IEEE Transactions on Power Systems, Vol. PWRS-2, No. 4, November 1987

ADMITTANCE MATRIX MODEL OF A SYNCHRONOUS MACHINEFOR HARMONIC ANALYSIS

A. SemlyenUniversity of Toronto

Toronto, Ontario, Canada

J.F. Eggleston J. ArrillagaUniversity of Canterbury

Christchurch, New Zealand

Abstract - The harmonic analysis of a power system requiresappropriate models of all system components. Synchronousmachines act as harmonic converters, sensitive to the sequence ofthe fundamental and harmonic frequencies. This paper describesthe derivation of a harmonic model of the machine in the form of athree-phase complex admittance matrix and its application to theharmonic behaviour of an asymmetrically loaded generator.

INTRODUCTION

The importance of harmonic analysis of power systems is onthe rise [1] due to an increased use of converters which are the pri-mary source of harmonics. Several papers have been published onthis topic, some describing the problem as harmonic power flow

[21,[3], some as harmonic penetration [4], and a recent book dealswith power system harmonics [5]. In addition to converters, alsoarc furnaces, fluorescent lamps and the magnetizing branches oftransformers [6] are recognized sources of harmonics. However, theharmonic behaviour of the synchronous generator has not beengiven serious consideration in harmonic analysis 14], even though itis well known that it converts negative sequence currents into thirdharmonic positive sequence and, in general, acts as a harmonic con-verter. This is so probably because of the complexity of the prob-lem, since conversion means coupling of harmonics which otherwisewould be examined separately. Therefore, generators are oftenrepresented by a single approximate impedance at each harmonic.However, three-phase transmission lines appear as strongly unbal-anced at harmonic frequencies and resonances may appear for singlemodes so that harmonic unbalance is created or strongly amplified[7]. The generator will pick up the unbalance and return other har-monics. Clearly, the complex interaction betweein generator andsystem cannot be ignored.

This paper describes the derivation of a synchronous machinemodel and its application to the analysis of an asymmetricallyloaded synchronous generator. It is a three-phase model because ofthe significance of negative sequence in harmonic conversion. It coIn-tains all harmonics (even and odd, as these turn out to be uncou-pled) and permits the calculation of the current vector

analysis are superimposed on the base load flow solution andrepresent an increment to it. The harmonic analysis will contain allsignificant frequencies, including d.c. and the fundamental fre-quency. The reference of the latter serves also as a phase referencefor the harmonics. Since the harmonic analysis is superimposed onthe base load flow, the generator will be assumed as having ashortcircuited field winding, which makes the harmonic model com-pletely passive. Equation (1) corresponds to this condition.

HARMONIC MODEL OF A SYNCHRONOUS MACHINE

Derivation of the Matrix Ydqh

The synchronous machine admittance model is derived fromthe d , q -axis differential equations. Generally accepted conventionsand notations are used, as detailed for instance in Ref. [8] and illus-trated in Fig. 1. With two damper windings in the rotor, s and t,the equations are:

Vd Rid + P (Ld id + Mdf i + Md, i, w(Lq iq +Mqt it)vq = Riq + p (Lq iq + M it ) +w(Ld id + Mdf if + Mdi

vf =R i + p (L i + Mdf id + Mf8 i,) 0°

0 (2")vs = Rs i, + p (L8 it + Mdq id + M0. if )vt = Rt it + P (Lt it + Mqt iq ) = O

In (2) vf has been set to zero because, as mentioned, the fieldvoltage has been considered in the base load flow. However, theparameters of the excitation circuit have to be included in Rf andLf I All variables are phasors of harmonic h:

wh= Re {WI e hwt } = W'hcos (hwt) W" h sin (h wt ) (3)

where

Wh WWkh + jW" h (4)

i = YabcV (1)

Data for synchronous machines are normally available interms of d-q axis and implicit in this information is the effect ofthe second harmonic terms of the inductances. Information on

fourth harmonic terms, absent from the d -q axis model, is not gen-erally available and is ignored in this investigation.

It is also assumed that a base load flow solution has beenobtained for the power system, balanced out at the fundamentalfrequency, so that a standard load flow program can be used. Theharmonic analysis is performed subsequently on the unbalanced sys-tem in a three-phase representation. The results of the harmoniic

S6 SM 35i0-3 A paper recomnended and approvedby the IEEE 'Power System Engineering Committee oftl'e IEFE 'Power Engineering Society for presentationat the IEEE/PES 1986 Suinmer Meeting, Mexico City,Ilexico, -July 20 - 25, 1936. Manuscript submitted.January 18, 1985; made avrailable for printingMay 5, 1986.

