IEEE Transactions on Power Delivery, Vol. 11, No. 1, January 1996

Phase Domain Modeling of Frequency-Dependent Transmission Lines by Means of an ARMA Model

T.Noda, Student Member, IEEE N.Nagaoka, Associate Member, IEEE A.Ametani, Fellow, IEEE Department of Electrical Engineering, Doshisha University

Tanabe-cho, Tsuzuki-gun, Kyoto-pref. 610-03, Japan

401

Abstract - This paper presents a method for time-domain transient calculation in which frequency-dependent transmis- sion lines and cables are modeled in the phase domain rather than in the modal domain. This avoids convolution due to the modal transformation, and possible numerical instability due to mode crossing. In the new approach, time domain convolu- tions are replaced by an ARMA (Auto-Regressive Moving Average) model that minimizes computation and is EMTP- compatible. A fast and stable method to produce the ARMA model is developed, and results are shown to agree well with both rigorous frequency-domain simulations and also field tests.

I. INTRODUCTION

It has become important to model the frequency dependence of transmission lines and cables [1-4] accurately in order to minimize the cost of construction. Also, because electric power systems include many nonlinearities, time-domain modeling is a practical necessity.

The Electro-Magnetic Transients Program (EMTP) developed by Bonneville Power Administration @PA) is the most widely used time-domain transient analysis program. Modal theory [5] is used in EMTP for the distributed representation of transmis- sion lines. For th is reason, weighting functions [6], exponential recursive convolution [7,10], linear recursive convolution [8], Z- transforms [ 91, and modified recursive convolution [ 1 1,121 all have been proposed. In those methods that relate the phase and modal domains by constant transformation matrices, the fre- quency dependence of the matrices is ignored. But it should not be ignored for untransposed lines or cables. In these cases, trans- formation matrices depend heavily on frequency. Theoretically, the frequency dependence of the matrices can be introduced into a time domain simulation by convolution [13]. However, for an n-phase transmission line, such an approach requires 2n(n-1) convolutions for the modal transformation at each time step. Another n convolutions are required for the modal propagation responses, and another n for the characteristic admittances. All together, 2n(n- 1)+2n convolutions make this approach a burden on both computer time and memory. Furthermore, mode crossing (at some fiequency, two or more eigenvalues become equal) enormously complicates practical implementation. On the other

95 WM 245-1 PWRD A paper recommended and approved by the IEEE Transmission and Distribution Committee of the IEEE Power Engineering Society f o r presentat- ion a t the 1995 IEEE/PES Winter Meeting, January 29, t o February 2, 1995, New Pork, NP. Manuscript sub- mitted July 25, 1994; made available f o r printing January 11, 1995.

hand, H.Nakanishi and one of the authors proposed a transient analysis method in the phase domain rather than in the modal domain so as to minimize the numerical effort and also avoid possible numerical instability due to mode crossing [14]. Also, a direct phasedomain calculation method based on two-sided recursion was recently proposed [ 151.

The present paper follows the above phase domain approach, but proposes a more efficient and sophisticated method based on an ARMA model. In this approach, time domain convolutions are replaced by the ARMA model, and the computation time is greatly reduced. Because of the nature of the Z-operator, an ARMA model can economically express a phase domain response that includes discontinuities due to modal propagation [16]. A fast and stable optimization method to identify the ARMA model also is developed, by applying Householder's transformation. Because this procedure is linearized, the previous time-consuming, sometimes-unstable, nonlinear optimizations no longer are required [17,18]. Moreover, two important problems have been resolved: 1) the determination of appropriate order of an ARMA model has been solved practically using the theory of Akaike's Infonnation Criterion (AIC) [19]; and 2) the stability of an ARMA model has been assured using Jury's method [20]. Finally, the equivalent circuit derived from the new method is compatible with programs such as EMTP, which are based on nodal admittance representation.

To confirm accuracy and practical use of the new method, transients on a 500-kV untransposed horizontal overhead line, a 500-kV untransposed vertical overhead line, and a 275-kV Pipe- type Oil-Filled @OF) cable are calculated and compared with field test results [21-231 and rigorous f?equency-domain solutions [ 241.

11. PHASE DOMAIN MODELING

A. Phase Domain Formulation A transmission line is represented as a multiphase distributed-

parameter line shown in Fig. 1 in a transient simulation. Consider a transmission line consisting of n conductors with length 1. In the frequency domain, voltages and currents at distance x from the sending end are expressed in the following column vectors, V(xo) = (VI, V2,...,VJT, Z(x,o) = (11, 12,...J,>T, where vi, Zj : voltage and current on the i-th conductor. Let Z(o) be the line series impedance matrix and Y(o) the line shunt admittance matrix per unit length, evaluated in the frequency domain [1-4]. The electromagnetic wave can be described by a pair of differential equations flelegrapher's Equation) solved as :

(1) V(X,O> = e-r(0)x ~ f ( o ) + e ~ ( ~ ) ~ ~ b ( o )

~ ( x , o ) = Yo(o>+z-r(0)x v ~ ( o ) ~ ~ ( o ) ) , (2) where Vko) : vector of forward traveling wave voltages, vb(o) : vector of backward traveling wave voltages, Yo(o) : characteris- tic admittance matnx, I-@) : propagation constant matnx.

0885-8977/96/$05.00 Q 1995 IEEE

402

Let a vector of the sending end voltages be Vl(o) and the receiving end q(o). Algebraic manipulation of (1) and (2) leads to (see Appendix A) :

(3)

(4)

where H(o)=d@Te-r(m)i : wave deformation matrix in the phase domain, z : minimum traveling time, Z~(o)=Yo-~(o) : characteris- tic impedance matrix. If the above equations are transformed into the time domain, they require 8 convolutions.

