Application of nonuniform-line theory to the simulation of electromagnetic transients in power systems

Electrical Power & Energy Systems, Vol. 20, No. 3, pp. 225-233, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain ELSEVIER

PII: S0142-0615(97)00043-4

0142-0615/98/$19.00+0.00

Application of nonuniform-line theory to the simulation of electromagnetic transients in power systems A S AIFuhaid, E A Oufi and M M Saied Electrical and Computer Engineering Department, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait

[5]. As another example, the sag in overhead lines causes the inductance per unit length, L, to increase with distance from the mid-span. As a third example, for a transition segment between an overhead line and an underground cable L is higher at points closer to the line whereas C, the capacitance per unit length, is lower compared to their corresponding values at points closer to the cable. In fact, L decreases and C increases according to a logarithmic function of the distance as measured from the line-cable junction. Thus, a nonuniform line can serve as useful tool for modeling several specialized transient phenomena. In a previous article [7], an s-domain distributed-parameter method was proposed to study the transient behavior of lossless single-phase nonuniform lines whose parameters can have arbitrary space variation. The method was applied to study electromagnetic transients along a high-voltage transmission tower caused by a direct lightning stroke [7]. The objectives of the present work are twofold: firstly, we extend the s-domain method to include lossy and untransposed three-phase nonuniform lines with provisions for frequency-dependent parameters. Secondly, we illustrate how the method can be applied to model electromagnetic phenomena of interest in power systems by presenting the simulation results of lightning transients on a transition segment joining an overhead line to an underground cable.

The objective of this article is to illustrate the applicability of the nonuniform transmission line as a modeling tool in electromagnetic transient studies of power systems. First, the s-domain method, that was developed previously by the authors for lossless single-phase nonuniform transmission lines, was generalized to include lossy untransposed threephase lines with frequency-dependent parameters. Second, the new generalized version of the s-domain method was used to obtain the transients on an actual transition segment between an overhead line and an underground cable. Third, the simulation results are compared with those obtained from the uniform-line model used by the well-known Electromagnetic Transients Program. © 1998 Elsevier Science Ltd Keywords: modeling of energy systems, eigenvalue analysis, digital simulation

I. Introduction A nonuniform transmission line is one whose inductance and capacitance per unit length have spatial variations along the line. Such spatial variations can be obtained if the characteristic impedance of the line is location-dependent. Power transmission lines are classified as uniform lines since their parameters are constant in the sense that they do not vary from one location to another. Nonuniform lines were found useful in modeling various electromagnetic transient phenomena in power systems [17]. For example, to obtain electromagnetic transients along a high-voltage transmission tower, when hit by a direct lightning stroke, the tower can be modeled as a nonuniform line

II. Review of previous work The s-domain method proposed in [7] models any lossless single-phase nonuniform line, which may have a general type of nonuniformity, using a cascade of distributedparameter exponential lines. An exponential line is specified by its length l and characteristic impedance Zc(x) = Z0~kx. From [7], the voltage and current of an exponential line are described, in the s-domain, by the following differential equations.

Received 15 June 1996; accepted 2 October 1996

225

Nonuniform-line simulation of electromagnetic transients in PS: A. S. AIFuhaid et al

226

d2V(s,x) k dV's'x'( ) dx2

dx

S2=o c2

(1)

elements of Zphase(X) are given by; t~0. 2(hn + p ) Znn =rio ~-~m r + Zcono,

(8a)

en

1 dV(s,x) sL(x) dx

l(s,x) --

(2)

and

• IZo. ~/(h~-t-hm+2p)2+d2nm . . . . . . .

Znm~-~-j~--ln

where, 1 C--

(8b)

v / ( h . - hm)= + -

where,

-

V(s, x) = A 1e x'x + A2 e~2x

(3)

where A1 and A2 are arbitrary constants, and the eigenvalues X~ and X2 can be computed from; k )kl, )k2 = 2 -4-

k~7@ s2 -[- c--~

(4)

The current l(s,x) can now be obtained by substituting from (3) into (2), and the arbitrary constants can be expressed in terms of the boundary conditions at the receiving or sending ends of the line. A nonuniform line, with a general type of nonuniforrnity, can be represented as a cascade of distributed-parameter single-phase exponential lines. Therefore, to model a nonuniform line of length 1and general characteristic impedance Zc(x), the line is divided into N exponential-line sections, which are assumed, for convenience, to be of equal length. For the jth section, which starts at location xj and ends at Xj+ 1 = Xj "~ l/N, we obtain;

