04111151-2

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No. I January 1982

ACCURATE MODELLING OF FREQUENCY-DEPENDENT TRANSMISSION LINES INELECTROMAGNETIC TRANSIENT SIMULATIONS

J.R. Marti, Member IEEE

University of British ColumbiaDepartment of Electrical Engineering

Vancouver, B.C. V6T 1W5

ABSTRACT references [2] to [8].

The parameters of transmission lines with groundreturn are highly dependent on the frequency. Accu-rate modelling of this frequency dependence over theentire frequency range of the signals is of essentialimportance for the correct simulation of electromag-netic transient conditions. Closed mathematical so-lutions of the frequency-dependent line equationsin the time domain are very difficult. Numericalapproximation techniques are thus required for prac-tical solutions. The oscillatory nature of the prob-lem, however, makes ordinary numerical techniques verysusceptible to instability and to accuracy errors.The,methods presented in this paper are aimed to over-comd these numerical difficulties.

I. INTRODUCTION

It has long been recognized that one of themost important aspects in the modelling of transmissionlines for electromagnetic transient studies is toaccount for the frequency dependence of the parame-ters and for the distributed nature of the losses.Models which assume constant parameters (e.g. at 60Hz) cannot adequately simulate the response of theline over the wide range of frequencies that are pre-sent in the signals during transient conditions. Inmost cases the constant-parameter representation pro-duces a magnification of the higher harmonics of thesignals and, as a consequence, a general distortionof the wave shapes and exaggerated magnitude peaks.

The magnification of the higher harmonics inconstant-parameter representations can readily be seenfrom figs. 13 and 14 (described in more detail inSection VIII). These figures show the frequency res-ponse of the zero sequence mode of a typical 100-mi,500 kV 3-phase transmission line under short-circuitand open-circuit conditions. Curves (I) correspond tothe "exact" response calculated analytically from fre-quency-dependent parameters obtained from Carson'sequations [1].. Curves (II) represent the responsewith constant, 60 Hz parameters.

Much effort has been devoted over the last tenyears to the development of frequency-dependent linemodels for digital computer transient simulations.Some of the most important contributions are listed in

In theory, many alternatives are possible forthe formulation of the solution to the exact lineequations. In practice, however, as it is illustra-ted in figs. 11 and 12, the nature of a transmissionline is such that its response as a function of fre-quency is highly oscillatory-. As a consequence, thenumerical problems that can be encountered in theprocess of solution are highly dependent- on the par-ticular approach.

The routines described in this paper avoid aseries of numerical difficulties encountered in pre-vious formulations. These routines are accurate,general, and have no stability problems. In the testsperformed, over a wide range of line lengths (5 to500 miles) for the zero and positive sequence modes,the same routines could accurately model the dif-ferent line lengths and modes over the entire fre-quency range, from 0 Hz (d.c. conditions) to, forinstance, 10b Hz. This is achieved without userintervention, that is, the user of these routines doesnot have to make value judgements to force a betterfit at certain frequencies, line lengths, or modes.In transient simulations, the frequency-dependent re-presentation of transmission lines required only 10-30% more computer time than the constant-parametersimulation.

II. TIME DOMAIN TRANSIENT SOLUTIONS

Even though the modelling of transmission linesis much easier when the solution is formulated in thefrequency domain, for the study of a complete systemwith switching operations, non-linear elements, andother phenomena, step by step time domain solutionsare much more flexible and general than frequency do-main formulations.

Probably the best known example of time domaintransient solutions is the Electromagnetic TransientsProgram (EMTP) first developed at Bonneville PowerAdministration (B.P.A.) from Dommel's basic work [9].The widespread use of this program has proven itsvalue and flexibility for the study of a large classof electromagnetic transient conditions.

The new frequency-dependent line model describedin this paper has been tested in the University ofBritish Columbia Version of the EMTP.

In the EMTP, multiphase lines are first decoupledthrough modal transformation matrices, so that eachmode can be studied separately as a single-phase cir-cuit. Frequency-independent transformation matricesare assumed in these decompositions. This procedureis exact in the case of balanced line configurationsand still very accurate for transposed lines. In themore general case of unbalanced, untransposed lines,however, the modal transformation matrices are fre-quency dependent. Nevertheless, as concluded by Mag-nusson [10] and Wasley [11], it seems that is stillpossible in this case to obtain a reasonably goodapproximation under the assumption of constant trans-

() 1981 IEEE

147

148

formation matrices.

Frequency-independent transformation matriceshave been assumed in the present work.