To reduce the computation time, the following relation (see Appendix B) :

(5) is applied to (3) and (4), and we obtain

Zl(@) =Yo(o)V,(0>

1 2 (0 ) = yo (a )V2 (0 1 -yo (o>e-J"'H(@){V, (0 1 + 2 0 (@)I2(@ >I

-yo (0 )e-"'H@ 1 {V,@ 1 + 2 0 (0 )Il(o 11,

YO (o ) ~ ( o = H* (a >yo (o

Il(@) = Yo(o)5(o> -e-/"'HT(o){y0(o)V2(o) + 12(" (6)

1 2 ( 0 > =yo(0>V2(0) -e-/""HT(o){Yo(o)V1(a) +11(". (7)

il(9 = yo(O*v1(t> - ipl(9

(T : transpose)

Transforming (6) and (7) into the time domain (lower case

(8) letters are used to denote their time domain counterparts),

ip2(t) = hT(r)*{yo(r)*vl(t -2) +il(t-z)), (1 1) where * : symbol to indicate matrix-vector convolution, h(t)=F-I {H(o)) (inverse Fourier transform). The above equations are compatible with Bergeron's expression [25]. As the terms underlined with - and = are the same, the number of convolutions is reduced to 4. Consequently, the computation time is greatly reduced using (8) to (1 1). The set of equations is called "Basic Equations of Phase Domain Method" in this paper. B. Equivalent Circuit for a Time Domain Simulation

The "Basic Equations of Phase Domain M e t h d indicate that the equivalent circuit for a time domain simulation is expressed as an n-terminal admittance paired with an n-terminal current source (Fig.2a). It can be seen that the first term of (8) and (10) corresponds to the n-terminal adrmttance, and the second term to the n-terminal current source corresponding to past history obtained &om (9) and (11). When including the eqyivalent circuit into a nodal admittance representation, how to model the ti-equency-dependent characteristic admittance YO(@) is important. To model Yo(@) with ARMA models, the convolution operation yo(t)*v(t) is easily decomposed as :

(12) Matrix yo0 is constant, and yol(t)*v(t-at) is a matrix-vector convolution without instantaneous responses. From (1 2), the "Basic Equations of Phase Domain Method" lead to :

yo(t)*v(t) = Yoov(t) + YOl(t)*V(t - W .

ilW = Yoovlw - i;lW

i;lW = iplW - YolW*vl(t - At>

(1 3)

(14)

i I '. Fig.1 A multiphase distributed-parameter lie

il i 2 2

Vl

Fig.2 Equivalent circuits for a time domain simulation by phase domain modeling

This set of equations corresponds to the equivalent circuit illustrated in Fig.2b. It can easily be introduced into programs based on nodal admittance representation, such as E m . First, the constant matrix yo0 is added directly to the node conductance matrix of the whole system before the transient calculation. During the simulation, the past history current source vectors il,l(t) and i>2(t) are then added to the current source vector of the system at each time step.

111. CONVOLUTION BY AN ARMA MODEL

A. ARMA model

An ARMA model essentially represents a discrete-time system. Input x(t) and output y(t) sampled at the calculation interval At are denoted as X(n)=X(f)lpnAt, y(n)=y(t)lt,dt (n=0,1,2;-). Using Z-transform theory, an ARMA model is defined by the following rational function of .rl in the z-domain :

403

initial value and a differential coefficient would be required, because the nonlinear optimization method searches for an optimum value using an iterative calculation. Moreover, assurance of convergence to an optimum value would be highly dependent on the choice of initial value.

On the other hand, the proposed Linearized LS Method chooses the following error function :

where an, b, : coefficients of the ARMA model, N : order of the ARMA model. Because the operator zn denotes a delay of n samples, (17) is transformed into the time domain as :

(18) y(n) = ax(n) + qx(n - 1)+. . +aNx(n - N>

The above equation is a time domain representation of an ARMA model, and is equivalent to applying the recursive convolution [7-121. Using this equation, output y(n) can be calculated by only 2N+I multiplications and 2Nadditions. B. Phuse Domain Matrix-Vector Convolution

-qy(n - l)-***-bNy(n - N ) .

A transient calculation based on the "Basic Equations of Phase Domain Method" requires matrix-vector convolutions in the following form :

where g(t) : transfer function matrix, x(t) : input vector, y(t) : output vector. The transfer function matrix g(t) corresponds to the wave deformation matrix h(t) or characteristic admittance matrixyo(t) in (8) to (1 1). Each element of At) is replaced by an ARMA model using an optimization method described in the following section. Once elements of C(z)=Z(g(t>} (Z{} denotes Z-transform) are replaced by an ARMA model, the relation between the zdomain input vector X(z) and the output vector Y(z) is expressed as Y(z)=C(z)X(z). Therefore, the i-th element of the output vector Y(z) is expressed as :

.?w = m * m , (19)

n

j= 1 4(z) = CGv(z)Xj(z). (20)

Substituting (17) into (20), the inverse Z-transformation gives

yi(n) = C{ aVoxj(n) n

j = 1

+a"lxi(n-l)+...+al~..x r J J . (n-Ny) (2 1) -bviuv(n-l)+..-+b@Nvuv(n-NV) },

where &, bii, : coefficients of (i, j ) element, Ng : order of (i, 1) element, u&)=Z-' { G&wj(z)}.

IV. OPTIMIZATION TECHNIQUES

A. Linearized Leust-Squares Method [ 16,171 This section describes a highly efficient method called

"Linearized Least-Squares (IS) Method", which approximates each element of the wave deformation matrix H(o) and of the characteristic admittance matrix Yo(@) in the phase domain by an ARMA model.

Let us represent each element of H(o) or YO(@) as a set of data (ok,Gk) (k=1,2;..,Q describing a transfer function defined at a discrete angular fiequency. The relation z=exp(ioAt) replaces the transfer function (ok,Gk) with a z-domain transfer function (zk,Gk). When approximating (zk,Gk) by an ARMA model using a conventional least-squares method, the error function for the k- th datum is errk=G(zk)-Gk. If we substitute (17) into erre we would obtain an error function that is nonlinear due to its rational polynomial form. In this case, we would need to apply a nonlinear optimization method to identifj the parameters. An

This error function includes all the parameters as linear parameters. This is why this method is called "Linearized LS Method", and a time consuming and sometimes unstable iterative calculation is no longer required.

