Finding all modes of nonlinear oscillations by

the Krawczyk-Moore-Jones algorithm

Kohshi Okumura The IRMACS Centre

Simon Fraser University Burnaby, British Columbia, Canada

Email: o.kohshi@gmail.com

Abstract- This paper presents the effective application of Moore and Jones's algorithm based on Krawczyk opera­tor (KMJ-algorithm) to seeking all the modes of nonlinear oscillations. To show the effectiveness, we take the problem of finding all modes of subharmonic oscillations in three-phase circuits. This problem has not long been resolved because all solutions of the determining equations of amplitudes and phases have not been guaranteed by such conventional numerical method as Newton method. Here in order to apply the KMJ algorithm the determining equations are symbolically derived by extending the asymptotic method. In all the processes of derivation the software Maple is used. The computational results are shown on the parameter space together with the number of solutions.

I. INTRODUCTION

In order to analyze periodic oscillations in nonlinear circuits, the determining equations of amplitudes and phases have been symbolically derived by harmonic balance or harmonic balance-like method (HB method) [1]. Even now the re­searchers of nonlinear electric and electronic circuits pay a special attention to the HB method and try to study the precise bifurcation analysis[2], [3].

Mode analysis of oscillations are finally reduced to finding out all solutions of the determining equation. As we know, Newton or Newton-like method can not seek all solutions because the setting of initial values is difficult task.

However, based on Krawczyk operator[4] Moore and Jones proposed an algorithm of searching the safe-starting region of Newton method[5], [6]. This algorithm uses the interval operation[7], [8], [9], [10] and can find all real solutions. We call this algorithm the KMJ algorithm.

On the other hand, as far as nonlinear oscillations in the three-phase circuits are concerned, the crucial problem of finding all modes of subharmonic oscillations has not been resolved for long time[15]. This is because the process of obtaining the determining equations are complicated and all solutions have not been found out. Unlike the case of finding operating points of nonlinear DC circuits[I1], [12], [13], the determining equations become the long simultaneous polyno­mials equations of type fi(X) = I.::k akx� lk . . . x�nk = O.

In this paper, with aid of the software Maple we derive the determining equations symbolically. And by means of the KMJ algorithm we find out all the modes of subharmonic oscillations of order 112 and show the numerical results. The

determining equations are derived by the extended asymp­totic method originally developed by Krilov, Bogoliubov and Mitropolsky[14].

II. OUTLINE OF THE KMJ ALGORITHM

We briefly describe the interval operation and the KMJ algorithm.

A. Interval operation

We denote by I(Rn) the set of n-dimensional rectangles. Let X E I(R), Y E I(R) be the real closed intervals X = [;f, x], Y = [U' y"]. The interval operation is defined by

X + Y = [l2. + U' x + y], X -Y = [;f -y,;f -u] x · Y = [min{;fu,;fy,xu,xy},max{;fu,;fY,xu,xy}]

X /Y = [minh/u, ;fly, x/u, x/y}, max{;f/U, ;fly, x/u, x/y}] where Y1l0.

The mid point m(X) of X are defined by m(X) = (;f + x)/2. We denote the set of interval matrices by I(Rnxn). The interval vector X E I(Rnx 1) and its midpoint m(X) E Rn x 1 are represented by

X=(X1,X2,···,xnf, XkE1(R) k=I,···,n m(X) = (m(X1)' m(X2)' ''' , m(Xn)f

where 'T' denotes the transposition. An interval matrix A E I(Rnxn) and its norm are defined by

m

A = (Aij), Aij E I(R), IIAII = max L IAijl· " j=l

B. Finding zero of f (x)

We use a natural interval extension of the real function performed by replacing the real variables with the correspond­ing interval variables. The natural interval extension F(X) of f(x) E Rnx1 is defined by

F(X) = (F1(X), F2(X),,,, , Fn(X))T.

where Fk(X) k = 1" " , n is a natural extension of h(x). The Jacobian matrix of f(x) and its interval extension are written by f'(x) Ulj(x)) and F'(X) (Flj(X)) i ,j

978-1-4673-0219-7112/$31.00 ©l012 IEEE 1143

1 , 2 " . . ,n, respectively. The Krawczyk operator for the equa­tion f (x) = 0 is defined by

K(X) = y � Y f(y) + [1 � Y F'(X)](X � y) } (1) Y = m(X), Y = [m(F'(X))]-l

where 1 is the n x n identity matrix. The conditions to confirm the existence of the solution in X is called Moore-test shown below: (1) If F(X)iJO, then no solution exists in X. (2) If K(X) n X = ¢, then no solution exists in X, where ¢ is empty set. (3) If K(X) c::; X and II 1 � Y F'( (X)) 11< 1 , then a unique solution exists in X. Starting from the initial region X ini, the bisection procedures of X are iterated until the safe-starting region X safe can be found by the Moore-test.

