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Wave Modal Technique for Switching Transients in Untransposed Transmission Lines
Electric Power Systems Research, 13 (1987) 49
- 57
Waye Modal Technique for Switching Transients in Untransposed Transmission Lines M. HAMED Faculty of Engineering, Suez Canal University, Port Said (Egypt)
(Received March 10, 1987)
SUMMARY
This paper presents a digital analysis for the calculation of internal overvoltages in untransposed transmission lines. This analysis is based on the Laplace transform as well as on the modal technique. The frequency dependent parameters due to the earth return effect are taken into account. A general transformation matrix is determined. Its elements are evaluated numerically and then formulated theoretically. The difference between surge and characteristic impedances of a line is formulated. A formula for the voltage in wave mode coordinates of an untransposed transmission line is deduced. The sequential pole switchings of circuit breaker contacts are considered. The voltage in phase coordinates is expressed mathematically. The potential coefficients are considered as constants. Key words: transients, modal, Laplace, frequency, earth return, transformation matrix, untransposed lines.
INTRODUCTION
UHV transmission appears to offer particular advantages principally to those countries where large generation sites are located far from load centers. In such sites either coal, hydro or nuclear fuels will undoubtedly be utilised. A very important factor in the future of UHV transmission, which has been presented in the past, will be the need to move large blocks of energy over long distances because of changes in the world’s fuel situation. An advantage of UHV transmission stems from the fact that, by its use, it may be possible to utilize existing concentrated energy 037%7796/87/$3.50
sources, or even to concentrate energy sources in a few areas particularly suitable for safety or environmental reasons. This would avoid spreading generation plants all over the territory [l]. The use of UHV may be influenced more by special requirements (environmental, nuisance) than by purely economic reasons so that the future of UHV will depend on the development of proper technical solutions to match these requirements. However, EHV and UHV transmission lines are frequently used in practice. As the nominal voltage of transmission is high, the cost of transposition towers will be expensive. So, from an economic point of view, the UHV transmission lines should be, mostly, untransposed. Also, when the length of UHV lines is increased, the parameters of the different phases will not remain the same, owing to the untransposition effect [l]. In addition, the frequency dependent parameters due to the earth return effect must be introduced in transient investigations [ 2’- 41. Hence, transient studies of UHV AC transmission lines are important because of the new tendency to increase the length of UHV lines. Transients in transmission lines have been investigated extensively by numerous researchers and different methods have been published for the calculation of electromagnetic transients in lines, especially in transposed systems [5 - 71. But a general digital model for this purpose in untransposed transmission lines under all possible conditions has not so far been published [8,9]. Also, the insulation level for networks in all countries is rising continuously. This level may reach a very high value, namely 1600 kV, as proposed in ref. .lO. Thus, the accurate insulation of a transmission line appears to be an important @ Elsevier Sequoia/Printed in The Netherlands
50
item. Hence, a general accurate method for the calculation of internal overvoltages in transmission line systems should be introduced in the field of transient investigations. This is important when the system insulation level becomes high. Therefore, the design of such a line will be based on the internal overvoltages which may be induced due to normal operation of a network [ 111. A general digital model for transients in untransposed lines may be based on the Laplace concept [l]. The frequency dependent parameters of a line can be taken into account. This process depends mainly on the modal technique, if it is possible in untransposed lines.
MODAL DECOMPOS~ION
Modal analysis appears to be the most commonly used method for calculation of transients in transposed transmission lines 151. It is based on transforming the three-phase system into a line in the three natural modes. This technique can be realized by using the (CI, fi, 0) matrix of Clark’s components [12]. This means that the representation of transposed lines in wave mode coordinates is possibIe. On the other hand, the elements of the transformation matrix for untransposed single-circuit transmission lines are frequency dependent, and thus the representation of untransposed lines in wave mode coordinates is possible only at each individual frequency. But if these elements vary with frequency in a weak manner, one general transformation matrix T can then be proposed. However, according to this variation the Fourier transform [13 - 161 is used, while Duhamel’s integral has been suggested 151. In this case the computational time and effort will be increased significantly. In the present research the frequency dependent transfo~ation matrix for the parameters of untransposed transmission lines must be replaced by a general transformation vector. Its elements should be frequency independe’nt and then the wave mode coordinates of the untransposed systems can be found. This concept is interesting since it reduces the computational effort and time significantly in the calculation of transients in untransposed transmission lines.
THE TRANSFORMATION MATRIX
The matrix of parameters H of a transmission line in phase coordinates can be transformed into a diagonal matrix H, which is called the matrix of parameters of a line in wave mode coordinates. This may be done using the abovementioned transformation matrix T. Therefore, the transformation matrix appears to be comprised of the mathematical eigenvalues of the original matrix of parameters in wave mode coordinates. However, it is known that the elements of the transformation matrix are generally frequency dependent. This means that for each individual frequency a certain transformation matrix may be deduced. On the other hand, it is required that a general form for the transformation matrix at any frequency is determined so that we may obtain a general transformation matrix which will be valid for the parameters of a transmission line at all values of frequency. So, the matrices of each parameter at different frequencies must be evaluated [4]. These matrices are computed for all fundamental parameters per unit length of a line, such as resistance R, series inductance L, parallel conductance G and parallel capacitance C. Here, it must be stated that the corona and the conductance of the studied lines have been neglected. Therefore the capacitance will be frequency independent. Since the earth return effect is included, the matrix of parameters H( f) can be formulated generally as
The elements a( f), b( f) and c(f) are the selfparameters of a line, while d( f), g( f) and m( f) are the mutual parameters. Also, it should be noted that symbol ( f) means that the parameter is frequency dependent and so is the matrix itself. The propagation coefficient matrix S and characteristic impedance matrix Z, will take the general form of matrix (1). Then, the eigenvectors for such a matrix H(f) c a n b e expressed as 1
1
TAf) T&f) Tz(f) Tdf) T,(f) 1
(2)
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It is seen that only six elements of the proposed transformation matrix T( f) of eqn. (2) are frequency dependent. Therefore, these elements at different values of frequency should be calculated. The transformation matrices for a typical 500 kV untransposed transmission line, with an average height above ground of 12 m and phaseto-phase spacing of 11 m, will be computed [ 171. The calculated transformation matrices for the resistance are listed in Table 1 [17]. The given matrices are determined using a special program since the normalization of the transformation matrix is not necessary at present. These matrices have also been found for inductance, as given in Table 2 [17]. Finally, as the capacitance is constant, its transformation matrix may be given in the form [17] T, =
1.000 1.000 -1.650 1.200 [ 0.999 0.999
1.000 -0.002 -0.999 1
(3)
Now, it is seen that the elements of the transformation matrices for the inductance matrix L vary weakly with frequency. Now, in the present study, all values of 0.999 or 0.998 will actually be taken as unity, and all values of 0.002 or 0.003 as zero. In this case only two elements will be frequency dependent, while all the other elements become constants. Then, one general form for the transformation matrix may be chosen [l]:
1
1
1
0
A(f)
-1
1
(4)