To obtain the least-squares solution of (22), the conventional normal equation method is well-known. But the accuracy will get worse as the order N increases. Therefore, Householder's transformation is applied to improve the accuracy. As a result, the linearization of the error function and the application of Householder's transformation make the calculation stable, fast and accurate. B. Samples in the z-domain

Angular frequency ok should be sampled logarithmically to cover a wide frequency range with a small number of samples K. The nature of the transfer function G(z) implies that G(z*)=G*(z) ("*" denotes complex conjugate) [ 171. This condition in turn determines the maximum frequency in the following equation :

z K =eJmmwu 3fmw = 1 / 2 ~ t . (23) This equation corresponds to the well-known sampling

theorem, and determines the maximum fiequency of the data set. C. Relative Error Evaluation

A relative type of error evaluation is selected, in order to improve the accuracy of the cut-off band [17]. This evaluation can easily be introduced into the least-squares calculation with weighting values. The weighting value for the k-th datum (o k,Gk) is given in the following equation (see Appendix C).

(24) In a practical calculation, the parameter identification must be

executed after the weighting values are calculated, although a calculation of D(zk) included in wk requires the parameters bl;*.,bN which are not calculated at this h e . But the parameters bl;**,bN for wk can be evaluated roughly without using relative error evaluation, as a high accuracy for wk is not required. Thus, the relative error evaluation method can be applied. D. Order and Stability of an ARMA Model

In [26], it is pointed out that the order of an ARMA model cannot be determined without trial and error. The proposed method basically increases the order one by one until a best fit is found. A simple error evaluation uses standard deviation (SD) :

wk = l/lD(zk>Gkr, D(z) = l+blZ-l+...+bNZ-N

SD(N) = - c I G ( z ~ , N ) - ~~1 . (25) i'" k=l

where, for a permitted constant error EA and maximum order Nmm, an order N meeting the condition SD(N)<EA can be

in the range of N <Nmm cannot be guaranteed. In this case, determined. But the existence of a model meeting this condition

404

Akaike's Information Criterion (AIC) [I91 provides a criterion for determining the order. The values provided in the data set (Ob, Gk) include approximation errors and numerical calculation errors. When the errors are small, the SD will decrease monotonically as the order increases, and achieve the condition SD(N)<EA at a comparatively small order. When the errors are large, the model will approximate not only the true electro- magnetic characteristics but also the effect of the introduced error, thus requiring too large an order. In this case, an order N which minimizes AIC(N) will give an appropriate order by estimating the error characteristics included in the data set. AIC(N) is given in the following equation.

(26) 1 K

{ k=l AIc(N) = Kh clG(Zk ,N) - Gk I" + 4 N + 2

Because a model for a transient calculation has to be stable, a stability check is also required. According to (20), an output element of the matrix-vector convolution is the sum of n discrete system responses. The sum, i.e. the output, should be stable when each of its elements is stable. In the proposed method, the stability of each ARMA element is evaluated using JUrys method [20] (see Appendix D), which is especially suitable for a discrete system, as it only requires simple algebraic computations.

As a result, a model meeting the following conditions is adopted: 1) the model has a minimum order which meets the condition SD(N)<EA, 2) the model has an order which mini- mizes AIC(N) when SD(N)<EA cannot be achieved, 3) the model is evaluated to be stable by Juryk method. E. Modeling of Traveling Time Differences

Because an element of a wave deformation matrix in the phase domain H,(o) consists of n modal compnents, its time domain response has discontinuities every ATk=TrT (7 is the traveling time of the fastest mode). But it is difficult to approximate this discontinuous response with a conventional ARMA model expressed in (17), as its impulse response with a denominator of order N is a sum of N complex exponentials. Therefore, a small modification is made in the expression of the ARMA model. The following condition determines whether the k-th mode component is dominant or not in H,(o) (see Appendix E) :

lalk(a-')h e-ykll > E M . (27) where a,k : ( i , k) element of transformation matrix A, ( c r l ) ~ : (k, 1) element of inverse transformation matrix A-1, 'fk : modal propagation constant of the k-th mode, EM : a small constant. The above a,k, (a1)&, and 'fk are obtained directly when evaluating the propagation response matrix e&@). When the above condition is met, the impulse response of H,(o) has a discontinuity at l=ATk. In this case, the ARMA model can be made to represent the discontinuity using the one sample delay nature of the Z-operator, by using the delay operator z-% (qk=integer(Aq/At)) which corresponds to the traveling time difference applied to the corresponding numerator term. For example, supposmg that H,l(o) has a discontinuity due to the k-th mode, the ARMA model with order N=5 is expressed as

where qk=integer(AfdAt), No(z)=uo+u~t'+u2t2, N.z)=uqk +aqhlz1+uqh2z2. This representation is extended to include each mode. Note that when both the kl-th mode and the kz-th

mode are involved dominantly in HG{o) and the traveling time differences ATk, and ATk2 are almost the same, the two modes can be represented with one q.

V. CALCULATED RESULTS

A. An !%transposed Horizontal Overhead Line Fig.3a shows a 500-kV untransposed horizontal overhead line

consisting of 3 phase wires and 2 ground wires. The voltages on the ground wires are not considered, reducing the order of the impedance and admittance matrices from 5 by 5 to 3 by 3 for the EMIT Cable Constants routine. Fig.4 shows the fitting of one element of the wave deformation matrix Hll(o) using the optimization method described above. The ARMA model of this element uses an order N=9, and is accurate in both amplitude and phase. The Symmetrical arrangement of the conductors reduces from 9 to 5 the number of different elements to be fitted. Table I compares the order of the elements with the order of the fitting obtained using a modal domain method 1121. The proposed optimization method uses lower orders to reproduce the given characteristics. To c o d i the computation efficiency, the execution time of both the optimization and the transient calculation are also compared in Table I1 with those of the modal domain method. Both simulation results were obtained using the ATP version of EMIF. Fig.3b illustrates an actual test circuit where the receiving end voltages were measured [21]. The calculated results agree well with the field test results and with an exact fiequencydomain simulation [24] as shown in Fig.5. B. An Untrmsposed Vertical Overhead Line

The fhxp"eguen dependence of a vertical overhead line is considerable. Fig.6a shows a 500-kV untransposed vertical overhead line consisting of 6 phase wires and 2 ground wires. The order reduction is also applied to neglect the ground wires. The n u m h of fitting is reduced from 36 to 18 by the symrnebical arrangement. Fig.6b shows an actual test circuit in which the sending and receiving end voltages were measured [21]. The calculated results agree well with the field test results and the frequencydomain simulation as shown in Fig.7.

