simulation of transients in unerground cables with frequency dependent modal

7/27/2019 Simulation of Transients in Unerground Cables With Frequency Dependent Modal

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IEEE Transact ions on Power Del ivery , Volume 3, No . 3, Ju ly 1988 1099SIMULATION OF TRANSIENTS IN UNDERGROUND CABLES

WITH FREQUENCY-DEPENDENT MODAL TRANSFORMATION MATRICES

L. Marti, Member IEEEThe University of British ColumbiaVancouver, Canada

AbstractThis paper presents a new mathematical model for thesimulation of electromagnetic transients in underground highvoltage cables. The solution is carried out in the time domain;therefore, this model is compatible with time domain solutionalgorithms, such as the one used in the EMTP. The frequencydependence of the cable parameters and of the modaltransformation matrices is accurately taken into accountComparisons with analytical and measured results are alsopresented.

1. IntroductionAccurate modelling of underground cables and transmissionlines plays an important part in the simulation of transientphenomena in power systems. A number of models have beenproposed to date. These models can be classified into two majorgroups, according to the solution techniques used in their hostprograms:

a) Time domain models.In this class of models, the solution is carried out in thetime domain without explicit use of inverse (Fourier or Laplace)transforms. Within this group, two tg e s of models deserveattention:

i) Lump ed-para meters models: The transmission system isrepresented by lumped elements (usually by several cascadedn-sections) evaluated at a single frequency [l]. A moresophisticated form of this type of model includes therepresentation of the ground return impedance using a suitablecombination of several R- L branches. This representation iswidely used in transient network analyzers [ 2 ] . The validity ofthese models is restricted to relatively short lines or cablesand, in general, their frequency response is only good in theneighbourhood of the frequency at which the parameters areevaluated.

ii) Distributed-parameters, frequency- dependent models: Thesolution is performed in the modal domain. The frequencydependence and the distributed nature of the line or cableparameters are taken into account [3]. The validity of thesemodels is restricted because the modal transformation matricesare assumed to be constant. This assumption can lead to poorresults in many cases of unbalanced overhead transmissionlines (especially multiple-circuit) when the simulation involvesa wide range of frequencies. In the case of undergroundcables, the modal transformation matrices depend strongly onfrequen cy, a nd constant- transformation- matrix models generallyproduce very poor results.

87 WM 154-8by the IEEE Transmission and Distribution Committeeof the IEEE Power Engineering Society for presenta-tion at the IEEE/PES 1987 Winter Meeting, Ne w Orleans,Louisiana, February 1 - 6, 1987.ted Janaury 30 , 1986; made available for printingDecember 2, 1986.

A paper recommended and approved

Manuscript submit-

0885-8977/88/0700,

b) Frequency domain models.In this class of models, the response of the transmissionsystem is evaluated in the frequency domain. The time domainsolution is then found using inverse transformation algorithmssuch as the FFT (Fast Fourier Transform) [4].The frequency dependence of the line or cable parametersand of the modal transformation matrices is taken into account.Even though inherent numerical problems such as aliasing andGibbs' oscillations have been alleviated (using windows and otheroscillation-suppressing techniqu es), the applicability of thesemodels is restricted by the limitations of their host programs.There are no general- purpose transient analysis pr ogram sin the frequency domain with the overall simulation capabilitiesof time domain programs such as the EMTP [5]. Suddenchanges in the network configuration (such as faults, opening andclosing of circuit breakers, etc.), and the modelling of non-lineareleme nts cannot be handle d easily with frequency- domain so lutionmethods.The model presented in this paper belongs to the classof time-domain, frequency- depen dent models. It overcom es themain limitation of existing time-domain line models; that is, ittakes into account the frequency dependence of the modaltransformation matrices. Also, by being compatible with thesolution algorithm of the EMTP, it enjoys all the implicitadvantages of a versatile host program.The new cable model is accurate, as demonstrated withcomparisons with analytical results. From a computational point ofview, the speed of this model is comparable to the speed offrequency- depend ent models with constant transformation matrices.All numerical results presented in this paper refer to thesimulation of underground high voltage cables. However, themodel itself is general, and given its computational speed, itshould also be very useful in the simulation of unbalanced and

multiple-circuit overhe ad transmission lines.2. Description of the model

The following conventions in notation will be used inthis paper:- Upper case letters indicate "frequency domain" quantities,whereas lower case letters are used to denote their "timedomain" counterparts (e.g., V in the frequency domain, andv(t) in the time domain).- Primed symbols refer to modal quantities; otherwise, allquantities represen t phase compo nents (e.g., v'(t) for modalvoltage, and v(t) for its phase domain equivalent).Consider an underground cable consisting of n conductorsof length 1. In the frequency domain, the relationship betweenvoltages and currents at sending and receiving ends can be

expressed as(1)c Vm + I m = F m = A F k

m (2)c V k - Ik = Bk = A Bwhere

-1099$01.00@1988 IEEE


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1looYc = 4yA = e x p ( - m r)

Yc is the characteristic admittance matrix; A is defined as thepropagation matrix; Y and Z are the shunt admittance andseries impedance matrices per unit length , respectively; V and Iare voltage and current vectors of dimension n; F and B areintermediate vector functions which, in the time domain. can beinterpreted as waves travelling in forward and backwarddirections, respectively. All matrices are dimensioned nxn, andsubscripts k and m are used to indicate sending andreceiving end quantities.

Eguations (1) an d (2) describe any transmission system,whether It is an overhead line or an underground cable. Onlythe values of Yc and A, as functions of frequency, determinetheir respective behaviours.Let Q be the eigenvector or modal transformation matrixwhich diagonalizes YZ. Equations (1) an d (2) can then betransformed into the m odal domain

Y; v; - 1 = B i = AB; (4)where Yc and A are diagonal matrices, and

V = Q - T V I = Q l F = Q F B = Q B A = Q - A QY, = Q - Y c Q - T

Note that the elements of Yc, A, and Q are complex-valuedfunctions of frequency. Also note, that Q T is the transpose ofQ, and Q - indicates the transpose of Q - .Transforming equations ( 3 ) an d (4 ) into the time domaingives

y,(t) vh(t) + i h ( t ) = f (t) = a(t) f;(t) ( 6 )my,(t) v;(t) - i;(t) = b;(t) = a(t) bh(t) (7)

where the voltages and currents, in modal components, are givenbyv(t) = qT(t) v(t)i(t) = q- t) i(t) (8)

Note that the symbol * is used to indicatematrix-vector convolutions. If t he eleme nts of matrices Yc, Aand Q are synthesized using rational functions, then yc(t), a(t)and q(t) become matrices whose elements are finite sums ofexponentials (see Appendix I). Therefore. the convolutions inequations (6 ) to (8) can be evaluated numerically usingwell- known recursive techn iques [4.Algebraic manipulation of these equations finally leads to

where yeq is a real, constant, symmetric matrix; hm(t) and hk(t)are defined as equivalent history current sources. At time t,these vectors are completely defined in terms of variables alreadyknown from previous time steps (see Appendix I).Equations (9) an d (10) can be represented by theequivalent circuit shown in Figure 1. This equivalent circuit iscompatible with the solution algorithm of the EMTP. In fact,the EMTP representation of lossless lines and offrequency-dependent lines with constant Q also share the sa meform. Only the updating of the history current sources at each

time step, and the value of yeq are different; therefore. takinginto account the variation with frequency of the modaltransformation matrices becomes transparent to the main core ofthe EMTP.$0 )h

Fig. 1: Equivalent circuit in the time domain.

