Implementation of a Transmission Line Model with the PEEC Method for Lightning Surge Analysis

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1 Implementation of a Transmission Line Model with the PEEC Method for Lightning Surge Analysis PEERAWUT YUTTHAGOWITH Department of Electrical Engineering, Faculty of Engineering King Mongkut s Institute of Technology Ladkrabang 1, Chalongkrung Road, Ladkrabang, Bangkok THAILAND kypeeraw@kmitl.ac.th Abstract: - This paper proposes a simple transmission line model adopted with the partial element equivalent circuit (PEEC) method for lightning surge analysis. The proposed approach is employed for calculating tower surge responses and voltages across insulators of an actual transmission tower. The proposed approach increases the efficiency of the method in terms of computation time because of reducing a number of elements in the PEEC method. Comparisons of the simulation results by the proposed approach in the time domain, and by the PEEC method without a transmission line model and available experimental data shows satisfactory agreement not only for amplitudes but also for waveshapes. Key-Words: -, Lightning surge analysis, Partial element equivalent circuit, Tower surge response, Transmission line model. 1 Introduction For an economical insulation design of the transmission and distribution systems, predictive calculations of overvoltages due to lightning in the transmission and distribution systems is most essential. Traditionally, circuit-based models and transmission line approaches assuming transverse electromagnetic (TEM) propagation mode were employed for the calculation of overvoltages due to a lightning strike to a transmission system [1]-[4]. The assumption of the TEM propagation mode might not be correct for a tall tower in EHV and UHV transmission systems, because of the nonuniform behavior of the tower, and reflections and scattering from the discontinuities and tower ends [5]-[8]. This behavior cannot be expressed by a time-independent model unless the rise time of the current is much longer than the round-trip time of the traveling wave. To obtain more accurate results, full-wave approaches such as the method of moment (MoM) [5]-[8], the finite-difference dime-domain (FDTD) technique [9]-[11], or a hybrid electromagnetic-circuit method such as the PEEC method [12]-[15] are more appropriate than a circuit and transmission line approach. The PEEC method is derived from a mixed potential integral equation (MPIE) and provides a full wave solution to Maxwell s equations. The method can be applied to both time and frequency domains. A major difference from the MoM is that the problem under consideration is transformed into the circuit domain where circuit analysis techniques can be applied. A detailed description of the approach can be found in [15], [16]. The calculated results by the full-wave PEEC method have shown good agreement in comparison with the results calculated by the MoM and the FDTD methods, as well as with experimental data [12]-[16]. Moreover, in some specific cases, the computational efficiency of the method was found to be better than that of the MoM and the FDTD method, because post processing for calculating voltages and currents along the studied system is not required. For the formulation of the full-wave PEEC method in the frequency domain, time responses are obtained by using the inverse Fourier's transform. As well known, a frequency domain analysis for transient studies might be time consuming because of the requirement of inversion of impedance matrix at every frequency step. A fullwave time-domain PEEC formulation can therefore be interesting, especially when dealing with nonlinearities. Voltages and currents along the structure should be memorized at every time step. To consider the presence of conducting ground, the image method can be employed but the solution matrix size is increased due to retardation effects. Therefore, the computational efficiency of the full-wave time-domain PEEC method is not always higher than that of a frequency domain PEEC implementation. Neglecting retardation effects of electromagnetic waves will increase the computational time efficiency of the PEEC method in the time domain because the past time of voltages and currents are memorized only at one previous time step. The inversion of impedance matrix is required only once and also the model of reduction (MOR) [17], [18] can be employed to reduce the equation size. However, in a typical configuration ISBN:

2 involving a few thousands of elements and a few hundred calculation points, over 8 percent of the computation time is spent for the formulation of the impedance matrix and inversion. Therefore, the MOR is not an effective way to increase time efficiency of the method in this case. As known [4], the transmission line approach assuming TEM propagation mode can be successfully applied to systems of horizontal conductors. Because the PEEC method transforms the MPIE to the circuit domain, circuit elements and also transmission lines can be implemented readily with the method. In the case of a lightning surge analysis in electrical power systems, most of elements (overhead lines) can be represented as horizontal wires. Therefore, a combination of the transmission line theory and the PEEC method would be an effective way to increase the efficiency of the method in terms of computation time and memory requirement. In this paper, the PEEC method neglecting the retardation effect (quasi-static approximation) in the time domain combined with the transmission line theory is proposed for lightning surge analysis. 2 The PEEC Method Combined with a Transmission Line Model Procedures for obtaining solution of the PEEC method start from discretizing geometry structure into small cells or elements which are composed of current cells and charge or potential cells. The current cells and potential cells are interleaved each other. The rectangular pulse is employed both charge and current basis functions. Then, Galerkin s method is applied to enforce the mixed potential integral equation which is interpreted as Kirchhoff's voltage law applied to a current cell, and the continuity equation or the charge conservation equation is applied via Kirchhoff's current law to a potential cell. Whole system equations in the frequency domain can be written in a matrix form corresponding to a modified nodal analysis (MNA) formulation as shown in (1). T jp 1 Y A IS (1) A R jl I U S where A is an incident matrix which expresses the cell connectivity, R is a matrix of series resistances of current cells, L is a matrix of partial inductances of current cells including the retardation effect, P is a matrix of partial potential coefficients of potential cells including the retardation effect, is a vector of potentials on potential cells, I is a vector of currents along current cells, U S is a vector of voltage sources, I S is a vector of external current sources, and Y is an additional admittance matrix of linear and non-linear elements. The equivalent circuit is extracted from threedimensional geometries of a considered structure. An appropriate solver is employed to obtain solution either in the time domain or in the frequency domain. Fig. 1 shows the procedures of the PEEC simulation. The detail of derivation and formulation of a PEEC for a thin wire structure is found in Reference [15]. Fig. 1. Procedures in the simulation of PEEC models. 2.1 Formulation of the quasi-static PEEC method in the Time domain The formulation of the quasi-static PEEC method in the time domain can be derived by using (1) and is given in equation (2); T P 1 d Y A I dt S (2) A R L d dt I U S where L is a matrix of partial inductances of current cells neglecting the retardation effect, and P is a matrix of partial potential coefficients of potential cells neglecting the retardation effect. The matrices, P and L, express electromagnetic coupling among the cells, and are dense. The solution of equation (2) based on quasi-static assumption can be obtained by employing an appropriate integration scheme. For simplicity, Backward Euler scheme is applied to equation (2) by discretizing in time. The following equation is obtained. 1 1 T n 1 n1 P Y A n U S L I t t (3) 1 n A R L I n n t I S P t 2.2 Transmission Line Model Formulation of a multi-transmission line in the frequency domain as multi-port network as illustrated in Fig. 2 is given as follow; ISBN:

To measure the insulator voltages, two current pulses characterized by different rise times were applied. The first pulse (Fig. 3b) has about.

To measure voltages across insulator strings a 1-k resistive voltage dividers were employed. Fig. 2. Multi-transmission line as multi-port network.

, Z, Y, and Z c are a propagation matrix, a series impedance matrix per unit length, a shunt admittance matrix per unit length, and a characteristic impedance, respectively.

I r (t) = V s (t-t d )+ Z c I s (t-t d ), (7-2) or it can be written in another form as the following; I s (t)-y c V s (t) = -Y c V r (t-t d )- I r (t-t d ), (8-1) I r (t)-y c V r (t) = -Y c V s (t-t

The transmission line model can be combined with the PEEC method by adding Y c as an additional admittance matrix on the matrix on the left hand side of (3) and adding additional delayed current

3 Verification of the PEEC Method and an Example Experimental results of transients on a transmission tower are presented in this section for comparison with simulated results by the quasi-static

