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Remark on the LumpedParameter Modeling ofTransmission LinesMehmet Salih Mamis, Muhammet KoksalPublished online: 30 Nov 2010.

To cite this article: Mehmet Salih Mamis, Muhammet Koksal (2000) Remark onthe Lumped Parameter Modeling of Transmission Lines, Electric Machines & PowerSystems, 28:6, 565-575, DOI: 10.1080/073135600268180

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Electric Machines and Power Systems, 28:565–575, 2000Copyright cs 2000 Taylor & Francis0731-356X / 00 $12.00 + .00

Remark on the Lumped Parameter Modelingof Transmission Lines

MEHMET SALIH MAMISMUHAMMET KOKSAL

Department of Electrical and Electronics EngineeringInonu University, 44069-Malatya, TURKEY

Transmission lines are distributed parameter systems. Instead of distributedparameter models, lumped parameter models are also used for steady-state andtransient analysis of transmission lines. Lumped parameter modeling for trans-mission lines is obviously an approximation. This paper aims to investigate thetheoretical error in the lumped parameter modeling of transmission lines forsteady-state and transient analysis. To determine the error, steady-state andtransient responses of the transmission line are calculated by using both dis-tributed and lumped parameter modeling. The eŒect of number of sections usedfor lumped parameter modeling is investigated. Obtained results can be used forthe determination of the number of sections.

Keywords transmission lines, modeling, steady-state response, transient re-sponse

1 Introduction

Transmission line is one of the most important parts of electrical power transmis-sion systems. For the accurate analysis of the transmission line, correct mathemat-ical models have to be used. In the subject of power system analysis, besides thedistributed parameter models, lumped parameter models are also used for bothsteady-state and transient analysis of power transmission lines [1–4]. Transmissionline is modeled by a cascade connection of many identical lumped parameter sec-tions in order to simulate its distribution nature. If the tolerable error level forlumped parameter approximation is initially known, it can be used to determinethe number of the sections used for modeling a transmission line. Although thenumber of sections can be taken as high as possible, it is not economical becauseof increase in the computer run time, and an additional error may accumulate dueto large number of numerical calculations. Also, special techniques are needed inthe case of large system dimension. A very limited number of theoretical studies onthis subject exist [4,5], and a complete research study including the computationalresults has not been done.

In this study, the results obtained by using inverse Laplace transform for dis-tributed parameter modeling [6] and state-space technique for lumped parametermodeling [7] are compared. Variation of the error with respect to the number of

Manuscript received in �nal form August 5, 1999.Address correspondence to Dr. Mehmet Salih Mamis.

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566 Mamis and Koksal

sections in the lumped parameter model is investigated. Both steady-state andtransient conditions are considered.

2 Transmission Line Equations

2.1 Distributed Parameter Modeling

In a uniform single-phase transmission line, four electrical characteristics, r, l , g,and c (resistance, inductance, conductance, and capacitance; all per unit length),are distributed perfectly along the line. The diŒerential length of a transmissionline is shown in Figure 1.

The steady-state behavior of a transmission line in frequency domain is de-scribed by the equations

dv

dx= zi, (1a)

di

dx= yv, (1b)

where z and y are the series impedance and shunt admittance per unit distance,and v and i are the voltage and current phasors in the conductor. DiŒerentiatingequations (1a) and (1b), again with respect to x , gives

d2v

dx 2 = zyv , (2a)

d2i

dx 2 = yzi . (2b)

The solution for the terminal voltages and currents can be obtained in frequencydomain as [8]

VS = VR cosh°`+ IR Z 0 sinh° , (3a)

IS = IR cosh°`+ VR Z1

0 sinh° , (3b)

Figure 1. DiŒerential length of a transmission line.

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Lumped Parameter Modeling of Transmission Lines 567

where ` is the total line length and the subscripts R and S stand for the receiving-and sending-ends, respectively. Z 0 and ° are the complex characteristic impedanceand propagation constant, respectively; they are de�ned by

Z 0 =p

z=y, (4a)

° = Ö zy . (4b)

In transform techniques, the j!- or s-domain expression for transient voltage atany point in the network is derived. It is generally impossible to obtain a closed-formtime-domain solution for a general input signal waveform. Therefore, a numericalFourier or Laplace Transform method is used [6,9,10].