3. Synthesis of input data functionsRecursive convolution can be used to update the historycurrent sources in equations (9) and (10) because the elementsof Yc, A and Q can he expressed in terms of rationalfunctions. The fitting algorithm used to synthesize these inputdata functions is a refined version of the one used in thefrequency-dependent line model of the EMTP. The originalalgorithm is described in [ 3 ] .The elements of Q can be synthesized with rationalfunctions when the following conditions are met:

a) The columns of Q (i.e., the eigenvectors of YZ) are scaledso that one of their elements becomes real and constantthroughout the entire frequency range. With this normalizationscheme, all the elements the eigenvectors becomeminimum-phase-shift functions.b) The eigenvectors of YZ are continuous functions of frequency.

This last requirement is met when a very stableeigenvalue/eigenvector algorithm, such as Jacobi [7 ] is used. Notethat the Jacobi algorithm is designed for symmetric matrices, andthe product YZ is not symmetric. However, the unsymmetriceigenvalue/eigenvector problem YZx= X x can be converted into asymmetric one if Zx=XY- x is solved instead [SI. Th econvergency rate and stability of the Jacobi algorithm can beimproved considerably if YZ at a given frequency is pre andpost multiplied by the transformation matrix obtained in thepreceding frequency step. If Q is normalized so that its elementsare minimum-pha se-shift functions, the elemen ts of Yc alsobecome minimum-phaseshift, and can be likewise synthesizedwith rational functions.The elements of A can he synthesized with rationalfunctions multiplied by exp(-jWT) [3]. The time delay constant

7 depends on the difference between the phase angle of agiven element of Aand the phase angle of its approxim ation byrational functions; also, r is numerically close to the travel timeof the fastest frequency component of a wave propagating on agiven mode. Note that the propagation matrix A (in the phasedomain) cannot be synthesized with rational functions because asingle time delay cannot be associated with each of its elements.

Figure 2 shows the magnitude of the elements ofeigenvector 3 for the 230 kV, three-ph ase undergr ound cable


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1101

I: 0 . 9 0.80

0 . 7 00 . 6 0

i; 0.500 0.402 0 . 3 0

0.100

< 0 . 2 0.. .. ~ Frequency (Hz)

Fig. 2: Magnitude of the elements of eigenvector 3 of Y Z------ Exact function.Synthesized function.- 0.045E 0 .0 4 04 0 . 0 3 5J2 0.030w 0 .025-0 0 . 020i; 0 . 0 1 5

22

v

>-

CY~ 0.010

Frequency (Hz)Fig. 3: Magnitude of element 3 of Yc:------ Exact function.Synthesized function.

1 . 1 7

$ 0. 3$ 0 . 20 . 1

0Frequency (Hz)

Fig. 4: Magnitude of element 3 of A :_----- Exact function.Synthesized function.described in Appendix 11 Note that there are twelve curvessuperimposed on this plot, corresponding to the magnitudes ofreal and synthesized elements of eigenvector 3. Also note thatthis eigenvector contains only four distinct elements; that is, themagnitudes of elements 1 and 2 are identical to the magnitudesof elements 5 and 6, respectively. Figures 3 an d 4 show Ycand A for mode 3. Their respective synthesized functions arealso superimposed. Note that the synthesized functions match theoriginal ones very closely. Errors are typically under 2% withinth e 0 to 1 MHz range.

The maximum number of functions that must beapproximated for a cable with n conductors is n(n+l). Fo rn = 6 , for example, the maximum number of elements to besynthesized would be 42. In practice, however, there is aconsiderable amount of symmetry within the elements of Q. Fo rthe 230 kV cable shown in Appendix 11, there are only 14distinct elements in Q. The total number of distinct functionsthat had to be synthesized in this case was 26.The order of a given approximation depends on th eshape of the curve. For example, in the case of the cable

mentioned above, the average number of terms needed toapproximate Yc, Aand Q from 0 to 1 MHz were 21, 15 an d7, respectively.4. Numerical results

In order to establish the accuracy of the new cablemodel, the analytical response of a three-phase undergroundcable is compared with the results obtained using theimplementation of the new model in UBCs version of theEMTP. The measured impulse response of a crossbonded cable isalso compared with its transient simulation. The effects of takinginto account the frequency dependence of Q are illustrated withthe simulation of a single-phas e line to grou nd fault on acrossbonded cable.4.1 Comparison with analytical results

Consider the 230 kV, three-phase underground cableshown in Figure 5, where a voltage source vs(t) IS connected tothe core of the first conductor (physical data for this cable, andthe characteristics of its series impedance matrix, are shown inAppendix 11).

s1

Fig. 5: Representation of a three-phase underground cableIf vgt) is a sinusoidal source of frequency WO , th eresponse will also be sinusoidal, and it can be found analytically(e.g., by solving equations (1) an d (2 ) evaluated at W = W O ) . Thisis the steady-state response at W = W .If vs(t) is not sinusoidal, it can still be represented asan infinite sum of sine and cosine functions

mv (t) = f(t) = a + C [a cos(nWot) + bn sin(nwot)]

O n = lwhere WO =2n/T. and T is the period over which the originalfunction is represented by its series equivalent

If a sufficiently large number of terms N is taken, theresulting series represents a reasonable approximation of theoriginal input function. The exact response to this input seriescan be obta ined by superimposing N steady-state respo nsesevaluated at w= n w o , for n=1,2, ...,N. This analytical solution canthen be compared with the numerical solution obtained with thenew model.

In the following simulation, vs(t) represents the Fourierseries approximation of a square wave of unit amplitude. Thisapproximation contains 1000 terms, and the period T is 40 ms(see Figure 6). The length of the cable is assumed to be 10km.