Printed in the U.S..

denotes any phasor. Bold is used to denote a complex number

and/or a matrix or vector. We have now

p = j Cw (5)

and equations (2) become algebraic operations in Vd,, Vq,, ldh, Iqh I

If,h, 1h It.. The last three can be eliminated from (2'), using (2")

so that we obtain

q

d

Figure 1 Basic machine representation

0885-8950/87/1100-0833$01.00© 1987 IEEE

833

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834

Vdh Zlh Z2A ]1

Vor Jn Z3m Z4c fqor

or, in more compact form

Vdqh - Zdq idqh

and

(6) V dh

v' h

2 ( dA+ V'q1) V- h 1 = (V dh -F Vq")21h 1 (12))2 (V" dh + V' q,) v" h I~(VId + V"q (11)

(7') These can be combined into phasors, as in (4), yielding:

ah-, 2 (dh I Vqh )

ah+, 2 (Vdh + Vqh )

For simplicity, we will denote the elements of the non-symmetric matrices Zeh or Ydjq by

ah bh1

Ch dh

A particular matrix Zdh of interest is for the rotor frequeh =0 . Then p becomes zero and equations (2') yield

R -ZL[Z dq. =

wLd R ]Der'ivation of Connection Matrix C

Although our aim is to derive an admittance matrix in phasequantities, the mathematical formulation is simplified by using an

intermediate connection matrix in terms of a,f3 quantities, intro-duced by the rotation

va,,.l [Cos wt -sin wt Vd1[VIj - sinwt cos wt v j (9')

where h' denotes the resultant harmonics. The inverse transforma-tion is

Vd - [-cswt sinwt VCA

Vq ~ -sin wt Cos wt V oh

and

(8) (13")V4- 1 - 2 (j Vd4 + gh )

h +j 2=-( iVdh + Vqx )

!ncyIn equations (11), (12) and (13) the superscript h indicates the

related d ,q -axis harmonic for a component of order h ±1 of a sta-tor (a,#-axis) harmonic. When two such components of the same

(8') order (related to a lower and a higher order rotor harmonic) com-bine, the superscript is dropped; see the vector (16').

Equations (13) should be written for all h. However if h 0=or h 1 = 0, then VO - VO; and if h 0O, then we obtaindirectly from (9'), converted into phasor relations,

Va,t Vd, + jvQlo

VPi, =-jVd + Vq.

(14')

(141")

By assembling equations (13) and taking (14) into account, weobtain

(15')va - CVdq

An identical transformation applies for currents:

(9")

Identical transformation applies for currents. If we write, as in

ia/i Cldq (15"1)

In (15')

VaQp = | V,I Vl1, Vi,, ya2.va2,... #Vctn 9 V 1rT (16')(10')

(10")

and substitute (10) into (9'), we obtain expressions for v,,, and vjh,in the form of (3), with h' -- h ± 1:

a == V' sh cos(h -1)wt - V" , sin(h - 1)wt +

V- ,h,sin(h + 1)wt

- VI Ph sin(h + 1)At

h-l 2-(V" V)

r" - + V' I)

Vd0-[ Vd, Vq., Vdl, Vl VVd2, Vq2, *.. ,Vd, VqT] (16")

and

(11')

C- 12

(11")

(12')

N2M N

M NM N

M NMM

N

(17')

with

(17"--)

Ij 1

and

idqh - Ydq Vdqb (7") (13')

(3),

VdA - V' dACOS (h wt )- V dhsin (h wt )

vqh - V qh cos (h wt ) V" qh sin (h wt )

-t V, h +lcos(h + 1)wt

Vup, '-V h -cos(h -1)wt - Ah sin(h - 1)wt +

whereVI h

(th -I

+ v'f,hcos(h + 1)wt

2(VAd +V2 Adi,

V,+ - 2 Adh V qX )

(1711)

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835

where the asterisk denotes the conjugate and n the highest har-monic investigated.