2.2 Lumped Parameter Modeling

An approximation for the distributed nature of a uniform transmission line is torepresent it as an interconnection of many lumped parameter identical sections.Each section may be in the form of T, ¼, and L and contains a series resistanceand inductance and a shunt conductance and capacitance, as seen in Figure 2. R ,L , G , and C are the total resistance, inductance, conductance, and capacitance ofeach section of the transmission line, respectively. The resistance R for each sectionis determined by dividing the total resistance of the line by the number of sectionsN ; L , C and G can be determined in the same manner [4,5].

A lumped parameter transmission line model can be obtained by connecting a�nite number of sections in cascade.

Linear, lumped parameter networks containing resistors, capacitors, inductors,and voltage and current sources can be represented by the following system of �rst-order ordinary diŒerential equations and a set of output equations written in theform [11]

Çx (t) = Ax (t) + B u(t), x (0) = x 0, (5a)

y(t) = C x (t) + D u(t), (5b)

respectively. In these equations the state vector x contains some of the capacitorvoltages and inductor currents, x 0 is the initial value of this vector, the excitationvector u represents the inputs, and the response vector y represents the output;A , B , C , and D are the constant real matrices with proper dimensions, and theirentries depend on the values of the lumped parameters of the network.

Figure 2. DiŒerent lumped parameter network models of each section of a uniformtransmission line; (a) T-network; (b) ¼-network; (c) L-network.

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The solution of equation (5a) can be evaluated in terms of the initial statevector x 0 and the excitation vector u, and is given by the expression [11]

x (t) = eA (t t 0 )x 0 +Z t

t 0

eA (t t¢ )B u(t ¢ )dt ¢ . (6)

Consider that a transmission system is excited by a sinusoidal voltage source, whichcan be expressed as

u(t) = Re{U ej !t }, (7)

where ! is the exiting frequency (r=s ) and the phasor U is de�ned as a complexnumber given by U = Up ej Á. Replacing j! in the above equation by the complexfrequency p, where p = ¾+ j! (¾ is the neper frequency), and inserting equation (7)into equation (6), by integration, the following equation is derived:

x (t) = Re{eA (t t 0 )x 0 + (pI A ) 1(ept B U eA (t t 0 )B U ept0 )}. (8)

This equation, through the use of equation (5b), gives the complete response of thetransmission line. In equation (8), the matrix (pI A ) is assumed to be nonsingular,which is a valid assumption when the excitation frequency p = j! is diŒerent fromall the eigenvalues of A (which are distinct in all practical applications other thana few exceptional cases [12]).

The steady-state solution x s s (t) is de�ned by limt ® ¥ x (t), and for an asymp-totically stable system it can be written as

x ss (t) = Re{(pI A ) 1B ept U }. (9)

3 Error in the Lumped Parameter Modeling

Lumped parameter representation is obviously an approximation, and selectingenough sections can minimize error due to lumped parameter approximation. Forequivalence between the true line and a T-, ¼-, or L-circuit to exist, and travelingwave eŒects can be reproduced satisfactorily, the terminal conditions of the T-, ¼-,or L-circuit must be coupled by an identical relationship to that of the true line. Ifa transmission line segment with length x can be represented by a single T-, ¼-, orL-circuit at one frequency, the error for other frequencies will be small if [4]

sin(! Ö lcx )! Ö lcx

»tan(0.5! Ö lcx )

0.5! Ö lcx» 1. (10)

This approximation is valid when the dimensionless term ! Ö lcx is small. It isobvious that this quantity decreases as the line length decreases, so the frequencyerror is decreased with the line length.

The error can be evaluated for any frequency and line length by expressing thefunctions in equation (10) in series form:

sin(! Ö lcx )! Ö lcx

= 1(! Ö lcx )2

3!+

(! Ö lcx )4

5!. . . (11a)

tan(0.5! Ö lcx )0.5! Ö lcx

= 1 +(0.5! Ö lcx )2

3+

2(0.5! Ö lcx )4

15+ . . . . (11b)

The lower the frequency and the shorter the line, the less will the error be.

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Lumped Parameter Modeling of Transmission Lines 569

Figure 3 indicates how sin(µ)=µ and tan(0.5µ), 0.5µ deviates from 1 with in-creasing µ = ! Ö lcx . While doing a lumped parameter modeling, the variation oferror with respect to µ in the curve should be considered for the selection of numberof equivalent circuits (N ) by using =N instead of x in equation (11), where ` is thetotal line length; i.e., in equation (11) ! Ö lcx is replaced by ! Ö lc =N .