3 To measure the insulator voltages, two current pulses characterized by different rise times were applied. The first pulse (Fig. 3b) has about.2 s rise time (fast rise time current) and the second one (Fig. 3c) has a time to crest of 3 s. The peak amplitude of the two pulses is 3.4 A. To measure voltages across insulator strings a 1-k resistive voltage dividers were employed. Fig. 2. Multi-transmission line as multi-port network. V s coth( ) Zc Vr cosech( ) cosech( ) Is, (4) coth( ) Zc Ir 2 ZY, (5) 1 Zc Z, (6) where V s, V r, I s, and I r are vectors of voltages and currents at sending ends and receiving ends, respectively., Z, Y, and Z c are a propagation matrix, a series impedance matrix per unit length, a shunt admittance matrix per unit length, and a characteristic impedance, respectively. From (4) to (6), the formulation in the time domain can be written in the matrix form with delay time (t d ) as the following; V s (t)-z c I s (t) = V r (t-t d )+ Z c I r (t-t d ), (7-1) V r (t)-z c I r (t) = V s (t-t d )+ Z c I s (t-t d ), (7-2) or it can be written in another form as the following; I s (t)-y c V s (t) = -Y c V r (t-t d )- I r (t-t d ), (8-1) I r (t)-y c V r (t) = -Y c V s (t-t d )-I s (t-t d ), (8-2) where Y c is an inversion matrix of Z c. The transmission line model can be combined with the PEEC method by adding Y c as an additional admittance matrix on the matrix on the left hand side of (3) and adding additional delayed current sources in (8) on the right hand side of (3). 3 Verification of the PEEC Method and an Example Experimental results of transients on a transmission tower are presented in this section for comparison with simulated results by the quasi-static PEEC method in the time domain, by the full-wave PEEC method in the frequency domain, and collected experimental data [1]. In the actual tower case, the towers composed of slant elements and cross arms including overhead ground wires and phase wires are considered in the simulation. The configuration of the experiment on an actual system of a 5 kv double circuit transmission line is illustrated in Fig. 3(a). The simulated system is composed of three towers with separation distances of 45 m and 56 m between towers, two overhead ground wires, and six phase wires. A current pulse is applied to the top of the middle tower. Current [A] (a) System configuration -1 Time [s] (b) Fast rise time current 1-1 Time [s] (c) Slow rise time current Fig. 3. System configuration and applied currents. The full-wave PEEC simulation in the frequency domain involves 256 frequencies upto 5 MHz and 5-m element length. The tower is composed of four main poles of which elements have.2 m radius; slant and horizontal elements have.1 m radius. The cross arms are composed of.2-m radius elements. The same conditions are employed in the quasi-static PEEC simulation in the time domain with a 16 ns time step. The overhead conductors are modeled using a lossless 8- ISBN:

4 conductor transmission line (6 phase wires and 2 ground wires). The towers and parts of the transmission lines close to the tower are modeled by the PEEC method for taking into account the retardation of electromagnetic fields around the tower by self and mutual impedances of the elements. The total number of elements without the use of the transmission line model is 2488 while the total number of elements when using the transmission line model is reduced to 64. simulated waveforms. Experimental waveforms were taken from [1]. Very good agreement of the results calculated by the full-wave PEEC method and experimental results are observed and a satisfactory agreement of the PEEC method neglecting retardation effects in the time domain with and without adopting the transmission line model are also observed. Voltage Time [s] (a) Upper phase -5 Time [s] (b) Middle phase -5 Time [s] (c) Lower phase PEEC with a TL model PEEC without a TL model Full-wave PEEC Measurement from [1] Fig. 4. Comparison between simulated and measured waveforms, fast rise time current injected. In the simulation, the tower-footing resistance is represented by a 17- resistance by connecting four 68 resistors at the bottom of four main poles of the tower. Figs. 4 and 5 show a comparison between measured and Time [s] (a) Upper phase Time [s] (b) Middle phase -2 Time [s] (c) Lower phase PEEC with a TL model PEEC without a TL model Full-wave PEEC Measurement from [1] Fig. 5. Comparison between simulated and measured waveforms, slow rise time current injected. Table 1 shows computation times relative to the time corresponding to the full-wave PEEC method in the frequency domain. From the results in Table 1, it can be seen that the computation efficiency of the PEEC ISBN:

5 method in the time domain is much better than that of the PEEC method in the frequency domain. Furthermore, a combination with the transmission line theory for the representation of overhead conductors is very effective to increase the computational efficiency of the method without losing accuracy. Table 1 Comparison of Computation Time Relative to the Computation Time of the Full-wave PEEC Method PEEC in the time domain without adopting transmission line model with adopting transmission line model Formulation of an impedance matrix and inversion (p.u.) Calculation of result by multiplication (p.u.) Total (p.u.) Conclusion The analysis and the results presented in this paper have shown that for applications involving long horizontal structures such as overhead power lines, a combination of the PEEC method with the transmission line theory is computationally very efficient while keeping the accuracy of a full PEEC approach. References: [1] M. Ishii, T. Kawamura, T. Kouno, E. Ohasaki, K. Murotani and T. Higushi, "Multistory transmission tower model for lishtning surge analysis," IEEE Trans. Power Del., vol. 6, no. 3, pp , July [2] N. Nagaoka, "A development of frequencydependent tower model," Trans. IEE Japan, vol. 111-B, no. 1, pp.51-56, 1991 (in Japanese). [3] A. Ametani, Y. Kasai, J. Sawada, A. Mochizuki and T. Yamada, "Frequency-dependent impedance and a multiconductor tower model," IEE Proc.- GTD., vol. 141, no. 4, pp , July [4] A. Ametani and T. Kawamura, "A method of a lightning surge analysis recommended in Japan using EMTP," IEEE Trans. Power Del., vol. 2, no. 2, pp , April 25. [5] Y. Baba and M. Ishii, "Numerical Electromagnetic Field analysis of tower surge response," IEEE Trans. Power Del., vol. 12, no. 1, pp , Jan [6] Y. Baba and M. Ishii, "Numerical Electromagnetic Field analysis on measuring methods of tower surge impedance," IEEE Trans. Power Del., vol. 12, no. 1, pp , Oct [7] Y. Baba and M. Ishii, "Numerical Electromagnetic Field analysis on lightning surge response of tower with shield wire," IEEE Trans. Power Del., vol. 15, no. 3, pp , July 2. [8] T. Mozumi, Y. Baba, M.Ishii, N. Nagaoka, and A. Ametani, "Numerical Electromagnetic Field analysis of archorn voltages during a backflashover on 5 kv twin-circuit line," IEEE Trans. Power Del., vol. 18, no. 1, pp , Jan. 23. [9] T. Noda and S. Yokoyama, "Thin wire representation in Finite DifferenceTime Domain surge simulation," IEEE Trans. Power Del., vol. 17, no. 3, pp , July 22. [1] T. Noda, "A tower model for lightning overvoltage studdies on the results of an FDTD simulation," Electrical Engineering in Japan., vol. 164, no. 1, pp.8-17, 28. [11] T. Noda, "A numerical simulation of transient electromagnetic fields for obtaining the step response of a transmission tower using the FDTD method," IEEE Trans. Power Del., vol. 23, no. 2, pp , April 28. [12] P. Yutthagowith, A. Ametani, N. Nagaoka, and Y. Baba, "Influence of a current lead wire and a voltage reference wire to a transient voltage on a vertical Conductor," IEEJ. TEEE., vol. 5, no. 1, pp.1-7, Jan. 21. [13] P. Yutthagowith, A. Ametani, N. Nagaoka and Y. Baba., " Influence of a measuring system to a transient voltage on a vertical Conductor," IEEJ. TEEE., vol. 5, no. 2, Mar. 21. [14] P. Yutthagowith, A. Ametani, N. Nagaoka, and Y. Baba., "Lightning induced voltage over lossy ground by a hybrid electromagnetic-circuit model method with Cooray-Rubinstein formula," IEEE Trans. Electromagn. Compat., vol. 51, no. 4, pp , Nov. 29. [15] P. Yutthagowith, Development of the PEEC Method and Its Application to a Lightning Surge Study., Ph.D. thesis, Doshisha Uinersity, Kyoto, Japan, June 21. [16] A. E. Ruehli., "Equivalent circuit models for threedimentional multiconductor systems," IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 3, pp , Mar [17] A. Odabasioglu, M. Celik, and L. Pileggi, Prima: Passive reduced-order interconnect macro modeling algorithm, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol.17, no.8, pp , Aug [18] S. Wang, J. He, B. Zhang, Rong Zeng and Z. Yu., "A time-domain multiport of thin-wire system for lightning transient Simulation," IEEE Trans. Electromagn. Compat., vol. 51, no. 4, pp , Nov. 29. ISBN:

Comparison of Transient Simulations on Overhead Cables by EMTP and FDTD

Comparison of Transient Simulations on Overhead Cables by EMTP and FDTD H. Xue, M. Natsui, A. Ametani, J. Mahseredjian, H. Tanaka, Y. Baba 1 Abstract--Transient simulations on an overhead cable are performed