Electrical Engineering in Japan, Vol. 97, No. 2, 1977 Translated from Denki Gakkai Ronbunshi, Vol. 97B, No. 4, April 1977, pp. 199-206

Subharmonic Oscillations in Three-Phase Circuits

KOSHI OKUMURA and AKIRA KISHIMA Kyoto University

1. Introduction

The subharmonic oscillation, particularly that caused by the nonlinear magnetization curve of transformer, has been studied intensively so f a r not only theoretically, but also experimentally [I 1. However, most of these studies have dealt with single-phase circuits and little study has been conducted on the subharmonic oscillation of three-phase circuits although the subharmonic oscillation of a power system has been studied theoretically and experimentally to some extent P, 31.

In this paper, we analyze the subharmonic oscillation (particularly the 1/3-subharmonic oscillation) of a delta-connected inductor circuit combined with series capacitors as shown in Fig. 1. The series-compensated transmission net- work connected to a no-load, three-phase trans- former can be regarded as an example of such a nonlinear circuit. In section 4, we show by analog simulation that two kinds of 1/3-subhar- monic oscillations, one with beat and the other with fixed amplitude, occur in the three-phase series resonance circuit stated above. In sec- tion 5, we analyze this subharmonic oscillation by the Krylov-Bogolyubov-Mitropol'skiy extended asymptotic method [4]. A s a result of this analy- sis, we show that the 1/3-subharmonic oscilla- tion with fixed amplitude may occur in two out of three phases when the resistance connected in series to each nonlinear inductor becomes large to some extent. In section 6, we demonstrate experimentally the existence of three kinds of 1/3-subharmonic oscillations.

2 . Circuit Equations

Consider a nonlinear inductor whose magne- tization curve is represented by

i' = f ($4) = CI $43 (1)

where if is the exciting current and $ is the flux interlinkage. Then the behavior of the circuit in

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Fig. 1 is described by

1 &! = Au- (r+AZR)i' + Ae( t ) d t

where

t$ = ($a, $b, $ c > ' ; magnetic flux interlinkage

u = (ua, ub, k)'; terminal voltage vector of

1' = (ia', ib', ic'); exciting current vector of

f (4) = [f($a),f($b), f($c)]'; magnetization vec-

e (t) = [ea(t), eb(t), ec(t)]'; symmetric three-

vector

capacitor

inductor

tor of inductor

phase source voltage vector

R = diag (R, R, R); series resistance matrix r = diag (r, r, r); delta-connected resistance

C = diag (C, C, C); capacitance matrix matrix

0 1 -1

Vectors with prime ", I ' are transposed vectors and diag [ ] is a diagonal matrix.

By applying o-d-q transformation to Eq. (Z), we

1SSN0036-9691/77/0200-0079$07.50/0 o 1978 Scripta Publishing Co.

Fig. 1. Three-phase series resonance circuit.

where O d ( B d , Bq, To,t)

= ( K d ' + I p p 3 I p d + 2 {(B+ K p 3 xcos3t-21p,IPosin 3r] T0+4Ip1Po*

Oq ( B d , Tpo, To, = ( ' P 2 + P p ' ) Tq-2{(Ipd~-Ipp3

x sin3t+2Cd?Tpqcos3t) To+4Ipp To' & ( q d , TPp, YOpr)

= (F2-3 Tpp3 B~lcos 3r +(Pq2-3 T2) qpp sin 3 t +6.(Fp,*+ TI') r0+4 Boa

In the above equation, 9dp 9q and 90 are d-, q- and o-components of flux interlinkages respec- tively and vd and vq are d- and q-components of capacitor terminal voltages, respectively. Parameters .5 , and 5 mpresent series resist- ance R, elastance 1/C of series capacitor and zero phase resistance (R + r/3), respectively; E represents the amplitude of the power source voltage.