III. SIMPLIFIED LINE MODEL

In Dommel's basic work it is assumed that theline has constant parameters and no losses. Underthese simplifying assumptions the line equations arewritten directly in the time domain. (To account forthe losses Dommel splits the total line resistanceinto three lumped parts, located at the middle and atthe ends of the line). From d'Alembert's solution ofthe simplified wave equations and Bergeron's conceptof the constant relationship between voltage and cur-rent waves travelling along the line, Dommel arrivesat the equivalent circuit shown in fig. 1 for theline as seen from node k. An analogous model is ob-tained for node m. In this model RC is the linecharacteristic impedance and Ikh(t) is a currentsource whose value at time step t is evaluated fromthe known history values of the current and voltageat node m T units of time earlier (-r is the travel-ling time).

ik(t )k

Vk (t)

im(t) ik(t)- m k

Vm Vk (t)

(a) (b)Fig. 1: Dommel's simplified line model. (a): Line

mode. (b): Equivalent circuit at node k.IV. FREQUENCY-DEPENDENT LINE MODEL:

HISTORICAL REVIEW

When the frequency dependence of the parametersand the distributed nature of the losses are takeninto account, it becomes very difficult, if not im-possible in a practical way, to write the solutionof the line equations directly in the time domain.This solution, however, can easily be obtained in thefrequency domain, and is given by the well-known re-lations (e.g. Woodruff [12])

V (w) = cosh[y(w)94V (w) --Z (w)sinh[y(w)Z]I (w) (1)k m c mand

I (w) = sinh[y(w)P]V (w) - cosh[y(w)9]I (w),(2)k z (w) m mc

where

Z (w) = characteristic impedance, (3)

y(w) - = propagation constant, (4)

Z'(w) = R'(w) + jwL'(w), Y'(w) = G'(w) + jwC'(w),R' = series resistance, LI = series inductance,G' = shunt conductance, C' = shunt capacitance(primed quantities are in per unit length).

One of the first frequency-dependent line modelsfor time-domain transient solutions was proposed byBudner [2], who used the concept of weighting func-tions in an admittance line model. The weight-

ing functions in this model are, however, highlyoscillatory and difficult to evaluate with accuracy.

In an effort to improve Budner's weighting-func-tions method, Snelson [31 introduced a change of vari-ables to relate currents and voltages in the time do-main in a way which is analogous to Bergeron's inter-pretation of the simplified wave equations. The newvariables are defined as follows:

forward travelling functions:

k(t) Vk(t) + Rlik(t),f (t) = v (t) + Rlim(t),m m

and backward travelling functions:

bk (t) = Vk (t) - Rlik(t),

bm(t) = vm(t) - R1i (t),

(5)

(6)

(7)

(8)where R1 is a real constant defined as R =Qim Zc(w)

Equations 5 to 8 are then transformed into thefrequency domain and compared with the line solutionas given by eqns. 1 and 2. This idea was further de-veloped by Meyer and Dommel [4] and resulted in theweighting functions a (t) and a2(t) shown in fig. 2,and the equivalent line representation shown in fig.3 for node k. In this circuit the backward travelling

t

7~~-5Fig. 2: Weighting functions in Meyer and Domunel's

formulation.

function bk(t) is obtained from the. "weighted" pasthistory of the currents and voltages at both ends ofthe line and is given by the convolution integral

bk (t) = r {fm(t-u)a1(u) + fk(t-u)a2(u)}du (9)An analogous equivalent circuit and convolution inte-gral are obtained for node m.

ik(t)k

Vk+t RrI Rl

Fig. 3: Meyer and Dommel's frequency-dependent linemodel at node k.

Meyer and Dommel's formulation of the weightingfunction technique represented a considerable im-provement over other weighting function methods, andhas given reliable results in many cases of transientstudies performed at B.P.A. This technique, however,still presents some numerical disadvantages. One ofthese disadvantages is the relatively time consum-ing process required to evaluate integral 9 ateach time step of the solution. In the case study

4-

CJ4

149

presented in [4], the running time per step for thecase with frequency dependence was about three timeslonger than the time with no frequency dependence.Another disadvantage is the difficulty in evaluatingthe contribution of the tail portions of al(t) anda2(t) to the convolution integral of eqn. 9. The suc-cessive peaks in these functions tend to become flat-ter and wider for increasing values along the t-axis.

Some of the main problems encountered with thismethod have been accuracy problems at low frequen-cies, including the normal 60 Hz steady state. Theseproblems seem to be related to the evaluation of thetail portions of the weighting functions. Also, anerror analysis seemed to indicate that the functiona2(t) is more difficult to evaluate with sufficientaccuracy than the function al(t).

