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1 21 Asia-Pacific International Symposium on Electromagnetic Compatibility, April 12-16, 21, Beijing, China FDTD Calculation of Lightning-Induced Voltages on an Overhead Two-Wire Distribution Line Toshiki Takeshima #1, Yoshihiro Baba #1, Naoto Nagaoka #1, Akihiro Ametani #1 Jun Takami #2, Shigemitsu Okabe #2, Vladimir A. Rakov #3 # 1 Doshisha University, Kyotanabe, Kyoto , Japan dtj151@mail4.doshisha.ac.jp, ybaba@mail.doshisha.ac.jp nnagaoka@mail.doshisha.ac.jp, aametani@mail.doshisha.ac.jp # 2 Tokyo Electric Power Company, R&D Center, Yokohama, Kanagawa , Japan takami.jun@tepco.co.jp, okabe.s@tepco.co.jp # 3 University of Florida,Gainesville, FL32611, USA rakov@ece.ufl.edu Abstract We have calculated lightning-induced voltages on a 68-m long overhead two-wire line using the finite-difference time-domain (FDTD) method for solving Maxwell s equations. The FDTD method employed here uses a three-dimensional nonuniform grid, which is fine (cell side length is.875 m) in the vicinity of overhead wires and coarse (maximum cell side length is 7 m) in the rest of the space. The overhead wires having radii of several millimetres are simulated by placing a wire having an equivalent radius of about.2 m ( m) in the center of an artificial rectangular prism having a cross-sectional area of (2.875 m) (2.875 m) and the modified electrical constants: low permittivity and high permeability. The inducedvoltage waveform, calculated for the condition that the returnstroke wavefront speed is 13 m/μs, the ground conductivity is 3.5 ms/m, and the grounding resistance ranges from 3 to 75 Ω, agrees well with the corresponding waveform measured by Barker et al. in a rocket-triggered lightning experiment. I. INTRODUCTION In order to optimize lightning protection means of telecommunication and power distribution lines, one needs to know voltages that can be induced on overhead wires by nearby lightning strikes. Lightning-induced voltages on an overhead multi-conductor line have been calculated reasonably accurately using field-to-wire electromagnetic coupling models (e.g., [1]) and approximate expressions for electric fields over lossy ground, such as the Norton approximate expressions [2] and the Cooray-Rubinstein formula [3]. Also, lightning-induced voltages have recently been calculated with a similar accuracy using the method of moments (MoM) (e.g., [4]) and the hybrid electromagnetic/circuit model (HEM) (e.g., [5], [6]). In [4] and [5], in order to consider the effects of lossy ground on electromagnetic fields, the Norton approximate expressions were employed, while in [6] the Cooray-Rubinstein formula was used. More recently, the finite-difference time-domain (FDTD) method [7] has been applied to analyzing lightning-induced voltages [8], [9]. In [8], the three-dimensional (3D) FDTD method was used for examining the influence of a nearby lightning strike object on the voltages induced on a single overhead wire above lossy ground. In [9], the 2D-FDTD method, instead of approximate expressions, was used for evaluating the electric fields over lossy ground, and the FDTD-calculated fields were used for evaluating the lightning-induced voltages with the Agrawal et al. field-towire coupling model [1]. One of the reasons for using the latter, hybrid approach is probably related to a difficulty of representing closely-spaced overhead thin wires in the 3D uniform-grid FDTD method. One of the advantages of the use of the FDTD method in analyzing lightning-induced voltages is that it yields electromagnetic fields in the presence of lossy ground directly in the time domain without using any approximate formula in the frequency domain. Another advantage is that it does not necessarily need a field-to-wire coupling model. One of the disadvantages is that it is computationally expensive. In this paper, we calculate lightning-induced voltages on a 68-m long overhead two-wire line using the FDTD method for different return-stroke wavefront speeds, 