Propagation of Travelling Waves on Transmission Lines

PROPAGATION OF TRAVELLING WAVES ON TRANSMISSION LINES

- FREQUENCY DEPENDENT PARAMETERS

J.K. Snelson

The Hydro-Electric Power Commission of Ontario

Toronto, Canada

Abstract-This paper describes an extension of Bergeron's Method ofCharacteristics that is applicable to the analysis of transmission lineswith frequency dependent parameters. This method can be inserted inexisting general purpose programs that use Bergeron's Method. Theapplication to single phase and multiphase transposed lines is discussedwith examples and the theory is developed for multiphase untransposedlines.

INTRODUCTION

General purpose computer programs for the calculation of tran-sients in electromagnetic systems have been written in recent years.Many of these use Bergeron's Method of Characteristics for the handlingof continuously distributed transmission lines. The program by Dommel1is an example which can handle large systems with a wide variety ofcomponents, including non-linear elements. In these programs thehandling of lossless transmission lines is easy but it is difficult to dealwith lines with frequency dependent resistance and inductance.

Hedman2 gives curves of resistance and inductance against fre-quency for typical transmission line configurations which show that theresistance can increase by a factor of 103 as the frequency varies from60 Hz to 1 MHz. In addition Battisson et a13 using a Fourier Transformtechnique showed that the frequency dependence of transmission lineparameters has significant effects on both the wave shape and the peakvoltages of some switching surges. It therefore seemed desirable todevelop a method of including frequency dependence in the generalpurpose programs based on Bergeron's Method.

Budner4 described a method for doing this which has the dis-advantage of considerably increasing the computation and storage re-quirements. The method described here can easily be included in pro-grams based on Bergeron's Method and should require much less com-putation time and storage than the method of reference 4.

BERGERON'S METHOD

A brief outline of Bergeron's Method is given here and ideas areintroduced that will be used later in the paper. The theory of Bergeron'sMethod is developed in more detail by Dommel1.

The transmission line equations are:

- -e = L A + RiOx &t

_0g = C --- + Geax Ot (1)

where e and i are voltage and current in the line at a distance x andL,C,R and G are respectively the line series inductance, shunt capac-itance, series resistance and shunt conductance per unit length.

Paper 71C 26-PWR-IV-B, recommended and approved by the Power SystemEngineering Committee of the IEEE Power Society for presentation at the 1971PICA Conference, Boston, Mass., May 24-26, 1971. Manuscript submitted January11, 1971; made available for printing June 9, 1971.

Bergeron's Method applies to lossless lines where R and G are zeroand L and C are independent of frequency.

Subject to these limitations there are relationships between theconditions at each end of the line at time t and at time t - T which existindependent of the terminating networks.

o K I N

eKf SU RG E I MPEDANCE-

Z eMO _ TRAVEL TIM E

TERMINAL K TERMINAL M

Fig. 1. Single Phase Lossless Line

Forethe line of Figure 1 these relationships are:

ek(t) - Z@ik(t) = em(t-r) + Z*im(t=T)

em(t) - Z.im(t) = ek(t-r) + Z.ik(t-7*) (2)

Equation 2 gives relationships between e and i at both ends of thetransmission line which, provided the conditions a travel time earlier areknown, enable the transmission line to be replaced by a current sourcein parallel with a resistance Z. This allows a solution to be obtained forthe voltages and currents at time t in the network consisting of the endsof the transmission lines and the components connected to them. Amethod for solving this network is described by Dommell.

The quantities e±Z.i are known as characteristics and are directlyrelated to the forward and backward travelling waves. In this paper thefollowing nomenclature is used:

Fk = ek + Z ik = 2 x forward travelling wave at end k

Bk = ek - Z ik = 2 x backward travelling wave at end k (3)

Equation 2 can be rewritten:

Bk(t) = F (t - r)

B(t) = Fk (t -r)(4)

i K R/4 R/2 R/4 ;M

eKf 0o4 j-4jZ

M----oOM

Fig. 2. Approximate Model For Line With Series Resistance

DommelI extends the basic equation (4) to include an approxima-tion for series losses. The model used for the transmission lines is shownin Figure 2. This results in equations:

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B (t)z FM(t- 7) + -

Bm)=7 F (t -r) + R4 Fm(t -r)Bmt Z+R/4 k Z+R,'4 (5)

In these equations the impedance used in defining the character-istics of equation (3) is modified to (Z + R/4). Equation (4) is a particu-lar case of equation (5) where R is zero. Physically equations (4) and(5) can be interpreted in terms of impulse responses in the followingway.