3. Equations for Abnormal Oscillation

To derive the circuit equations describing the abnormal oscillation, let us notice that zero phase sequence flux interlinkage go is very small in comparison with 9 d and 4q , respectively. Therefore, lettiig \ko NN 0, the singular point of Eq. (3) can be represented by

(Pro, BPO, VIP, uqo)

=(p0coa80, posin80, o0sin80-o~cose~)

where PO, uo and 80 are determined from

(4) I (S2+7*) p0'-2 77p0~+p0~- E2=0 &I= tan-' {v po2-1)/~po2} ao= 77 pOs

Denoting the variations of \kd, qq, Vd and vq by A%& A@q, AVd and Avq, respectively, we de- fine new variables xk (k = 1, 2, .. ., 5) by

(5) I t ~ + j x ~ = ( A B ~ + j A B q ) e - ~ ~ e ' TS +js4 = (4 ud + jduq) e-je* I r5=yO

Substitution of Eq. (5) into Eq. (3) gives the cir- cuit equations describing the abnormal oscilla- tion such that

tion such that

4. Analysis by Analog Simulation We have applied 0-(Y-P transformation to Eq.

(2) and solved the equation thus obtained by analog computer. The results are shown in Fig. 2, ac- cording to which two kinds of 1/3-subharmonic oscillations occur. One, say, oscillation A, is the 1/3-subharmonic oscillation with beat and the other, say, oscillation B, is the 1/3-subharmonic oscillation with fixed amplitude. The regions of 1/3-subharmonic oscillations are shown in Fig. 3; oscillation A occurs in the region shaded with solid lines and oscillation B occurs in the region shaded with dashed lines.

5. Analysis of 1/3-Subharmonic Oscillations by Extended Asymptotic Method

We assume that the internal resonance condi- tion 2w1 = u2 holds. We consider two cases!

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Further, k(x1, x2, T) and h(x1, x2) are expressed as

1 / 3 - subharmonic oscillation with fixed amplitude

Fig. 2. Oscillograms of 1/3-harmonic oscilla- t ions.

Fig. 3. Regions of 1/3-harmonic oscillations (analog computer).

1) the case where external force gi (x i , x2, x5, T) has the same frequency as the natural frequency; 2) the case where the frequency of the external force is different from the natural frequency.

5.1 1/3-subharmonic oscillation with beat

Let the solution of the unperturbed system de- scribed by Eq. (6)1 be

~ ~ ( 0 ) =a (or' ei#+a pkl* e - j $

+ (s+jy) ( o k 2 e j 2 # + (s- j y) p2*e-'+ 1 (7) (l=olr, j = l ' q , k = l , 2 , 3 , 4 j

where wkl and u s are characteristic functions of the unperturbed system corresponding to eigen- values j w l and jw2, respectively. Symbols marked with * represent conjugate values.

On the other hand, Eq. ( 6 ) ~ gives

where

K a = U n ( a , s , y ) + j U n ( a , t , Y )

H m = P m h 2, Y ) + j Q m b , Y) Ho = Ho (a, z, Y) Qa = (an- 1)/3, 0m=2 nr/3

Since x5(0) contains the components with appraxi- mately the same angular frequencies as k(xl(O), x2(0), T), we can let

L= N, Pi ~ ( 2 I - 1)/3 (10)

Substituting Eqs. (8) and (9) into Eq. ( 6 ) ~ and pick- ing up the terms with the same angular frequency, we obtain

T ( u , s, y)z=u (11)

where 9 is a 2L x 2L-matrix and z and u are vec- tors such that

Therefore if det 9 f. 0, then

which gives the component with the same angular frequency as x5(0). The detailed discussion about this matter is given in [5] ; xk(0) (k = 1, 2 , 3, 4) can be obtained by applying the extended asympto- t ic method to Eq. (6) i . More precisely, we ex- pand the solution of Eq. (6)1 as follows:

X I = X&(Q (a, X , y, 4) + E X k ' " ( U , Z,y, $!J)+...... (13)

k = 1.2,s. 4

where 9 and a, x, y and @ are determined from

Xk(l), . . . are periodic functions of

Substituting Eqs. (7), (81, (131, (14) and (15) into Eq. (611 and comparing the te rms of E , ~ 2 , E3, . . . , we obtain the differential equations with

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respect to sary and sufficient conditlons for xl$l to be

x@ , . . . . Hence the neces- expressed as

periodic are given by 4 =~(uI +EDI (ao, ZO, YO)] ds++o (20)