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1102

- 0 . 20-04L , , , , , , . . . . . . . . . . . . . . . . . . . .- 0 . 6 0 0 10 20 30 40 50 60Time (mil l iseconds)Fig. 6: Fourier series approximation of a square wave.N=1000, T = 4 0 ms.

5 O- 4 . 0U 3.0d

s 2 . 0& 1 . 0$ 0

5 -2.0

v

m

- 1 . 0-c2 - 3 . 0' 4 .0

-5 .0 0 10 20 30 40 50 60Time (mil l iseconds)Fig. 9: Relative error func tion, receiving-end vdt age . Core I :Analytical response.Ne w model.

------

these errors do not accumulate with time. In fact, this simulationwas allowed to proceed up to 120 ms (12000 time steps) andno deterioration in the transient response was observed.

4.2. Comparison with a field testThe field test presented here has been reproduced from

[l]. which in turn quotes [9 ] as the original source. Figure 10shows the circuit diagram of the test, and Appendix I1summarizes the physical data of the crossbonded cable used.

4p 42 44 46 48 50 52 54 56 58 60Time (mil l iseconds)Fig. 7: Square- wave response, receivingend vdtage. Core 1.--_-__ nalytical response.Mew model.,+ 0 . 5 0 2$10.10 40 42 44 46 48 50 52 5 4 56 50 60

Time (mil l iseconds)Fig. 8: Square-wave response, recei vinge nd v dtage. Sheath 1:-_____nalytical response.Mew model.

Figure 7 shows the' voltage at th e receiving end of core1, and Figure 8 shows the voltage at the receiving end ofsheath 1. The analytical solution is indicated with a dashed line,and the EMTP simulation is shown in solid trace. Figure 9shows the relative differences between the two curves plotted inFigure 7.

From these results it can be seen that the agreementbetween analytical and transient solutions is very good (thesuperimposed curves in Figures 7 an d 8 are practicallyindistinguishable). Note that th e largest erro rs (less than 2%)occur in the first sharp peaks of the response. Also note, that

Fig. 1 0 Field test connection diagram.An impu lse of waveshape 0 x 4 0 ~ s (i.e., negligible fronttime) with a peak magnitude of 7.3 kV was applied between thecentre core and earth. The sheaths are connected together andgrounded through a 10 Q resistance at the sending and receivingends.Figure 11 shows the voltage at the sending end of core2, and Figure 12 shows the voltage of sheath 2 measured atthe second crossbonding point. Simulation results are shown insolid trace, while the measured response is shown in dashedtrace.The response obtained with the new model agrees wellwith the simulation results shown in [l] . The aut hors ofreference [I ] indicate that the available test data regardingcomplex permittivity, earth resistivity, etc., was not well known.

Also, it appears that the description of the input waveform mayhave been oversimplified or loosely described in [ 9 ] . When thetime-tc-half-value of the input voltage impu lse is chang ed fromMUS to 4 5 p s . the agreement between calculated and measuredresults improves considerably (see Figure 13). The sheathvoltages, however, are not affected much by this change in theinput waveform.


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I . 5 0 7

1103

C I

0 20 40 60 80 100 120 140 16C 180 20 0Time (microseconds)Fig. 11: Sendingend vdtage. Core 2:Field test.New model.

------

' 0 . 15 1 . . . I . . . I . . . I . . . . I . . . . I . . . . . . I . . . . . ,0 20 40 60 80 100 I20 140 160 t8 0 200Time (microseconds)Fig. 12: Second crossbonding-point vdtage. Sheath I :Field test.New model.

------h2v

0 20 40 60 80 100 120 140 160 180 200Time (microseconds)Fig. 13: Sendingend vdtage (waveshape 0 x 4 5 ~ s ) .Core 2:Field testNew model.

------Overall, the agreement between the waveshapes ofcalculated and simulated results are reasonably good. consideringthe many uncertainties involved in such a test (e.g.. earth not

homogeneous, ground ing resistances not well known, ex ) .4.3 Simulation of a single-phase, line to ground fault

The effects of taking into account the variation withfrequency of the modal transformation matrix Q will beillustrated with the simulation of a line to ground fault Theconnection diagram for this simulation is shown in Figure 14.

Fig. 14: Simulation of a line to ground fault.The peak magnitude of the voltage sources is 1.0 P.u.,and their phas e angles are set 120' apar t (i.e., -120, 0' an d+120). The cores of this 230 kV underground cable are openat the receiving end; the sheaths have been crossbonded andgrounded at the sending and receiving ends; the length of thecable is 3 km, and the crossbonding points are evenly spaced(i.e.. each minor section is 1 km long). The simulation startsfrom 60 Hz steady-state initial conditions, and the receiving endof core 1 is connected to ground at t = 5 ms.Figure 15 shows the receiving-end voltage of core 2, andFigure 16 shows the fault current (receiving end of core 1). Theresponse obtained with the new cable model is shown in solidtrace. The response obtained when Q is assumed to be constant(evaluated at 5 kHz) is shown in dashed trace.These results clearly indicate that the frequencydependence of Q has a very significant effect on transientsimulations of this type. Note that during the first 5 ms of thesimulation, both solutions coincide. However, after the Occurrenceof the fault, the differences in the first peak of the voltageresponse are approximately 30%. while the differences in the firstpeak of the fault current exceed 60%.Figure 17 shows the receiving-end voltage of core 2when the simulation starts from zero initial conditions at t = 0.The response when Q is constant is quite good during the first2 ms of the simulation. As the high frequency transientsattenuate, the differences increase. After the fault occurs, thedominant frequency component is 60 Hz, and the response whenQ is constant deteriorates considerably.It has been found that the accuracy offrequency -dependen t models with constant Q. depends on th etype of transient situation being simulated, as well as thefrequency at which Q is evaluated. For example, reasonably good

I .80N 1 . 4 0 j

0 . 2 0'c) - 0 . 2 0

0 5 IO 15 20 25Time (milliseconds)Fig. 15: Receiving-end vd tage . Core 2:------ Fre que ncy dependent model w ith constant Q .New model.


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11041.80 7

1.80 71.40 7NE2 1.00:

& 0.60165 0.20:>0 0.20:

- 1 . 80 1 . , . . , . . . . I . . . . I 1 , . , , , . . , ,0 5 10 15 20 25Time (mi l l i seconds)

M'g - 1 .oo:alE

-1.40:-1.80

Fig. 16: Receivin g- end current. Core 1:------ Fre que ncy dependent model with constant Q .New model .