The matrix C is singular because it contains pairs of rows (orcolumns) with matrices M or N only, which are singular due to thefixed relation between the a and /5, or the d and q components,respectively. This singularity can be removed by deleting the first(or the second) row and column of the matrix and by leting its sizeincrease indefinitely (or, alternatively, deleting also the last row andcolumn). In this way, only one of VU, V#, and of Vd4,, VY willremain in each of the vectors vr and vdq of (16) (and only one ofthe last two elements in these vectors if the truncated matrix C isconsidered). Keeping only one of V,,,, VP. does not result in lossof information since VP - ,jV,. Similarly, it can be seen from(17') that the sum Vd. +jVq. itself affects the value of Val and ofVP rather than its individual components Vd, and Vq}. Any of thelatter, having assigned an appropriate complex value, can replacethe sum Vd, +IjVq.

Let C' denote the reduced matrix C, and let C' and C* betheir conjugate, respectively. Then it can be easily verified, bydirect substitution and taking (17") into account, that C'C'* -U,where U is the unity matrix. This does not mean however that Cis the inverse of C. Still, we prefer to use the full matrix C of (17')because of its symmetry and simplicity, with the understandingthat a reference to an inverse relationship to (15) is interpreted toapply to the reduced matrix C' and the reduced set of variables.The inverse relationship to (15) can be derived directly startingwith equations (9") and performing calculations as in equations (10)to (14). We obtain

Vdq - C . Xa

idq = la#

Derivation of Matrix Y.#We assemble equations (7"), for all h, into the relation:

iiq -Ydq Vdq

where

AO = NYdqN / 2

A1 - (2MYdq,M + NYdq2N) / 4

A = (MYdq,_,M + NYdlh,+N) / 4 (for A - 2,...n -1)(23)

B'1 = MYdqIN/2B h -= MYdq,,N / 4

B'^ = NYdqAM/4(for h -2,...,n -1)(for h 1.n- 1)

An -ZMYdqg,_M / 4

These expressions can be calculated from:MYdq,M _((ai +di ) - j (b, -s ) )M

NYdqN -((ai+di) + j(bi-c))N

MYdq, N ((ai -di ) + i(bi ci ) P

NYdqM- ((a-d, ) -j(bjci) Q

where

(24)

(25)11P -= [_ -1 ' Q -(18X)

From equations (23) we can see that the self admittance AA(181) for the harmonic of order h depends only on the rotor frequenciesof order h d1, and the mutuals B'A and B' ' between harmonics

h+1 and h -1 depend on the rotor frequency of order h. Theseresults are consistent with the well known pertinent physicalphenomena. An intuitive block diagram of the voltage and currentharmonic relationships in the armature and rotor is shown in Figure2.

(19)

where Ydq is block diagonal with blocks of the form (8).For given vj we obtain Vdq from (18') then id from (19) and

i,# from (15"). Alternatively, we can write

iap1Y3.1fie (20)

The matrix Y,,,# has one non-zero block band on each side ofthe diagonal and a zero block separating band. Therefore even andodd harmonics do not interact in the generator.

The impedance matrix Z,,,f can be obtained similarly to (21):

Za,"'- CZdq C' (26)

where

Y:-CYdq C'

The non-symmetric matrix Yafj has the form

AO

B'I

B, 1

B' ' ' '

A1 B"A2

B'2Ah

B' t h

B, '-

B'n -1

I \\Figure 2 Harmonic interaction 1.e,tween generator and system

Derivation of Matrix YabcWe first augment Y,1, by a zero sequence diagonal matrix,

(22) comprising all harmonics:

ta/oo =Y

(27)YojJIt is generally assumed that zero sequence currents produce no

significant resultant flux in the air-gap and accordingly equation(27) shows no coupling between the zero sequence and the a,/5 com-

ponents. At least one author [9J disv u('s briefly such an effect, butstresses that it is negligible and subsequently removes the couplingfrom the analysis.

(21)

Y -

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836

For any harmonic we can write

Vapol, - Th Vabch

iaflo# -= Th iab,h

where

1

Th = 0

1-3

From equation (32)

o- Y v + io - gens (35)(28)

However, when the excitation is perfectly smooth, i, is onlyaffected by positive sequence fundamental frequency current. There-fore equation (35) becomes

io -Ygenv + i+ (36)1 12 2

2 21 13 3

(29)

Equations (27), (28) and (29) can be completed and assembledto yield the following:

where vr and i+ are vectors in the phase frame of reference contain-ing only the positive sequence fundamental components of v and irespectively. While v+ is kept fixed at 1 p.u., i+ is not known untilthe final solution. However i+ can be expressed as a function of iusing

if T i (37)

a/Jo Tv abc

i T-111abc =T flo

An admittance matrix

Yabc YT-0Y(o T

where(30)

(31) T=1

3

could be calculated, but the sequence (30) is equivalent to theintended use in (1) (where the subscripts abc have been omittedfrom v and i).