For the lossless case described by equation (4), if an impulse offorward (ie, into the line) travelling wave is injected at one end at timet = 0, then the backward (ie out of the line) travelling wave at the otherend at time r is equal to it.

For the transmission line model with series losses of Figure 2 andequation (5), if an impulse of forward travelling wave is injected at endm at time t = 0, then part of it is transmitted and contributes to Bk(r)and part of it is reflected by the resistor at the centre and contributesto Bm(r). The responses of Bk and Bm are shown in Figure 3.

tBK

tBm

0 T _ TIME(A) FORWARD RESPONSE

70_r-- TIM E(B) BACKWARD RESPONSE

Fig. 3. Response Functions For Lumped Resistance Model

These impulse responses are approximations to the impulse re-

sponses of a line with continuously distributed resistance, and fre-quency dependent resistance and inductance. If the representation ofcontinuously distributed resistance was improved by including a largenumber of smaller lumped resistances connected by short lossless trans-mission lines, then the reflection from the resistance in the centre of theline shown in Figure 3b would be replaced by many smaller reflections.These reflected pulses would not arrive only at time r but would occur

over a range of time. The reflections from resistances near the sendingend would start arriving soon after time zero. The reflections fromresistors near the far end would arrive shortly before time 2r and wouldbe smaller as the pulse will be attenuated as it must pass down the linetwice. If the effects of frequency dependence were to be included thenthe travel time and attenuation of different frequency componentswould be different and this will further modify the shape of the re-

sponses. The principle effect would be, that the pulse of Figure 3awould be broadened.

These effects result (as shown in the next section) in responses ofthe form shown in Figure 4. With responses of this form Bk(t) andBm(t) are no longer determined by Fk(t - r), and Fm(t - r) alone. How-ever, if Fk and Fm are considered to be made up of a series of impulsesof varying amplitude but all of a duration At then Bk and Bm can befound by summing the effects at time t of a number of these impulses.The shape of the responses would only need to be determined once foreach line and so could be a complex calculation without greatly in-creasing the computation time. The summing of the effects of a number

0.1

WI

0

fI (B) BACKWARDRESPONSE

0.01

0 I2 3

TIME IN MILLISECONDSA= 50 ,lSEC

Fig. 4. Response Functions of Frequency Dependent 200-Mile Line

of impulses must be carried out many times, once at each time step, butis only a simple calculation and so should not greatly increase the com-putation time.

DERIVATION OF IMPULSE RESPONSE FUNCTIONS

This section derives from the transmission line equations expres-sions for the impulse response functions discussed in the previoussection. As frequency dependence is included the problem is trans-formed into the frequency domain using Fourier Transforms. For thepurposes of this paper the Fourier Transform of time function X(t)will be denoted by X (co).

The Fourier Transform of equation 1 is:

-(re = jwLi., + -i.

dx = JwOe + i (6)

The complex surge impedance and propagation constant are de-fined by:

Z = s(R + jwL)/(jcC + G) (7)

(8)'Y = 4(R+ jwL) (:JwC+ G)

The solution to equation 6 is known to be:

e =em osh ( Yd) - %Z sit (Yd)

ik =- cosh (Yd) +i;/Zsinh(Yd) (9)

Before a transformation can be made from currents and voltagesas variables to forward and backward characteristics it is necessary todetermine the value of ZI to use in the definition of characteristics(equation 3). ZI must be the impedance the line represents to the out-side network at the instant an impulse is applied. This response will begoverned by the high frequency effects. Therefore we define:

(10)ZI = is (Z)W- 0o

86

---Io- I Mf m


This limit exists for transmission lines because as c tends toinfinity, Rav/Jand L tends to a constant Loo

Therefore = 29limk:j + 0w L. =(11)

Z1 is a real number and so represents a pure resistance. Using Z1 asthe surge impedance we define the transform of the characteristics as:

Fk = ek + Z1 k

(12)Bk% - Zl Tk

The time step of u is chosen to be At (the time step of the maincalculation) and so equation (15) becomes:

B/t) = E W1(n At) (t - n At)

nnxln=n

+ > W2(nAt) Fk(t - not)

n1=1

and similar expressions for Fm and BmiThe variables ek, ik, em Tm can be eliminated from equations 9

and 12 to give:

Bk A1 F A2 Fki m 2 k

B = A1 Fk+ 2F (13)

where 1 =c

A c - Y/2(Z1/Z - z/z1z)eihi(Yd)A~~~~~ - -/ sin- - d)-

2 cosh Yd) + 1J2 Z1/Z + Z/z1 sinh YVd

(14)

Equations 13 are in the frequency domain. To transform into thetime domain it is necessary to remember that multiplication of functionsin the frequency domain corresponds to convolution of the transformedfunctions in the time domain.

Transformed into the time domain equations 13 become:

Bet) = fAi(U) Fm(t - u) du + fA2(u) Fk(t -)k) 2 t

(15)co ,co

B3(t) (u) Fk(t - u) du + f A2(u) Fm(t - u) du

The significance of A1 and A2 can be seen by examining equation(13). If Fk is zero ie, ek + ik ZI = 0 and Fm is a Dirac impulse functionthen the Fourier transforms of Fk and Fm are 0 and 1 respectively.Then from equation (13) Bk = A1 and B. = A2.

This gives in the time domain Bk = A1 and Bm = A2Therefore Al is the backward characteristic at the remote end of

the line and A2 is the backward characteristic at the sending end due toan impulse of forward characteristic.

Certain properties of the response functions A1 and A2 can bededuced from this. For real systems no response can occur before theexcitation is applied; therefore, A1 and A2 are zero for all negativetime. There can be no response in a real system at the remote end at atime less than the length of the line divided by the speed of light.Therefore A1 is zero up to this time. The condition that ek + ikZ1 = 0implies that the line is terminated in an impedance ZI, the surge im-pedance. Therefore, the injected impulse is not reflected from the endand the response at the remote end consists of a single pulse. These con-siderations enable the infinite integrals of equation (15) to be replacedby integrals over the short time that A1 and A2 are sensibly non zero. Inaddition for computation purposes it is convenient to approximate theintegrals by summations.

n92 =

B (t) = EW1 (n At) Fk( t

n-n3+ > W2(nAt) Fm(t - nAt)

n=1

(16)

where W1 = A1IAt and W2 = A2At. The choice of limits, n1, n2 and n3is discussed in Appendix I.

Equation (16) can replace equation (4) or (5) of a program basedon Bergeron'sMethod and the rest of the program can remain the same.In developing these expressions no assumptions have been made as tothe nature of the terminating networks and so this change need notrestrict the generality of the program.

This equation is the summation of the effects of impulses origi-nating over a period of time that was suggested in the previous section.Provided the line parameters can be found at any frequency W1 and W2can be calculated using equation (14) and a numerical inverse Fouriertransform technique. Figure 4 shows the response functions of a typical200-mile line calculated using these expressions. These show the generalfeatures predicted by the general argument of the previous section.

APPLICATION TO SINGLE PHASE LINES

The equations as developed in the previous section can be applieddirectly to single phase lines. There are two main difficulties, thedetermination of the line constants at a wide range of frequencies andfinding the numerical inverse Fourier Transforms of the weightingfunctions.

The line constants may be calculated using the formulae developedby Carson5, the graphs published by Hedman2 or they may be measured.The method does not require the variation with frequency to follow anyparticular analytical form. However, care must be taken to ensure thatthe constants are physically realizable or else the response functionsmay show physically impossible features.

The results quoted in this paper were all obtained using thetechnique of Modified Fourier Transforms6'7'8 to obtain the inverseFourier Transforms of the weighing functions. A brief discussion of thistechnique as used in this paper is given in Appendix I.

Figure 5 shows the response of a 200-mile open ended line withzero shunt conductance and frequency dependent series resistance andinductance to a step function. The line constants used were calculatedwith a program based on Carson's method for the configuration of a230-kv line when excited in the ground mode. This shows an initialresponse approximating to a square wave which rapidly changes tosomething approximating a damped sine wave.