6, 13, and 2 m/μs, ground conductivity values,.35, 3.5, and 35 ms/m, and grounding resistance values, 3, 5, and 75 Ω. The employed FDTD method uses a three-dimensional nonuniform grid, which is fine (cell side length is.875 m) in the vicinity of overhead wires and coarse (maximum cell side length is 7 m) in the rest of the space. The overhead thin wires are simulated using arbitrary-radius-wire representations, which place a wire having an equivalent radius of about.2 m ( m) in the center of an artificial rectangular prism having a cross-sectional area of (2.875 m) (2.875 m) and the modified electrical constants: low permittivity and high permeability. We compare the FDTDcalculated waveforms of lightning-induced voltages with the corresponding waveforms measured by Barker et al. [11] in a rocket-triggered lightning experiment. II. METHODOLOGY Fig. 1 (a) shows the experimental configuration, including a 682-m long two-wire test distribution line and a rocket /1/$ IEEE 1317

2 triggered lightning channel located 145 m away from the distribution line, that was employed by Barker et al. [11]. Fig. 1 (b) shows plan and side views of its representation for FDTD calculations. The two horizontal wires at heights m and m are parallel to the y axis. The working volume of 425 m 966 m 56 m for FDTD calculations is divided nonuniformly into rectangular cells, and is surrounded by six planes of Liao s second-order absorbing boundary condition [12] to avoid reflections there. The side length in y direction of all the cells is 7 m (constant). x- and z-directed sides of the cells are not constant:.875 m in the vicinity of the two horizontal (y-directed) wires, and the side length increases gradually (to 1.75, 3.5, and 7 m) with increasing distance from the wires. The total number of cells in the working volume is = 1,225,44. The 7.6-mm-radius upper and 5.9-mm-radius lower horizontal wires are represented by using the thin-wire representation of Noda and Yokoyama [13], which places a wire having an equivalent radius of about.2 m ( m) in the center of an artificial rectangular prism having a cross-sectional area of (2.875 m) (2.875 m) and the modified electrical constants: low permittivity and high permeability. At the ends of the horizontal wires, the upper wire is connected to the lower wire via a 455-Ω resistor. At P1, P9 (located approximately at the middle of the distribution line), and P15, the lower wire is connected via a vertical wire (down conductor) to the surface of the ground (the thickness from the ground surface to the bottom absorbing boundary is set to 7 m) having relative permittivity ε r =1 and conductivity, σ =.35, 3.5 or 35 ms/m, via a vertical wire in series with a lumped resistor of R g =3, 5 or 75 Ω. Note that the ground conductivity was not given by Barker et al. [11], but the grounding impedance values were measured at P1, P9, and P15 at a frequency of 4 khz, and ranged from 3 to 75 Ω. Since the cross-section of the cells closest to the vertical wire is not a square (.875 m 7 m), we cannot employ the thinwire representation of Noda and Yokoyama [13]. Instead, we use the representation of Railton et al. s [14]. The radius of the down conductors is set to 7.5 mm. The vertical lightning channel is represented by a vertical array of current sources [15]. Each current source is activated by the arrival of an upward-propagating return-stroke wavefront whose speed is v = 6, 13 or 2 m/μs. Note that lightning return-stroke wavefront speed was not measured by Barker et al. [11]. The time increment is set to 1.3 ns. (a) III. ANALYSIS AND RESULTS Fig. 2 shows the waveform of lightning channel-base current measured by Barker et al. [11], and its approximate waveform used for the FDTD calculations. Fig. 3 (a) shows FDTD-calculated waveforms of lightning-induced voltages between the upper wire and the ground surface at P9 for the return-stroke wavefront speed v = 13 m/μs, the grounding resistance value R g = 5 Ω, and different values of ground conductivity σ =.35, 