Fundamental Harmonics1 a a2a 2 1 a 0 Fundamentala a2 1

0 0 Harmonics

APPLICATION OF THE MODELTO AN ASYMMETRICALLY LOADED GENERATOR

The test system of Figure 3 is used to illustrate the effect ofthe harmonic admittance model of the generator. It consists of a

single circuit line of flat configuration corresponding to the Islingtonto Kikiwa 220 kV transmission line in the New Zealand system [71.The transformer is Yg /Y. connected, and its rating is 100 MVA,14kV/220kV, with a leakage reactance of 0.112 p.u.

Equation (1) is modified to include the effect of the fundamen-tal power flow solution. This is achieved by adding a positivesequence fundamental current injection i, which maintains a con-

stant (1 p.u.) positive sequence component of fundamental voltageat the generator terminals. Taking into account the sign conventionindicated in Figure 3, the generator current becomes

Combining equations (32), (34), (36) and

expression for the generator current(37) yields the following

i= -YgenZcI +Ygenv + Ti (38)

or

Ai b (39)

where

A = U + YgenZ9 - T

i -Ygenv + io (32)

Because there is no coupling between odd and even harmonics,vectors i and v and the generator admittance matrix Ygen containonly the odd harmonics of Yabc-

Figure 3 Test system

The input impedance matrix of the system (Z, ), as seen from

the generator bus, can be derived from existing programs 141, [71, so

that for each harmonic the following relationship applies

VAh ZSi ih (33)

and for all harmonics

v- Zi (34)

and

b Ygen +

Equation (39) is solved for i by Gaussian elimination and vh isobtained from (33) for each harmonic.

The transmission line introduces resonance and standing wave

effects which somewhat obscure the understanding of the generatorharmonic contribution. Thus the problem is analyzed in threestages, i.e. with the generator connected to (i) an unbalanced resis-tive dummy load, (ii) an unbalanced tuned load and (iii) an untran-sposed transmission line via a transformer. Details of the generatorused in the test example are given in the Appendix.

Unbalanced Resist'ive Load

The load is represented by the following admittance matrix

(40)Yo +-== Y+ KY1

KY+ Y+

where Z. is a block diagonal matrix.

(37')

un*ansposed Ike Iinnm

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where

Y0 is a zero sequence conductance

Y+ is a positive (and negative) sequence conductance

K is a coupling coefficient between the positive and nega-tive sequences

Since the generator produces no zero sequence voltage, Y0 canbe chosen arbitrarily; to prevent a singularity its value is madeequal to Y+ in our example. Thus the three-phase admittancematrix becomes

(a) (b)

Figure 5 Simplified equivalent circuit of the machine(a) direct axis(b) quadrature axis

1 + 2K

3

Y.bc = Y+ KK3

-K

L 3

K3

K3

1- K 2K3 3

2K K1-3

3 3

(41)

This matrix, although not physically realizable, can be used toexamine the effect of generator loading and degree of unbalance.

Voltage (%)10

8

6

4

2

0

1.1 0.2coupling coefficient

C

Figure 4 Variation of positive sequence third harmonic voltagewith coupling and loading

Figure 4 shows the effect of coupling and loading on the levelof 3rd harmonic voltage distortion. The results indicate that forloads within the nominal rating (i.e. 1 p.u.) the 3rd harmonic vol-tage is almost directly proportional to the load admittance, andhence to current or power. Also, the 3rd harmonic voltage is seen tobe directly proportional to the level of coupling. The 5th harmonicvoltage, not shown in the figure, varies approximately in proportionto K2.