Zj(x)=Zoje kjx forxj < x
1

p =

is the speed of light. The voltage solution is given by;

(8c)

is the complex depth for earth return, a is the earth's conductivity, re. is the external radius of the n th conductor, hn and hm are the heights of the n th and m th conductors, respectively, and dnm is their horizontal separation, The n th conductor's internal impedance Zcond., which includes skin effect, is approximated by; Zcond"= ;RZc,, + Z~2f.

where the n th conductor dc resistance Roe" and its high frequency internal impedance Zhe. are; 1 Rdc" = 7r(re2,,_ r2)acond"

=

Zhfn

1 27rO-cond" re.Pcond.

1

where,

and,

kj ~ Nln[ zc(x~Xj+l)] /

(9d)

being the conductor complex depth, ri. its internal radius, and acond" its conductivity• To compute Yphase(X),we first obtain Maxwell's potentialcoefficient matrix, Mphase(X), whose elements are given by

(7)

M.. = 1-~-ln 2h. 27re0 re.

L ZAx=xA J

withj = 1,2..... N. Equations (5)-(7) together with the voltage and current solutions of each exponential-line section give a complete s-domain description of any type of lossless nonuniform lines. Using this model and the loading conditions at the receiving end of the line and its excitation at its sending end, ladder-network theory is applied to compute the sdomain voltages and currents at any point along the original nonuniform line. The time-domain solution for the variables of interest can be obtained for any specific location x and time point t by using an efficient numerical algorithm, developed by Hosono [8], to obtain the inverse of the Laplace transform.

(9c)

with

(5)

(6)

(9b)

and

Pcond. = v/Jw#0acond.

Zo, & Zc(x = xj)

(9a)

(10a)

and

1 . Dnk M.k = 2--~e0m ~

(10b)

where D.k is the distance between the n th conductor and the image of the k th conductor [10]. Then, Yph,se(X) is given by; Yphase(X) -- jwM~lase(X)

(10c)

For a nonuniform line, and unlike ordinary uniform lines, all the geometric distances used in (8)-(10) are location dependent, and therefore, the line parameters must be computed for each desired location. At location x, the three-phase characteristic-impedance matrix of a nonuniform three-phase line, Zc_ph~e(X),is defined as;

III. Method of analysis In this section we generalize the method to include lossy untransposed three-phase nonuniform lines with frequencydependent parameters. Let Zphase(X) and Yphas¢(X) be the senes matrix (in fl m - ) and the shunt admittance • ~mpedance ' matrix (in S m-~), respectively, at a given position x along an untransposed three-phase line. According to [13], the

Zc_phase(X ) =

v/Zphase(X)r~le(X)

(11)

To use the exponential nonuniform-line model described in the previous section, we resort to modal analysis and consider, for now, a lossless line. We will then show how to modify the analysis to take line losses into account. For a

Nonuniform-line simulation of electromagnetic transients in PS: A. S. AIFuhaid et a l lossless line, Zphase(X) iS purely imaginary and, therefore, Zc-pha~(X) becomes a real matrix. Zc-phaso(X)can be transformed to a diagonal matrix Z~-mode(X)using the following modal transformations; Vmode(X)= Tv(x)Vphase(x)

(12a)

lmoae(X) = Tl(X)lphase(X)

(12b)

Using (12) to diagonalize;

d2Vphasc(x) dx 2 =Zphase(X)Yphase(X)Vphase(X

)

(13)

yields,

d2Vmode(X) dx 2 = AVmode(X)

(14a)

where A is a diagonal matrix given by; A "-- T V l ( x ) Z p h a s e ( x ) Y p h a s e ( x ) Z ~ ( x )

(14b)

In (14b) we have used the fact that [14]; Tt = ( T v l ) ' Using the modal transformations Tv and TI, and the matrix A, we can now compute the modal parameters of the line at a given location x as follows [10]; first we obtain Yrnode(X) from; Ymode(X) ~---T~ (x) Yphase(x)TV(x)