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aZS

S(2

1--I0

Fig. 5. Step Response of 200-Mile Line

16 mH

200 M ILE

_ _O TRANSMISSION- O B

LINE C

(A) SYSTEM REPRESENTED

PHASE AZ PHASE B AND C---

W (C) FREQUENCY INDEPENDENTEL PARAM ETERS

* *(60 HZ VALUE_S)

0 A

0 1 2 3 4 5 6 7 a 9 10 I 12TIME IN MILLISECONDS

Fig. 6. Single Phase Energisation of 200 Mile Line

APPLICATION TO CONTINUOUSLY TRANSPOSEDMULTIPHASE LINES

An approximation which greatly simplifies the study of multi-phase lines is to assume the line to be continuously transposed. In thiscase the series impedance and shunt conductance matrices both haveequal diagonal elements and equal off diagonal elements. Dommellgives a simple transformation matrix which is independent of the actual

line parameters. This matrix diagonalizes such matrices and separates theproblem into a number of modes. These consist of a ground mode and(m-i) line modes (where m is the number of phases). The line modes allhave the same parameters which are relatively independent of frequency.The ground mode involves currents returning through the earth and itsparameters are highly frequency dependent. With this transformationeach mode can be solved as a separate single phase problem using theequations developed for the single phase case.

As it is principally the ground mode that is frequency dependentit is a reasonable compromise between accuracy and computation timeto consider the ground mode frequency dependent using equation ( 16)and to neglect the frequency dependence of the line modes usingequations (4) or (5).

The program developed by Dommel1 has been extended to includecontinuously transposed lines where the ground mode or both theground mode and the line modes are frequency dependent. The mod-ification has been done without restricting the generality of theoriginal program. This modified program has been used for a number ofstudies. As an example of the use of the program Figure 6 shows thesingle phase energization of a 200-mile, 230-kv line from a source of 16mH inductance both with and without frequency dependent groundmode parameters.

APPLICATION TO UNTRANSPOSED MULTIPHASE LINES

The theory developed'so far cannot be directly applied to untrans-posed multiphase lines as modal transformations will be different ateach frequency and so the modal transformations must be included inthe frequency dependent part of the calculation. However, the ideasused in the single phase case can be applied to the multiphase untrans-posed case. Appendix II develops, using matrix algebra, the equivalentexpressions for this case. This results in equations of the same form asequation (16) except that the quantities must be interpreted as matrixquantities. However, the amount of computation required to calculatethe matrix response functions will be many times larger than for singlephase or transposed lines.

There is no theoretical reason why this theory should not beimplemented but so far it has not been considered that the gain inaccuracy of representation justifies the programming effort involved.

COMPARISON BETWEEN NEW METHOD ANDBUDNER'S METHOD

Budner's Method4 is mathematically similar to the method pre-sented in this paper. It differs, however, in that equation (9) in the fre-quency domain is manipulated into the following form:

k= Y + Y2 em

im = Y1 em + Y2 ek (17)

The admittance functions Y I and Y2 are then transformed into thetime domain to give a relationship expressing the currents in terms ofconvolution integrals of the admittance functions and the voltages. Thephysical interpretation of the admittance functions Y1 and Y2 is thatthey represent the transform of the currents at the sending and receivingends when an impulse of voltage is applied with both ends subsequentlyshort circuited. When transfprmed into the time domain these functionshave many peaks as the pulse is reflected up and down the line by theshort-circuited ends. Eventually the pulse dies away as it is attenuatedby the losses in the line. Therefore, the range of time during whichthese admittance functions are non zero is many travel times of the line.This means that the admittance functions in the time domain can onlybe adequately described by a large number of points. Budner uses 2048points in his paper for his numerical example. This number could

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possibly be reduced a little if the frequency dependence of the line modewere ignored. However, the number of points would still have to belarge to describe the multiple peaks adequately.

The method used in this paper uses in effect the impulse responsesof the line when terminated in it's surge impedance at both ends. Thisprevents reflections and ensures that the response at each end consistsof a single pulse which generally lasts less than two travel times of theline. These single pulses can be adequately described by relatively fewpoints. The example shown in Figure 6b used 48 points to describeeach response function.