EAl+juEDi=- C9r1* 2r k = l 1 1 4

X ~ E X ~ ( Z ~ ' ~ ) , Z Z ( O ) , z ~ ( O ) ) e - ~ + d 4 ]

lip1* W1

6 B 1 - 2 y ~ DI+ ~ ( E C I + ~ z E D I )

="( 2 K h = l 5 pr'*~Exr(Zl(o) , Z P , ZP)

x e-''+d (I C grz* prz ]/kll

where 90 is the initial phase angle. Stability of the singular point is tested by the Routh-Hurwitz method (refer to [ 71 for the detailed discussion).

The existence region of the stable singular ) (16) point of Eq. (17) is shown in Fig. 4. Theoreti-

cally speaking, this region contains the region where the 1/3-subharmonic oscillation with fixed amplitude occurs. Comparing the regions in Figs. 3 and 4, we see that the two regions coincide rather well with each other.

where cPkl ( 1 = 1, 2) are the characteristic func- tions of the adjoint system of the unperturbed system as described by Eq. (6). Equation (16) gives E A ~ , . . . , &D1 and therefore Eqs. (14) and (15) give

where coefficients a i l , b l i , . . . , are given in [6]; U and V in Eqs. (17) and (18) are complex- valued functions of z and are expressed as

U = u ( a , 2, Y) = - ( 5 + j3v)(Z1So + ZI*SI + ZZ*SZ

I = V ( a , Z, Y) (19) +Z3*St+Z4*S4) I = - (4 - j 37)) (2, SI + zz so+ ZI* sz

+ Zz*S3+ ZS*S4)

where Si (i = 0, . . . , 4) are complex-valued func- tions of a, x and y and the detailed discussion on Si is given in [5]. From the singular point (ao, xo, yo) of Eq. (17), phase angle 4 can be

5.2 1/3-subharmonic oscillation with fixed amplitude

Let the solution of the unperturbed system de- scribed by Eq. (6)1 be represented by

We derive x5(0) in the same manner 89 we did in section 5.1. In the present case, Z l , Kn and Hm all are complex-valued functions of u, v, x and y. We aLo let Wm = 2 m/3 and Neglecting those components of the zero phase sequence magnetic flux which have angular fre- quencies higher than 5/3, matrix and both z and u are four-dimensional vec- tors (see [5] for the detailed discussion).

To derive the solution of Eq. (6)1 by the ex- tended asymptotic method, let the solution of Eq. (611 be expanded as

1 = (2Z-1)/3.

becomes a 4 x 4

X k = f 1 ( 0 ) ( 2 ( , v, I , y)

+ E Z ~ " ' ( U , v, Z, y) + ... ( k = 1 , 2 , 3 , 4 ) (22)

where a(@, %(I), . . . are periodic functions and u, v, x and y are determined from the simultane- ous differential equations such that

(23)

Substituting Eqs. (211, (22), (23) and (8) into Eq. (6)1 and comparing the terms of &, &2, &3, . . .

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Fig. 4. Region of 1/3-harmonic oscillation with Fig. 5. Region of pure 1/3-harmonic oscillation beat (extended asymptotic method). (extended asymptotic method).

on both sides, we obtain t

necessary and sufficient conditions for xk(l) to be periodic a re expressed as

differential equations . . . . Hence, the with respect toxk(O),

x r E X h ( Z 1 ( 0 ) , xz(o), r , (o ) ) e - jm~c i r ]

Equation (24) gives &A1, . . . , &D1 and therefore

(25)

Fig. 6 . Effect of zem phase sequence resist- ance f .

where

Si (i = 0, 1, 2, 3) are complex-valued functions of u, v, x and y (see [5] for the detailed discussion).

The existence region of the stable singular point of Eq. (25) is shown in Fig. 5. Comparing the two regions in Figs. 3 and 5, we see that they coincide rather well with each other.