As suggested by Meyer and Dommel, the meaning ofthe weighting functions al(t) and a2(t) can be vis-ualized physically from the model shown in fig. 4.In this model the line is excited with a voltage im-pulse 6(t) and is terminated at both ends by the re-sistance RI of eqns. 5 to 8. Under these conditionsal(t) is directly related to the voltage at node m anda2(t) to the voltage at node k. From this model, itcan be seen that the successive peaks in these func-tions (fig. 2) are produced by successive reflectionsat both ends of the line.

ik(t ) im(t)k

ti~~~~~~~~~~~~~~~~~~

(t)

Vk (t ) _a2(t 1

m

r

o1(t)..Vm(t) fR1

a K1,t

a2(t) =0

tFig. 6: Weightingfunctions al(t) and a2(t) in the new

formulation.

VI. MATHEMATICAL DEVELOPMENT OF THE NEW MODEL

In order to replace R1 by Zeq for the generationof the new weighting functions, the forward and back-ward travelling functions (eqns. 5 to 8) can be de-fined in the frequency domain as

Fk(w) = Vk(w) + Zeq(w) k(w)F (w) = V (w) + z (w)I (w)m n eq m

(10)

(11)and

Bk(w) = Vk(W) - Zeq(L))Ik(w)B (w) = V (w) - Z (w)I (w)m m eq m

(12)

(13)

where Zeq(-w) = impedance of linear network approxi-mating Zc().

Comparing eqns. 10 to 13 with the general linesolution in the frequency domain (eqns. 1 and 2), itfollows that

Bk(w) = Al(w)FmF()

Fig. 4: Physical interpretation of Meyer and Dommellsweighting functions.

V. FREQUENCY-DEPENDENT LINE MODEL:NEW FORMULATION

The development of this model can be best ex-plained from the physical interpretation of the con-cept of the weighting functions developed by Meyerand Dommel.

From the system shown in fig. 4 it can be seenthat if the resistance Rl is replaced by an equiva-lent network whose frequency response is the same asthe characteristic impedance of the line Zc (L),there will be no reflections at either end of theline. If such an equivalent network can be found,the new al(t) weighting function will have onlythe first spike and the function a2(t) will becomezero. This is shown in fig. 5. The form of the newweighting functions is shown in fig. 6. With thisnew model the problem of the tail portions and ofthe accurate determination of a2(t) are thus elimina-ted.

ik(t)_

$(t )

Vk (t)

Zeq

im(t)m

,a,l(t)*-Vm t) Zq

Fig. 5: Physical interpretation of the function al(t)in the new formulation.

Bm (w) = A1 (w) k(w), (15)where

Y(w) 1

A1(w) = e = cosh[y(w)k] + sinh[y(w)9]j16)The time domain form of A1(w) is the function al(t)shown in fig. 6. The time domain form of eqns. 14and 15 is given by the convolution integrals

bk(t) = fT fm(t-u)a (u)du (17)and

(18)

The lower limit of these integrals is T because, asit can be seen from fig. 6, al(t.)=O for t

150

where ek(t) and em(t) are the voltages across thenetwork Zeq. After converting to a modal representa-tion, eqns. 21 and 22 give at each time step t theequivalent line models shown in fig. 7.

ik (t ) im(t)

Fig. 7: New frequency-dependent line models at nodesk and m.

Synthesis of the Characteristic Impedance

The network Ze representing the line character-istic impedance Zc

151

reference line for the zero and positive sequencemodes and for different lengths.

Table 1: Number of exponentials for the simulation ofal(t) for reference line.

VII. NUMERICAL TECHNIQUES

As indicated earlier, in order to simulate thecharacteristic impedance by an R-C equivalent networkand to allow recursive evaluations'of the past his-tory convolution integrals, the frequency domain func-tions Zc(w) and Al(w) are approximated by -rationalfunctions.

The problem of finding a rational function tosimulate the response of a network is studied in net-work synthesis theory. There are different numericaltechniques to approximate a tabular function of fre-quency by means of a rational fraction of polynomials(e.g. Karni [14]).

However, most of the traditional techniques (forexample, Butterworth's, Chebyshev's, Lagrange's) havemainly been applied to particular classes of pro-blems, such as ideal filter responses. Of more re-cent development are more general numerical tech-niques, such as least-square optimizations and opti-mum search (e.g. gradient) algorithms.

Despite their merit for rational approximationsof specific functions, programs using these routinesrequire a series of control parameters and adjustmentsthat depend on the particular function approximated.One of the main reasons for this is that the degree ofthe approximating polynomials is established before-hand and then the rational function is "forced" to fitthe given curve. Specification of polynomials oflarger or smaller degrees than actually required forthe given function often results in numerical insta-bility and accuracy problems. These problems are men-tioned by Semlyen [8], who applies a least squarestechnique to simulate the system response function.

In the modelling of frequency-dependent transmis-sion lines the form of the\functions to be approxima-ted depends on the particulAr line, its length, 'andthe particular mode. An approximating function "tai-lor-cut" for a specific case will not generally re-present the best solution for other cases.