The two methods both solve the same equations by mathematicallycorrect techniques. However, the new method is more convenient tonumerically compute because the response functions can be representedby fewer points. This reduces the computation time as fewer pointsmust be found by numerical inverse Fourier transformation and fewermultiplications are required to perform the convolution at each timestep. It also reduces the amount of information that must be stored inthe computer memory as fewer locations are needed to store the re-sponse functions, and fewer locations are needed to store the history ofthe line voltages and currents.

CONCLUSIONS

An extension of Bergeron's Method of Characteristics to analysefrequency dependent lines has been presented. This method is rela-tively easy to insert in existing general purpose programs usingBergeron's Method.

The application of the method to single phase and transposedmultiphase lines has been discussed with examples of its use and thetheory has been extended to cover untransposed multiphase lines.

ACKNOWLEDGEMENTS

The author wishes to thank the Bonneville Power Authority formaking the original program available.

APPENDIX I

Method of Performing Numerical Inverse Fourier Transform

The results quoted in this paper were all obtained using thetechnique of Modified Fourier Transforms. This technique was devel-oped by Day et al697¼8. Here the convergence of the Fourier Integrals isaccelerated by the use of a complex frequency.

In using the method certain parameters must be specified. Theseare the maximum angular frequency, the angular frequency step sizeand the real part of the complex frequency. Day et al6,7T8 investigatedthese factors in some detail. For this paper the following values havebeen used which follow closely the values recommended by Day et al.

Maximum angular frequency= 40/rAngular frequency step size = 40/(500r)Real part of frequency = 11(1.5r)

These values seem adequate to handle ground modes of propaga-tion where high frequencies are considerably attenuated. For line modeswhere the frequency dependence is slight a considerably higher max-imum angular frequency would be required.

The range of time for which the response functions must be calcu-lated depends to some extent on the degree of frequency dependenceof the parameters. However, the response function at the receiving endshould not be required outside the range 0.95T - 3r and at the sendingend should not be required outside the range 0 - 2r (ie, in equation ( 16)nl need not be less than 0.95T/At, n2 need not be greater than 3r/Atand n3 need not be greater than 2r/At).

A useful check on the accuracy can be made for lines with zeroshunt conductance. In this case a line with unit trapped charge in thesteady state should remain charged with no change. In this case all cur-rents are zero and all voltages are unity for all time. For these conditionsequation (16) reduces to:

n=E_

1 = E ,W(n,&t) + .2 w2(nAt)

n--=(18)

The difference between the sums of the response functions andunity is a measure of the error of the numerical inverse Fourier Trans-form calculation. For the cases tested with the values specified above,the error has usually been less than 1 per cent. There is always a smallerror as there are many approximations in the numerical calculation ofW1 and W2. This error, if it is allowed to remain, can lead to trappedcharge either growing or decaying exponentially over many time steps.This problem can be overcome for zero shunt conductance lines byscaling the response functions to make their sum equal to 1.

An alternate technique might be to use the Fast Fourier Trans-former; however, in this application it would have certain limitationsdue to the requirement that the number of points in the time domainmust equal the number of points in the frequency domain. In calcu-lating the response functions the number of points needed in the timedomain is governed by the number of points needed to adequately de-fine the response function. As the response function is a single pulse itcan be defined by relatively few points eg, 48 points were used for theexample shown in Figure 6. However, to obtain adequate accuracy inthe calculation of these points requires the consideration of a largenumber of points in the frequency domain, eg, in the example shownin Figure 6, 500 points were used. If these numbers of points had to beequal as in the Fast Fourier Transform then either the number of pointsin the frequency domain must be reduced with a consequent loss ofaccuracy or many more points than necessary must be calculated inthe time domain.

In using the method described in this paper the value of the re-sponse functions must be known at intervals of one time step of themain calculation for one or two travel times of the line. When the timestep is small, this, may lead to a much larger number of points beingrequired than are needed to adequately describe the response functions.In this case the program can be arranged to calculate some of the pointsby numerical inverse Fourier transformation and to interpolate betweenthese points to find the other points needed.

APPENDIX II

Response Functions of Untransposed Multiphase Line

Wedepohl9 performed the analysis of untransposed long m-phaselines in the steady state. The following analysis uses many of the resultsof that paper.