TABLE 1 The calculated transformation matrices for the resistance matrix at different frequencies Frequency &Hz) 3
25
40
50
Transformation matrix T,
1.000 -1.400 0.608 1.000 - 1.810 0.820 1.000 - 1.850 0.850 1.000 - 1.860 0.870
1.009 0.999 0.657 1.000 1.000 0.990 1.000 1.000 0.996 1.000 1.000 0.996
1.000 -1.601 -0.567 1.009 - 1.202 -0.201 1.000 -0.609 - 0.430 1.000 -0.461 -0.611
TABLE 2 The calculated transformation matrices for the inductance matrix at different frequencies Frequency W-W 3
Transformation matrix T,
1.000 - 1.830 0.999 1.000 - 1.790 0.999 1.000 - 1.780 0.999 1.000 - 1.780 0.999
25
40
50
1.000 1.090 0.999 1.000 1.116 0.999 1.000 1.120 0.999 1.000 1.120 0.999
1.000 -0.003 -0.996 1.009 -0.003 -0.997 1.000 -0.003 - 0.997 1.000 - 0.003 - 0.998
Fortunately, this transformation matrix is similar to the matrix of Clark’s components a, j?, and 0 for symmetrical transmission lines, which takes the form
1
1
1 1
T = - 2 [1
-101 1
Therefore, the elements B( f) and A(f) f o r transposed lines are constants equal to 2 and 1, respectively. This phenomenon leads us to the second step in order to compute their values at different frequencies. Since the transformation matrix of Clark’s components appears to be a special case, the transformation matrix T(f) becomes the general one for either transposed or untransposed transmission lines. It will also be called the (a, p, 0) transformation matrix. In other words, this means that the wave modes of a line will be a, /I and 0. The coefficients of the general transformation matrix It is concluded above that the two elements A( f) and B( f) in the transformation matrix of eqn. (4) are frequency dependent. It is required now to determine the amount of variation in the value of each coefficient, either A(f) o r B( f ). These coefficients are computed for different values of the frequencies, then the approximated values of coefficients A( f) a n d B( f ) according to the mentioned computations can be chosen. Before this process, they can be determined theoretically without going into
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any empiricism. The matrix of the phase parameters of a transmission line H(f), eqn. (l), can be transposed into a matrix of parameters in wave mode coordinates H( f), using the proposed transformation matrix T(f):
H(f)w=T(f)-‘H(f) T(f)
(5)
Since the matrix H( f), of parameters in wave mode coordinates should have a diagonal form, the following relations must hold: A2(f)+2k(f)A(f)-2=0 P(f)-2k(f)B(f)-2=0
(6)
The effect of frequency on the variation of the coefficient k(f) may now be required. The coefficient k( f) depends on the phase geometry as well as on the earth return effect. The general form for the coefficient k( f) can be found to be [l]
k(f)_4f)+Mf)+4f)-2~(f) 2Mf)+df)l
(7)
If the line phases are arranged horizontally, expression (7) will be simply formulated as [l]
k(f) =g(f)/Wf)
(8)
Now it can be seen that the coefficient k( f) for transposed transmission systems becomes constant and equal to i. The results of calculations for various values of coefficients A(f) and B(f) according to changes in the value of coefficient k( f) are Coefficient
of transfowtien
0.5 .0 0.25 Fig. 1. Dependence of the coefficients of the transformation matrix T on the coefficient k( f ).
plotted in Fig. 1. Both calculated curves can be approximated by two straight lines [ 11, in spite of the quadratic relation between them, as shown in eqn. (6). This approach will simplify the evaluation of the studied coefficients A( f) and B( f) for a certain transmission line. The coefficient k(f) has a value of zero if the mutual effect between the two outside phases is zero. In this case the spacing between the outer phases must be quite large. This condition represents the ultimate condition of asymmetry, that is, untransposed transmission lines. However, for transposed lines the coefficient k(f) appears to be a constant equal to f. This means that the coefficient k( f) varies, in practice, between zero and f. An indication as to a practical explanation for the physical meaning of a mutual parameter between the phases is necessary. In most practical cases the mutual parameter between the two edge phases is approximately half that between the middle phase and each outer phase. This can be expressed mathematically by
m(f)=%(f)
(9)
Then, the values of the coefficients of the transformation matrix for such untransposed transmission lines, A( f) and B( f ), will be 1.2 and 1.7, respectively. Here it must be noted that only coefficient B(f) differs from the value (1.8) which was published previously [l]. However, these values of the coefficients A( f ) and B( f ) will be constants if they become frequency independent. The variation of coefficient k(f) is very limited with respect to change in frequency within the range due to the earth return effect. Since the variation of coefficient k( f ) is limited, the range of change in its value will be small. Accordingly, the region of change of both coefficients of the transformation matrix, A(f) and B( f ), should be small. Referring to Fig. 1 we find that the variation of both coefficients A( f) and B( f ) will be less than the change of coefficients k( f) due to the slope of the approximated straight lines between them. This proves that the change in the values of the coefficients of the transformation matrix will be smaller. Therefore, the elements A( f) and B( f) of the transformation matrix T(f) can be considered as constants A and B respectively, and the transformation matrix may take the form of matrix T. This is the required evidence for this treatment of transient
53
processes in single-circuit transmission lines of the untransposed type. As the above transformation matrix for the parameters of untransposed transmission lines is the only one at all values of frequency, the simulation of transients in such untransposed lines in the natural modes becomes possible. Thus, only one general frequency independent transformation matrix T needs to be proposed in the form [l] 1 T=-B [1
1 1 -1O 1A
1
(10)
This equation is similar to that deduced before in ref. 1. This general matrix will simplify significantly the digital modelling of transients in untransposed transmission lines. This approach can also be applied to compute the transients in transposed transmission lines. In the case of symmetrical lines it becomes the Clark’s components matrix. Generally, this concept is known as modal analysis, which is based on the transformation matrix technique. It must be repeated that the transposition of long UHV transmission lines is difficult and expensive to accomplish. Therefore, the study of untransposed lines is mandatory and important for determination of the insulation level of untransposed tranmission lines. The insulation level of UHV transmission lines can also be calculated according to computation of the switching transients. Although it is concluded that for three-wire systems the general transformation matrix is frequency dependent, the presence of earth wires which are normally installed on towers should be included as a practical condition of transmission lines. It has been concluded [18] that the transformation matrix of three-phase transmission line parameters is still unchanged when single earth wires or even double earth wires are installed on the towers. Hence, the proposed general transformation matrix T becomes the main operator in the chosen modal analysis for the calculation of transients in untransposed lines, even with earth wires on towers. TELEGRAPHIC EQUATIONS
The main difficulty in the process of transient calculations for overhead transmission
lines appears to be the general solution of the known differential equations of lines. These equations give the relation between the voltage at a distance x along a line, u(x, t), and current i(x, t). These equations are expressed in matrix form as [l]
$ [i(x, t)] + R[i(x, t)] -& [i(x, t>l = C $ [u(x, 01 + f.34~ 01 -; [u(x, t)] = L
(114 (lib)
The solution of eqns. (11) in phase coordinates may be impossible because the parameters of transmission lines are frequency dependent due to the earth return effect as well as to the mutual effect between phases. It is known that the fundamental parameters of a line are frequency dependent and they vary widely with frequency in phase coordinates, while they are approximately constant in the c1 and /? wave modes [l]. They vary greatly in the third (zero) wave mode. Both propagation coefficient and velocity of propagation can be represented as a change in inductance. The surge impedance z, will also vary widely in the zero wave mode [ 11. Although the parameters of untransposed lines change with frequency, the elements of their transformation matrix T have been deduced to be frequency independent, as expressed by eqn. (10). Then, the original equations (11) can be transformed into wave mode coordinates (CI, B, 0) using the general frequency independent transformation matrix T [l] : -d[ u,, fi,&, dx
Gl = L, 8, 0 $[iII, &0(X9 t)l +
-d [i% 8,0(X, dx
R,, B.