. . . . . . . " " " ' , " " " . , . . . . . . . ~ ' . ~ ' ~ . ~

Fig. 17: Receivin g-end vdt age (zero initial conditions). Core 2:------ Fre que ncy dependent model with constant Q.New model .results can be obtained during simulations where the currentsflowing in the cable are very small. Also, good results can beobtained in transient simulations where the frequency range ofinterest is relatively high (e.g., above 1 kHz), or relatively low(e.g., below 1 Hz), and Q is evaluated at a frequency withinthis range (as long as the columns of Q can be normalized sothat their imaginary parts are very small).5. Computational speed

The elapsed C PU time in the time-step loop of UBC'sEMTP was measured for a case where the energization of athree-phase undergr ound cable was simulated. Using aconstant-p aramete rs model as a reference, th e relative speeds ofthe new cable model and the frequency-dependent model withconstant Q were determined. The order of the rational functionapproximations used was the same for both models. The resultsare shown in Table I.Model Relative timing

Constant- parameters 1.0Frequency- dependent 6.0New Model 6.14

Table I: Relative CPU t imes . EMTP t ime-s t ep loop.

It can be seen from these results that the computationalspeed of the new cable model is comparable to that of thefrequency-dependent model with constant Q.

Although these results are probably influenced bydifferences in the programming of the algorithms, they indicatethat the additional computational effort required to take intoaccount the frequency dependence of Q is relatively low.Work is currently being carried ou t at UBC to improvethe speed of frequency-dep endent models by reducing the orderof the synthesized functions at running time. The maximumfrequency that can be reproduced in a numerical solution islimited by the sampling rate At (fmax=1 /(2At); therefore, thesynthesized functions need not reproduce the original ones abovethis frequency. If poles above fmax are dropped in anappropiate manner at running time, the accuracy of thesimulation is not affected, and considerable savings in computertime can be obtained.

6. ConclusionsThe underground cable model presented here, accuratelytakes into account the variation with frequency of the parametersand of the modal transformation matrices. This overcomes themain limitation of existing frequency- dep end ent line models fortime-dom ain solution algorithms.The accuracy of the synthesized functions Y'c, A'and Qis very high, and generally higher than the accuracy with whichearth resistivity, dielectric losses, and other cable parameters canbe estimated. Therefore, from a practical point of view, theaccuracy of the new cable model is only limited by the accuracyof the input data and by the simulation capabilities of the host

program.Computational speed is very high, and of the same orderof magnitude as that of frequency-de penden t models withconstant transformation matrices. High computational speed makesthe detailed simulation of crossbonded cables attractive. Also,given that minor sections are usually of the same length. onlythe parameters of a single section need to be synthesized.Furthermore, by modelling each section separately (making thecrossbonding connections explicitly), non-linear voltage limiters atthe crossbonding points can be easily taken into accountThe model is numerically stable. Relatively longsimulations have been made (12000 time steps) and there hasbeen no deviation from the correct answers.The model is general. Its application to multiple-circuitoverhead transmission lines should be of considerable practicalimportance.

AcknowledgementsThe author would like to thank Professor H. W. Dommelfor his encouragement and advice; Professor J. R. Marti forproviding the rational-functions fitting routines used in thedevelopment of the new cable model, and for many stimulatingdiscussions; Professor A. Ametani for providing the cable dataused in the comparison with the field test; the reviewers of thispaper, for their valuable comments and suggestions. Also, thefinancial support of Bonneville Power Administration, BritishColumbia Hydro and Power Authority, and the University ofBritish Columbia is gratefully acknowledged.

References[l ] Nagaoka. N. an d Ametani, A.. "T ransient Calculation s on

Crossbonded Cables. '' IEEE Transactions on Power Apparatusand Systems, April 1983, pp. 779-787.[2] CIGRE Working Group 13.05, "The Calculation of SwitchingSurges: 11.- Network Representation for Energization andRe-Energization Studies on Line Fed by an InductiveSource". Electra, No. 32, 1974, pp. 17-42.


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Marti, J. R., "Accurate M odelling of Frequency- Depe nden tTransmission Lines in Electromagnetic Transient Calculations."IEEE Transactions on Power Apparatus and Systems, JanuaryWedepohl L. M. and Indulkar, C. S ., "Switching Overvoltagesin Long Crossbonded Cable Systems Using the FourierTransform." IEEE Transactions on Power Apparatus an dSystems, July IAugust 1979, pp. 1476- 1480.Dommel H. W. and Meyer W. S . , "Computations ofElectromagnetic Transients". IEEE Proceedings, vol. 62(7), pp.Semlyen A. and Dabuleanu A. , "Fast and Accurate SwitchingTransient Calculations on Transmission Lines with GroundReturn using Recursive Convolutions." IEEE Transactions onPower Apparatus and Systems, March/April 1975, pp.Strang, G., Linear Algebra and its Applications. New York:Academic Press, 1976, pp. 283-288.Hornbeck R. W., Numerical Methods. Quantum PublishersINC, 1985, pp. 229-231.Shinozaki. H., et al., "Abnormal Voltages of a Core at aCrossbonding Point.", J. Tech. Lab. Chugoku Electric Power

1982. pp. 147-157.

983-993, July 1974.

561-571.

CO., VO I 39, 1971, pp. 175-198.[lo ] Wedepohl L. M. and Wilcox D. J., "Transient Analysis ofUndergrou nd Power-Tran smission Systems". IEE Proceedings,Vol. 120, No. 2, February 1973.

APPENDIX IThe elements of Q and Y'c, when normalized asminimum-phaseshift functions, can be approximated by rationalfunctions P(w) of the form,

m k .P(w) = ko + Zwhere ko, ki and pi are real constants.

-= l j w + p .

In the time domain this equation becomesm

i= 1where 6(t) is the Dirac impulse function, and u(t) is the unitstep function.

p(t) = kO 6( t ) + C hi exp(-pf) u(t) (1.2)

The convolution of p(t) with a given time function f(t)can be expressed as

g(t) = f(t) p(t) = d f(t) + h(t) (1.3)where d is a real constant, and h(t) depends on past history ofg(t) an d Rt) [31.