5.3 Single phase 1/3-subharmonic oscillation

In this section, we study the effect of zero phase sequence resistance 5 on the 1/3-subhar- monic oscillation with fixed am litude in Fig. 6, amplitude \I r l = (= Ju* of the component with angular frequency w l becomes approximately equal to amplitude of the component with angular frequency w2, if b is large to some extent. This means that capaci- t o r terminal voltages, which are given by

A s shown

(= &02 + yo21

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t . : , I I I

Exciting current (effective value) (mA)

Fig. 7. Magnetizing characteristics of saturable reactors (60 Hz).

Fig. 8 . Regions of 1/3-harmonic oscillations.

O-0 140 Line voltage (effectlve

value) (v)

(c) Single phase 1/3 - suanamonic

V,, Vb, Vc : Terminal voltages of reactors La, h, L,, U,, Ub, Uc: Terminal voltages of capacitors, E: phase voltage of power source

Fig. 9. Oscillograms and frequency spectra of 1/3-har- monic oscillations.

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Table 1. 1/3-harmonic components of capacitor voltages

Acknowledgement

The authors thank Wakabayashi Laboratory of Nuclear Energy Research Laboratory of Kyoto University for allowing them to use the analog computer. They also thank Mr. Y. Shima (pres- ently with Hitachi Ltd.), Mr. A. Niwa (presently with Kinki Nippon Railway Co.) and Mr. M. Matsu- mot0 (presently graduate student of Kyoto Univers- ity) for their cooperation.

1 A. 1 Ab

__--___ -0.1450 -0.2869 -0.1457 0.637 0.637 0.637

-0.1461 -0. I409 0.0287 0,986 0.986 0,986

hi. 0.0000 -0.2485 0.2485 ,/ 0.812 0 812

~ - _ _ _ _ _ _ ~ ~ ~ ~ - _ _ ~ ~ _ _

have the 1/3-subharmonic components as de- scribed in Table 1 for L = 1.40. Accoding to this table, modes M i and M4 seldom appear in phase a but they appear dominantly in phase b and phase c. This indicates that the single- phase 1/3-subharmonic oscillation occurs when zero phase sequence resistance L is large to some extent. In Fig. 6, rl and n are the ampli- tudes of 1/3-subharmonic component and funda- mental component of zero phase sequence flux, respectively .

6. Experimental Study

We have tested experimentally whether the 1/3-subharmonic oscillations discussed in the previous sections occur. The ac magnetization curves of the saturable recctors used for the ex- periment are shown in Fig. 7. The series re- sistance of each saturable reactor is 1.6lQ. The measured 1/3-subharmonic oscillations regions are shown in Fig. 8. The 1/3-subharmonic oscil- lation with beat occurs in the region shaded with oblique solid lines and the 1/3-subharmonic oscil- lation with fixed amplitude occurs in the region shaded with dashed lines. The single-phase 1/3- subharmonic oscillation occurs in the region shaded with horizontal solid lines. Oscillograms and frequency spectra of respective s ubharmonic oscillations are shown in Fig. 9. The above ex- perimental results confirm the occurrence of the three kinds of 1/3-subharmonic oscillations.

7. Conclus ions

We have confirmed theoretically and experi- mentally that the three kinds of 1/3-subharmonic oscillations occur in the three-phase resonant circuit containing nonlinear inductors.

1.

2.

3.

4.

5.

6.

7.

mFERENCES

For instance, see C. Hayashi. Nonlinear Oscillations in Physical Systems, McGraw- Hill Co., 1964. Nagamura. Tech. Report. Electrotechnical Lab. No. 603, 1961. G.C. Kothari et al. Analysis of ferro-oscil- lations in power systems, Proc. Inst. Elec. Engrs., Vol. 121, No. 7, p. 616, 1974. Mitropol'skiy, Krylov and Bogolyubov (trans- lated into Japanese by M. Mashiko). Non- linear Oscillations, Kyoritsu Publishing Co., 1958. Ohmura and Kishima. Material presented to Nonlinear Problem Study Committee, NLP

Okumura and Kishima. Ibid., NLP 74-22, 1974. Okumura and Kishima. Ibid., NLP 70-18, 1970.

75-2, 1975.

Submitted November 25, 1975; resubmitted September 13, 1976

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