o[i,,
B,
o(xv 01
(1%)
t)l = c,, p, 0 $ [v,, p,o(x, t>l +
G, p,
o[u,, p, ok t)l
( 12b)
CHARACTERISTIC IMPEDANCE
In order to simplify the solution of eqns. (12) we assume that all conductors of a line are horizontal, spacing and height are constants, loss by radiation as well as corona and conductance are neglected, and earth is homogeneous [3,41. For a loaded untransposed line with a load 2 at the receiving end, the voltage transform
54 V&p) at a distance x can be formulated for each wave mode as [l]
R/L 60
V(%P) = 27 + 2, m-t exp[ -ZS(Z - x)] exp( - SX)
z+“.z e+exp(-ZSZ) c
WY P> 100
(13)
where p is the Laplace operator and V(O,p) is the voltage transform at the sending end. In this case S, 2 and 2, correspond to each wave mode. Sometimes the characteristic impedance 2, may be approximated as the surge impedance z,. This approach has been investigated in the present research. The results prove that the surge impedance z, must not be used instead of the characteristic impedance. There is a large difference between the two terms. The difference between the frequency dependent characteristic impedance and the frequency independent surge impedance must now be justified. The percentage error D(p) in this difference is defined by D(P) =
z
-2 -
2,
x 100%
(14)
The original al(t) of this error may be formulated as
10
1 Fig. 2. Percentage error due to consideration of the surge impedance instead of the characteristic impedance for different values of the ratio R/L.
chosen. Also, it is seen that the percentage error is a damped function with time so that the exact characteristic impedance must be introduced during the calculation of transients in transmission lines. Also, all parameters of a line should be exact frequency dependent parameters. At this moment the transformation matrix is frequency independent,
VOLTAGE ANALYSIS
where I, and &, are the known modified Bessel functions. Then, formula (14) for different values of ratio R/2L is applied in order to determine the percentage error cl(t) as shown in Fig. 2. It is seen that the minimum difference occurs at a small time referred to the surge impedance z,,, and the maximum value at a short time relative to the characteristic impedance 2,. This may be related as D’(P) = WP)/D +
WPII
(16)
The original can be determined using the convolution theorem as d(O) 1 n--l 1 &4 f(n - 4 d’c*): = T,fo - f(o) m = o
(17)
where f(d = Cl- O/T* + m The time interval To for the calculations can be
Equation (13) gives the general form of the voltage transform, in wave mode coordinates, at a distance x along a load line. On the other hand, for unloaded transmission lines, the impedance of load 2 should tend to infinity so that the voltage transform in wave mode coordinates can be simplified as [l]
Equation (18) may be transformed into phase coordinates using the deduced transformation matrix T. In this case, the voltage function in the time domain should be determined in wave mode coordinates. Then the voltage in phase coordinates can be simplified in the form [l]
%I(% t> =
2(A :B)
W,(x, 0 + (A + ~)~jh, t)
55
u&, t) = --& [BU,(Z, t) + Au& t)l
(19b)
%(X7 t) = 2(A : B) [Au&, t) + (A + B)%k t) + &I(~,
(194
e1
Equations (19) give the phase voltages at a point x of the untransposed transmission lines, while they may be formulated for transposed systems as [l] u(x, 0 = ;[2%(? 0 + u,(x,
When switching the first phase a of an unloaded line at a moment t, = 0, the voltage transform at a point x: of this line in phase coordinates can be determined according to ref. 20 as follows:
01
V&P) =
+
AE,(p) A cosh[&,(Z -x)1 Vb(% P) = 2(A + B) cosh(S,Z) i - B cosh[S,(Z - x)] cosh(S,Z) 1 VC(X, P) =
SEQUENTIAL POLE SWITCHING
2B cosh[S,# -x)] cosh(S&
+ A cosh[S,,(Z - x)] cosh(S,Z)
(20)
The above analysis proves that for transposed lines there are only two wave modes while the third one will appear if the line is untransposed. This may be explained as the two wave modes being overlapped or mixed into only one mode owing to the symmetry of such lines.
- x)] 2(y$ IA cosh[S# cosh(S,Z)
J%(P) 2(A + B)
(214
(2lb)
A cosh[S,(Z - x)] cosh(S,Z)
+2B cosh[SP(Z -x)] cosh(S8Z)
Since the poles of a circuit breaker should not, in practice, be switched on or off simultaneously, their sequential closure must be considered. In this case, the voltage will be higher than that of simultaneous switching conditions. To deal with this problem the opened pole switch may be simulated by an artificial electromotive force to be short-circuited when the switch is closed. This approach is known as the direct method [7]. When a pole switch is closed, voltages will be induced across the other phases. These induced voltages must be added to the already existing voltage.
Fig. 3. The receiving-end voltage of a single-circuit untransposed transmission line erected on a double-circuit tower [ 191.
+ A cosh[S,JZ - x)] cosh(S,,Z) 1
(2Ic)
The source voltages or the voltages at the sending end of phases a, b and c are expressed as E,(p), E,(p) and E,(p), respectively. Figure 3 gives the evaluated voltage at the receiving end of a 330 kV, 200 km single-circuit untransposed transmission line [19]. A unit step function is applied as a voltage source at the sending end of phase a. Now in order to demonstrate the possible errors which may result from the proposed formula, the voltage at the receiving end of a 150 km line is computed (see Fig. 4). The circuit
Fig. 4. The calculated receiving-end voltage of a 150 km transmission line: curve 1, L = 25mH; 2, L = 200mH; -, using the proposed algorithm; - - -, using the method of ref. 16.