(15)

Next, we obtain Zmo~o(X)from;

Zmode(X) = Ayn~le (x)

(16)

Then, we compute the, modal characteristic-impedance matrix Zc-mod~(x)from; Z~- mo~e(X)= v/Zmoo~(x)Y~o~e(X)

(17)

It is worth noting that all of the matrices in (8)-(17) are frequency dependent :in addition to being location dependent. Further, observe that Zc-mode(X)is a real diagonal matrix containing the characteristic impedances of all the propagation modes. To model a line section as an exponential-line section, the elements of Zc-~ode(X) are evaluated at the two ends of the section, and the parameters k and Z0 are computed for each mode using (6) and (7) as described in the previous section. Once these parameters were obtained, the A B C D generalized circmt constants can be readily calculated for each line section and for each mode; thus for the i th mode, we compute; A i ( s ) ~-

~kile -

xi2 l _ ~i 2 e - xil I

- hit l]

G(s) = ki, hi2 e - x~tt _ e - x'21 sLi o

(18b)

(18c)

hil -- }~.i2

Di(s) = ek'l[hil e - x,11_ hi2e - xi2l] hi, - k, 2

where ~il, Xi2 are the eigenvalues of the i th mode which are computed from (4), 1 is the length of the line's section, and L0i is the inductance corresponding to Z0r From (18), observe that Ai(s ) :/: Di(s) and Ai(s)Di(s ) - B i ( s ) C i ( s ) --/: 1 which follow from the fact that the nonuniform line is not a bilateral element. Using the inverse modal transformations given in (12), the modal A B C D constants for all line sections are transformed to the phase domain. The A B C D constants for the cascaded line sections and the equivalent shunt lines representing the towers (more on this later) are combined, in the phase domain, using the properties of the A B C D constants [12]. Using the A B C D constants together with the loading conditions at the receiving end and the source's impedance and excitation, s-domain expressions can be obtained for the desired voltages and currents. We now consider the line's resistance. We choose the model proposed by H. W. Dommel [9,10], which was implemented successfully in the EMTP to model lossy uniform lines. Thus for a lossy nonuniform line, its total resistance matrix, obtained from Zphase(X), is split into three lumped series resistances which are placed in the middle, and at the two ends of the line. Since the resistance is obtained from the matrix Zphase(X), then it is frequency dependent as well as being location dependent. The A B C D constants of these lumped resistances must also be computed and combined with the exponential-line sections. We now consider the transmission towers which directly affect the ground mode. For lightning-surge studies, a transmission tower may be modeled accurately as an equivalent vertical transmission line with a given characteristic impedance and propagation velocity. The length of the line is equal to the height of the tower and the propagation velocity v is usually chosen as 0.85 of the speed of light [4,5]. For a cylindrical tower the characteristic impedance is not constant, and the tower should be modeled as a nonuniform line [11]. For the purposes of this study, a conical tower is assumed, which can be represented as a uniform line whose characteristic impedance is given by; [2(h 2 + r2)] Zc = 301n [ r~

(19)

where h is the tower's height and r the radius of its base. Formulae for Zc for other classes of towers are given in [11]. The voltage at the top of the tower is of interest when the line is hit by a direct lightning stroke. We compute this voltage using its input impedance, which is given by; Zc sinh(sr) + Rf cosh(sr)] Zinput(S)~- Zc Zc cosh(sT")-~-R e sinh(sr)J

(20)

(18a)

~i~ - ~ki2

Bi(s) = sLio ekil[e - ?~21 - e ~kil _ ~ki2

227

(18d)

where Rf is the tower's footing resistance, Zc and r = h/v are the tower's characteristic impedance and travel time, respectively. The A B C D constants for the tower can be readily computed, using standard formulas [12], and then combined with the A B C D constants of the ground mode of the exponential-line sections. It is worth emphasizing that (19) and (20) enter the calculations of the ground mode only.