III. FUNDAMENTAL EQUATION

The three-phase circuit has the �-connected nonlinear inductors and series capacitors in between the Y-connected voltage sources and nonlinear inductors. The mathematical model of the three-phase circuit is given by a simultaneous nonlinear differential equations.

dcjJ ( 2) ., } - =Pv� r�RP 'l �Pe dt

C �� = Pi', i' = f(cjJ) (2)

where the variables and parameters are given by the following notation: T means the transpose.

cjJ = (¢a,¢b,¢e)T : magnetic flux vector v = (Va,Vb,Ve)T : capacitor voltage vector e(t) = (ea, eb, ee)T : source voltage vector i' = (i�, i�, i�) T : current vector in �-winding R = diag(R, R, R) : series resistance matrix C = diag( C, C, C) : series capacitance matrix r = diag(r, r, r) : resistance matrix in �-winding.

The vector valued function f (cjJ) = (f( ¢a), f( ¢b), f( ¢e))T specifies nonlinear inductors. Each element of f ( cjJ) is given by a polynomial

(3)

where C1 > 0, Cn > 0 and n(? 3) is an odd positive integer. The matrix P is given by

1 o

�1

�1 1 1 . o

The three-phase voltage sources are given by three trigono­metric functions with an angular frequency wand a phase difference 27f /3

ea(t) = Em cos(wt + 'P)

} eb(t) = Em cos(wt � 27f /3 + 'P) ee(t) = Em cos(wt + 27f/3 + 'P)'

(4)

Here we normalize the variables and the parameters of Eq.(2) by introducing scale factors a'lj; and av as follows:

T = wt + 'P + 7f /2 V = V3avv E = V3aq,/wEm � = rav/ai C1 = c1adwav

(5)

where C1 + cn = 1 and aq, = wav. Then we have normalized differential equation of Eq.(2). In reality the resistance r is far smaller than resistance R, we put the corresponding parameter � to zero.

IV. TRANSFORM ATION AND STATIONARY POINT

We transform normalized Eq.(2) into the differential equa­tions represented by the zero-direct-quadrature (Odq) compo­nents that simplify the analysis of three-phase circuits. The transformation is called the Odq transformation defined by the matrix

� [ �os T �os (T � 2;) �os (T + 2;) 1 3 �sinT �sin(T � 2;) �sin(T + 2;) (6)

We have to notice that by the Odq transformation the three­phase abc components (wa, Wb, we) = (W COST, W COS(T � 27f/3), WCOS(T + 27f/3)) are transformed into the Odq com­ponents (wo, Wd, wq) = (0, W, O) , which indicates that there exists only one time-invariant d-axis component W. By the Odq transformation and normalization, Eq.(2) is transformed into a set of non-autonomous differential equations

d<Po � 0 dVo � dT � ,

dT � 0 d<p d , ( ) � = � Vd � (Ind <Po, <Pd, <Pq, T + <Pq + E

d<p q , ( ) � = � Vq � (Inq <Po, <Pd, <Pq, T � <Pd d;d = Vq + 1')I�d(<po, <Pd, <Pq, T) dVq , ( ) dT

= � Vd + 1')Inq <Po, <Pd, <Pq, T

(7)

We have easily the solutions <Po, Vo = canst.. We assume that the zero-phase capacitor voltages Vo is equal to zero. On the other hand,<Po is not necessarily equal to zero because there exists the permanent magnetisation of nonlinear inductors. Hence, we treat <Po as a parameter. THe subharmonic com­ponents are superposed on the source frequency components. We expand the right-hand side of Eq.(7) and take the zero­frequency components, which becomes zero in the steady sate. Hence, we have

d<Pd 1 r27r( , ( ) ) � = 27r Jo �Vd � (Ind T + <Pq + E dT = 0 d<pq 1 r27r( , ( ) ) � = 27r Jo � Vq � (Inq T � <Pd dT = 0 dVd 1 r27r , dr = 27r Jo (Vq + 1')Ind(T)dT = 0 dVq 1 r27r ( , ( ) dT

= 27r JO �Vd + 1')Inq T dT= O

(8)

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where

I�d(T) = I�d(<PO, <Pd, <Pq,T) I�q(T) = I�q(<po, <Pd, <Pq,T).