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breaker is closed at the peak value of a sinusoidal source. The calculated voltage is compared with that given before. It is shown that the maximum error does not exceed 5%. This proves that the accuracy of the proposed approach is quite sufficient. At the time t,, the second phase b of the circuit breaker at the sending end of a line is switched on and then the voltage transform in phase coordinates at a point x for the time period after switching the second phase can be determined in the form [20,21]
X
A[ 1
+ A[ 1
-
+
;(A - B)] cosh[S,(Z - x)] cosh(S,Z)
i(A - B)] cosh[S,(Z - x)] cosh(S,Z)
+2B cosh[S& - 41 + cosh(SBZ)
X
LE (p)
2 b
cosh[S,(Z - x)] cosh[S,(Z - x)] cosh(S,Z) cosh(&Z)
(224 vb(xvP)
=
- B cosh[S,(Z -x)] 2 cosh(S,Z)
VckP) =
( 22b)
2(y$;) X
=
’
2(A + B)’
[,,(,+A)(~-g)
+A+ +;)-B’(l+y)]E,(p) A + B + B(A - B) E,(P) - 2 Eb(P) 2(A + B)
(2W Vc(O,P) =
A2 - 2AB - 1 E,(p) + (A +
B)2
s E,(P) (23~)
The voltage transform can be calculated in phase coordinates with the help of eqns. (23) and the transformation matrix T. APPLICATION
The voltages at the receiving end of a typical horizontal single-circuit untransposed transmission line of 750 kV, 300 km, have been computed. The first phase a is assumed to be switched at time t, = 0 when the phase angle of the sending-end voltage is 60”. The second phase b is closed at tb = 3ms and finally the third phase will be switched at t, = 6 ms. The calculated voltages at the receiving end are shown in Fig. 5. This proves that the proposed algorithm can be easily programmed. It reduces the computational time as well as the computational effort. CONCLUSIONS
A[ 1 + ;(A - B)] cosh[S,(Z - x)] cosh(S,Z)
+A[ 1 - ;(A - B)] cosh[So(Z - x)] cosh(S,,Z) - 2B cosh[SB(Z -x)] cosh(S8Z) X
vb(o,P)
+
Only two elements of the transformation matrix of the transmission line parameters are
$b(d
cosh[S,(Z -x)] cosh[S,(Z - x)] cosh(S,Z) - cosh(S,Z)
(22c) At the time t, the third phase c is switched. The voltage in phase coordinates at the sending end for t > t, can be determined as [20,21]
v.(‘,d = (l+AY (A +
B)2
E(p)+ g E,(P) *
(234
Fig. 5. The receiving-end voltage of a single-circuit untransposed transmission line.
57
frequency dependent. One general transformation matrix may be chosen as a good approach. Two constant coefficients of the transformation matrix are proposed. Use of the given method is recommended for the design of long UHV untransposed transmission lines. The expressions deduced simplify the calculations and reduce the computational time and effort. Use of the convolution theorem simplifies the analysis when considering the sequential switching of circuit breaker poles. The exact characteristic impedance must be introduced in the determination of switching overvoltages in transmission lines. REFERENCES M. Hamed, On the possibility of transient calculations in single circuit untransposed transmission lines using Laplace transforms, Proc. Jordan. Int. Electr. Electron. Eng. Conf., Jordan, Amman, Vol. 1, Jordan Univ., Amman, 1983, pp. 39 - 43. M. Hamed and R. S. Momtaz, Wave propagation on overhead transmission lines under non-zero initial conditions, Proc. 27th Midwest Symposium on Circuits and Systems, West Virginia University, Morgantown, U.S.A., June 11 - 12, 1984, Western Periodicals, North Hollywood, CA, pp. 341 - 343. P. C. Magnusson, Travelling waves on multi-conductor open wire lines - a numerical survey of the effects of frequency dependence of model composition, IEEE Trans., 91 (1972) 999 - 1008. J. R. Carson, Wave propagation in overhead wires with ground return, Bell Syst. Tech. J., 5 (1926) 539 - 554. S. Hayashi, Surges on Transmission Systems, DenkiShoin, Kyoto, Japan, 1955. L. F. Dmokhoviskaya, Engineering Calculations of Internal Overvoltages in Transmission Lines, Energia, Moscow, 1972. J. P. Bickford, N. Mullineux and J. R. Reed, Computation of Power System Transients, IEE Monogr. Ser. No. 18, Peter Peregrinus, London, 1976.
8 C. Menemenlis and Z. T. Chun, Wave propagation on nonuniform lines, IEEE Trans., PAS-101 (1982) 833 839. 9 M. Hamed and D. P. Papadopoulos, Electromagnetic transients in non-uniform untransposed three phase transmission lines, J. Franklin Inst., 319 (1985) 507 511. 10 M. Sforzini, Discussion on power transmission, Electra, 73 (1980) 34 - 42. 11 J. R. Stewart and D. D. Wilson, High phase order transmission, Part I - Steady state considerations, Part II Overvoltages and insulation requirements, IEEE Trans., PAS-97 (1978) 2300 - 2317. 12 A. E. Dolgenov et al., Transients Calculations in Electric Systems Using Computers, Energia, Moscow, 1968. 13 D. P. Carrol and F. Nozari, An efficient computer method for simulating the transients on transmission lines with frequency dependent parameters, IEEE Trans., PAS-94 (1975) 1167 - 1176. 14 R. Uram and R. W. Miller, Mathematical analysis and solution of transmission line transients, IEEE Trans., PAS-83 (1964) 1116 - 1137. 15 D. E. Hedman, Propagation on overhead transmission lines, IEEE Trans., PAS-84 (1965) 200 - 211. 16 R. G. Wasley and S. Selvavinayagamoorthy, Forward and backward response functions for transmission line transient analysis, IEEE Trans., PAS-92 (1973) 665 692. 17 M. M. Ahmed, Transformation matrix of unsymmetrical transmission line parameters, Energitica, 4 (1978) 114 117. 18 M. Hamed, D. P. Papadopoulos and D. Ismail, Transformation matrix determination of transmission line parameters for various transmission system configurations, J. Franklin Inst., 319 (1985) 513 - 519. 19 Transmission Line Reference Book, 345 kV and Above, EPRI, Palo Alta, CA, 1975. 20 M. M. Ahmed, Transients in transmission lines, Proc. Fifth Znt. Congr. for Statistics and Computer Science, Cairo, Egypt, Vol. 3, Ain Shams Univ., Cairo, 1980, pp. 83 - 95. 21 D. A. Esmail, Internal overvoltages in high voltage transmission lines, Thesis, Suez Canal University, Port Said, Egypt, 1986.