Effect of Saliency

It is apparent from the analysis that saliency is the maindetermining factor in the process of harmonic conversion. To exam-ine the effect of saliency, the simplified equivalent circuits in Figure5 are used for the machine's direct and quadrature axes respec-tively. These circuits include only one winding in each axis of therotor (i.e. Md, = Mf, -0). Further simplification is achieved bysetting the direct axis magnetizing inductance Mdf to 1 p.u. andmaking Ld = L1 , Lq = Lt and Rf = Rt and assuming equalleakage in both axes (i.e. Ld - Mdf = Lq - Mqt ). Finally saliency,defined as the ratio of quadrature to direct axis rotor fluxes, can beexpressed by

S =I1-MMdf

6

Voltage (%)

5

4

13

2

-3

25.0 50.0 75.0 100saliency (%)

Figure B Effect of saliency on generator harmonic voltages

The effect of saliency has been tested on a purely resistiveload of 484[ (1 p.u.) with a coupling coefficient of 10%. Cases ofperfect saliency (Mqj = 0) and zero saliency (M5j = 1) producedno noticeable effect with perfect coupling (i.e. zero leakages). How-ever, the addition of some leakage (0.2 p.u.) showed considerabledifference in the resulting levels of harmonic voltage distortion, as

showni in Figure 6.

Unbalanced Tuned Load

This case introduces the effect of resonance by replacing theresistive load with the delta connected circuit of Figure 7, the reso-nance frequencies approximating those of the open circuited line ofthe test system. The machine data are given in the Appendix.

The sequence components admittance matrix of the delta cir-cuit is

Yo+- = O

O

0

(3 + 6) Y

-Sa 2y

7 B

0

-baY,

3 + 61

( 13)

(42)

Thus from specified leakage, saliency and resistances R and Rf,the machine parameters of equations (2) can be calculated.

While this model is not representative of an actual machine, itdoes provide a simple way of varying and displaying the effect ofsaliency on voltage distortion.

c

C

Figure 7 Tuned delta load

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838

where

a = 1 [4200

Y2 == (1 + 6)Y1, valid at a particular frequency

and, using the previous definition of coupling coefficient,

3K-I - (44)|3 + 6 |(4

The values of L1, C1, L 2 and C2 are chosen to satisfy threeconditions, namely, to give the same positive sequence admittanceat the fundamental frequency, to produce specified resonance fre-quencies f o 1 and f o 2 for Y1 and Y2 respectively, and to maintaina specified degree of unbalance (determined by the couplingcoefficient) at fundamental frequency. The three resistances areassumed equal.

From the test system with a line length of 200 km, approxi-mate values are derived for the delta branch parameters; at 50 Hzthe reactance (mostly capacitive) and resistance are 1555 and 12ohms, respectively.

In a real transmission line of flat construction the asymmetri-cal input impedance gives rise to two line modes which resonate atslightly different frequencies. This effect is simulated in the dummyload by maintaining a centre resonance frequency f 0 and varyingthe actual resonant frequencies of Y1 and Y2 symmetrically onboth sides of fo. Figure 8 shows the variation of harmonic voltageswith f o 1-f 0 2 while keeping f , constant at the third harmonic.

By maintaining a constant unbalance, the amount of negativesequence voltage and current at fundamental frequency is approxi-mately constant with varying f , 1- f o 2-

With t oi > f o 2 Y1 is capacitive and Y2 inductive and viceversa with f o 2 > f o l. This results in a strong unbalance at 3rd

Voltage (p.u.)

-34.3

1.0

,.-l

50

harmonic, leading to increasing negative sequence 3rd harmonic vol-tage and current with f o l-f o 2 This is followed by correspondingincreases of 5th harmonic voltage and current. The ratio of 5th har-monic positive sequence to 3rd harmonic negative sequence isapproximately constant.

Untransposed Open Circuit Line

In the test system of Figure 3 the length of the untransposedline was varied from 50 to 800 km, again with the machine dat ofthe Appendix.

Figures 9a and b show the positive and negative sequencethird harmonic voltages at the machine terminals. While themachine cannot generate harmonics directly, these appear as aresult of the unbalance produced by the untransposed line and aretherefore very dependent on the length of line.

In the range of line lengths between 165 to 200 km each har-monic voltage shows a double peak. These peaks correspond to thedifferent resonant lengths of the a and /3 propagation modes presentin a line of flat construction. The peaks occur at different lengthsfor the different harmonics (e.g. at 170 and 195 km for positivesequence third and at 175 and 190 km for negative sequence third).This effect is due to the impedances and degree of unbalance vary-ing greatly near the resonances of the two modes.