IV. Case study For the remainder of this article, we consider the following case study which deals with a transition segment between a high-voltage overhead line and an underground cable. Observe that the inductance per unit length of the line L is higher at points closer to the line whereas the capacitance per

228

Nonuniform-line simulation of electromagnetic transients in PS: A. S. AIFuhaid et al

unit length C is lower compared to their corresponding values at points closer to the cable. In fact, L decreases and C increases according to a logarithmic function of the distance as measured from the line-cable junction. Figure 1 and Figure 2 illustrate an actual transition segment between a double-circuit 275 kV overhead line and the corresponding underground cables, which is typical of a major substation in Kuwait high-voltage transmission network. As shown in the figures, the horizontal spacings between the conductors as well as their heights are decreasing continuously from the high-voltage tower toward the gantry. The horizontal distance between the tower and the gantry is 50 m. At the tower: the height of the lowest phase is 21.6 m, the vertical phase spacing is 8.4 m, the horizontal spacing between the any two phases of the two circuits is 19.5 m and the height of the ground wire is 55.8 m. At the gantry: the height of the phase conductors is 8.0 m, the horizontal distance between the two circuits is 6.5 m, and II

II

the height of the ground wire is 14.0 m. The phase conductors are Zebra 54/7 with an outside diameter of 28.6 mm, 4 bundle and spacing between bundles is 0.45 m. The ground wire is made of galvanized steel with an outside diameter of 11.88 mm. The tower is now assumed to be hit by a direct lightning stroke which is modeled as a double-exponential current source with an assumed surge impedance of 250 f~. Its current is given by i(t)= 30.397[e -t/rt - e - t / r 2 ] where i(t) is in kA, Tl = 17.63/~s and T2 = 0.0316/~s and t is/~s. This waveform represents a 0.2/25/zs impulse with a 30 kA peak value. To obtain the transient behavior of the transition segment, and since it is only 50 m long, a single exponential-line section is used to model it; its parameters are computed from (6) and (7). The tower and gantry are modeled as uniform lines using (19) with a footing resistance of 10 ~ for each. Figure 3(a) and (b) show the voltages at the tower and

It-

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31

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.4

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42.0

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Figure 1. Top view of the transition between the overhead line and the underground cables. Shown are the horizontal positions of the gantry, surge diverter (SD), wave trap (WT), coupling condenser (CC) and the sailing end (SE)

Nonuniform-line simulation of electromagnetic transients in PS: A. S. AIFuhaid et al

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,.6

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SPACER CONNECTOR (BIM£TALJC) CU CONDUCTOR CU CONDUCTOR500 m m 2 +'TWIN

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Figure 2. Front view of the transition between the overhead line and the underground cables from a point near the gantry gantry, respectively, using the proposed nonuniform-line model, when a lightning stroke hits the top of the tower. Also shown in these t~o figures are the corresponding voltages obtained from the more familiar uniform-line model. The figures show a close match between the simulation results of both line models, indicating that the nonunifortuity of the transition segment is not significant in practical terms. This conclusion was confirmed when the input impedance for both line models, as seen from a point just slightly to the right of the top of the tower, were computed and were found to be very close. Another contributing factor to this close agreement is the modeling of the tower and gantry as uniform lines. Figure 4(a) and (b) depict the current in the ground wire from the tower towards at the tower and at the gantry, respectively, using both line models. Again, the simulation results of both line model,; are in good agreement. Figure 5(a) and (b) give the induced voltages on the conductor of the phase closest to the ground wire at the tower and at the gantry. Here, the two line models produce different waveforms but the peak voltages are nearly the same. This difference is because the physical nonuniformity of the line is more pronounced for the phase conductors than the ground wire as can be seen from Figure (1). It is well known that the voltages induced on the phase conductors are much smaller than the voltages produced in the ground mode (on the ground wire, the tower, or the gantry), and our simulation results clearly reflect this observed fact. Therefore, it can be concluded that the differences in the phase voltages between the two Jine models are not significant in practical applications. At this point, it is worth pausing to reflect on the test results presented in the figures. In view of the close agreement between the results of the nonuniform- and uniform-line models for tihe ground conductors, one may be tempted to question the need for the nonuniform-line

model. Actually one can only prove that the nonuniformity studied here is insignificant only after performing the detailed study using the nonuniform-line model and then comparing the results with those obtained from the uniformline model. Moreover, it is entirely conceivable that other investigators may encounter some types of nonuniformity, such as the modeling of high-voltage transmission towers, which can be examined using the methodology presented here. It is also worth pointing out that we have only considered transients caused by a lightning stroke, which is a high-frequency phenomenon; whether the nonuniformity of the line may have a more pronounced effect in other types of transients remains an open question. The important point is that this case study illustrates the use of the proposed nonuniform-line methodology as a modeling tool for studying transient phenomena in power systems.