The transmission line equations can be written in the frequencydomain as:

-2 =[R + JL] [G + J<}C e =Pe

(19)

where e and i are vectors and R,L,G, C and P are square matrices. Eigenvector matrices S and Q are chosen such that

72 = s-1 p S = Q-1 PtQ

89

G + jwc R + J.L 1 = P It


where y is a diagonal matrix being the matrix of propagation constantsof a system of independent modes. The surge impedance matrix at agiven frequency is:

Z =S 'Y-I S- [R +jw@L]

ie, a line teUSinated in this set of impedances is reflection free at thespecified frequency.

The solution of this system of equations is:

k =S cosh (Yd) S81 em - Z Q sinh(vYcd)Q- m

(20)

k = AQ cosh(Yd) QcC1I + Q sinhh(Yd)Q 1 Z1 em

Characteristics can be defined in terms of Z1 the limit of Z as thefrequency approaches infinity. Z1 is a real matrix as at very high fre-quencies the reactive terms predominates over the resistive:

Fk ek= + Z1 Xk

B e (21)

ek, em, 'k, and im can be eliminated between equations (20) and (21)to give:

Bk K1F +A2FcX 1 Fm A2.k

Bm A1 F.k + A2 (22)

where

A1 =2[F cosh(Yd) S + Z1 Q cosh(Yd) Q 1 Z1

+ Z1Q sinh(yd) Qc1 rz

+ Z Q sinh(Yd) Q1 z(](23)

A2 -1/2 K1 oosh( rd) S1

-Z1Q cosh(Xd)Q71 z-1 + Z Q sinh(Yd)Qf1Z

- Z Q sinh( Yd) Q1 z 1]Equations 22 are in the frequency domain and are identical to

equations 13 except that the quantities are now matrices. Equations 23define the response functions and are considerably more complex thanequations 14 of the single phase case. However, equations 23 reduce toequations 14 if the matrices are of order 1. These equations can betransformed into the time domain giving convolution integrals and theintegrals approximated by summations to give:

n,,n2B1jt) = Wi(nAt)Fm(t-nfAt)

n=:n

n=n=3+ E W2(nAt)Fk(t-nAt)

n2Bm(t) =Z Wp(nAt)Fk(tn "t)

11=133n3,

+ > W2(n At)Fm(t-n At)n=l

This is identical with equation 16 except that W1 and W2 are nowsquare matrices of the order of the number of phases and Bk, Bm, Fmand Fk are vectors.

REFERENCES

[11 H.W. Dommel, "Digital computer solution of electromagnetictransients in single - and multiphase networks", IEEE Trans.Power Apparatus and Systems, vol PAS-88, pp 388-399, April1969.

[21 D.E. Hedman, "Propagation on overhead transmission lines, Pt II -

earth conduction effects and practical results", IBID, vol PAS 84,pp 205-21 1, May 1965.

[31 M.J. Battisson, S.J. Day, N. Mullineux, K.C. Parton and J.R. Reed,"Some effects of the frequency dependence of transmission lineparameters", Proc IEE, vol 116, pp 1209-1216, July 1969.

[4] A. Budner, "Introduction of frequency-dependent line parametersinto an electromagnetic transients program", IEEE Trans. PowerApparatus and Systems, vol PAS-89, pp 88-97, January 1970.

[5] J.R. Carson, "Wave propagation in overhead wires with groundreturn", Bell Sys. Tech..J., vol 5, pp 539-554, 1926.

[6] S.J. Day, N. Mullineux and R.J. Reed, "Developments in obtainingtransient response using Fourier transforms, Pt I: Gibbs phe-nomena and Fourier integrals", Int. J. Elec. Engng. Educ., vol 3,pp 501-506, 1965.

[7] S.J. Day, N. Mullineux and J.R. Reed, "Developments in obtainingtransient response using Fourier transforms, Pt II: use of modifiedFourier transform", IBID, vol 4, pp 31-40, 1969.

[8] S.J. Day, M.J. Battisson, N. Mullineux and J.R. Reed, "Develop-ments in obtaining transient response using Fourier transforms, PtIII: global responses", IBID, vol, 6, pp 259-265, 1968.

[9] L.M. Wedepohl, "Application of matrix methods to the solution oftravelling-wave phenomena in polyphase systems", Proc IEE, vol110, pp 2200-2212, December 1963.

