Transient analysis of nonuniform lossy transmission lines with frequency dependent parameters

Electric Power Systems Research 52 (1999) 223 – 228 www.elsevier.com/locate/epsr

Transient analysis of nonuniform lossy transmission lines with frequency dependent parameters M.S. Mamis *, M. Koksal Electrical and Electronics Engineering Department, Inonu Uni6ersity, 44100 Malatya, Turkey Received 25 November 1998; accepted 4 January 1999

Abstract An s-domain method for transient analysis of lossy, frequency dependent nonuniform transmission lines having parameters arbitrary varying with space is presented. Nonuniform line is divided into adequate number of sections which are then assumed uniform. The terminal equation for the nonuniform line is calculated by using the terminal equations of the uniform line sections. By using the boundary conditions, total response in s-domain is obtained. Fast inverse Laplace transform is used for frequency to time domain conversion. In one of the examples, the step response of an open ended nonuniform line is obtained and the results are compared with those obtained by lattice-diagram technique. The effect of number of sections is investigated and it is concluded that for sufficiently large number of sections, the results approach to the analytical solutions which can be obtained only for very special nonuniform lines. In the second example, the method is applied for the calculation of transmission tower lightning surge response. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Transient analysis; Nonuniform lines; Fast inverse Laplace transform; Tower surge response

1. Introduction In power transmission and distribution networks, if the distance between the line and ground varies, the change of line parameters is nonuniform. Overhead line-cable connections, sagging overhead lines and steel line towers are examples of the lines having nonuniform parameters. In these lines, the inductance per unit length is higher at points far to the ground whereas the capacitance per unit length is lower compared to the corresponding values close to the ground. Modeling and analysis of nonuniform lines in the complex frequency (s) or time (t) domains have recently gained interest and very little work has appeared in the literature [1–5]. Excluding the well known conventional methods using travelling wave technique [1], s-domain analysis of nonuniform lines having exponential space variations was proposed by Saied et al. [2]. The technique they used has limitations on implementing transmission line losses and frequency dependence of parameters. This method was further developed by Oufi * Corresponding author.

et al. [3]. In their work, the restrictions on the space variations of the characteristic impedance were removed by dividing the line into sections appropriate for exponential approximation. However, the limitations of this method as well as the others such as the neglect of the frequency dependence of the line parameters and the consideration of the lossless lines do still exist as its drawbacks. A time domain method based on finite difference approximation of the wave propagation equation was recently proposed by Correia de Barros and Almeida [4], which can be interfaced with Electromagnetic Transients Program (EMTP). Later, Nguyen et al. introduced an EMTP compatible time domain characteristic model of a nonuniform line for electromagnetic transient simulation [5]. Despite the improvements in this field, the fact is that an efficient method for transient analysis of nonuniform transmission lines that include line losses and frequency-dependent parameters has not been developed yet. The method presented in this paper uses s-domain for the analysis of nonuniform lines. The line having nonuniform parameters is divided into adequate number of sections and the parameters of each section are approximated by uniform parameters. For each section,

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Fig. 1. Nonuniform transmission line constructed by the cascaded connection of uniform line sections.

instead of the traditional lumped parameter modeling, distributed parameter uniform line modeling is used. By this approach, terminal equations of uniform sections involving hyperbolic functions are obtained in matrix form. These equations are used to calculate a terminal equation for the whole nonuniform line in s-domain. The proposed method offers two advantages: One, nonuniform lines having arbitrary variation can be easily simulated. Second, transmission line losses and frequency dependence of parameters can be also included into the analysis.

where Z= r+sl and Y=g+ sc are the series impedance and shunt admittance per unit distance, and V and I are the voltage and current phasors. The solution for terminal voltages and currents of the uniform line shown in Fig. 2 can be easily obtained in frequency domain as VS = IR cosh gd+ VRZ0 sinh gd,

(3a)

1 IS = IR cosh gd+ VRZ − sinh gd, 0

(3b)

2. Mathematical modeling of nonuniform lines

where the subscripts R and S stand for the receiving and the sending ends, respectively, and d is the total line length [6]. Z0 and g are complex characteristic impedance and propagation constant, respectively, and they are defined by

2.1. Terminal equations of uniform line

Z0 = Z/Y,

(4a)

g= ZY.