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Waye Modal Technique for Switching Transients in Untransposed Transmission Lines M. HAMED Faculty of Engineering, Suez Canal University, Port Said (Egypt)
(Received March 10, 1987)
SUMMARY
This paper presents a digital analysis for the calculation of internal overvoltages in untransposed transmission lines. This analysis is based on the Laplace transform as well as on the modal technique. The frequency dependent parameters due to the earth return effect are taken into account. A general transformation matrix is determined. Its elements are evaluated numerically and then formulated theoretically. The difference between surge and characteristic impedances of a line is formulated. A formula for the voltage in wave mode coordinates of an untransposed transmission line is deduced. The sequential pole switchings of circuit breaker contacts are considered. The voltage in phase coordinates is expressed mathematically. The potential coefficients are considered as constants. Key words: transients, modal, Laplace, frequency, earth return, transformation matrix, untransposed lines.
INTRODUCTION
UHV transmission appears to offer particular advantages principally to those countries where large generation sites are located far from load centers. In such sites either coal, hydro or nuclear fuels will undoubtedly be utilised. A very important factor in the future of UHV transmission, which has been presented in the past, will be the need to move large blocks of energy over long distances because of changes in the world’s fuel situation. An advantage of UHV transmission stems from the fact that, by its use, it may be possible to utilize existing concentrated energy 037%7796/87/$3.50
sources, or even to concentrate energy sources in a few areas particularly suitable for safety or environmental reasons. This would avoid spreading generation plants all over the territory [l]. The use of UHV may be influenced more by special requirements (environmental, nuisance) than by purely economic reasons so that the future of UHV will depend on the development of proper technical solutions to match these requirements. However, EHV and UHV transmission lines are frequently used in practice. As the nominal voltage of transmission is high, the cost of transposition towers will be expensive. So, from an economic point of view, the UHV transmission lines should be, mostly, untransposed. Also, when the length of UHV lines is increased, the parameters of the different phases will not remain the same, owing to the untransposition effect [l]. In addition, the frequency dependent parameters due to the earth return effect must be introduced in transient investigations [ 2’- 41. Hence, transient studies of UHV AC transmission lines are important because of the new tendency to increase the length of UHV lines. Transients in transmission lines have been investigated extensively by numerous researchers and different methods have been published for the calculation of electromagnetic transients in lines, especially in transposed systems [5 - 71. But a general digital model for this purpose in untransposed transmission lines under all possible conditions has not so far been published [8,9]. Also, the insulation level for networks in all countries is rising continuously. This level may reach a very high value, namely 1600 kV, as proposed in ref. .lO. Thus, the accurate insulation of a transmission line appears to be an important @ Elsevier Sequoia/Printed in The Netherlands
50
item. Hence, a general accurate method for the calculation of internal overvoltages in transmission line systems should be introduced in the field of transient investigations. This is important when the system insulation level becomes high. Therefore, the design of such a line will be based on the internal overvoltages which may be induced due to normal operation of a network [ 111. A general digital model for transients in untransposed lines may be based on the Laplace concept [l]. The frequency dependent parameters of a line can be taken into account. This process depends mainly on the modal technique, if it is possible in untransposed lines.
MODAL DECOMPOS~ION
Modal analysis appears to be the most commonly used method for calculation of transients in transposed transmission lines 151. It is based on transforming the three-phase system into a line in the three natural modes. This technique can be realized by using the (CI, fi, 0) matrix of Clark’s components [12]. This means that the representation of transposed lines in wave mode coordinates is possibIe. On the other hand, the elements of the transformation matrix for untransposed single-circuit transmission lines are frequency dependent, and thus the representation of untransposed lines in wave mode coordinates is possible only at each individual frequency. But if these elements vary with frequency in a weak manner, one general transformation matrix T can then be proposed. However, according to this variation the Fourier transform [13 - 161 is used, while Duhamel’s integral has been suggested 151. In this case the computational time and effort will be increased significantly. In the present research the frequency dependent transfo~ation matrix for the parameters of untransposed transmission lines must be replaced by a general transformation vector. Its elements should be frequency independe’nt and then the wave mode coordinates of the untransposed systems can be found. This concept is interesting since it reduces the computational effort and time significantly in the calculation of transients in untransposed transmission lines.
THE TRANSFORMATION MATRIX
The matrix of parameters H of a transmission line in phase coordinates can be transformed into a diagonal matrix H, which is called the matrix of parameters of a line in wave mode coordinates. This may be done using the abovementioned transformation matrix T. Therefore, the transformation matrix appears to be comprised of the mathematical eigenvalues of the original matrix of parameters in wave mode coordinates. However, it is known that the elements of the transformation matrix are generally frequency dependent. This means that for each individual frequency a certain transformation matrix may be deduced. On the other hand, it is required that a general form for the transformation matrix at any frequency is determined so that we may obtain a general transformation matrix which will be valid for the parameters of a transmission line at all values of frequency. So, the matrices of each parameter at different frequencies must be evaluated [4]. These matrices are computed for all fundamental parameters per unit length of a line, such as resistance R, series inductance L, parallel conductance G and parallel capacitance C. Here, it must be stated that the corona and the conductance of the studied lines have been neglected. Therefore the capacitance will be frequency independent. Since the earth return effect is included, the matrix of parameters H( f) can be formulated generally as
The elements a( f), b( f) and c(f) are the selfparameters of a line, while d( f), g( f) and m( f) are the mutual parameters. Also, it should be noted that symbol ( f) means that the parameter is frequency dependent and so is the matrix itself. The propagation coefficient matrix S and characteristic impedance matrix Z, will take the general form of matrix (1). Then, the eigenvectors for such a matrix H(f) c a n b e expressed as 1
1
TAf) T&f) Tz(f) Tdf) T,(f) 1
(2)
51
It is seen that only six elements of the proposed transformation matrix T( f) of eqn. (2) are frequency dependent. Therefore, these elements at different values of frequency should be calculated. The transformation matrices for a typical 500 kV untransposed transmission line, with an average height above ground of 12 m and phaseto-phase spacing of 11 m, will be computed [ 171. The calculated transformation matrices for the resistance are listed in Table 1 [17]. The given matrices are determined using a special program since the normalization of the transformation matrix is not necessary at present. These matrices have also been found for inductance, as given in Table 2 [17]. Finally, as the capacitance is constant, its transformation matrix may be given in the form [17] T, =
1.000 1.000 -1.650 1.200 [ 0.999 0.999
1.000 -0.002 -0.999 1
(3)
Now, it is seen that the elements of the transformation matrices for the inductance matrix L vary weakly with frequency. Now, in the present study, all values of 0.999 or 0.998 will actually be taken as unity, and all values of 0.002 or 0.003 as zero. In this case only two elements will be frequency dependent, while all the other elements become constants. Then, one general form for the transformation matrix may be chosen [l]:
1
1
1
0
A(f)
-1
1
(4)
TABLE 1 The calculated transformation matrices for the resistance matrix at different frequencies Frequency &Hz) 3