C. A Pipe-Type Oil-Filled Cable Fig.8a shows a 275-kV pipe-type oil-filled cable. Because of

its thin sheaths, the frequency dependence of the cable is also considerable. As the sheaths and the enclosing pipe touch each other, the order reduction can be applied. Fig.8b shows the test circuit in which the sending and receiving end voltages were measured 1231. The calculated results agree well with the field test results and the exact solution as shown in Fig.9.

25m P.W.

(a) conductor [z] arrangement '& 0

~antl ACSR

t = O 415611 83km

200611m (b) test circuit

Fig.3 An untransposed horizontal overhead line

405

0

8 -10

o theoretical ARMA

I I I 1000 1E+M 1Et05 1E+06

FBEQUMCI I B A -40 I

I I lEt04 lEt05 1E+06 -1EO'

1000 FBEPUENCY [Ezl

Fig4 Fitted results of wave deformation H, ,(a)

I I I I I I I 1 0.8 -

frequency transform

-

260 zao 300 320 340 360 380 400 TIM [ u s 1

(a) Calculated results

gbb @) Field test results

9

Fig.5 Transients on the horizontal overhead l i e

/ 22'4m - G.W.

Fig6 An untransposed vertical overhead line

- 2 I SendinP uhasea I I 1 ""r=*. .* I frequency transform

I 0.2 0.4 0.6 0.8 1

TIM OO'

I

- - freq trans

';d 2 Receivinp uhaea

0

t; - - freq. trans. ! ? Y o

' =-

?io0 350 400 450 500 - f O O 350 400 450 500 TIHE (us1 TIHE Ius1

freq trans

TIKE Ius1 TINE [as1

2 - 0

ARMA f!eq. trans. t;

0 - 2 -

SO0 350 400 450 500 300 o , ~ ~ ~

2-Rec

5 e.

P

TIHE I ssl (a) Calculated results

receiving (b) Field test results

Fig7 Transients on the vertical overhead line

POF cable lube

(a) conductor arrangement a (b) test circuit

Fig.8 A pipe-type oil-filled cable

406

TABLE I ORDERCOMPARISON

Proposed Phase Domain Method

PHASE d-ro yo MODE exp(-yl) Z,

(12). (32) 12 4 2 16 8

Modal Domain Method [I21

(lJb(3,3) 9 4 1 12 22

( 1 3 , (3J) 11 4 3 21 21 (11X(13) 10 * (12) 10 4 * these elements are identml to (1,2) and

(3,2), because of the symmetry of Yo

(a) Calculated results (b) Field test results Fig.9 Transients on the pipe-type oil-filled cable

VI. CONCLUSIONS

1) This paper presents a method for representing fkequency- dependent transmission lines in the phase domain rather than in the modal domain, for use in time domain transient calculations. This approach avoids convolution due to the modal transforma- tion, and possible numerical instability due to mode crossing.

2) The approach is made efficient by replacing time domain convolution with ARMA model representation. Using the Z- operator, an ARMA model can economically express a phase domain response which has discontinuities due to modal propagation. A fast and stable method to identify the ARMA model is also developed.

3) Simulation results of actual transmission lines using the proposed method agree very well with exact frequencydomain simulations and field test measurements.

REFERENCES

[l] J.R.Carson, "Wave propagation in overhead wires with ground return," Bell Sysf. Tech., J., Vol. 5 , pp.539-554, 1926. [2] F.Pollaczek, 'ilber das Feld einer unendlich langen wechsel stromdurcMossen Einfachleitung,"E.N.T., Band 3 (Heft 9), pp.339-360, 1926. [3] S.ASchellcunoff, "The electromagnetic theory of coaxial transmission line and cylindrical shields,"BeZlSysf. Tech. J., Vol. 13, pp.532-579, 1934. [4] AAmetani, "A general formulation of impedance and admittance of cables," IEEE Trans., Power Apparatus and Systems, Vol. PAS-99 (3), pp.902- 910,1980. [5] L.M. WedepohI and S.E.T.Mohamed, "Multiconductor transmission lines : Theory of natural modes and Fourier integrals applied to transient analysis," Proc. IEE, Vol. 116, pp.1553-1563,1969. [6] W.S.Meyer and H.W.Domme1, "Numerical modeling of fkequency- dependent transmission parameters in an electromagnetic transient program,

TABLE I1 COMPUTATION TIME COMPARISON

Proposed Phase Modal Domain Domain Method Method [12]

optimization 203.2 sec 374.2 sec transient calculation 38.4 sec 32.6 sec

(IBM compatible computer 486SLC-50 + 387SX)

" lEEE Trans., Power Apparatus and Systems, Vol. PAS-93, pp.1401-1409, 1974. [7] ASemlyen and A Dabuleau, "Fast and accurate switching transient calculations on transmission lines with ground retum using recursive convolutions," IEEE Trans., Power Apparatus and Systems, Vol. PAS-94(2),

[SI Ahetani, "A highly efficient method for calculating transmission line transients," IEEE Trans., Power Apparatus and Systems, Vol. PAS-95 (9,

[9] W.D.Humpage, K.P. Won& and T.T.Nguyen, "2-transform electromagnetic transient analysis in power systems,'' IEEProc., Vol. 127, F't. C, No.6, pp.370- 378,1980. [lo] ASemlyeu, "Contributions to the theory of calculation of electromagnetic transients on w s s i o n line with fiequency dependent parameters," IEEE Trans., Power Apparatus and Systems, Vol. PAS-100 (2), pp.848-856,1981. [ll] J.F.Hauer, "State-space modeling of transmission line dynamics via nonlinear optimization," IEEE Trans., Power Apparatus and Systems, Vol.