The right hand-side is the simultaneous polynomial equations. By introducing two polar coordinates(po, 80) and (ao, 80), the stationary points can be expressed by

(<PdO, <PqO, VdO, Vqo) = (Po sin80,po cos80,aosin80,ao cos 8o). V. DETERMINING EQUATIONS FOR SUBHARMONIC

OSCILLATIONS

A. Extended asymptotic method We deal with the subharmonic oscillations whose funda­

mental frequency is equal to a fraction 1/m(m = 2, 3, 4 , ... ) of the normalized source frequency 1. The subharmonic com­ponents denoted by � <P d, � <P q, � Vd, � Vq are superposed on the stationary point (<PdO, <PqO, VdO, Vqo). Putting

<Pd = <PdO + �<Pd <Pq = <Pqo + �<Pq } (9) Vd=VdO+�Vd Vq=Vqo+�Vq we introduce another coordinate (Xl, X2, X3, X4 ) defined by

Xl + jX2 = (�<Pd + j�<Pq) exp�j!lo } 10 X3 + jX4 = (�Vd + j�Vq) exp-JOO j = A. () Substituting Eq.(lO) into Eq.(8), we have the simultaneous differential equations with variables xk(k = 1, . . . , 4)

(11)

where x = (X1,X2,X3,X4 )T, Xo = <Po and E is a small parameter. Because this equation is represented by very many terms, we discard its representation.

Here, in order to have the 11m subharmonic solutions, we apply the extended asymptotic method to Eq.(11). By introducing new parameters h1 and h2, we modify A of Eq.(ll) as

where

0 1 �1 0

.4= �1 0 0 �1

h1 0 0 1

0 h2 �1 0 and the nonlinear part as

EY = EAx + EY (X1' X2, XD, T) .

Hence Eq.(ll) is represented by

dx A A

-= Ax + EY (X1, X2, XD, T) dT

(12)

(13)

(14)

(15)

We denote the eigenvalues of.4 by ±jw1, ±jw2. If the 1/m­subharmonic oscillation occurs, the angular frequency 11m is decomposed into two components W1 = 1 � 1 1m and W2 = 1 +

11m on the dq-plane. Here, we assume that 1/m-subharmonic

oscillation occurs in an internal resonance condition VW1 = W2 where v and new parameters h1 and h2 are determined by

v = (m + l)/(m � 1),

The periodic solution of Eq.(15) is assumed to be given by the asymptotic expansion of the small parameter E

(17)

The first term of the expansion is given by

2 xeD) = 2 )ak 'Pk ej(WkTHk) + ak 'Pk e-j(WkTHk)) (18) k=l

where 'P1 and 'P2 are the eigen vectors corresponding to the eigenvalues jW1 and jW2 , respectively. The amplitudes a1 and a2, and the phase differences 81 and 82 are assumed to be varying slowly and to be determined by the simultaneous nonlinear differential equations

da1 = EA1(a) + E2 A2(a) + ... dT da2 = EB1 (a) + E2 B2(a) + ... dT d81 2 dT = EC1(a) + E C2(a) + .. . d82 = ED1 (a) + E2 D2 (a) + .. . dT

(19)

where a = (a1 ' a2, 81,82). Substituting Eqs.(19), (18) and (17) into Eq.(15) and equating the same power of E, we have the following series of the differential equations:

ax (0) A

EO : W1 ------a;- � AX(D) = 0 (20)