25
40
50
Transformation matrix T,
1.000 -1.400 0.608 1.000 - 1.810 0.820 1.000 - 1.850 0.850 1.000 - 1.860 0.870
1.009 0.999 0.657 1.000 1.000 0.990 1.000 1.000 0.996 1.000 1.000 0.996
1.000 -1.601 -0.567 1.009 - 1.202 -0.201 1.000 -0.609 - 0.430 1.000 -0.461 -0.611
TABLE 2 The calculated transformation matrices for the inductance matrix at different frequencies Frequency W-W 3
Transformation matrix T,
1.000 - 1.830 0.999 1.000 - 1.790 0.999 1.000 - 1.780 0.999 1.000 - 1.780 0.999
25
40
50
1.000 1.090 0.999 1.000 1.116 0.999 1.000 1.120 0.999 1.000 1.120 0.999
1.000 -0.003 -0.996 1.009 -0.003 -0.997 1.000 -0.003 - 0.997 1.000 - 0.003 - 0.998
Fortunately, this transformation matrix is similar to the matrix of Clark’s components a, j?, and 0 for symmetrical transmission lines, which takes the form
1
1
1 1
T = - 2 [1
-101 1
Therefore, the elements B( f) and A(f) f o r transposed lines are constants equal to 2 and 1, respectively. This phenomenon leads us to the second step in order to compute their values at different frequencies. Since the transformation matrix of Clark’s components appears to be a special case, the transformation matrix T(f) becomes the general one for either transposed or untransposed transmission lines. It will also be called the (a, p, 0) transformation matrix. In other words, this means that the wave modes of a line will be a, /I and 0. The coefficients of the general transformation matrix It is concluded above that the two elements A( f) and B( f) in the transformation matrix of eqn. (4) are frequency dependent. It is required now to determine the amount of variation in the value of each coefficient, either A(f) o r B( f ). These coefficients are computed for different values of the frequencies, then the approximated values of coefficients A( f) a n d B( f ) according to the mentioned computations can be chosen. Before this process, they can be determined theoretically without going into
52
any empiricism. The matrix of the phase parameters of a transmission line H(f), eqn. (l), can be transposed into a matrix of parameters in wave mode coordinates H( f), using the proposed transformation matrix T(f):
H(f)w=T(f)-‘H(f) T(f)
(5)
Since the matrix H( f), of parameters in wave mode coordinates should have a diagonal form, the following relations must hold: A2(f)+2k(f)A(f)-2=0 P(f)-2k(f)B(f)-2=0
(6)
The effect of frequency on the variation of the coefficient k(f) may now be required. The coefficient k( f) depends on the phase geometry as well as on the earth return effect. The general form for the coefficient k( f) can be found to be [l]
k(f)_4f)+Mf)+4f)-2~(f) 2Mf)+df)l
(7)
If the line phases are arranged horizontally, expression (7) will be simply formulated as [l]
k(f) =g(f)/Wf)
(8)
Now it can be seen that the coefficient k( f) for transposed transmission systems becomes constant and equal to i. The results of calculations for various values of coefficients A(f) and B(f) according to changes in the value of coefficient k( f) are Coefficient
of transfowtien
0.5 .0 0.25 Fig. 1. Dependence of the coefficients of the transformation matrix T on the coefficient k( f ).
plotted in Fig. 1. Both calculated curves can be approximated by two straight lines [ 11, in spite of the quadratic relation between them, as shown in eqn. (6). This approach will simplify the evaluation of the studied coefficients A( f) and B( f) for a certain transmission line. The coefficient k(f) has a value of zero if the mutual effect between the two outside phases is zero. In this case the spacing between the outer phases must be quite large. This condition represents the ultimate condition of asymmetry, that is, untransposed transmission lines. However, for transposed lines the coefficient k(f) appears to be a constant equal to f. This means that the coefficient k( f) varies, in practice, between zero and f. An indication as to a practical explanation for the physical meaning of a mutual parameter between the phases is necessary. In most practical cases the mutual parameter between the two edge phases is approximately half that between the middle phase and each outer phase. This can be expressed mathematically by
m(f)=%(f)
(9)
Then, the values of the coefficients of the transformation matrix for such untransposed transmission lines, A( f) and B( f ), will be 1.2 and 1.7, respectively. Here it must be noted that only coefficient B(f) differs from the value (1.8) which was published previously [l]. However, these values of the coefficients A( f ) and B( f ) will be constants if they become frequency independent. The variation of coefficient k(f) is very limited with respect to change in frequency within the range due to the earth return effect. Since the variation of coefficient k( f ) is limited, the range of change in its value will be small. Accordingly, the region of change of both coefficients of the transformation matrix, A(f) and B( f ), should be small. Referring to Fig. 1 we find that the variation of both coefficients A( f) and B( f ) will be less than the change of coefficients k( f) due to the slope of the approximated straight lines between them. This proves that the change in the values of the coefficients of the transformation matrix will be smaller. Therefore, the elements A( f) and B( f) of the transformation matrix T(f) can be considered as constants A and B respectively, and the transformation matrix may take the form of matrix T. This is the required evidence for this treatment of transient
53
processes in single-circuit transmission lines of the untransposed type. As the above transformation matrix for the parameters of untransposed transmission lines is the only one at all values of frequency, the simulation of transients in such untransposed lines in the natural modes becomes possible. Thus, only one general frequency independent transformation matrix T needs to be proposed in the form [l] 1 T=-B [1
1 1 -1O 1A
1
(10)
This equation is similar to that deduced before in ref. 1. This general matrix will simplify significantly the digital modelling of transients in untransposed transmission lines. This approach can also be applied to compute the transients in transposed transmission lines. In the case of symmetrical lines it becomes the Clark’s components matrix. Generally, this concept is known as modal analysis, which is based on the transformation matrix technique. It must be repeated that the transposition of long UHV transmission lines is difficult and expensive to accomplish. Therefore, the study of untransposed lines is mandatory and important for determination of the insulation level of untransposed tranmission lines. The insulation level of UHV transmission lines can also be calculated according to computation of the switching transients. Although it is concluded that for three-wire systems the general transformation matrix is frequency dependent, the presence of earth wires which are normally installed on towers should be included as a practical condition of transmission lines. It has been concluded [18] that the transformation matrix of three-phase transmission line parameters is still unchanged when single earth wires or even double earth wires are installed on the towers. Hence, the proposed general transformation matrix T becomes the main operator in the chosen modal analysis for the calculation of transients in untransposed lines, even with earth wires on towers. TELEGRAPHIC EQUATIONS
The main difficulty in the process of transient calculations for overhead transmission
lines appears to be the general solution of the known differential equations of lines. These equations give the relation between the voltage at a distance x along a line, u(x, t), and current i(x, t). These equations are expressed in matrix form as [l]
$ [i(x, t)] + R[i(x, t)] -& [i(x, t>l = C $ [u(x, 01 + f.34~ 01 -; [u(x, t)] = L
(114 (lib)
The solution of eqns. (11) in phase coordinates may be impossible because the parameters of transmission lines are frequency dependent due to the earth return effect as well as to the mutual effect between phases. It is known that the fundamental parameters of a line are frequency dependent and they vary widely with frequency in phase coordinates, while they are approximately constant in the c1 and /? wave modes [l]. They vary greatly in the third (zero) wave mode. Both propagation coefficient and velocity of propagation can be represented as a change in inductance. The surge impedance z, will also vary widely in the zero wave mode [ 11. Although the parameters of untransposed lines change with frequency, the elements of their transformation matrix T have been deduced to be frequency independent, as expressed by eqn. (10). Then, the original equations (11) can be transformed into wave mode coordinates (CI, B, 0) using the general frequency independent transformation matrix T [l] : -d[ u,, fi,&, dx
Gl = L, 8, 0 $[iII, &0(X9 t)l +
-d [i% 8,0(X, dx
R,, B.