(4b)

To overcome the difficulties in implementing the losses and frequency dependence, nonuniform transmission line is considered as cascaded connection of uniform line sections as shown in Fig. 1. Therefore, we shall give the terminal equations for uniform line which are thereafter used to determine the terminal equations of nonuniform line. Four electrical characteristics (resistance, inductance, conductance and capacitance) of a uniform transmission line are distributed uniformly along the line. The voltage and current wave propagation along the line (at a point x) are related to the line’s distributed resistance (per unit length) r, inductance l, conductance g and capacitance c by the generalized telegrapher’s equations −

(n(x, t) (i(x, t) = ri(x, t)+ l , (x (t

(1a)

−

(n(x, t) (n(x, t) = g6(x, t) +c . (x (t

(1b)

Note that Eqs. (3a) and (3b) are also valid in the case that the series parameters of transmission line (r and l) are frequency dependent. In this case, the series impedance Z= r+ sl is replaced by an adequate expression yielding the frequency dependence.

2.2. Terminal equations for nonuniform line Since the terminal equations of each section can be found even for the lossy and frequency dependent cases [7,8], the terminal equations of the whole line shown in Fig. 1 and hence the response of the system involving such a line can be calculated. Rewriting Eqs. (3a) and (3b) for ith section having series impedance Zi = ri = Sli and shunt admittance Yi = gi + SCi, we obtain

Frequency domain behavior of a transmission line can be obtained by taking the Laplace transform of Eqs. (1a) and (1b) with respect to time as −

dV =ZI, dx

(2a)

−

dI =YV, dx

(2b)

Fig. 2. Uniform transmission line.

M.S. Mamis, M. Koksal / Electric Power Systems Research 52 (1999) 223–228

 n 

V Si cosh g i di Z oi sinh gi di = −1 I Si Z oi sinh gi di cosh g i di

'

n n V Ri I Ri

(5)

Zi and goi = Zi, Yi are the characteriswhere Zoi = Yi tic impedance and the propagation coefficient of the ith section, respectively; and di is the length of this section starting from x= xi and ending at x = xi + 1. The constant resistance parameter ri of the ith section is chosen as the arithmetic mean of the values of the space dependent resistance r(x) of the line at the end points of this section, i.e. ri =

r(xi )+r(xi + 1) . 2

(6)

Other parameters li, gi and ci can be calculated in the same manner. Eq. (5) can be written in the compact form as (7)

Si = Hi Ri

H= H1H2 …Hn

V Si , Si = I Si

(8a)

cosh g i di Z oi sinh gi di Hi = − 1 Z oi sinh gi di cosh g i di and

 n

V Ri = Ri . I Ri

n

(8b)

2.3. Fast in6erse of Laplace transform Many algorithms for the numerical computation of the inverse Laplace transform are found in the literature. For frequency to time domain conversion, the algorithm developed by Hosono [9] has been selected for its accuracy, efficiency, and ease of implementation; due to these properties, this method is known to be Fast Inverse Laplace Transform (FILT) [2,9]. Laplace inversion formula for a function F(s) is f(t)=

1 2pj

(8c)

(9)

By using the relations in Eqs. (7) and (9), the terminal equation S1 = (H1H2 … Hn )Rn.

(10)

can be written for the whole line. Since the sending-end (receiving-end) of the first (last) section is the sending-end (receiving-end) of the actual line, changing the notation accordingly Eq. (10) can be written as (11)

S =HR; where

 n  n

V S =S1 = S , IS R =Rn =

VR IR

&

g + j

(13)

F(s)exp(st)ds.

g − j

For evaluation of the inverse Laplace transform numerically, Hosono used an approximation for exp(s) as es=

ea , 2 cosh(a − s)

=e s − e − 2ae 3s + e − 4ae 5s − …,

Since the receiving-end quantities of the ith section are the sending-end quantities for the (i +1)-th section, we can write Ri = Si + 1 =Hi + 1Ri + 1.

(12c)

is the terminal coefficient matrix of the line. Once Eq. (11) is calculated for any frequency s= s + jv, voltages and currents at the terminals of the line at that frequency can be calculated by considering the boundary conditions at both terminations.

where

 n 

225

(14)

which is obviously valid for sufficiently large a, i.e. a 1. Using Eq. (14), the inverse Laplace transform in Eq. (13) is approximated by f(t)=(ea/t)(F1 + F2 + F3 + …),

(15)

where Fn = (−1)n Im F{[a + j(n− 0.5)p]/t}.