[I21 J.RMarti, "Accurate modelling of frequency-dependent transmission lines in electromagnetic transient simulations," IEEE Trans., Power Apparatus and

[131 Ahetani, "Refiaction coefficient method for switchingsurge calculatims on untransposed transmission lines (Accurate and approximate inclusion of fiequency dependency)," IEEEPES Summer Meehng, C 73444- 7, 1973. [14] H.Nakanishi, and AAmetani, "Transient calculation of a transmission line usingsuperpositionlaw," IEEProc., Vol. 133, Pt. C, No. 5,pp.263-269, 1986. [I51 G.Angelidis and ASemlyen, "Direct Phase-Domain Calculation of Transmission Line Transients Using Two-sided Recursions," IEEE PES SummerMeeirng, 94 SM 465-5 PWRD, 1994. [16] T.Noda and N.Nagaoka, "A time domain surge calculation method with fiquencydependent modal transformation matrices," IEE Japan, Elec. Power Sysr. Tech. Record, PE-93-153,1993. [17] T.Noda and N.Nagaoka, "Development of ARMA models for a transient calculation using linearized least-squares method," Trans. IEE Japan, Vol. 114-B, No.4,1994. [IS] T.Noda, T.Sawada, K.Fujii, and N.Nagaoka, "ARMAmodels for transient calculations and their identification (identification of coefficients and order)," IEE Japan, AnnualMeehng Record, Paper No. 1373,1994. [19] J.S.Lm and AV.Oppenheim, Advanced Topics In Signal Processing, Rentice-Hall, 1988. [20] E.LJuty, Theory and Applicahon of the z-Transform Method, Wieley, New York, 1964. [21] AAmetani, T.Ono, and AHonga, "Surge propagation on Japanese 500kV untransposedtransmission line," Proc. IEE, Vol. 121, No.2, 1974. [22] AAmetani, E.Osaki, and Y.Honaga, "Surge characteristics on an untransposed vertical lme," Trans. IEE Japan, Vol. 103-B, pp.117-124,1983. [23] N.Nagaoka, M.Yamamoto, and AAmetani, "Surge propagation character- istics of a POF cable," Trans. IEEJapan, Vol. 105-B, pp.645-652,1985. [24] N.Nagaoka and AAmetani, "A development of a generalized fiequency- domain transient program-FTP," IEEE Trans., Power Delivery, Vol. PWRD-

pp.561-571,1975.

p~.1545-1549,1976.

PAS-100(12), pp.49184925,1981.

Systems, Vol. PAS-101 (l), pp.147-155,1982.

407

3(4), p~.1996-2004,1988. [25] H.W.Domme1, "Digital mmputer solution of electromagnetic transients in single- and multi-phase networks," IEEE Trans., Power Apparatus and Systems,

[26] T.Yahagi, Tbeory of Digital Signal P r o c e s s i n g , vol. 1-3, Corona Publishing, 1986.

Vol. PAS-88 (4), w.388-398, 1969.

ACKNOWLEDGMENTS

The authors are grateful to N.Mori for his assistance, and to L.Dub6, W.S.Meyer, and T.H.Liu for their valuable discussions.

APPENDICES

A. The Derivation of ( 3 ) and (4)

Eliminating Vb(o) from (1) and (2), we obtain h(o)~(x ,o> + ~(x ,o ) = 2~~(w)e-~("Wf(w).

x = 0 : Y~(o)v,(o)+ Zl(0) = 2Yo(o)vf(o)

(A-1) At the sending end (1, A) and at the receiving end (2, ~ l ) , the following two equations are obtained fiom (A-1).

(A-2)

x = 1 : yO(o)V2(o) + Z2(o) = 2&(o)e-r(")'V~(o) (A-3) Equation (A-2) can be modified as

Substituting (A-4) into (A-3), we obtain (3). In the same manner, we obtain (4) by eliminating VAo) fiom (1) and (2). B. The Proof of Yo(o)H(o)=fl(o)Yo(o)

Let A and B be a voltage and current modal transformation matrix respectively, and a symbol diag(a,) denote a diagonal matrix with the k-th diagonal element a,.

2Vf(@) = V.(o)+Zo(o>Z1(~). (A-4)

Yo(w)e-r(")' = {B &ag(yok) A-'}{A diag(e-Yk') A-'} = B diag(yok) diag(e-'k') A-' = B diag(e-'k') diag(yok) A-' = {B diag(e-'k') B-l}{B diag(y0k) A-'}

= {(A-l)Tdiag(e-'k') AT}{B diag(yok) A-l} = {A diag(e-'k') A-l}T{B diag(y0k) A-'}

Using the relation B=(A-')T, B-l=AT,

-{e-rwi - } T &(a).

Therefore, Yo(o)H(o)=HT(o)Y,(o).

C. Weighting Value for the Relative Error Evaluation

Let N(z) and D(z) be the numerator and denominator polynomial of a transfer function G(z). The error function for the k-th datum of the Linearized LS Method is expressed as err'p N(z,)-D(z,)G,. For relative error evaluation, the following equation is used for the weighting value of the k-th datum w,.

(A-5)

From the above equation, a weighting value for the relative error evaluation is w,=l/ID(zi)G,I. D. Stabiliw Evaluation by July's Method

Let D(z)=bo+blzl+***+bhnN be the denominator polynomial of an ,4RMA model. The following table called "Juryk table" denotes this method.

line number 1 bo bl b2 "' bN 2 bN bN.1 b ~ - 2 ." bi 3 CO C1 4 "' CN-1 4 CN.1 CN-2 CN.3 "' CO 5 d o dl d2 ... dnr.2 6 dh1-2 d ~ - 3 dN4 '.' do

m - 3 40 41 42

where Ck= bobp b N & , r , dk= Cock- CN-k-lCN-1, ".

Jury proved that the conditions D( l)>O, (-lrJD(-l)>O, bo>lbrJI, cplcrJI, do>ldd. . . . , and qo>Iqd correspond to a stable ARMA model. E. Evaluation ofModa1 Components

Phase domain wave deformation is expressed as V,o)=H(o) V,(o), where Vdo) : deformed voltage vector at distance I , V,(o) : applied voltage at the sending end. Thus, an element of H(o) can be expressed by the following equation (e, : thej-th unit vector).

Hi/ (O)= {Vd(@))j-th, When K ( O ) = e j (A-6) On the other hand, as the applied modal voltage is given as P,(o>=A-'(o) V,(o)=A-l(o)q, where A(o) is the modal transformation matrix, the deformed modal voltage is expressed by the following modal propagation.