o[i,,
B,
o(xv 01
(1%)
t)l = c,, p, 0 $ [v,, p,o(x, t>l +
G, p,
o[u,, p, ok t)l
( 12b)
CHARACTERISTIC IMPEDANCE
In order to simplify the solution of eqns. (12) we assume that all conductors of a line are horizontal, spacing and height are constants, loss by radiation as well as corona and conductance are neglected, and earth is homogeneous [3,41. For a loaded untransposed line with a load 2 at the receiving end, the voltage transform
54 V&p) at a distance x can be formulated for each wave mode as [l]
R/L 60
V(%P) = 27 + 2, m-t exp[ -ZS(Z - x)] exp( - SX)
z+“.z e+exp(-ZSZ) c
WY P> 100
(13)
where p is the Laplace operator and V(O,p) is the voltage transform at the sending end. In this case S, 2 and 2, correspond to each wave mode. Sometimes the characteristic impedance 2, may be approximated as the surge impedance z,. This approach has been investigated in the present research. The results prove that the surge impedance z, must not be used instead of the characteristic impedance. There is a large difference between the two terms. The difference between the frequency dependent characteristic impedance and the frequency independent surge impedance must now be justified. The percentage error D(p) in this difference is defined by D(P) =
z
-2 -
2,
x 100%
(14)
The original al(t) of this error may be formulated as
10
1 Fig. 2. Percentage error due to consideration of the surge impedance instead of the characteristic impedance for different values of the ratio R/L.
chosen. Also, it is seen that the percentage error is a damped function with time so that the exact characteristic impedance must be introduced during the calculation of transients in transmission lines. Also, all parameters of a line should be exact frequency dependent parameters. At this moment the transformation matrix is frequency independent,
VOLTAGE ANALYSIS
where I, and &, are the known modified Bessel functions. Then, formula (14) for different values of ratio R/2L is applied in order to determine the percentage error cl(t) as shown in Fig. 2. It is seen that the minimum difference occurs at a small time referred to the surge impedance z,,, and the maximum value at a short time relative to the characteristic impedance 2,. This may be related as D’(P) = WP)/D +
WPII
(16)
The original can be determined using the convolution theorem as d(O) 1 n--l 1 &4 f(n - 4 d’c*): = T,fo - f(o) m = o
(17)
where f(d = Cl- O/T* + m The time interval To for the calculations can be
Equation (13) gives the general form of the voltage transform, in wave mode coordinates, at a distance x along a load line. On the other hand, for unloaded transmission lines, the impedance of load 2 should tend to infinity so that the voltage transform in wave mode coordinates can be simplified as [l]
Equation (18) may be transformed into phase coordinates using the deduced transformation matrix T. In this case, the voltage function in the time domain should be determined in wave mode coordinates. Then the voltage in phase coordinates can be simplified in the form [l]
%I(% t> =
2(A :B)
W,(x, 0 + (A + ~)~jh, t)
55
u&, t) = --& [BU,(Z, t) + Au& t)l
(19b)
%(X7 t) = 2(A : B) [Au&, t) + (A + B)%k t) + &I(~,
(194
e1
Equations (19) give the phase voltages at a point x of the untransposed transmission lines, while they may be formulated for transposed systems as [l] u(x, 0 = ;[2%(? 0 + u,(x,
When switching the first phase a of an unloaded line at a moment t, = 0, the voltage transform at a point x: of this line in phase coordinates can be determined according to ref. 20 as follows:
01
V&P) =
+
AE,(p) A cosh[&,(Z -x)1 Vb(% P) = 2(A + B) cosh(S,Z) i - B cosh[S,(Z - x)] cosh(S,Z) 1 VC(X, P) =
SEQUENTIAL POLE SWITCHING
2B cosh[S,# -x)] cosh(S&
+ A cosh[S,,(Z - x)] cosh(S,Z)
(20)
The above analysis proves that for transposed lines there are only two wave modes while the third one will appear if the line is untransposed. This may be explained as the two wave modes being overlapped or mixed into only one mode owing to the symmetry of such lines.
- x)] 2(y$ IA cosh[S# cosh(S,Z)
J%(P) 2(A + B)
(214
(2lb)
A cosh[S,(Z - x)] cosh(S,Z)
+2B cosh[SP(Z -x)] cosh(S8Z)
Since the poles of a circuit breaker should not, in practice, be switched on or off simultaneously, their sequential closure must be considered. In this case, the voltage will be higher than that of simultaneous switching conditions. To deal with this problem the opened pole switch may be simulated by an artificial electromotive force to be short-circuited when the switch is closed. This approach is known as the direct method [7]. When a pole switch is closed, voltages will be induced across the other phases. These induced voltages must be added to the already existing voltage.
Fig. 3. The receiving-end voltage of a single-circuit untransposed transmission line erected on a double-circuit tower [ 191.
+ A cosh[S,JZ - x)] cosh(S,,Z) 1
(2Ic)
The source voltages or the voltages at the sending end of phases a, b and c are expressed as E,(p), E,(p) and E,(p), respectively. Figure 3 gives the evaluated voltage at the receiving end of a 330 kV, 200 km single-circuit untransposed transmission line [19]. A unit step function is applied as a voltage source at the sending end of phase a. Now in order to demonstrate the possible errors which may result from the proposed formula, the voltage at the receiving end of a 150 km line is computed (see Fig. 4). The circuit
Fig. 4. The calculated receiving-end voltage of a 150 km transmission line: curve 1, L = 25mH; 2, L = 200mH; -, using the proposed algorithm; - - -, using the method of ref. 16.