(16)

Retaining the first (k−1) terms of the infinite series in Eq. (15) and applying the Euler transformation on the rest, the following series which is used for fast inverse Laplace transform is obtained;



k−1

p

n=1

n=0

n

f(t)= (ea/t) % Fn + (1/2p + 1) % ApnFk + n .

(17)

The truncation coefficients Apn are defined recursively by App = 1,

Apn − 1 = Apn +

  p+ 1 n

(18)

(12a) (12b)

are the sending-end and receiving-end terminal variables, and

where the last term denotes combinations. To obtain the time domain solution of the system involving nonuniform transmission line, the frequency domain results obtained in the previous section are transformed to time domain by the use of FILT shortly described in this section.

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Fig. 3. Step response of an exponential line obtained by: — , proposed method; and —, lattice method.

2.4. Computer implementation Based on the experience gained on FILT when applied to transmission lines, choose proper values for a, k, p; satisfactory results are obtained if a is chosen to be between 3 and 5 [2,3,9,10]. p and k ( \p) must be chosen sufficiently high to achieve the required accuracy [9,10]; otherwise considerable truncation errors may cause misleading results. To start the computation of any response of the system in the interval (Dt, tmax), specify the above parameters Dt and which is the time difference between the subsequent time instants at which the response is desired to be known. In order to compute f(ti ), ti =iDt, i= 1,2, …; 1. Compute sn = (a+ j(n −0.5)p/ti for n = 1, 2, …, k+ p, 2. Determine transfer matrix H in Eq. (11) for each sn and calculate s-domain output variable (voltage or current) as described in Section 2.2, 3. Substituting the s-domain voltages or currents computed in Step 2 in Eqs. (16) and (15), compute time domain solution at ti. 4. If ti 5tmax, set i= i+ 1, go to step 1; otherwise stop.

Fig. 4. Effect of the number of sections on the calculated step response: — , step response with five sections; and — , with ten sections.

Fig. 4 shows the effect of number of uniform line sections used for representing the nonuniform line. In the figure, the step responses calculated for five and ten sections are plotted. As seen from the figure, the curve obtained by using ten sections has less reflection effects due to smaller differences in characteristic impedances of cascaded uniform line sections. These effects are further reduced as in the plot of Fig. 3 where 20 sections are used. Example 2. Transients along a transmission tower In order to be able to make comparisons, the same example studied by Menemenlis and Chun is considered here. In their work, the transients along the highvoltage transmission tower shown in Fig. 5 were calculated by using lattice diagram technique [11].

3. Results and discussions Example 1. Step response of open ended line In this example, an exponential line with characteristic impedance (Zo =150e qx V (q =0.00766 m − 1) and propagation constant g =1/c (c is the velocity of light) is considered. The line is 50 m long and its receivingend is open. Step responses of the line calculated by the proposed method with FILT and by the time domain lattice diagram technique [11] are shown in Fig. 3. In both methods, 20 uniform distributed parameter line sections are used for the simulation of the nonuniform line. Except the decaying weak local oscillations due to use of FILT, the results are well agreed as seen from the figure.

Fig. 5. A high voltage transmission tower hit by a direct lightning stroke.

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227

spectively) at the tower top after the lightning stroke. The receiving-end voltage at the bottom of the tower is also computed and plotted in Fig. 6(c). As seen from the plot in Fig. 6(a), the sending-end (tower top) voltage reaches to its peak at t= 0.11 ms and it is nearly 8.5 MV at this time. The approximate theoretical value of this voltage is 9.2 MV; this is calculated by multiplying the value of the sending-end current at t= 0.11 ms, which is  26 kA, by the surge impedance which is equal to 353 V at the tower top. The difference between the theoretical and computed values is due to the fact that the line impedance changes from top to the bottom. The sending-end voltage drops suddenly due to reflected wave at receiving-end at  t=0.5 ms which is twice the travel time of line t= 75/3 ×108 s. The sending-end current increases at the same time due to arrival of the reflected current wave. The arrival of the incident wave to the line end at t= 0.25 ms is also observed in Fig. 6(c). Fig. 6 compares favorably with the results given in [1,3] where the methods given do not work in the case of frequency dependence and losses. In case of consideration of losses, the resistance of the tower is introduced into Eq. (19) which is now takes the form

Zo (x)= (50+ 35 x)

Fig. 6. Transient response of the transmission tower: (a) sending-end voltage; (b) sending-end current; and (c) receiving-end voltage.