(A-7) { ~ d i ( o ) ) k - t h = {A-'(o)ej}k-the-Yk' = (a-l)k e-yk' Transforming the above equation into the phase domain, the phase domain deformed voltage is expressed as

{Vd(o))& = {A(o)Vf(m))i.~ = &lk(a-l)$e-Yk'. (A-8) k=l

Substituting (A-8) into (A-6), we obtain

~, , (o ) = ialk(a-l)ke-yk'. (A-9) k=l

Consequently, it is possible to evaluate whether the k-th mode component is dominant or not in H,(o) with (27).

BIOGRAPHIES

Taka Noda was born in Osaka, Japan, on July 4, 1969. He received the B.Sc. and M.Sc. degrees from Doshisha University, Kyoto, Japan in 1992 and 1994. Presently, he is a Ph.D. student at Doshisha University. Mr. Noda is a Member of IEE of Japan.

Naoto Nagaoka was born inNagoya, Japan, on October 21, 1957. He received the B.Sc., M.Sc. and Ph.D. degrees from Doshisha University, Kyoto, Japan in 1980, 1982 and 1993. Presently, he is an Associate Professor at Doshisha University. Dr. Nagaoka is a Member of IEE of Japan.

Akihiro Ametani (M'71,SM'84,F92) was born in Nagasaki, Japan, on February 14, 1944. He received the B.Sc. and M.Sc. degrees from Doshisha University, Kyoto, Japan, in 1966 and 1968, and the Ph.D. degree from University of Manchester, England in 1973. He was employed by Doshisha University from 1968 to 1971, the University of Manchester (UMIST) fiwm 1971 to 1974, and also Bonneville Power Administration for summers from 1976 to 1981. He is currently a Professor at Doshisha University. His teaching and research responsibilities involve electromagnetic theory, transients, power systems and computer analysis. Dr. Ametani is a Fellow of IEE and a member of CIGRE and IEE of Japan and is a Chartered Engineer in the United Kingdom.

408

Discussion

Adam Semlyen (University of Toronto): I would like to congratulate the authors for their interesting contribution to the computation of electromagnetic transients. The most fundamentally interesting feature of the described methodology is the fact that the procedure, while fully reflecting the frequency dependence of the components involved and of the system, is entirely in the time domain. In principle, the discrete- time modeling adopted is the most natural representation of any linear subsystem for time domain simulation. Indeed, in this approach, just as in reference [15] of the paper, no frequency domain model has first to be built and then adequately approximated for time domain calculations

The paper comes in close succession to our study [15] on the same topic. The computational methodology of the two papers is essentially the same but is designated by different names: nYo-Sided Recursions (TSR) in [15] and Auto-Regressive Moving Average (ARMA) in this paper. The fact that these studies have been performed independently and almost simultaneously, with many strong and essential simiiarities both in the fundamental ideas and in some details, suggests that the new methodology (TSR or ARMA) is not simply a minor variation of existing methods but a well structured, rationally built procedure developed in response to widely perceived needs in the calculation of electromagnetic transients.

The main similarities in the two methods, in addition to the basic idea of using a small number of time domain samples in both input and output, are the following:

(a) The input-output relations are written in matrix form in the phase domain rather than for individual modes.

(b) The coefficients of the recursive process are identified directly in the frequency domain over a wide range of data points using robust linear algebra procedures for least squares problems.

(c) The requirement of testing for the stability of the procedures has been recognized, addressed, and satisfactorily solved.

There are of course also differences between the two methods. One, perhaps, pertains to the strong reliance in this paper on the z-transform itself which is not used in [15]. It is also noteworthy that the paper presents an application to the calculation of cable transients where the usefulness of the new method is most evident. The authors' comments on these issues would be highly appreciated.

Manuscript received February 22, 1995.

H W E N V. NGUYEN, HERMANN W. DOMMEL and JOSE R. MART1 (The University of British Columbia, Vancouver, B C ,

The authors should be congratulated for developing a sophisticated method which takes into account the complete fie- quency-dependent nature of untransposed transmission lines for time-domain programs such as the EMTP The discussers would appreciate it if the authors could comment on the following issues 1) Realizing the reduction in the number of convolution operations

at each time step, the discussers have also been working in modelling the untransposed transmission lines directly in the phase-domain in a Ph D project since early of 1994 For modal parameters calculations, a diagonalization technique proposed in [A] has been used. In addition, the fitting is performed directly in the phase-domain using a modified version of the rational approx- imation method developed in [B] Our experience has shown that the off-diagonal elements of the wave deformation matrix H(o) are not easily fitted The difficulty increases for lines with strong asymmetry We wonder if the authors have encountered the

Canada)

same experience. If possible, could the authors show the com- parisons of the fitted and the exact hnctions of one of the off-diagonal elements (for example element 56) of their vertical line (Fig 6), in both frequency- and time-domains The graph of the elements of the wave deformation matrix in the time-domain might show the discontinuities caused by different modes as pointed out in section E of the paper Incidentally, what is the typical value of sM in this section? Is it an arbitrary constant or is there a criterion for selecting it7 Does it vary for different line geometries? For a typical element of the H(o) matrix of the vertical line in Fig 6, how many q's are needed to represent accu- rately the travelling time differences caused by all the modes?

2) Have the authors applied their modelling technique for lines with strong asymmetry? An example is a six phase line where the left three phases are vertical while the right ones are horizontal Another case which is very common in practice is when two or more different voltage lines share the same right-of-way and may be located one underneath the other

3) Lastly, did the authors assume zero conductance value in their line shunt admittance matrix Y(w)7 If not, what was the typical value used for it?

[A] L M Wedepohl, H V Nguyen and G D Irwin, "Frequency-Dependent Transformaoon Matnces for Untransposed Transmission Lines using New- ton-Raphson Method," Submitted for the 1995 IEEE Summer Power Meeting, Portland, Oregon [B] J R M m , "Accurate Modelling of Frequency-Dependent Transmission Lines in Electromagnetic Transient Simulations," IEEE Trans on Power Ap- paratus and Systems, Vol PAS-101, pp 147-157, 1982

Manuscript received February 28, 1995.