Figure 3. Function (a) sin(µ)=µ and (b) tan(0.5µ)=0.5µ plotted against µ to indi-cate error when lines are replaced by lumped circuits.

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Variation of ln(E ) with respect to ln(N ¢ ), where E is the relative errorand N ¢ is the normalized number of sections, is shown in Figure 4. E equals to|1 N ¢ sin(1=N ¢ )| for equation (11a) and to |1 2N ¢ tan(0.5=N ¢ )| for equation (11b);N is normalized by the value 2¼ f Ö lc` for both equations (11a) and (11b). Notethat with increasing µ (decreasing N ), the error increases.

Figure 4. Change of ln(E ) with respect to ln(N ¢ ); a) E = |1 N ¢ sin(1=N ¢ )|,b) E = |1 2N ¢ tan(0.5=N ¢ )|. Crosses on the curve correspond to discrete values ofN = 1, 2, 3, . . . .

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Lumped Parameter Modeling of Transmission Lines 571

4 Computational Results

Example 1: Steady-State Analysis

This example is studied to illustrate the performance of lumped parameter mod-eling for the steady-state analysis of a transmission line. A 160.9 km (100 mile)single-phase transmission line with resistance r = 0.032 W / km, inductance l =0.88 mH/ km, and capacitance c = 13 nF/ km is considered. It is assumed that theline is terminated by a load composed of a resistance R L = 193.0 W and an in-ductance L L = 0.466 H connected in series. Maximum of the receiving-end voltagecalculated theoretically by using the transmission line terminal equations involvinghyperbolic functions is vr = 0.892269 p.u. By taking this value as a reference, errorfrom the lumped parameter representation of the transmission line with diŒerentnumber of T-sections is given in Table 1. As seen from the table, the percentageerror is 0.251% in the case of �ve number of sections, and it decreases as the numberof sections increases.

Variation of ln(E ) wrt ln(N ), where E is the percentage error and N is thenumber of sections, is shown in Figure 5. As seen from the �gure, the variation islinear as in Figure 4; the least square linear �t for the curve is computed to be

ln(E ) = 0.38104 1.054107 ln(N ) . (12)

From this equation, E can be expressed explicitly in terms of N as

E = 1.464N1.054107

. (13)

The power of N in this equation is approximately equal to 1.0, and therefore theerror is simply inversely proportional with the number of sections N .

Example 2: Transient Analysis

In this example, transient overvoltages due to transmission line energization arestudied. A lossy three-phase, 380-kV transmission line with length 271 km is con-sidered [1]. It is assumed that this line has an open circuit at its receiving-end,

Table 1Error with respect to N

Max. ofNumber of receiving-end Absolute

sections voltage v ¢r error %Error(N ) (pu) |v v ¢r | E

5 0.894509 22.4E-4 0.25110 0.893387 11.2E-4 0.12620 0.892823 5.54E-4 0.06230 0.892634 3.65E-4 0.04140 0.892540 2.71E-4 0.03050 0.892483 2.14E-4 0.02460 0.892445 1.76E-4 0.02080 0.892398 1.29E-4 0.014

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Figure 5. Change of ln(E ) with respect to ln(N ).

and it is energized by the sinusoidal voltage source with series inductance 50 mH.Phase-A of the sinusoidal source at the instant of energization is at its peak value.Sixteen L-sections are used to represent the three-phase transmission line for thisstudy. For the simpli�cation of the analysis, the three-phase system is decoupled byapplying modal analysis techniques, and each mode is solved in the same manneras single-phase case [13].

For the variation of the voltage of phase-A, comparison of the results obtainedby fast inverse Laplace transform (FILT) [10], which are veri�ed by the results ofElectromagnetic Transients Program (EMTP), using distributed parameter repre-sentation of transmission line and state-space analysis using the lumped parameterapproximation, are shown in Figure 6. The curves are similar in wave shapes, butas time increases, there appears a phase diŒerence between them. This phase diŒer-ence is due to the lumped parameter approximation of the transmission line, andits amount depends on the number of sections N used to represent the line. Forexample, if a step input is applied to the sending-end of a lossless line that is ap-proximated by one L-section, the receiving-end voltage (capacitor voltage) oscillateswith a period of 2¼ (lc)1/ 2s; however, when a distributed parameter transmissionline is used, the period of square voltage wave at the receiving-end is 4 Ö lc s[4], which is ¼=2 times less than the period calculated by the lumped parameterapproximation. If a higher number of lumped parameter sections is used to rep-resent the line, the model will be closer to the distributed parameter modeling.Further, for each N with the proper time-scaling techniques, this discrepancy be-tween the lumped and distributed parameter models can practically be reducedto zero while considerable computer time is saved by using lumped parametermodel [7].