Discussion

L. Carlsson and J. Riubrugent (ASEA, Vasteras, Sweden): The authoris to be commended for this excellent paper, which to a rigoroustreatment adds a very clear presentation of the conceptual principlesunderlying one of the chief problems of accurate digital simulation ofswitching transients.

At ASEA we have been using a similar technique for the groundmode simulation in our transients program for more than two years.This program is based on Bewley's treatment of the travelling wavesrather than Bergeron's, and the method used to simulate the frequencydependent characteristics of lines implies some simplified assumptions(10). We have found, however, that when some modifications which arenow being introduced are completed, our simulation technique will beidentical to that described by the author.

Our experience with the actual version allows us to make thefollowing remarks:a) for switching overvoltage calculations only the influence of thefrequency dependent parameters on the propagation of the groundmode waves need be taken into account.b) the additional computer time required to simulate the frequencydependent characteristics of lines is, in most cases, unimportant.c) the results obtained for typical cases of switching overvoltagesagree very well with both field and TNA records, specially when fullytransposed lines are considered.

We would like the author to comment on the intluence of the fre-quency dependent modal transformation which corresponds to a

Manuscript received June 8, 1971.

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rigorous treatment of the transmission characteristics of a non-trans-posed line.

Also could the method outlined be used to account for frequencydependent parameters on lumped two-ports (eg., generator and trans-former equivalent inductances), which could eventually be simulated asshort line stubs?

REFERENCES

[10] H. Bertil Thoren and K. Lennart Carlsson, "A Digital ComputerProgram for the Calculation of Switching and Lightning Surges onPower Systems", IEEE Trans. Power Apparatus and Systems, vol.PAS-89, pp. 212-218, February 1970.

D.E. Hedman and W.R. B'Rells (Power Technologies, Inc. Schenectady,N.Y. 12301): This is an excellent paper which describes a realistic andaccurate method for incorporating frequency dependent parametersinto the simulation of ground mode distortion. Many earlier papers haveused a much simplified attenuation model such as a delayed distortion-less attenuation or frequency independent resistance. Other modelsusing real-time convolution have required an excessive amount of com-putation at each time step. The method described in the current paperapparently requires about 200 multiplications and additions for everyline at each time step. Since the number of calculations seems to beindependent of line length, it would seem that even in the currentmethod the amount of computer time might become excessive whenstudying a system containing a large number of lines. What are theauthor's comments on this point?

The author suggests that an involved calculation of the actual lineresponse function is not prohibitive since that calculation is performedoutside of the time loop. Nonetheless, calculation of Carson's con-nection terms, etc., for 500 (complex) frequency points for each linewould seem to require a not insignificant amount of computation time,especially in the case of a transmission line with ground wires. Further,obtaining reasonably accurate frequency response data for the phaseconductors and the ground wires would seem to pose somewhat of aproblem. Does the author have some estimate of the ratio of the pre-calculation to running calculation for this program?

The method outlined in the paper emphasizes the fact that theexternal circuit sees a constant surge impedance Z1, which is the highfrequency surge impedance. Will the author please comment in moredetail on the way in which the frequency dependent surge impedanceinfluence is incorporated into the solution.

It is noted in Appendix I that the weighting factors calculatedfrom the inverse Fourier transform may not sum to exactly 1.0, due tonumerical inaccuracy. The author's suggestion that these factors shouldbe scaled so that their sum is exactly 1.0 should be emphasized. Cal-culations made on our programs suggest that it takes only a short whilefor even a small error such as this to result in a large amount of trappedcharge being inadvertently lost.

Again, the author should be commended on a fine paper. Thequality of this paper results both from a presentation of a well thoughtout new analytical procedure as well as the emphasis on the practicalaspects of the problem.


Alan Budner, (Bonneville Power Administration, Portland, Oreg. 97208):The author has proposed a simplification of the methods developed inreference [4]. However, serious questions exist about the author'sprocedure. The assumption of the method of characteristics in equation(3) as a description of a lossy line is open to serious doubt, since, as theauthor notes, it is derived on the basis of a lossless line. Conclusionsdrawn about the shape of Al and A2 in equation (15) do not seemcorrect. Zl is a surge impedance only valid at high frequencies. How-ever, the spectrum of an impulse contains a uniform distribution of allfrequencies. Hence, for lower frequencies, the line cannot be matchedand reflections exist. Therefore, the existance of only single peaks in AIand A2 upon which the major simplification of this method depends,does not seem substantiated. These apparently basic contradictionsshould be clarified.