Thor Henriksen and B j ~ r n Gustavsen (Norwegian Electric Power Research Institute, Trondheim, Norway) : The authors are to be commended for presenting an efficient method for time domain calculation of transients on transmission lines and cables which takes into account the frequency dependency of the transfer function and the characteristic admittance. We have the following remarks and questions :

The paper deals with a general problem often encountered in transient calculations : How to include in time domain calculations an admittance or transfer function whose characteristics are known in the frequency domain? Numerical convolution can in principle always be applied, but the evaluation of the convolution integral can be very time consuming.

The above mentioned problem can be avoided by approximating the actual function in the frequency domain by an analytical function which gives the convolutlon integral a recursive formulation in the time domain. The authors have selected the ARMA function (17) which gives a two sided recursion formula (18). The latter seems to be identical to (4) in [15]. An altemative approach is to use a rational function (with the complex frequency s as variable). The coefficients of the polynomials can be considered as unknowns, which are found in a similar way as the coefficients in (22) in the paper. Another altemative is to use poles and zeroes as parameters.

The various approaches give differences in accuracy and computational requirements. In our opinion, differences in computational requirements are in most cases of minor importance.

409

Stability can be regarded as a part of the accuracy problem since the error due to the approximation in the frequency domain should be measured in the time domain. This error depends in general also on the actual transients. Have the authors compared their approach to the two alternative approaches regarding accuracy (and stability)? If so, have they also made the comparison for other applications than the ones presented in the paper?

Another point of general interest is the frequencies used when determining the parameters of the approximating function. The authors claim that these frequencies should be logarithmically spaced, but how many points are needed? Is it sufficient to use a relatively small number, or is it in general advisable to use as high number as possible? We expect that problems may arise if some elements of the transfer function contain contributions from modes having widely different velocities. (This will be the case when doing calculations on crossbonded cables as the sheath voltages cannot be neglected). Such elements will oscillate very fast in the frequency domain, and we therefore wonder what will happen if these oscillations are not resolved at high frequencies due to the use of logarithmically spaced samples.

The authors present an approach to this problem, given by (28). However, is seems to us that the numerical examples in the paper do not involve modes having widely different propagation velocities. Have the authors tried the ARMA model on such cases?

This problem could have been solved by separating the contribution from the different modes in the frequency domain, although it requires the eigenvectors to be calculated as smooth functions of frequency. The authours claim that mode crossing "enormously complicates practical implementation". However, it is according to our experience possible to overcome that problem by taking the direction of the eigenvectors into account.

Suppose that H&o) in (28) is decomposed in the frequency domain and that the ARh4A model (28) is established separately for each mode. The coefficients b,, in eq. (28) would then be different for each mode. It is therefore suprising to observe that it is seems sufficient to use a low order as 5 in (28). Does this imply that the coefficients are forced to be the same for all modes? If so, could one expect to improve the accuracy by performing the above mentioned decomposition in the frequency domain?

The authors check the resulting ARMA model for stability by applying Jury's method. But what is done if Jury's method predicts instability?

In order to obtain a linearized Least Squares Method the authors optimize the fitting function using (22) instead of the original function G(z) (17). How does this affect the accuracy of the resulting fit?

The authors' comments would be appreciated very much.

Manuscript received February 28, 1995.

T.Noda, N.Nagaoka, and A.Ametani: The authors would like to thank the discussers for their interesting and useful comments.

In reply to Prof. ASemlven :

Because the proposed ARMA model methodology is based on the Z-transform theory, we were able to use two methods, which have been developed and fully investigated in the field of signal processing. One is the theory of AIC (Akaike's Information Criterion) for the order determination and the other is the Jury's method for the stability evaluation These methods have been approved to be useful and actually used in practice. Therefore, we were able to rely upon these methods and were also able to save time to develop reliable corresponding methods. Moreover, the agreements between calculation results and field tests shown in the paper approve that these methods are useful. Especially, tables I and I1 explain that the proposed order determination algorithm described in section IV-D is efficient by the combination use of SD (Standard Deviation), AIC, and the Jury's method.

The traveling time differences due to modal propagation should be modeled adequately, because the phase-domain propagation response of a transmission line consists of modal components. Especially, in case of a cable, it is almost impossible to model the traveling time differences by a sum of exponentials, namely, by a rational function of s in the frequency domain, because the traveling time difference between coaxial and sheath modes is considerable. To solve this problem, we proposed the modification of an ARMA model, in section IV-E, to express the phase-domain propagation response with an ARMA model, using that the operator z-1 is a delay operator. It should be noted that the modification is not a problem of the proposed linearized procedure of the least-squares parameter estimation. Therefore, it can be said that the usefulness of the proposed method is evident by the accuracy of cable transients shown in Fig. 7 , as pointed out by the discusser.

In reply to Mr. H.V.Nguyen, Prof. H.W.Domme1, and Prof. J.R.Marti :

1) The proposed linearized least-squares (LS) method performs the fitting directly in the z-domain rather than in the Laplace s-domain. Therefore, operator z-", which denotes a delay of n samples, facilitates the approximation of the off-diagonal elements of the wave deformation matrix H(w), which include discontinuities due to modal propagation as described in section IV-E. On the other hand, using a rational function of s, it is difficult to approximate an element of the off-diagonal elements, because a rational function of s essentially expresses a continuous response in the time domain. Moreover, the off-diagonal element is not always a minimum-phase function, although the rational fitting method developed in ref. [B] assumes the minimum- phase condition. On the other hand, the proposed linearized LS method does not assume the condition. Thus, we have no difficulty for fitting of the off-diagonal elements.

Fig. A shows the fitted results of an off-diagonal element H56(o) of the untransposed vertical overhead line described in section V-B. The figure shows (a) order determination

410

procedure, namely, AIC and SD - order characteristic, (b) the fitted results in the frequency domain, and (c) step response in the time domain. Not only in the frequency domain but also in the time domain, the proposed linearized LS method shows high accuracy. Additionally, Table A shows the determined coefficients.