The tower was modeled by a nonuniform line with the characteristic impedance Zo (x) =50 +35 x V

(19)

where x (m) is the height measured from the ground level. The propagation constant was assumed to be g= 1/c. The lightning surge was characterized by a current expressed as i(t)= 30.397(e − t/t1 −e − t/t2), where i(t) is in kA, t is in ms, t1 =17.63 ms and t2 = 0.0316 ms. This current represents a 0.2/25 ms impulse with a 30 kA peak value. They assumed that this current was fed by a current source having a 250 V source impedance. For the application of the proposed method to the tower problem in concern, the nonuniform line is divided into 5, 10, 20 uniform distributed parameter sections all equal in length. Satisfactory results are obtained by FILT with a =5, k = 20 and p = 10. As in the first example, it is observed that ten sections are enough to obtain sufficiently accurate results. For ten sections, the results shown in Fig. 6 are obtained. Fig. 6(a,b) shows voltage and current waveforms (re-

'

a 1+ V s

(20)

where a is responsible for the ohmic loss of the tower. For a= 3× 106, 6× 106 s − 1, the tower response for the receiving-end voltage is computed and the comparison of the variations with the lossless case is shown in Fig. 7. Consideration of the other cases such as frequency dependence of parameters and skin effects, only the form of last term in the square root in Eq. (20) will be changed; the solution method is not effected by these sophistications and it equally applies.

Fig. 7. Effect of tower loss on receiving-end voltage: — , lossless. Lossy cases: – – – , a = 2 ×106; and ----- -, a =6 ×106.

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4. Conclusions A numerical method for the transient analysis of nonuniform lines is presented. A nonuniform line is simulated by adequate number of uniform line sections. By this way, the terminal equation describing the whole line, and the response of the system it is involved are calculated in s-domain. To find the real time solutions FILT is used. Since uniform lines are used to represent the nonuniform line, frequency-dependent parameters and transmission line losses can easily be included in the analysis. Mathematical formulation and computer implementation of the proposed technique is quite easy comparing with the other s-domain methods. It is shown that accurate results can be obtained by using sufficiently large number of sections.

References [1] C. Menemenlis, Z.T. Chun, Wave propagation on nonuniform lines, IEEE Trans. Power Syst. PAS-101 (4) (1982) 833– 839. [2] M.M. Saied, A.S. AlFuhaid, M.E. Elshanelwily, s-domain analysis of electromagnetic transients on nonuniform lines, IEEE Trans. Power Deliv. 5 (4) (1990) 2072–2083.

.

[3] E.A. Oufi, A.S. AlFuhaid, M.M. Saied, Transient analysis of lossless single-phase nonuniform transmission lines, IEEE Trans. Power Deliv. 9 (3) (1994) 1694 – 1701. [4] M.T. Correia de Barros, M.E. Almeida, Computation of electromagnetic transients on nonuniform transmission lines, IEEE Paper No. 397-0 PWRD, Summer Power Meeting, Portland, Oregon, 1995. [5] H.V. Nguyen, H.W. Dommel, J.R. Marti, Modelling of singlephase nonuniform transmission lines in electromagnetic transient simulations, IEEE Trans. Power Deliv. 12 (2) (1997) 916 – 921. [6] A.E. Guile, W. Paterson, Electrical Power Systems, vol. 1, 2nd ed., Pergamon Press, Oxford, UK, 1982. [7] J.P. Bickford, J.V.H. Sanderson, M.M. Abdelsalem, S.E.T. Mohamed, S.A. Morais, O. Olipade, Developments in the calculation of waveforms and frequency spectra for transient fault currents and voltages, IEE Proc. 127 (3) (1980) 145 – 152. [8] P. Moreno, R. Rosa, J.L. Naredo, Frequency domain computation of transmission line closing transients, IEEE Trans. Power Deliv. 6 (1) (1991) 275 – 281. [9] T. Hosono, Numerical inversion of Laplace-transform and some applications to wave optics, Radio Sci. 16 (1981) 1015– 1019. [10] M.S. Mamis, Steady-state and transient analysis of power transmission lines by using state-space techniques, Ph.D. Thesis, University of Gaziantep, Gaziantep, Turkey, 1997. [11] L.V. Bewley, Travelling Waves on Transmission Systems, 2nd ed., New York, Dover, 1963.