To determine the error, the phase diŒerence is measured for diŒerent number ofsections, and the change of percent of phase diŒerence with respect to the number

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Lumped Parameter Modeling of Transmission Lines 573

Figure 6. Sending-end voltage for phase-A computed by (a) FILT, (b) State-spacemethod.

of sections N is shown in Figure 7. This �gure shows that the phase diŒerencedecreases as the number of sections increases. Variation of ln(E ) with respect toln(N ) for this example is plotted in Figure 8. The least squares approximation forthe curve in this �gure is

ln(E ) = 0.949 + 1.098 ln(N ) . (14)

Figure 7. Variation of the phase shift Á with respect to the number of sections N .

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Figure 8. Change of ln(E ) with respect to ln(N ).

Error E can be obtained from the above equation, and it is

E = 0.387N1.098

, (15)

which shows that the error is simply inversely proportional to the number of sec-tions.

5 Conclusions

Error due to the lumped parameter approximation of transmission lines is exam-ined theoretically by comparing the results obtained by inverse Laplace transformfor distributed parameter model and state-space technique for lumped parametermodel. It has been shown that the percentage error variation is inversely propor-tional with the number of sections used. Given a speci�ed tolerable error level, the�gures obtained can be used for the determination of the number of sections forlumped parameter modeling of transmission lines. This way, the use of an unneces-sary high number of sections and the resulting computation eŒorts and accumulativeerrors are eliminated.

References

[1] Iliceto, F., Cinieri, E., and Di Vita, A., 1984, “ Overvoltages due to Open-PhaseOccurrence in Reactor Compensated EHV Lines,” IEEE Trans. Power Apparatusand Systems, Vol. PAS-103, pp. 474–482.

[2] Jeyasurya, B., Vu, T. H., and Smolinski, W. J., 1983, “ Determination of TransientApparent Impedances of Faulted Transmission Lines,” IEEE Trans. Power Apparatusand Systems, Vol. PAS-102, pp. 3370–3378.

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Lumped Parameter Modeling of Transmission Lines 575

[3] Morcos, M. M., and Anis, H., 1996, “On Rondom Line-to-Neutral Switching Tran-sients in Transmission Systems,” Electrical Machines and Power Systems, Vol. 24,pp. 303–311.

[4] Greenwood, A., 1991, Electrical Transients In Power Systems, New York, John Wiley& Sons.

[5] Smith, J. R., and Bell, W. W., 1984, “ Selection of Transmission Circuit Models forPower-System Transient Studies,” IEE Proceedings, Vol. 131, Pt. C, pp. 1–4.

[6] Al Fuhaid, A. S., and Saied, M. M., 1988, “A Method for the Computation of FaultTransients in Transmission Lines,” IEEE Trans. Power Delivery, Vol. 3, pp. 288–297.

[7] Mamis, M. S., and Koksal, M., 1996, “ Numerical Solution of Partial DiŒerential Equa-tions for Transmission Lines Terminated by Lumped Components,” Proc. of the 5t h

International Colloquium on Numerical Analysis, August 13–17, Plovdiv, Bulgaria,pp. 87–98.

[8] Stevenson, W. D., Jr., 1982, Elements of Power System Analysis, New York, McGraw-Hill.

[9] Moreno, P., Rosa, R., and Naredo, J. L., 1991, “Frequency Domain Computation ofTransmission Line Closing Transients,” IEEE Trans. Power Delivery, Vol. 6, pp. 275–281.

[10] Hosono, T., 1981, “Numerical Inversion of Laplace-Transform and Some Applicationsto Wave Optics,” Radio Science, Vol. 16, pp. 1015–1019.

[11] Swisher, G. W., 1976, Introduction to Linear System Analysis, Champaign, MatrixPublishers Inc.

[12] Mamis, M. S., 1997, Steady-State and Transient Analysis of Multi-Phase PowerTransmission Lines by Using State-Space Techniques, Ph.D. Thesis, University ofGaziantep, Gaziantep.

[13] Hedman, D. E., 1965, “Propagation on Overhead Transmission Lines I-Theory ofModal Analysis,” IEEE Trans. Power Apparatus and Systems, Vol. PAS-84, pp. 200–205.

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