56
breaker is closed at the peak value of a sinusoidal source. The calculated voltage is compared with that given before. It is shown that the maximum error does not exceed 5%. This proves that the accuracy of the proposed approach is quite sufficient. At the time t,, the second phase b of the circuit breaker at the sending end of a line is switched on and then the voltage transform in phase coordinates at a point x for the time period after switching the second phase can be determined in the form [20,21]
X
A[ 1
+ A[ 1
-
+
;(A - B)] cosh[S,(Z - x)] cosh(S,Z)
i(A - B)] cosh[S,(Z - x)] cosh(S,Z)
+2B cosh[S& - 41 + cosh(SBZ)
X
LE (p)
2 b
cosh[S,(Z - x)] cosh[S,(Z - x)] cosh(S,Z) cosh(&Z)
(224 vb(xvP)
=
- B cosh[S,(Z -x)] 2 cosh(S,Z)
VckP) =
( 22b)
2(y$;) X
=
’
2(A + B)’
[,,(,+A)(~-g)
+A+ +;)-B’(l+y)]E,(p) A + B + B(A - B) E,(P) - 2 Eb(P) 2(A + B)
(2W Vc(O,P) =
A2 - 2AB - 1 E,(p) + (A +
B)2
s E,(P) (23~)
The voltage transform can be calculated in phase coordinates with the help of eqns. (23) and the transformation matrix T. APPLICATION
The voltages at the receiving end of a typical horizontal single-circuit untransposed transmission line of 750 kV, 300 km, have been computed. The first phase a is assumed to be switched at time t, = 0 when the phase angle of the sending-end voltage is 60”. The second phase b is closed at tb = 3ms and finally the third phase will be switched at t, = 6 ms. The calculated voltages at the receiving end are shown in Fig. 5. This proves that the proposed algorithm can be easily programmed. It reduces the computational time as well as the computational effort. CONCLUSIONS
A[ 1 + ;(A - B)] cosh[S,(Z - x)] cosh(S,Z)
+A[ 1 - ;(A - B)] cosh[So(Z - x)] cosh(S,,Z) - 2B cosh[SB(Z -x)] cosh(S8Z) X
vb(o,P)
+
Only two elements of the transformation matrix of the transmission line parameters are
$b(d
cosh[S,(Z -x)] cosh[S,(Z - x)] cosh(S,Z) - cosh(S,Z)
(22c) At the time t, the third phase c is switched. The voltage in phase coordinates at the sending end for t > t, can be determined as [20,21]
v.(‘,d = (l+AY (A +
B)2
E(p)+ g E,(P) *
(234
Fig. 5. The receiving-end voltage of a single-circuit untransposed transmission line.
57
frequency dependent. One general transformation matrix may be chosen as a good approach. Two constant coefficients of the transformation matrix are proposed. Use of the given method is recommended for the design of long UHV untransposed transmission lines. The expressions deduced simplify the calculations and reduce the computational time and effort. Use of the convolution theorem simplifies the analysis when considering the sequential switching of circuit breaker poles. The exact characteristic impedance must be introduced in the determination of switching overvoltages in transmission lines. REFERENCES M. Hamed, On the possibility of transient calculations in single circuit untransposed transmission lines using Laplace transforms, Proc. Jordan. Int. Electr. Electron. Eng. Conf., Jordan, Amman, Vol. 1, Jordan Univ., Amman, 1983, pp. 39 - 43. M. Hamed and R. S. Momtaz, Wave propagation on overhead transmission lines under non-zero initial conditions, Proc. 27th Midwest Symposium on Circuits and Systems, West Virginia University, Morgantown, U.S.A., June 11 - 12, 1984, Western Periodicals, North Hollywood, CA, pp. 341 - 343. P. C. Magnusson, Travelling waves on multi-conductor open wire lines - a numerical survey of the effects of frequency dependence of model composition, IEEE Trans., 91 (1972) 999 - 1008. J. R. Carson, Wave propagation in overhead wires with ground return, Bell Syst. Tech. J., 5 (1926) 539 - 554. S. Hayashi, Surges on Transmission Systems, DenkiShoin, Kyoto, Japan, 1955. L. F. Dmokhoviskaya, Engineering Calculations of Internal Overvoltages in Transmission Lines, Energia, Moscow, 1972. J. P. Bickford, N. Mullineux and J. R. Reed, Computation of Power System Transients, IEE Monogr. Ser. No. 18, Peter Peregrinus, London, 1976.
8 C. Menemenlis and Z. T. Chun, Wave propagation on nonuniform lines, IEEE Trans., PAS-101 (1982) 833 839. 9 M. Hamed and D. P. Papadopoulos, Electromagnetic transients in non-uniform untransposed three phase transmission lines, J. Franklin Inst., 319 (1985) 507 511. 10 M. Sforzini, Discussion on power transmission, Electra, 73 (1980) 34 - 42. 11 J. R. Stewart and D. D. Wilson, High phase order transmission, Part I - Steady state considerations, Part II Overvoltages and insulation requirements, IEEE Trans., PAS-97 (1978) 2300 - 2317. 12 A. E. Dolgenov et al., Transients Calculations in Electric Systems Using Computers, Energia, Moscow, 1968. 13 D. P. Carrol and F. Nozari, An efficient computer method for simulating the transients on transmission lines with frequency dependent parameters, IEEE Trans., PAS-94 (1975) 1167 - 1176. 14 R. Uram and R. W. Miller, Mathematical analysis and solution of transmission line transients, IEEE Trans., PAS-83 (1964) 1116 - 1137. 15 D. E. Hedman, Propagation on overhead transmission lines, IEEE Trans., PAS-84 (1965) 200 - 211. 16 R. G. Wasley and S. Selvavinayagamoorthy, Forward and backward response functions for transmission line transient analysis, IEEE Trans., PAS-92 (1973) 665 692. 17 M. M. Ahmed, Transformation matrix of unsymmetrical transmission line parameters, Energitica, 4 (1978) 114 117. 18 M. Hamed, D. P. Papadopoulos and D. Ismail, Transformation matrix determination of transmission line parameters for various transmission system configurations, J. Franklin Inst., 319 (1985) 513 - 519. 19 Transmission Line Reference Book, 345 kV and Above, EPRI, Palo Alta, CA, 1975. 20 M. M. Ahmed, Transients in transmission lines, Proc. Fifth Znt. Congr. for Statistics and Computer Science, Cairo, Egypt, Vol. 3, Ain Shams Univ., Cairo, 1980, pp. 83 - 95. 21 D. A. Esmail, Internal overvoltages in high voltage transmission lines, Thesis, Suez Canal University, Port Said, Egypt, 1986.