On the other hand, the elements of A' can beapproximated by rational functions of the formR(w) = P(w) exp (-jwr) (1.4)

where P(w) is a rational function of the same form shown in(1.1) with ko=O, and T is a real constant associated to modaltime delay. Equation (1.4). in the time domain, becomes

mi = Ir(t) = C ki exp(-p{t-.r)) u( t-T ) (1.5)

Unlike the numerical convolution f(t)*p(t), the convolutionof f(t) with r(t) is given by

1105

where h(t) only depends on past history values of b(t) and f(t).Let us now consider equation (6).

where the elements of matrices a'(t) and y'c(t) are sum s ofexponentials, as indicated in equations (1.2) and (1.5). Note thatmatrix- vector convolutions, rather that scalar convolutions. arenow being indicated by the symbol "*".The numerical convolution of y'c(t) with vh(t) can thenbe expressed as

y',(t) vh(t) = gh( t ) = Y ' , ~ vh( t ) + h ' (t) (1.7)m lwhere y'co is a real, diagonal matrix, and h'm l(t) is a functionof past history values of gh(t) and vh(t) (see equation (1.3)).

T o obtain the phase voltages,

Similarly, the phase currents will be given byim(t) = q(t) * i;(t)i,(t) = qo + h;,,(t)i h ( t ) = q; ' im(t) - q; ' h h g ( t ) (1.9)

where qo is a real, nxn matrix; h'm z(t) is a function of pasthistory terms of vm and v h ; h'm3(t) is a function of pasthistory terms of im and ih.After introducing (1.7). (I.@, and (1.9) into (6). algebraicmanipulation leads to

yeq v,(t) + i,(t) = h,(t) (1.10)where,

Since fh(t) depends on past history values of f h and fkonly, hm(t) is completely determined at time tFollowing a similar procedure for equation (7)

Equations (1.10) and (1.11) represent an n-conductortransmission system in the time domain, when the transient issolved at discrete time steps. Equivalent history sources hm(t)and h q t ) are updated continuously throughout the solution.APPENDIX I1

Physical data for the 230 kV underground cable used inthe analytical comparisons of section 4.1 and 4.3 (Cable 1). andfor the 110 kV crossbonded cable used in the field test ofsection 4.2 (Cable 2) a re given below.


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1106

rl ( c m )r 2 ( c m )r3 ( c m )Tu ( c m )r5 ( c m )

Core Resistivity ( a m )Shea th Resis tiv ity (a m )Inner insulation tan&Outer insulation tan&Inner insulation EOuter insulation e rEarth resistivity ( a m )

Cable 1 Cable 20.002.343.854.134.84

0.0170 10 - E-0.2100 10 - E-

0.0010.001

3.58.050 7..........................

1.0 m

1.051.783.083.264.08

0.0183 IO -0.0280 10- =

0.0400.1003.33.810 0

.35 ma - - - -+35 mFig. n.1:Cable I configuration.

Figures 11.3 and 11.4 show some of the elements of theimpedance matrix for Cable 1. Note that these impedances areexpressed as loop quantities; therefore, they do not depend onbonding or grounding connections [ lo] . Also note, that zero andpositive sequence impedances can be derived directly from theseloop impedances. In the captions of Figures 11.3 an d 11.4,"core-sheath" corresponds to the loop formed by the core withretum through the sheath; "sheath-ground'' corresponds to the-loop formed by the sheath with return through the ground;"sheath-ground-sheath'' corresponds to the mutual impedancebetween two sheath-groun d loops. Ther e is also a mutualimpedance between the core-sheath and sheath-ground loops,which is shown here.

/F r e que nc y (Hz)

Fig. 11.3: Series impedance matrix in Imp quantities. Resistance:+ oreshea th .--t- Sheath-ground.--4-- Sheath- ground- sheath.

0Frequency (Hz)

Fig. 11.4: Series impedance matrix in Imp quantities. Inductance:+ ore-sheath.+ heath-ground.A heath- ground- sheath.Fig . 11.2: Cable 2 configuration


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Discussion1 IO7

A. Am etani and N. Nagaoka (Doshisha University, Kyoto, Japan): Theauthor is commended for presenting an interesting paper. Inclusion of afrequency-dependent (FD) transformation matrix (Qmatrix) into a transientcalculation by a time-domain technique su ch as the EMTP has been o ne ofthe important problems to be solved to achieve a higher accuracy of thecalculation. To overcome this problem, at least two different approacheshave been proposed. One is to apply a real-time convolution to take intoaccount the FD effect of the Q matrix at every step of transforming phasorcomp onents to modal comp onents and vice versa within a modal framework[A]. The other one is to carry the transient calculation in the actual phasedomain so as to avoid a usage of the FD Q matrix between the phase domainand the modal domain [B]. Because the EMTP is widely used all over theworld as a transient analysis program and is structured to carry the transientcalculation within a modal framework, the former approach has a practicalsignificance.The method to deal with the FD Q matrix in the paper is based on theformer approach, of which the original idea was proposed by one of thediscussers in [A] more than 10years ago. (Eq. (8) in the present paper issame as (19) in [A].) Because there was no idea of recursive convolution atthat time (and thus the mod al approach requires a large computation time, asshown in Table 2 of [A]) the approach was not regarded practical at all. Theauthor has completed the approach in a sophisticated manner for the case ofa single-core coaxial (SC) cable. The discussers believe that the work make san important contribution to the field of a transient analysis in an electricpower system.However, there are several questions which are not clear in the paper.The authors comments on the following questions are appreciated.1) Eigenvalues and vectors calculation (Sec. 3(b) and Fig. 2)i) The discussers have proposed a similar approach to improve theaccuracy and stability of eigenvalues and vectors calculations and also toavoid a mode crossing or exchange phenomenon appearing during anumerical evaluation of the eigenvalues and vectors [C]. To confirm theaccuracy of the calculated eigenvectors (Q matrix), it is necessary to re-diagonalize the original YZ matrix by the given Q matrix and measure theerror of the re-diagonalization. This process is always carried out in theeigen-calculation program developed by the discussers. It is interesting tosee the re-diagonalization error of the Q matrix, of which the third columnelements are shown in Fig. 2 , calculated by the author. For a reference, therediagonalization error of the Q matrix calculated by the discussers was0.045 percent at maximum for the same cable as that in the present paper(Cable 1 in Appen dix 11).ii) In general, the admittance matrix of an SC cable, which is a mainconcern of the present paper, is of a very simple form with many zeroelements, and thus it is easily diagonalized. The admittance matrices of apipe-type (PT) cable, a cable which is a multiphase SC cable with thinsheaths enclosed within a pipe, and an untransposed vertical overhead lineare of a more complicated form. Thus, it is a matter of question if theauthors method is applicable to the PT cable and the untransposed overheadline.2) Fourier series approximation (Sec. 4.1, Figs. 6 to 9)As a usual practice of a transient analyst, numerical Laplace transformwith a weighting function is adopted to handle a transient with an ac sourceor a time varying source, to avoid a num erical instability due to poles alongimaginary (ju) xis and well-known Gibbs oscillation. The discusserswonder why the author adopted Fourier series approximation rather thanLaplace transform. Figures 6 to 8 clearly show Gibbs oscillation, whichdecreases the accuracy of the calculated results.3) Comparison with a field test (Sec. 4.2, Figs. 11 to 13)Because the dominant frequency of the transient on the crossbonded cablewith I-km length is quite high, the transformation matrix in this frequencyregion is almost constant and does not affect a transient calculation. In otherwords, a constant transformation matrix can give almost the same result asthat with the frequency-dependent transformation matrix. Therefore, it ishard to discuss the accuracy or appropriateness of the proposed methodfrom the results.The source wavefo rm (originally 40-1s wave tail, less than 0.1-pswavefront) was deformed to have a 45-ps wave tail in Fig. 13. The changeresults in the decrease of the peak voltage and the phase shift of thewaveform after the peak, and thus the calculated result shows a betteragreement with the field test result. It is interesting to see if the change alsoresults in a better agreement with the field test on the sheath voltages, forexample, on the result corresponding to Fig. 12.To clarify the inaccuracy of the source waveform , the impulse generatorcircuit which was connected to the left-hand side of the 5004 esistance inthe field test is illustrated in Fig. A. It would be appreciated if the author