The technique employed in this work avoids theabove-mentioned problems by allowing the approximatingfunction to "freely" adapt itself to the form of thefunction being approximated. This technique is basedon an adaptation of the simple concept of asymptoticfitting of the magnitude function, first introduced byBode [15]. During the process of approximation, thepoles and zeros of the rational approximating functionare successively allocated, as needed, while followingthe approximated function from zero frequency to thehighest frequency at which the magnitude of the ap-proximated function becomes practically zero or con-stant. The entire frequency range is'thus consideredand'a uniformly accurate -approximation is obtained.Since the poles and zeros are allocated when needed,the degree of the approximating polynomials is notpre-established, but' determined automatically by theroutine.

currence of ripples or local peaks in the approxima-ting function. This problem is avoided here by allow-ing only real poles and zeros.

Some Analytical Considerations

Phase Functions:

The rational functions (23) and (29) determinedby the method of asymptotic approximation have nozeros in the right-hand side of the complex plane.Under these conditions, it is shown in Fourier Trans-form Theory (e.g. Papoulis [16]) that the phase func-tion is uniquely determined from the magnitude func-tion and that the rational function belongs to theclass of minimum-phase-shift functions. The agreementbetween the phases of P(w) and Zc(w), and the phasesof the corresponding rational approximations obtainedin the present work shows the correctness of'the mini-mum-phase-shift approximations.

Causality Condition:

The rational approximations P in eqn. 29 andZeq(s) in eqn. 23 tend to a constant for s=jw whenw-+, and have no poles in the right-hand side of thecomplex plane. These conditions are enough (e.g.Popoulis [16]) to assure that the corresponding timedomain functions are causal (function=O for t

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FREQUENCY (HZ)

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O I Ifrtr T11T1IITT1IIrlx 11T1 1 11 1'111 11 111111 II[1 1 1 1111,T , rrr10 10-l 1 10 r10 10 10" 10 10

FREQUE14CY (HZ)

Fig. 9: Simulation of the characteristic impedance.Curves (I): Exact parameters. Curves (II):New model parameters.

2A1

m A12 (32)

It is interesting to note from this last equationthat the open circuit voltage is independent of thecharacteristic impedance. This explains why some fre-quency dependence models that neglect the frequencydependence of the characteristic impedance can giveacceptable results if they are only tested for open-circuit conditions. On the other hand, as can be seenfrom eqn. 31, the correct modelling of Zc is very im-portant for short-circuit conditions.

The results of these comparisons are shown infigs. 11 and 12 for the zero sequence mode and alength of 100 miles. These comparisons were also madefor other line lengths (from 5 to 500 miles), as wellas for the positive sequence mode, with'similarly goodagreements. The same agreement was also found for thecorresponding phase angles.

Figs. 13 and 14 show the comparison between theresponses obtained using exact parameters and thoseassuming constant 60 Hz parameters. The limitationsof the constant-parameter model for the simulation ofthe lower and higher frequencies is clearly illustra-ted in these figures.

The magnification of the higher harmonics by the

Fig. 10: Simulation of the weighting function. Curves(I): Exact parameters. Curves (II): New mo-del parameters.

constant-parameter model can also clearly be seen inthe transient simulations shown in figs. 15 and 16.These figures compare the simulations using the newline model and the constant-parameter model for twocases of open-circuited line energizations. In fig.15 the zero sequence mode of the 100-mi reference lineis energized with a sinusoidal, 60 Hz, voltage source,with the peak voltage applied at t=0. In fig. 16 theline mode is energized with a unit voltage step.

ii) Time Domain Tests:

The validity and accuracy of the new line modelin time domain simulations can also be assessed fromsingle frequency open and short circuit conditions.For this purpose, the line represented by its fre-quency-dependent transient model was energized by asingle frequency sinusoidal voltage source. Startingfrom the correct a.c. initial conditions (so that nodisturbances exist) transient simulations using theEMTP were run.' Under the indicated conditions, thetime domain solutions must be perfectly sinusoidalwaves with magnitude and phase as given by eqns. 31and 32. These tests were performed for the differentline lengths and modes and for frequencies along theentire frequency range. The results had the correctsinusoidal waveforms and were in complete agreementwith the magnitude and phase values previously ob-tained in the frequency tests.

152

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500

400

700

600

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300

200

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a 40000

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20000

7000

6000

5000

C3i( 40000Li' 3000

2000

1000

2 3 4 6 10 2 3t4 6 103 2 34 16 04 2 3 4 6 1i0FREQUENCY (HZ)

6 ]03 2 3 46 2 3 4 6 105UENCY (HZ)

Fig. 11: Short-circuit frequency response. (Sourcevoltage = 100 kV, Ik in amperes) Curves (I):Exact parameters. Curves (II): New modelparameters.

