapman andHall, London, 1975)

14 LAHOUD, M.A., and HARLEY, R.G.: 'Analysis techniques for thedynamic behaviour of a multigenerator system', Trans. S. Afr. Inst.Electr. Eng., 1983, 74, (4), pp. 53-63

15 HUANG, T.S., and PARKER, R.R.: 'Network theory: An intro-ductory course' (Addison-Wesley, 1971)

7 Appendix

7.1 System parameters

System base values

base time tb, sbase armature power PjJ, MVAbase armature voltage Vb

a, kVbase armature current Ib, kAbase armature impedance Zb

a, ohmbase field power Pb

f, MVAbase field voltage V*, kVbase field current Ib

f, kA

42

Singlegenerator1107224(1 - 1)25.790.5375361882.85

Equivalentgenerator1214424(1 - 1)51.580.26910721885.70

base field impedance Zbf, ohm

base electrical angle 6be, rad

(electrical)base electrical speed cob,

rad/s (electrical)base electrical acceleration

ccbe, rad/s2 (electrical)

base electrical torque Tb,Nm/rad (electrical)

base mechanical angle #£,,rad (mechanical)

base mechanical speed cobm,

rad/s (mechanical)base mechanical acceleration

cLbm, rad/s2 (mechanical)

base mechanical torque Tbm,

Nm/rad (mechanical)base inertia Jb, kgm2/(rad)2

(mechanical)base stiffness Kb, Nm/(rad)2

(mechanical)

Generator electrical parameters

Xd = 2.466 p.u. Xkd = 0.13 p.uXq = 2.28 p.u.Xa = 0.22 p.u.Ra = 0.003 p.u.

Xfd = 0.19 p.u.Rfd = 0.000984 p.u.

Singlegenerator65.941

1

1

3.412

1

1

1

6.825

1.365

1.365

X

X

X

X

106

106

106

106

Equivalentgenerator37.971

1

1

6.825

1

1

1

1.365

2.730

2.730

X

X

X

X

106

107

107

107

J\. hA v - A -J Yf* *•*•

Rkd = 0.015 p.u.Xkql = 0.59 p.u.Rkql = 0.00545 p.u.Xkq2 = 0.06 p.u.Rkq2 = 0.011 p.u.

Mechanical parametersThe values of the inertia constants are as follows:

# ! = 0.158 sH2 = 1.598 sH2 = 1.593 s

H 4 = 1.625 sH5 = 0.674 sH6 = 0.034 s

The per-unit values of shaft stiffness are

Kl2 = 10.63 p.u. K4 5 = 26.93 p.u.K23 = 25.34 p.u. K56 = 70.53 p.u.K34 = 23.42 p.u.

For the purpose of this study, the per-unit values ofdamping £>, were taken as 1% of Hf [1].

The turbine torque distribution is assumed to be asfollows:

HP LP133.7% 21.6%

LP218.8%

ansmi'ssion system parameters

Transformer:

Transmission line

Xx =0.15Rx = 0.0001

: XL = 1.0097R = 0.0845

XL = 2.0194

LP325.9%

p.u.p.u.p.u.p.u. (1072 MVA base)p.u.

R = 0.169 p.u. (2144 MVA base)

7.2 Linearised system equationsThe linearisation of eqn. 1 yields the linearised generatoraxis currents as

Api = -lXI + Ai - [G]/o Aco + Av}}(14)

where the subscript '0' indicates the steady-state values ofthe variable and A indicates a small change. The linearisedelectrical torque is obtained from eqn. 4 as:

ATe = -o ) 0 [ [G] T i0 Ai/2 (15)

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986

while the linearisation of the mechanical torque yields

ATm = APm - (Pm0/(o0) Aco (16)

where Pm0 is the steady-state power transmitted along theturbine shaft. The two-axis equations of the transmissionline have been stated in Section 3.2. The voltage across thecapacitor is a state variable, hence a linearised form of itsdifferential equations is required. From eqn. 6

Apvcd = AiJC - co0 Avc

q - vcq0

Apvcq = AiJC + co0 Avc

d

Aco

vcd0 Aco

(17)

The linearisation of the generator's electrical torques andof the nonlinear sub-matrices contained in [S] and [£ ] ofeqn. 10 results in new matrices [P ' ] , [S'] and [£ ' ] suchthat [14]

Apx = Ax + AM (18)

7.3 Symmetrical systemsIn order to illustrate the common mode and anti-modeeffect in symmetrical systems this appendix considers themode shapes and mode frequencies of two separate reson-ant systems when they are coupled together. Fig. 13A

ss/////////

Fig. 13 A Single degree of freedom system

////////s///// >

shows a single degree of freedom (SDOF) system consist-ing of a mass m connected to a rigid frame by a spring ofstiffness k. This system has a single natural frequency givenby

Sl = Jk/m (19)The two-degree of freedom symmetrical system shown inFig. 13B consists of two identical SDOF systems of thekind in Fig. 13A coupled together by a spring of stiffnesskc. The equations describing the unforced motion of thetwo masses are written in matrix form as

Lo-K i ro-i

J IAIThe natural frequencies of this symmetrical system can befound by solving for the eigenvalues of the system. Theeigenvalues are given by the solution of the scalar equation

det = 0 (21)

where [M] and [K] are the mass and stiffness matrices,respectively, of eqn. 20 and [/] is the identity matrix. Sub-stituting for [M] and [K~\ from eqn. 20 yields

detX — (k 4- kc)/m

kjm

kjm

kc)/m= 0 (22)

or

X2 - Xl{k + kc)/m +(k + kc)2/m2 - k2/m2 = 0 (23)

and solving this for X gives

Xx = k/m or X2 = {k + 2kc)/m (24)

Thus the natural frequencies of the coupled system are

(25)and s2 = y/(k + 2kc)/lmThe mode shape vectors can be found by solving the equa-tions

= XH> (26)

(27)

for each X, to yield

Wy = [1 , 1]T and

Fig. 13B Symmetrical two-degree of freedom system

Thus the two modes of the coupled system are not just theSDOF system mode repeated twice, but are in fact twonew modes shifted slightly apart.

These new modes are the so-called common mode andanti-mode associated with symmetrical systems. The differ-ence in frequency between the common mode and anti-mode depends on the strength of the coupling between thetwo systems and increases with stronger coupling.

The common mode frequency is the same as that of theSDOF system and represents the two masses oscillating inphase. For this mode the coupling spring kc does notstretch at all and the masses move as though they wereuncoupled. The anti-mode frequency is slightly higher thanthat of the common mode and represents an out of phasemovement of the masses.

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986 43