Fig. A . The source circuit in the field test.

would calculate again using this circuit and show the results.4) Fault calculation (Sec. 4.3, Figs. 15 to 17)Why ar e the calculated results by tw o metho ds almost identical for timesmaller than 5 ms in Fig. 15? Since the transformation matrices used for themethods are different, the steady-state solutions seem to be different.Also, it is a question why the calculated result only by constant Q matrixat 5 kHz is shown. To observe a steady state of a fault, Q matrix at 60 Hzshould be used, and for a transient, Q matrix at a dominant transientfrequency (maybe 5 kHz) should be used. In other words, if Q matrix at 60Hz is used, a better agreement may be obtained by the constant Q matrixmethod for Fig. 15 to 17. However, the accuracy of a transient solution bythe constant Q matrix at 60 Hz is, in general, poor. It might be better toshow calculated results by the constant Q matrix at 60 Hz for comparison.Once again, the author is commended for his interesting and timelypaper.

References[A] A. Ametani, Refraction Coefficient Method for Switching-SurgeCalculations on Untransposed Transmission Lines-Accurate andApproximate Inclusion of Frequency-Dependence, presented atIEEE 1973 PES Summ er Meeting, C73-144-7, 1973.[B] N. Nakanishi and A. Ametani, Transient Calculation of a Transmis-sion Line Using Superposition Law, ZEE Proc., vol. 133, Pt.C(5),[C] N. Nagaoka, M. Yamamoto, and A. Ametani, Surge PropagationCharacteristics of a POF Cable, Electr. Eng. Jpn. (LISA), vol.105(5), pp. 67-75, 1985.

pp. 263-269, 1986.

Manuscript received February 9, 1987.Adam Semlyen and H. Hamadanizadeh (University of Toronto): Wewould like to commend the author for having implemented the repres ita-tion of the frequency dependence of transformation matrices in time do aincalculations. The application shown is for cable transients but, a s the authorcorrectly points out, the effect of frequency dependen ce of the transforma-tion matrix may be significant in the case of multicircuit overhead lines.Could the author provide some results related to this application?The nice thing about the procedure is that an important part of thecalculations, i.e., fitting the transformation matrix with rational functions,is performed in a preprocessing mode. Only the convolutions and updatingof the past history vector in (1.8) and (1.9) requires additional computingtime when calculating a transient, although this time may not be asinsignificant as indicated by the author (increase of 2. 3 percent from 6 .00 to6.14 in CPU time). According to the figures given in the paper for theaverage number of terms in the fitted YI ,A an d Q, the number of firstorder differential equations to be solved for 6-modal Y an d A ar e 6 x(21 + 15 ) = 216 and fo r 30 elements of Q this number% 30 x 7 = 210.Since at each end of the line the modal quantities have to be transformed tophase quantities for the solution of node voltages and then back to modalquantities again for updating the history vector, this nonconstant transfor-mation matrix will increase the number of equations from 216 to 636. D oesthis increase not require much m ore computation time than 2.3 percent? If itdoes, then should one not try to model Q with a lower order approximationthat will not result in a significant decrease in the accuracy of the model?

Manuscript received February 17, 1987.

Olov Einarsson (Asea Research): The author shou ld be congratulated for aclearly written paper describing an efficient method for handling thefrequency-dependent modal transformation matrices in time-domain mo del-ing of cables and overhead transmission lines. I have the followingquestions and comments.


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1108I ) A property of the Jacobi eigensystem algorithm is that the eigenvec-tors are nearly orthogonal (using the metric defined by U-') lso foreigenvectors belonging to the same eigenvalue. It seems likely that thisproperty is vital in order t o meet the condition that the eigenvectors of YZare continuous functions of frequency.2) In the comparison with analytical results of section 4.1, the Fourierseries expansion of the square wave of Fig. 6 exhibits the well-knownovershot due to Gibb's phenomenon. Is this overshot also included in theinput time function of the EMTP simulation using the new model, or ca nsome part of the relative error show n in Fig. 9 be attributed to the overshot?3) Are Figs. 15 and 16 correct? Fig. 15 seems to indicate that there is a

longitudinal voltage of about 0.5 pu over core 2 in the steady state, orrelated to a transient having a time constant much longer than 25 ms.According to Fig. 16, this longitudinal voltage is roughly in phase with thecurrent of core 1. Consequently the voltage cannot be induced by magneticcoupling between the two cable cores.Manuscript received February 19, 1987.

B. R. Shperling (New York Power Authori ty, New York): A newtransmission line time-domain model, which takes into account frequencydependent parameters, was developed by the author. From the transientanalysis point of view, the suggested model represents a logical develop-ment of L. M. Wede pohl's results on the modal transformation matrices formultiphase systems and their application to the BPA's ElectromagneticTransient Analysis Program.The author should be complimented for achieving very high accuracy insimulating system transients using the developed cable model. Indeed, thesystem frequency characteristics obtained with the help of the suggestedmethod practically coincide with analytical results. At the same time,underground or submarine cables represent only one element of aconventional network. Thus, for example, in a simplest case of anunderground cable energization, a quite precise representation of a sendingend system, including its frequency response characteristics, is required totake full advantage of the suggested model. In addition, presence ofnonlinear elements might also influence the results. Keeping these factors inmind, and realizing the inherent inaccuracy of system data, what are theaccuracy requirements for the calculations of transient overvoltages innetworks with underground or submarine cables?Analysis of the transients during single phase-to-ground faults, which areillustrated by Figs. 15 and 16, needs some clarification. According to thesefigures, before fault application the voltages and currents calculated withthe help of the new model and the frequency dependent model w ith constantQ at 5 kHz, re identical. After attenuation of the high frequency transientsfrom fault application, the steady-state voltages and currents are signifi-cantly different for both methods. It is obvious that different cableparameters play different roles before and after fault application. Thus aftera fault application, cable inductances, with their frequency dependencies,become dominant for transient and steady-state processes. At the sametime, for cases without a line fault, the cable currents are relatively small,and precise simulation of the inductance frequency dependency becomesless important. Is it possible to generalize the discussed comparison andconclude that for cable switching operations without line faults a frequencydependent model with constant Q is sufficient?