10

9.8

.7

a05- (I) and X})

XT- 4]

2

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Fig. 13: Short-circuit frequency response. (Sourcevoltage = 100 kV, Ik in amperes) Curves(I):Exact Parameters. Curves (II); Constant,60 Hz parameters.

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0I. 3... ... I I..... IF163 4 6 102 2 3 4 6 02 2 3 46 0FREQUENCY (HZ1

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FREQUENCY (HZ)

Fig. 12: Open-circuit frequency response. (Sourcevoltage = 1.0). Curve (I): Exact parametersCurve (II): New model parameters.

Fig. 14: Open-circuit frequency response. (Sourcevoltage = 1.0). Curve (I): Exact para-meters. Curve (II): constant, 60 Hz para-meters.

60000

50000

E 40000

sr. 30000Lur-i

20000

10000

0

800 -

600 -a:

C~

0 400 -

200 -

0I0

0. 010 0.020TIME (SEC)

--,-I0 .040

Fig. 15: Sinusoidal energization of open-circuitedline (peak voltage at t=O). Curve (I) Con-stant, 60 Hz parameters. Curve(II):New mod-el parameters.

--312.5 > TI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I(a)

312.5 kV

I~ --

0.010 0.020 0.030 0.040 0.050 0.060TIME (SEC)

(b)

312.5 kV

0.010 0.020 0.030 0.040 O.050 0.060TIME (SEC)

(c)

Fig. 16: Step function energization of open-circuit-ed;line. Curve (I): Constant, 60 Hz para-meters. Curve (I): New model parameters.

Comparison with Field Test:

The new line model was used to simulate the BPAfield test described in reference [4]. This testsimulates a single line to ground short circuit on anopen-ended 222 km, 500 kV, 3-phase transmission line.The short circuit was applied to phase-c. The fieldtest 'oscillograph for the voltage at phase-b at theend of the line is shown in fig. 17(a). To comparewith BPA's digital simulation in ref. [4],the same in-tegration step At=50 psec was used, and the zero se-quence mode of the line was represented by the newmodel d*escribed in this paper. The re'sult of thissimulation is shown in fig. 17(c). This result com-pares well with the field test and with BPA's simu-lation (fig. 17(b)). In BPA's simulation the averagetime per step, as compared' with'the solution withconstant parameters, was 3.13 times longer. In thesimulation with the new model this time was only 1.19times longer.

IX. CONCLUSIONS

A new, fast, and reliable approach has been de-

Fig. 17: Field test simulation. (a): BPA field testoscillograph. (b) BPA simulation. (c) Newmodel simulation.

veloped for the accurate modelling of transmissionlines over the entire frequency range. The routinesfor obtaining the parameters of the model do not pre-sent the numerical difficulties encountered with pre-vious formulations. These routines are easy to usebecause they do not require value judgements on thepart of the user. Further work is needed in connec-tion with the representation of unbalanced, untrans-posed lines with frequency-dependent modal transforma-tion matrices.

X. ACKNOWLEDGEMENTS

The author would like to express his gratitude toDr. H.W. Dommel, whose clear and practical thinkingare always the best encouragement; to the Universityof British Columbia Computer Centre for its convenientand easy to use facilities; to'Central University ofVenezuela for their financial support during theauthor's leave of absence at U.B.C.; and to the Bonne-ville Power Administration for their constant coopera-tion, andT for allowing the reproduction of the. fieldtest result used in this paper.

154

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155

XI. REFERENCES

[1] The University of British Columbia, "Line para-meters Program." Vancouver, B.C.

[2] A. Budner, "Introduction of Frequency-DependentLine Parameters into an Electromagnetic Tran-sients Program." IEEE Trans. Power Apparatus andSystems, vol. PAS-89, pp. 88-97, Jan. 1970.

[3] J.K. Snelson, "Propagation of Travelling Waves onTransmission Lines- -Frequency Dependent Para-

- meters." IEEE Trans. Power Apparatus and Systems,vol. PAS-91, pp. 85-91, Jan/Feb. 1972.

[4] W.S. Meyer and H.W. Dommel, "Numerical Modellingof Frequency-Dependent Transmission-Line Para-meters in an Electromagnetic Transients Program."IEEE Trans. Power Apparatus and Systems, vol.PAS-93, pp. 1401-1409, Sept/Oct. 1974.

[5] A. Semlyen and A. Dabuleanu, "Fast and AccurateSwitching Transient Calculations on TransmissionLines with Ground Return Using Recursive Convolu-tions." IEEE Trans. Power Apparatus and Systems,vol. PAS-94, pp. 561-571, March/April 1975.(6] A. Ametani, "A Highly Efficient Method for Cal-culating Transmission Line Transients." IEEETrans. Power Apparatus and Systems, Vol. PAS-95,pp. 1545-1551, Sept/Oct. 1976.