The value alk (a-1)b e-6 1 of eq. (27) indicates the ratio of the k-th mode in the (i, j ) phase component. When I a,k(a-l)b e-d I is small, the corresponding discontinuity due to the k-th mode is small, and an independent qk term in (28) is not necessary for an approximation of the small discontinuity. Therefore, E~ determines the maximum error due to the modeling of traveling time differences. When a user gives the fitting error constant of the linearized LS process described in section IV-D, the modal error constant EM

should be selected to a little greater value than EA, because the discontinuity which is determined not to have an independent qk term by (27) is also approximately taken into account in the linearized LS process. E , ~ = 1% and E~ = 3% were used to perform the fitting of the three cases described in section V. Considering the above, EM depends only on the fitting error constant

In case of the untransposed vertical overhead line in section V-B, only one qk(=48) is used as indicated in Table A. UsuaIly, in case of an overhead line, only one qk is required, because the propagation velocity of the earth return mode is much smaller than the other modes, namely, aerial modes, and the velocities of the aerial modes are almost the same. On the other hand, in case of a cable, more qk terms should be required.

provided by a user.

2) Calculation examples in the text concern existing lines and cables in Japan, and we have not applied the proposed method to lines with strong asymmetry which seem not to be existing. It can be found that the asymmetry and a multi- circuit line do not make the fitting difficult, because of the reasons described in the above.

0.2

0 - e o E -0 1 -

G -0 0

c

3) We assumed zero conductance value. Because the proposed method uses a least-squares fitting instead of a local asymptotic fitting, the zero conductance value does not affect the fitting procedure.

I I I I

1 - -

- 2 -

3 - frequency transform

In reply to Dr.T.Henriksen and Dr. B.Gustavsen :

As explained in the discussion by Mr. Nguyen et al., it is very difficult to approximate the off-diagonal elements of the wave deformation matrix H(w) with a rational function of s. Therefore, it is Micult to compare the accuracy of the proposed ARMA method with the rational function methods. But the time-domain accuracy of the proposed method can be confirmed by a comparison with results using the rigorous frequency domain simulation in Fig 5(a), Fig 7(a), and Fig. 9(aX

shows the fitted results of NIZ(O) of an underground single- core coaxial cable illustrated in Fig. B(a). The element Hlz(w) of the cable is a typical example which has widely different velocities. As pointed out by the discussers, the element oscillates very fast in the frequency domain as shown in Fig B@). But the fitting by the proposed linearized LS method with a small number of samples is accurate enough both in the frequency domain and in the time domain as is clear in Fig. B(c). Our experience indicates that the number of samples is enough to be 10-20 points per decade. The reason, for the small number of samples, is that the,

0 I I I I 1 I I I I

- l o o t x >' Y -1

-200 t - -400 ---I , I I m y 1 500

3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 ORDER

0 . 2 1

,

"3 4 5 E 7 8 9 10 11 12 13 ORDER

(a) AIC and SD - order characteristic

0, I I I I

9 -10 I - 1

-40 L I I I I 100 1000 1EtG4 1Et05 1EtOE

FRE'JUENCP [Hzl

-400 I I I 100 1000 1Et04 1E+05 E+06

FERUMCI [Hzl

-0 .4 l I I I I 0 0.2 0.4 0 .6 0.8 1

TIME [nsl

I

Fig. 8 shows a pipe-type oil-filled cable which involves widely different propagation velocities. Also, Fig. B(b)

(c) Step response

Fig A Fitting results of a off-diagonal element H 5 6 ( ~ )

411

TABLE A Coefficients ofH56(w)

At 0 . 5 p N 1 2 (order) a0 1 . 4 1 9 0 3 7 6 2 7 6 0 5 4 0 8 ~ 1 0 ~ ~ ~ a1 -3.707989297151094~10-~~ a2 1.817490407109451 a3 -3.591784826682045 0 4 3.557198723072919 a5 -1.763529792817600 a6 3 . 5 0 0 0 5 5 7 9 5 1 6 3 5 4 9 ~ 1 0 - ~ ~ a48 -3 . 073333098105003x10~05 a4 9 -1.617214031477093~10~~' U50 8 . 3 7 8 2 4 1 3 4 8 6 8 0 7 0 0 ~ 1 0 ' ~ ~ 051 -1.270130798932017~10~~~ U52 8 . 2 3 6 5 0 6 5 2 2 7 2 8 6 2 2 ~ 1 0 - ~ ' as3 -1.990873656694414~10-~~ b i -4 .988118230362191 bZ 1 .01187198 9 2 7 4 0 9 2 ~ 1 0 ' ~ ' bo -1.068433629862729~10*~' br 6.332858647194881 b5 -2.276268007811602 b, 6 . 2 0 9 32 7 9 7 8 7 5851 6 xl0-'I b7 -5.610115304335307~10-~~ b, -1.923184241665601~10-~' bs 2 . 2 0 4 3 1 2 3 6 8 0 6 1 5 6 0 ~ 1 0 ~ ~ ~ bio -1.385472395859206~10-~~ b i i 5 . 2 1 1 8 6 4 7 6 4 6 4 4 8 5 1 ~ 1 0 - ~ ~ b12 -9.371868655154460~10~~~

frequency-domain oscillating nature is inherently taken into account by the term z-qk in (28).

The discussers suggest a fitting methodology, which performs the fitting for each mode. This methodology requires the convolutions due to modal transformation matrices. But it is pointed out that an element of the transformation matrices is not a causal function, which has a response in the negative time in ref. [A]. We doubt if it is possible to carry the convolution of a non-causal function in the time-domain simulation.

In the order determination process described in IV-D, the order is increased one by one until a best fit is found. Thus, an order evaluated to be unstable by Jury's method is not used, and the order is increased for the next fitting.

The influence using (22) instead of the original function is almost canceled by using a relative type of error evaluation described in section IV-C.

p ,=la.

l e n g t h

0 ,=loon

lm

1.05cm 1.78cm

3 08cm 4 08cm

, m

26cm p ,=28nQ/m

(a) An underground single-core cable

10, I I I

-20 I I I I /

1000 1E+04 1E+05 1E+06 FRE!JUEECP [A21

-300' I I I 1000 1E+M 1Et05 1E+06

FEEBUMCY [Hzl

(b) Fitted results in the frequency-domain

a t; ARMA I

frequency transform -1

5

(c) Step response

Fig.B Fitted results ofHlz(o) of an underground single-core cable

[A] T.Ino and C.Uenosono, "An examination of highly accurate equivalent circuit for double-circuit DC transmis- sion lines," Trans. IEE Japan, Vol. 114-B, No.5, 1994.

Manuscript received April 12. 1995.