Manuscript received February 19, 1987.

L. Marti: The author would like to thank the discussers for their interestingand useful comments. First of all, I would like to address the openingremarks made by Messrs. Ametani and Nagaoka. Th e importance of takinginto account the frequency dependence of the modal transformationmatrices of transmission systems has been recognized for a number ofyears. One of the first attempts to handle the problem within the fra mewor kof time-domain solution algorithms, was proposed in 1973 (reference [A]by the above discussers). In this reference, the equations describing asingle-circuit untransposed transmission line are solved using modalanalysis. The transformation between phase and modal voltages andcurrents in the time domain is obtained using d irect numerical convolutionbetween q( t ) nd the corresponding voltage and current vectors (whe re q ( t )is obtained from Q ( w ) using FFT techniques); that is,ipha\e(f)= q ( t )* imcde( f )

Umdde(t) =q' ( t ) * Uphase( t ) ,The me thods proposed in [A] and the cable model proposed in this paper usethis basic relationship between modal and phase quantities. The cable mo delin this paper differs from those proposed in [A], however, in the way it is

interfaced with its host program (e. g., the EM TP), and in the way in whichthe convolutions are evaluated numerically. Essentially, the interface isdone directly in phase quantities by splitting the convolutions into a partcontaining variables at instant t only, and into another part containingknown history. This splitting leads toiphase(t) = 40node ( t )+histi-phase

wherei m d e (0 ~ ~ ~ - ~ o d ~mode t )+ hist,.,,, I

an dumcwje(t) = q&haae( l ) + histv-mde'

which, after some algebraic manipulation, produces the final formiphase(t) = [qoyco-modeqb]Uphaae

+ { qo hist,.,d, + ~ ~ ~ ~ ~ c d ~ h i ~ t ~ - ~ o d ~ l +ist,.pbg1.In this form, the matrix [qoyco~modeq'Js real and constant and the secondterm is known history which is evaluated by recursive convolution (thehistory term hm3 n Appendix I should not have been primed because it is inphase rather than in mode quantities),The remaining questions posed by Messrs. Ametani and Nagaoka willnow be addressed in the same order in which they appear in theirdiscussion.I) Regarding the calculation of eigenvalues and eigenvectors.an eigenvalue-separation procedure proposed in 1964 [C 11, wherei) The approach proposed by the discussers in reference [C] is based on

P - [ Z Y - w / c U ] Pis diagonalized instead of Z Y ; c is the speed of light; and matrix P s givenby

1 1 - 1 - 1 / 2[: 0 + ]As the discussers correctly point out in [C], this indirect approachincreases the accuracy of the calculation of the eigenvalues of Y Z byincreasing the numerical separation of almost coalescing eigenvalues.However, it is unclear why the discussers feel that this approach is similarto the use of seeding in combination with the modified Jacobi methodpresented in this paper, and why this approach prevents the eigenvectorswitchover phenomenon encountered with standard eigenvalue/eigenvectormethods.The accuracy of the eigenvalues obtained using seeded Jacobi is onlylimited by the m achine accuracy, given that the method is completely stable.Default settings in the eigenvalueleigenvector routines written by the authorforce the re-diagonalization errors to be less than that is, theEuclidean norm of the off-diagonal terms of Q- Y Z Q is 10 - timessmaller than the norm o l ts diagonal entries.ii) It is unclear to the author why the discussers feel that the particularform of the admittance matrix is a concern of this paper, since it is thematrix product Y Z rather than Y alone that must be diagonalized in order toevaluate A I , Y : , an d Q. The softwa re needed to process the parameters ofpipe-type cables and transmission lines (in a fo rm compatible with the newmodel) will be written in the near future as part of the ongoingimplementation of the new cable model in the DCG/EPRI version of theEMT P (preliminary results seem to indicate that the EMT P support routinesCABLE CONSTANTS and LINE CONSTANTS do not produce thesmooth eigenvector functions required by the cable model). Therefore, itwill soon be possible to verify the applicability of the new model to otherforms of transmission systems such as pipe-type cables and overheadtransmission lines.2) Regarding the comparison with analytical results shown in section4.1, it was not the purpose of the simulation to calculate the exact response

of a square wave. As indicated in the paper, only a comparison with anexact analytical answer was intended. Thus the use of a Fourier seriesexpansion. The use of either Laplace or Fourier transforms would not haveproduced analytically exact answers, since numerical transformations arealways subjected to a certain amount of error, and the comparisons wouldnot have been rigorous from a mathematical standpoint.3) With regards to the comparison with the field test shown in section4.2, it is true that this test alone may not establish the accuracy of the model


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1109completely. The author intends to do mo re comparisons with field tests, asthey become available.By changing the time-to-half value of the input waveform in thecomparison with the field test, the peak value of the voltages at the sendingend of the cable increased, thus giving a better agreement with theexperimental results. However, as indicated in section 4.2, this change inthe input waveform does not affect significantly the sheath voltages.Figure C .l below sh ows the sending-end voltage of core 2 when theequivalent circuit supplied by th e discussers is used instead (in absence o fadditional data it was assumed that the 0.5- pF capacitor was ch arged to 7.3kV). t can be seen from this plot that the response obtained with the newsource representation is actually worse than the responses calculated withthe 0 x 40 ps an d 0 x 45 ps waveforms.

Fig. C. 2.ground fault when Z , < Z , .Steady-state voltages on a lossless three-phase system during a line-to-

0 20 40 60 80 100 120 140 160 180 200T i me ( mi c r ose c onds )Fig. C. 1. Comparison of Core 2 voltages for the field test simulation shown insection 4.2. a) 0 x 40 ps input function. b) 0 x 45 ps input function. c) Newsource representation.