[7] A. Semlyen and R.A. Roth, "Calculation of Expo-nential Step Responses - Accurately for threeBase Frequencies." IEEE Trans. Power Apparatusand Systems, vol. PAS-96, pp. 667-672, March/April 1977.

[8] A. Semlyen, "Contributions to the Theory of Cal-culation of Electromagnetic Transients on Trans-mission Lines with Frequency Dependent Para-meters." IEEE PES Summer Meeting, Vancouver, B.C.July 1979.

[9] H.W. Dommel, "Digital Computer Solution of Elec-tromagnetic Transients in Single-and MultiphaseNetworks." IEEE Trans. Power Apparatus and Sys-tems, vol. PAS-88, pp. 388-399, April 1969.

[10] P.C. Magnusson, "Travelling Waves on Multi-con-ductor Open-Wire Lines-A Numerical Survey of theEffects of Frequency Dependence of Modal Composi-tion." IEEE Trans. Power Apparatus and Systems,vol. PAS-92, pp. 999-1008, May/June 1973.

[11] R.G. Wasley and S. Selvavinayagamoorthy, "Approx-imate Frequency-Response Values for Transmission-Line Transient Analysis," Proc. IEE, vol. 121,no. 4, pp. 281-286, April 1974.

[12] L.F. Woodruff, "Principles of Electric PowerTransmission." 2nd Edition. New York:Wiley, 1938,pp. 105-106.

[13] E. Groschupf, "Simulation transienter Vorgangeauf Leitungssystemen der Hochspannungs-Gleich-strom-und-Drehstrom-Ubertragung", Dr. -Ing, gene-hmigte Dissertation, Feb. 23, 1976.

[14] S. Karni, "Network Theory: Analysis and Syn-thesis." Boston: Allyn and Bacon, 1966, pp. 343-390.

[15] H.W. Bode, "Network Analysis and Feedback Ampli-fier Design." New York: Van Nostrand, 1945.

[16] A. Papoulis, "The Fourier Integral and its Appli-cations." New York: McGraw-Hill, pp. 204-217,1962.

Jose R. Marti (M'71) was bornin Spain on June 15, 1948. Hereceived the degree of Electri-cal Engineer from Central Uni-versity of Venezuela, Caracas,Venezuela, in 1971, and the de-gree of M.E. in Electric PowerEngineering from RensselaerPolytechnic Institute, N.Y., in1974. He is presently a Ph.D.candidate at the University of.British Columbia, Canada.

From 1970 to 1971 he worked for Exxon in Venezue-la in coordination of protective relays. From 1971 to1972 he worked for a Consulting Engineering firm inCaracas, Venezuela, in relaying and substation designprojects. In 1974 he joined the Central University ofVenezuela as a professor in Power System Analysis. Heis presently at U.B.C. on a leave of absence from Cen-tral University of Venezuela.

156

DiscussionAdam Semlyen (University of Toronto, Ontario, Canada): This is atimely and interesting paper. It shows the path the author has taken toperfect the weighting function method4 so that the impulse responses ofFig. 2 are replaced by something more manageable. The end result isidentical to the approach which uses a wave propagation transfer func-tion and a frequency dependent characteristic impedance."8A I wouldlike to make the following remarks related to this fact.a. Fundamentally, there exists a single set of transfer functions for

transmission line transient analysis which yield smooth step or im-pulse responses. There are e-Y(Ac)I and Z(c(w). This fact is related tothe decomposition of voltages and currents into travelling wavesand has led us to use, since 1970, these transfer functions for simula-tion of transients on transmission lines with frequency dependentparameters.B It is therefore not surprising that the author is now us-ing this same approach as the result of perfecting a differentmethod.

b. There are several sets of transfer functions which one can use intransient analysis on a transmission line. They represent relationsbetween the two terminal voltages and currents, and can be express-ed by 2x2 transfer function matrices with only two independentelements. Examples are the two-port transfer functions of eqns. (1)and (2), driving port and transfer admittances, and the transferfunctions related to the weighting function a,(t) and a2(t) of Fig. 2.All these contain hyperbolic functions of y(co)i and, consequently,have complex poles in the s-plane. Therefore, an oscillatingbehaviour is unavoidable unless the hyperbolic functions are com-bined in an exponential function.

c. The forward and backward travelling functions of eqns. (5) to (8)are not very different from the travelling wave components VI andVI' (VI + V" = V). They are related by the factor 2; if R, is replac-ed by Z, as in eqns. (10) to (13):

F = twice the voltage V' of the Qutgoing waveB = twice the incident wave voltage V"

it is then clear that replacing R1 by Z, actually replaces the methodwhich uses the weighting functions a(t) anda2(t) of Fig. 2 by the ap-proach which uses the propagation transfer function.

d. Often, in previous calculations, the characteristic impedance hasbeen considered constant. This appears to be justified for positivesequence, according to Fig. 9. When the approximation is accep-table the weighting function approach and the travelling wavetransfer function approach become identical, except for numericalprocedures.A useful contribution of this paper is the RC realization of the

characteristic impedance. It permits easy implementation in the EMTPand reflects the fact that the rational approximation (23) satisfies allessential physical requirements.