1 ax(1) A (1 ) A (0) aX(D) E : W1 -- � Ax = Y(x ,T) � A1 --aT aa1

aX(D) aX(D) ax(O) �B1 -- � C1 -- � D1 -- (21) aa2 081 082

Eq.(21) can be expressed by substituting Eqs.(19) and (18) into Eq.(21) as

-oo<m<oo,m#±l,m#±v +{ <J>(1 )(a) � (A1 + ja1C1)ejO''P1 }ejw,T

+{<J>(V) (a) � (B1 + ja2D1)ej!l2'P2}ejvwlT + (conjugate termsX22)

where <J>(m) is the Fourier coefficient defined by

(23)

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o o 0.1

Ss 0.2

8 solutions 32 solutions 56 solutions 80 solutions

o

•

Fig. I. The region where solutions exist. (5 and '/)5 are ( and '/), respectively in the case of n = 5 and <Po = O. The number of solution is given to the right side of figure with marks as indicated.

B. Determining Equations The condition that Eq.(22) has no secular terms can be

written by

(c,oi, {<I>(1 ) (a) - (AI + jalCde-jlil'PI}) = 0 } (24) (c,02' {<I>(v)(a) - (Bl + ja2Dl)e-JIi2'P2}) = 0

where ( ) denotes the inner product and the notation *

implies the complex conjugate. c,ol and c,02 are eigen vectors corresponding to the eigen values jWl and jW2 of the matrix - At in the adjoint system, respectively. From Eq .(24) we have

(c,oi,<I>(1 )(a)e-jli,) } Al + j a 1 C 1 = -'--'--"'-..,.-

--'---'-:--

-'-

('Pi, 'PI) (25)

Bl + ja2Dl = (c,02' <I>

( (V)(a)

)e-jIi2)

. 'P2' 'P2 Equating both hands-sides, we have cAl,'" ,cDl. In order to have polynomial expression of cAl,'" ,cDl, we change variables of polar form to those of rectangular form as al ejlil = a + jb and a2ejli2 = c + jd. Hence, in the first approximation Eq.(l9) can be written as

da db } dt = EAl(a, b, c, d) dt = EBl(a, b, c, d) & � � dt = ECI (a, b, c, d) dt = EDI (a, b, c, d).

The determining equations are given by EAl = Lk p�l) aalk ... da4k = 0 } EBI = Lk p�) aalk ... da4k = 0 ECI = Lk p�3) aa'k ... da4k = 0

(27)

EDI = LkP�4) aalk ... da4k = O.

where p�l) (l = 1, . . . , 4) is determined by parameters E, (,7) and so on. oozk(l = 1"" , 4) is integer. Because Eq.(27) has very long expression, we omit it here. In order to find all the real solutions of Eq.(27), we use the KMJ algorithm.

VI. ANALYTICAL RESULTS

We pay a particular attention to all the modes of 112-subharmonic oscillations. Furthermore, we set n = 5, namely

the nonlinear function is given by f (¢) = Cl ¢ + C5 ¢5. The parameter region where all solutions are obtained is shown in Fig.l. The utmost 80 solutions are found in the region with the mark •. The 28 among 80 solutions are found to be asymptotically stable and 52 unstable. The 28 stable solutions are classified into 3 groups: The first is 4 solutions with al = 0 , a2 = 0. 2181. The second is 12 solutions with al = 0. 1187, a2 = 0. 1246. The third is also 12 solutions with al = 0. 1119, a2 = 0. 1324. All stable 28 solutions have 7r /2 phase difference.

VII. CONCLUSION

We have shown the effectiveness of the KMJ algorithm by finding all modes of the subharmonic oscillations of order 112 in three-phase circuits. As the result, the precise modes of the subharmonic oscillations of order 112 has become evident. The KMJ algorithm could be very useful in the systems where the determining equations are symbolically obtained by the harmonic balance or harmonic balance-like method.

ACKNOWLEDGMENT

The author gives his thanks to the graduate student Mr. Masaki Konishi and the associate professor Takashi Hisakado at Kyoto University in Japan for their collaboration.

REFERENCES

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[5] R. E. Moore, "A test for existence of solutions to nonlinear systems," SIAM J. Numer. Anal., Vol. 14, pp. 611-615, 1977.

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[8] R. E. Moore, 'Interval Analysis,' Prentice-Hall, Inc., Englewood Cliffs, 1966.

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[12] L. V. Kolev and V. M. Mladenov, "An interval method for global non­linear DC circuit analysis," IntJ.Cir.Theor.and Appl., Vol. 22, pp.233-241. 1994.

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