J.K. Snelson: I should like to thank the discussers for the interest theyhave shown in the paper and the interesting points they have raised.

The experience of Messrs. Carlsson and Riubrugent using a similarmethod provides confirmation of the usefulness of techniques of thistype. As mentioned in the paper the equations for non-transposed lineshave not yet been programmed and so it is not yet possible to commenton the influence of the frequency dependence of the modal transforma-tions of this type of line. The suggestion that the frequency dependenceof other components could be represented in the same way is interestingand is theoretically possible. Another interesting possibility is to repre-sent a section of a complex network by its impulse response. This mightprove useful in line energization studies where the impulse response ofthe source network could be computed once and then used many timesin the calculation of different switching conditions. For all these meth-ods it must be borne in mind that only linear components can berepresented in this way. The non-linear effects in transformers orgenerators might cause difficulty and the resulting representation mayor may not be amenable to computation in a reasonable time.

Messrs. Hedman and B'Rells raise the question of the computationtime involved. Comparison of computation times between differentcomputer systems is difficult as machines have different speeds anddifferent types of operating systems. These differences can have largeeffects on the computation time. However, some indication of therelative increase in computer time is that using a UNIVAC 1108 thesolution without frequency dependence given in Fig 6c took 9 secondswhile the solution with frequency dependence given in Fig 6b took 14seconds. These times include the calculation of the response functionsbut exclude the calculation of the line parameters as a function of fre-quency which is performed by a separate program. This increase in timeis not very large. However if in studying a large system with many linescomputation time becomes excessive, it may be advantageous to repre-sent the full frequency dependent characteristics of the most significantlines and to use a frequency independent representation of the otherlines.

As suggested by Messrs. Hedman and B'Rells the line parametersare required at a large number of frequencies (eg 500). However, it isnot necessary to calculate the line parameters at all frequencies usingCarson's equations as the resistance and inductance are relatively smoothfunctions of frequency (2). The technique used by the author is tocalculate the parameters at 10-20 representative frequencies and tointerpolate between these points to find the values at other frequencies.

The paper maintains a distinction between the high frequencysurge impedance Zl and the complex surge impedance Z(c). The surgeimpedance Z 1 is used in the external circuit to account for the imme-diate effects at a line terminal due to changes at that terminal. Toaccount for this effect the high frequency surge impedance is the propervalue to use. The differences between the true surge impedance Z(co)and the high frequency surge impedance become apparent at later timesand are accounted for by the functions W2 and A2. These functions canbe considered to be correction factors to account for the differencesbetween Z and Z 1. It can be seen from equation 14 that if Z were equalto Z1 at all frequencies these functions would be zero.

Dr. Budner raises some basic questions which require some furtherexplanation. The definitions of Fk and Bk given in equation 3 were usedfor the purposes of the physical argument and as Dr. Budner suggests,are not precise for a lossy frequency dependent line. For the purposesof analysis precise definitions are given in equation 12. These quantitiesare similar to the characteristics of a lossless line although due to losses,etc, their properties are a little different. The same name has been usedin order to maintain the analogy. The manipulation from equation 9 toequation 13 merely constitutes a transformation to a new set of vari-ables. At this stage no assumptions are made as to the properties of thenew variables.

As suggested by Dr. Budner the surge impedance ZI only exactlymatches the line at high frequencies and so some reflections occur atlow frequencies and contribute to the functions Al and A2. However,as the variation of surge impedance with frequency is not large (the ratioof surge impedance at 60 Hz to surge impedance at 1 MHz is about1.5: 1), even the low frequencies are close to being matched. Therefore,these frequencies, although not perfectly matched, are severely attenu-ated at each reflection. Calculation of the functions W1 and W2 beyondthe range shown in Fig 4 shows that the amplitude decreases quiterapidly and is very small compared with the peak. This extended re-sponse is usually negligible, although it has been observed to lead tosome inaccuracies in the apparent 60 Hz parameters of very short lineswhere the line natural frequency is many times greater than 60 Hz.

Manuscript received July 7, 1971.

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