It is interesting-to note that the d.c. value ofZc, as shown in Fig. 9, isnot very large (r- 620Q). It indicates a relatively large shunt conduc-tance G adopted in the calculation. Could the author please commenton the way it has been selected; whether it is related to attenuation oftrapped charges or to losses?If G # 0 is the assumption in the calcula-tion of Z,, has it been considered in the calculation of the propagationtransfer function (16) as well? Does its magnitude affect significantlythe calculated overvoltages?

in previous calculations5 we have considered G = 0. This has had theeffect that a step voltage arrived at the other end distorted but in fullmagnitude (at t = 00). The computational effect was that if a line hasbeen disconnected at both ends in some cases the trapped voltage tend-ed to take off. This indicates some instability resulting, apparently,from the basic difference equation (26). A numerical analysis of thestability of (26) in a closed loop condition indicates indeed the possibili-ty of a slow numerical instability if G = 0. The remedy consists in adop-ting non-zero conductance. The author's comment on this topic wouldbe appreciated. One should of course mention that this type of unstablebehaviour is related to the nature of information transfer from one endof the line to the other, as shown in the recursion formula (26). Thisshows that new values of s are related to values of f, and therefore of sitself, many time steps back; and only two past values are used tocalculate the new value. A method using full, i.e. non-recursive, or par-tial convolutions is therefore expected to have better stabilitycharacteristics.

For the fitting of the propagation transfer function the author uses a"backwinding" of A, by eJWT. IS T based on light velocity? We used toadd a AT to take into account the toe portion of the propagation stepresponse curves. The result was that our original time domain fitting-'was good even with only two exponentials. Later, for frequency domain

fitting and very accurate (and smooth) results we went up to six (realand/or complex) exponentials. This is less than half of the numbersshown in the present paper.The "toe" of the propagation step response is not just an empirical

fact but it is a theoretically expected extension AT of r, intrinsicallyrelated to the concept of penetration depth, and, as discussed, impor-tant for efficient rational approximation. This flat portion of the stepresponse is due to the fact that f(t) and all its derivatives of finite orderare zero at t = 0. In the frequency domain the transfer function P(s)and also slP(s) for n = 1, 2, ...N must all be zero for co = oo ins = jco.P(s) is given in equation (28) of the paper:

P(s) -- e-Y(s)1 x eST = e-(Y(s)i-sTThe expression in the exponent is:

G(s) = y(s)I - STwhere

y(s)= \fZ(s)Y(s)Z(s) = Z,(s) + sL(s)

(a)

(b)

(c)Y(s) = sC

In (c), Z, pertains to the conductor and L(s) is the complex inductanceof the earth return.C We express it for a single conductor:

L(s) = Po n 2(h +P')L~~s, 2TV r (d)1

where p is the complex penetration depth:

At high frequencies p - 0, and equation (d) becomes:

L(s) P (in h+ -h)Then Z,(s) of (c) has the simple expression:

Z (s) 1c cctcpc

where 1PC =

VSIOI

and I, is the perimeter of the conductor along the surface of the outsidewires.Consequently G(s) becomes:

G(s) = 2( o p + s 2 (rn r + ph) ) -2h - St

s1.i0C 2pT -- 1)c r

= SE( + 2h +

(-+ ) =

.tn 72h i; Qc i

This expression shows that the value of G(joo)= (joc) (joo) of equation(b) is in fact itself infinite. Then, of course, P(s) of (a) and all productss"P(s) become zero for s =jco,co = oo.

Finally, sincef(O),f'(0), f"(0), ...fN(0) are all zero, a Taylor series ex-pansion of f(t) around t = 0 yields f(t) 0. This will be valid for a smallvalue of t(

REFERENCES

[Al A. Morched and A. Semlyen, "Transmission Line Step ResponseCalculation by Least Square Frequency Domain Fitting", IEEEPaper No. A 76 394-7 presented at the 1976 Summer PowerMeeting in Portland, Oregon.

[B] A. Semlyen, "Accurate Calculation of Switching Transients inPower Systems", IEEE Paper No. 71 CP 87-PWR, presented atthe 1971 Winter Power Meeting in New York, N.Y.