4) In the type of cables studied in this paper, it has been observed thatwhen the currents ar e relatively sm all, the use of a constant transformationmatrix evaluated at frequencies above 1 lcHz produce s very goo d results. Assoon as the curren ts are significant (i.e. , after the o ccurrence of the line-to-ground fault) the response with Q constant deteriorates considerably, asshown in Figs. 15 to 17. Ha d Q been evaluated at 60 Hz, then the open-circuit response of the cable would hav e been poorer than the on e shown inthe paper because the magnitude of the imaginary part of Q at 60 Hz is notnegligible, and the frequency-dependent line models currently used in theEMT P require that Q be real.The comments made by Mr. Einarsson will be addressed next.1) The use of the modified Jacobi method to solve the symmetriceigenproblem Z x = AY- x is indeed an important factor in the calculationof smooth eigenvector functions of frequency. However, it should bepointed out that pre- and post-multiplication by the transformation matrixobtained in the previous frequency step (or seeding) is just as im portant. Infact, the use of the modified Jacobi method without seeding does notproduce smooth eigenvector functions. Similarly, the use of seeding incombination with algorithms such as the Q R decomposition, also fail toproduce smooth eigenvector functions.2 ) In the comp arison with the analytical results shown in section 4.1, theinput voltage used in the EM TP simulation is the Fourier series approxima -tion of a square wave. T herefore, the oversho ot due to Gibbs phenomenonis included in the EMT P results. The discrepancies between the analyticaland numerical results shown in Fig. 9 seem to be consistent with the error sin the approximations by rational functions of Y i ,A , and Q.3) The results obtained in the simulation of the single-phase line-to-ground fault may seem unusual, but they are indeed correct. The apparentcontradiction pointed out by Mr. Einarsson can be explained using phasoranalysis, assuming that the three-phase cable system is balanced andlossless. If the system is balanced the voltages induced in the unfaultedphases are directly proportional to the ratio Z , , , / Z , , where

z, = ( Z ,- 1 ) / 3z,= ( Z ,+ 2 2 , /3 .

In the simulation shown in section 4.3, the voltage in core 2 ( V b in thephasor diagram shown in Fig. C.2) will be given by

v b = v s b - ( z m / z s ) vs awhere V,,, V,,, and V , are the source voltages.As illustrated in the phasor diagram shown below, if Z m / Z s s negative,then Vb will be roughly in phase with the fault current Z,. The seriesimpedance matrix Z per unit length (evaluated at 60 Hz) for theunderground cable under consideration is shown below. The sheaths havebeen eliminated because in this example they are grounded. 10.1600+j0.1530 ) (0.0827 -j0.0198) (0.0581 -j0.0354)(0.0581 -j0.0354) (0.0827-jO.019 8) (0.160O+jO.1530)Z = (0.0827-jO.0198) (0.1430+jO.137 0) (0.0827 - j 0 . 0 1 9 8 ) ohmsikm[The steady-state voltages and currents calculated using the correct value ofZ also agree well with the results of the transient simulation shown insection 4.3.The comments made by Messrs. Semlyen and Hamadanizadeh will beaddressed next. The cable model presented in this paper should beapplicable to the case of multiple-circuit transmission lines, where themodal transformation matrix Q is known to depend on frequency. At thispoint in time, however, the software needed to generate Q as a smoothfunction of frequency has not yet been adapted to existing line constantsprograms.As indicated in section 5, the relative computational speed of the newcable model was measured using UBCs version of the EMTP. Theseresults are strongly influenced by the programming techniques used in theimplementation of the new cable model at UBC. The additional computa-tional burden of taking into account the frequency dependence of Q is notdirectly proportional to the number of recursive convolutions evaluated.Howev er, with programm ing considerations being equal, the differences inCPU time should be higher than those reflected in Table 1. This has beenverified by forcing the approximations by rational functions of Q to be oforder zero; that is, by assuming Q constant, but using the solution algorithmof the new cable model. The relative CPU time in this case is 3.6.These results suggest that reducing the ord er of the approximations of theelements of Q would result in considerable reductions in computationaltime, as Messrs. Semlyen and Hamadanizadeh suggest. However, thelargest savings that could be expected would be of the orde r of 4 0 percent inthe limiting case when Q is constant. It is not clear, however, if

computational savings of this order of magnitude would alw ays justify thepotential loss of accuracy.Mr. Shperling raises an interesting question: If the system data and themodels which represent other network components are only accurate to, fo rexample, 10 or 15 percent, is it justifiable to try to model an undergroundcable within a 2-percent accuracy range?.It is difficult to give a d efinite an swer to this question because accuracyrequirements depend on the type of simulation, and on the configuration ofthe system itself. There are studies where it is sufficient to model an


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1110underground cable as a pi-circuit or even as a simple shunt capacitance.When the main concern of a simulation is the transient response of the cableitself, it is probably best to model the cable with the highest accuracypossible, even if the basic data such as earth resistivity, dielectricpermittivity, and other factors are not known very accurately. Knowing thata model is nearly as accurate as the input data available remove s one sourceof uncertainty when the results of a transient simulation are interpreted.Recent studies on the validity of the constant transformation matrixassumption in the case of overhead transmission lines [CZ], [C3] seem toindicate that a constant transformation matrix Q evaluated at frequenciesbetween 500 Hz an d 5 lcHz produces good results when the currents arerelatively small. The same studies also indicate that significant loss ofaccuracy may occur in cases of strong asymmetry (e.g., when two circuitsof different voltage levels share the same tower).In the case of the type of underground cables studied at UBC, it appearsthat Q can also be assumed to be constant in situations where the currentsare very small. It may not be wise to generalize these observations beforefurther study of a larger variety of cable constructions and configurations ismade. It might be better to say that in absence of a better model, the best

answers can be obtained when Q is real, and eva luated at high frequenc ies,and when studies involve small currents.References

R. H . Galloway, W . B. Shorrocks, and L. M. W edepohl, Calcula-tion of Electrical Parameters for Short and Long Polyphase Trans-mission L ines, Proc . IEE, vol. 111, pp. 2051-2059, Decem ber1964.J. R. Marti, Validation of Transmission Line Models in theEMTP , Presented to the Power System Planning and OperatingSection of the Canadian Electric Association at Vancouver, BC,March 1987.J. R. Marti, H . W. Dommel , L. Marti, and V. Brandwajn,Approximate Transformation Matrices for Unbalanced Transmis-sion Lines, to be published in the Proc . Ninth P o w e r SystemsComputation Conference , Lisbon, August 30-September 4 , 1987.

Manuscript received April 10, 1987.

simulation of transients in unerground cables with frequency dependent modal

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