V. S u m m a r y and conclusions (1) The s-domain method developed in [7], which dealt exclusively with lossless single-phase lines, was extended so that untransposed three-phase lines and lossy lines, with frequency-dependent parameters, can be analyzed. (2) An actual transmission configuration was considered to illustrate the application of the nonuniform transmission line model to the study of electromagnetic transients in power systems. (3) For the particular case study considered here, and for the practically important ground mode, the simulation results of the uniform-line model are almost indistinguishable from the more complex nonuniform-line model indicating the adequacy of the uniform (EMTP) line model.

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Figure 5. (a) The induced voltages (in kV) on the phase conductor nearest the ground wire at the tower vs time (in s) for (a) the nonuniform-line model and (b) the uniform-line model. (b) The ir~duced voltages (in kV) on the phase conductor nearest the ground wire at the gantry vs time (in s) for (a) the nonuniform-line model and (b) the uniform-line model

Nonuniform-line simulation of electromagnetic transients in PS: A. S. AIFuhaid et al (4) Other types of tran,;ient phenomena, or transmission configurations may lead to different conclusions. The nonuniform-line model can, thus, serve as a useful tool to determine when a particular type of nonuniformity is significant or not.

6. 7. 8.

VI. Acknowledgements The authors thank the staff of the Ministry of Electricity and Water of Kuwait for providing the data for the case study.

9. 10.

VII. References 1. Walker, L. R. and Wax, N., Nonuniform transmission lines and reflection coefficients. Journal of Applied Physics, 1946, 17(Dec.), 1043-1045. 2. Wills, J. and Sinha, N. K., Nonuniform transmission lines as impedance transformers. IEE Proceedi.~gs, 1956, 103B(March), 166-172. 3. Barthold, L. O., An approximate transient solution of the tapered transmission line. A1EE Trans. on Power Apparatus and Systems, 1958, PAS-77(Feb.), 1556--1561. 4. Sargent, M. and Darveniza, M., Tower surge impedance. AIEE Trans. on Power Apparatus and ~'stems, 1969, PAS-88(May), 680-687. 5. Menemenlis, C. and Chun, Z. T., Wave propagation on nonuniform

11.

12. 13. 14.

233

lines. IEEE Trans. on Power Apparatus and Systems, 1982, PAS101(April), 833-839. Saied, M. M., A1Fuhaid, A. S. and Elshandwily, M. E., s-domain analysis of electromagnetic transients on nonuniform lines. 1EEE Trans. Power Delivery, 1990, PWRD-5(Oct.), 2072-2083. Oufi, E. A., A1Fuhaid, A. S. and Saied, M. M., Transient analysis of lossless single-phase nonuniform transmission lines. IEEE Trans. Power Delivery, 1994, PWRD-9(3, July). Hosono, T., Numerical inversion of Laplace transform. Electrical Engineering in Japan, 1979, 99(5), 43-49. Phadke, A. G. Editor, Digital Simulation of Electrical Transient Phenomena, 1EEE Tutorial Course # 81 EHO173-5-PWR, New Jersey: IEEE Service Center, 1980. Dommel, H. W., Electromagnetic Transients Program Reference Manual (EMTP Theory Book), Bonneville Power Administration, Portland, Oregon, August 1986, pp. 4.50-4.64. IEEE Working Group on Lightning Performance of Transmission Lines, A simplified method for estimating lightning performance of transmission lines". 1EEE Trans. Power Apparatus and Systems, 1985, PAS-104(4, April), 919-932. Stevenson, W. D., Elements of Power System Analysis, 2nd edn, McGraw-Hill, New York, 1962, pp. 116-132. Semlyen, A. and Deft, A., Time domain modelling of frequency dependent three-phase transmission line impedance. IEEE Trans. Power Apparatus and Systems, 1985, PAS-104(6, June), 1549-1555. Wedepohl, L. M., Application of matrix methods to the solution of travelling-wave phenomena in polyphase systems. Proceedings lEE, 1963, ll0(Dec.), 2200-2212.