[C] A. Deri, G. Tevan, A. Semlyen and-A. Castanheira, "The Com-plex Ground Return Plane, a Simplified Model for Homogeneousand Multi-Layer Earth Return", IEEE Paper No. 81 WM 222-9presented at the 1981 Winter Power Meeting in Atlanta, Georgia.

Manuscript received June 3, 1981.

J. R. Marti: The author would like to express his appreciation for thedetailed review contributed by Prof. Semlyen and for the opportunityto comment on some important aspects of the formulation presented inthe paper.As Prof. Semlyen notes in the first part of his Discussion, the for-

mulation of the transient problem in transmission lines in terms of the"natural" system functions, Z,(co) and e-y(c'4E, greatly simplifies thenumerical manipulations involved in the solution of the problem. Themain reason why our approach was developed from the concept ofweighting functions resulting from Bergeron's type of relationships(eqns. 10 to 13) was because these relationships allowed us to more easi-ly visualize the physical significance (fig. 5) of the functions involved inthe formulation of the problem, and, as a result, to arrive at very im-portant conclusions regarding the nature of their mathematical syn-thesis (e.g. R-C realization of Z,(o)).As the Discussor notes, the use of a constant value (simple resistance)

instead of a more complete model (e.g. R-C network) to simulate theline characteristic impedance would appear to be justified for the aerialmode (fig. 9, botton), though only if the frequencies of interest arehigher than about 20 Hz for the line studied in the paper. However, inmost cases of transient studies (asymmetric conditions in the system),the accurate modelling of the ground return mode is the most criticalone. The strong frequency dependence of the characteristic impedancefor the ground mode is clearly seen in fig. 9 (top).The need for a finite value of the shunt conductance G (G 0) arises

from the fundamental mathematical description of the travelling-wavephenomena. In its most basic form, the current travelling wave isrelated to the voltage travelling wave through the characteristic im-pedance Z,; for instance, for a forward-travelling wave

I+ (x,w) V+(x,)z (w)C

and similarly, for a backward-travelling wave. If these relations are tobe valid for any frequency, they should also be valid for Cw = 0 (dc condi-tions). But, for w=0, z R(dc) If G(dc) is taken to

/G(dc)

157be zero, then Z,(dc) = oo, that is, the basic relationship between the cur-rent and voltage waves will present a singularity at co = 0. It can then beunderstood that a numerical solution based on the fundamental travell-ing wave equations can give extraneous results when trying to simulatedc conditions (e.g. trapped charge) if G is assumed to be zero. With theformulation presented in the paper, which considers a finite value of G,no problems have been encountered in the simulation of trapped chargeor other dc conditions (e.g. exponentially decaying dc components inasymmetric short-circuit currents and final dc levels in open-circuit orshort-circuit step responses).

For most cases of transient studies, as long as it is chosen within areasonable order of magnitude, the actual finite value used for G is notvery critical and its effect upon the simulation is practically negligible.(A possible exception would be specific studies of the coronaphenomenon, where a more detailed representation of the non-linearcorona characteristics would be necessary.) The value of G used in thesimulations presented in the paper was 0.3 x 10-'mho/km, whichrepresents a "rule of thumb" average value for shunt losses (leakagethrough the insulation plus corona losses) in high voltage overheadlines. Since G is considered as one of the line parameters, the same valueis used for both the evaluation of the characteristic impedance and theevaluation of the propagation function.

In the formulation presented in the paper, the phase displacementfactor T in eqn. 28 is not evaluated from the travelling time at the speedof light, but it is directly obtained in the frequency domain by compar-ing the phase angle of the rational function Pa(s) in eqn. 29 (which is aminimum-phase-shift function) with the phase angle of the propagationfunction Aj(co). It can be seen from eqn, 28 that P In

* - ~~~~~T=

this way, the effect of the particular line configuration, propagationmode, and what Prof. Semlyen refers to as the "toe" of the propaga-tion function are automatically taken into account, without the needfor additional calculations (since all the information is intrinsically con-tained in A,(co)). (For 100 miles of the line studied in the paper, thevalue of T was 0.598 ms for the zero sequence mode and 0.539 ms forthe positive sequence mode.) Also, in connection with this point, wewould like to emphazise that the method of asymptotic tracing used toobtain the rational function Pa(s) in eqn. 29 only requires the magnitudeftinction of Aj(co) (IA,(c)l = IP,(w)1). Due to the analytical propertiesof P(s), the phase angle of this function is "fixed" by its magnitude.Therefore, the order of the approximation is not determined by thephase angle or by T, but only by the shape of the magnitude of A,(co)and by the accuracy with which the approximation is desired.

Finally, we would like, in closing these comments, to reiterate our ap-preciation of Prof. Semlyen's extensive discussion on a subject in whichhe has been one of the main pioneers.

Manuscript received July 24, 1981.