Voltage (%)

(a) negative sequence

l7 line length (km)

6

5

4

3

2

1

(b) positive sequence

100 200 300 400 500 600 700 800line length (kim)

Figure 9 Variation of third haline length(a) negative sequence(b) positive sequence

Lrmonic voltage with transmissionfif-if0 (Hz)01 02 )

Figure 8 Harmonic voltage distortion versus f ol1-f o 2 with centreresonant frequency f0 = 150 Hz

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The harmonic voltage levels in this range of distances are toolarge to be ignored, viz. 6w % of positive sequence third and 4% ofnegative sequence third harmonics respectively. This clearly demon-strates the need for detailed generator representation as proposed inthis paper. The levels of distortion are also substantially higherthan the levels calculated in the dummy loads, as a consequence ofthe standing wave effect of the line.

CONCLUSIONS

A generalized steady state model of the synchronous machinehas been developed which can take into account any asymmetry ordistortion present in the armature voltages. It has been shown thatwhen the field voltage is perfect d.c. the harmonic model of themachine becomes a passive admittance matrix. The level ofinterharmonic coupling has proved to be partly load related (i.e.affected by the size of load current and degree of asymmetry) andpartly generator related (affected by saliency). Such harmonic cou-pling cannot be detected with present harmonic models, where thegenerator is short-circuited behind the suhtransient reactance.

Computer results, with the machine connected to a dummy(asymmetrical) load and to an untransposed transmission line, havebeen obtained to corroborate the theory. They indicate that theharmonics generated by the machine may often exceed the levelsprescribed by harmonic legislation and they need to be assessedaccurately. The effect of two different resonant modes have beendemonstrated, leading to a strong unbalance and thus high voltagedistortion. The main harmonic contributions from the generator are

positive and negative sequence third harmonic currents, whichtherefore cannot be eliminated by generator or transformer connec-

tions.

ACKNOWLEDGEMENTS

The authors would like to express their appreciation to theNatural Sciences and Research Council of Canada and to the NewZealand Energy Research and Development Committee for financialsupport.

REFERENCES

839

[31 D. Xia and G.T. Heydt, "Harmonic Power Flow Studies,Parts I and II", ibidem, Vol. PAS-101, No. 6, June 1982, pp.1257-1270.

[4] T.J. Densem, P.S. Bodger, and J. Arrillaga, "Three PhaseTransmission System Modeling for Harmonic PenetrationStudies", ibidem, Vol. PAS-103, No. 2, Feb. 1984, pp. 310-317.

[5] J. Arrillaga, D.A. Bradley, and P.S. Bodger, "Power SystemHarmonies", book, John Wiley, to appear in 1985.

[61 H.W. Dommel, A. Yan, and Shi Wei, "Harmonics fromTransformer Saturation", IEEE Paper No. 85 SM 381-9,presented at the 1985 IEEE Summer Power Meeting, Van-couver, B.C.

(7] J. Arrillaga, T.J. Densem, and B.J. Harker, "Zero SequenceHarmonic Current Generation in Transmission Lines Con-nected to Large Converter Plant", IEEE Trans. on PowerApparatus and Systems, Vol. PAS-102, No. 7, July 1983, pp.2357-2363.

[81 D. O'Kelly and S. Simmons, "Introductionj to GeneralizedMachine Theory", book, McGraw-Hill, London, 1968.

[91 C.V. Jones, "The Unified Theory of Electrical Machines",book, Butterworths, London, 1967.

[101 H.H. Hwang, "Unbalanced Operation of Three-PhaseMachines with Damper Circuits", IEEE Trans. on PowerApparatus and Systems, Vol. PAS-88, No. 11, November1969, pp. 1585-1593.

APPENDIX

Generator Data

The generator data, based on that of Hwang [10], is as follows:

Nominal power rating = 100 MVANominal voltage = 14 kVNominal frequency = 50 Hz

[1] IEEE Working Group on Power System Harmonics, "PowerSystem Harmonics: An Overview", IEEE Trans. on PowerApparatus and Systems, Vol. PAS-102, No. 8, Aug. 1983, pp.2455-2460.

[2] W. Song, G.T. Heydt, and W.M. Grady, "The Integration ofHVDC Subsystems into the Harmonic Power Flow Algo-rithm", ibidem, Vol. PAS-103, No. 8, Aug. 1984, pp. 1953-1961.

R = 0.005 p.u.Rf = 0.0005 p.u.R, = R -0.02 p.u.Ld = 1.2 p.u.Lg = 0.8 p.u.Lf = 1.2 p.u.Ls -1.0p.u.Li = 0.831 p.u.

Mdf = 1.0 p.u.Md, = 1.0 p.u.Mqj = 0.6 p.u.Mf, -1.0 p.u.Ro = Lo -o00

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