3.5, and 35 ms/m. Fig. 3 (b) shows FDTD-calculated waveforms for R g = 5 Ω, σ = 3.5 ms/m, and different values of v = 6, 13, and 2 m/μs. Fig. 3 (c) shows FDTD-calculated waveforms for v = 13 m/μs, σ = 3.5 ms/m, and different values of R g = 3, 5, and 75 Ω. In each part of Fig. 3, the voltage waveform measured at P9 by Barker et al. [11] is also shown (b) Fig. 1. (a) Experimental configuration employed by Barker et al. [11], including a 682-m long two-wire test distribution line and a rocket-triggered lightning channel located 145 m away from the distribution line and (b) plan and side views of its representation for FDTD calculations. Current [ ka ] Approximated Fig. 2. Waveform of a lightning channel-base current measured by Barker et al. [11], and its approximate waveform used for FDTD calculations. 1318

3 Induced voltage [ kv ] Induced voltage [ kv ] Induced voltage [ kv ] σ =.35 ms/m 3.5 ms/m 35 ms/m (a) (b) v = 2 m/μs 13 m/μs 6 m/μs R g = 75 Ω Ω 3 Ω (c) Fig. 3. FDTD-calculated waveforms of lightning-induced voltage at point P9 for (a) v=13 m/μs, R g =5 Ω, σ =.35, 3.5 or 35 ms/m, (b) R g =5 Ω, σ =3.5 ms/m, v=6, 13 or 2 m/μs, and (c) v=13 m/μs, σ =3.5 ms/m, R g =3, 5 or 75 Ω. Also shown is the measured voltage waveform. It is clear from Figs. 3 (a), (b), and (c) that the magnitude of lightning-induced voltage increases with decreasing the ground conductivity, increasing the return-stroke wavefront speed, and increasing the grounding resistance. When σ =3.5 ms/m and v=13 m/μs are assumed, the FDTD-calculated waveform agrees best with the corresponding measured waveform. This conclusion also agrees with that of Ren et al. [9] who used the 2D-FDTD calculations with the Agrawal et al. field-to-wire coupling model, and that of Yutthagowith et al. [6] who used the HEM calculations with the Cooray-Rubinstein formula. Figs. 4 (a) and (b) show waveforms of vertical electric field and horizontal magnetic field, respectively, calculated for σ = 3.5 m/ms and v= 13 m/ms. The corresponding waveforms measured by Barker et al. [11] are also shown. The calculated field waveforms agree reasonably well with the measured ones. Vertical electric field [ kv/m ] Horizontal magnetic field [ A/m ] d = 11 m Calculated for σ = 3.5 ms/m and v = 13 m/μs (a) d = 11 m Calculated for σ = 3.5 ms/m and v = 13 m/μs (b) Fig. 4. FDTD-calculated waveforms of (a) vertical electric field and (b) horizontal magnetic field at horizontal distance 11 m from the lightning channel for σ =3.5 ms/m and v=13 m/μs. Also shown are the measured vertical electric and horizontal magnetic field waveforms. IV. CONCLUSIONS We have calculated lightning-induced voltages on an overhead two-wire line using the FDTD method. The FDTD method employed here uses a 3D nonuniform grid, which is fine in the vicinity of overhead wires and coarse in the rest of the space. The overhead wires having radii of several millimetres are simulated using thin-wire representations. The FDTD-calculated waveform of lightning-induced voltage for the condition that the return-stroke wavefront speed is

4 m/μs, the ground conductivity is 3.5 ms/m, and the grounding resistance ranges from 3 to 75 Ω, agrees well with the corresponding rocket-triggered lightning voltage waveform measured by Barker et al. [11]. This indicates that lightninginduced voltages on realistic multi-wire distribution lines can be analyzed reasonably accurately using the FDTD method without invoking a field-to-wire coupling model. REFERENCES [1] M. Paolone, F. Rachidi, A. Borghetti, C. A. Nucci, M. Rubinstein, V. A. Rakov, and M. A. Uman, Lightning electromagnetic field coupling to overhead lines: theory, numerical simulations, and experimental validation, IEEE Trans. Electromagn. Compat., vol. 51, no. 3, pp , Aug. 29. [2] K. A. Norton, The propagation of radio waves over the surface of the earth and in the upper atmosphere, part II --- the propagation from vertical, horizontal, and loop antennas over a plane earth of finite conductivity, Proc. IRE, vol. 25, no. 9, pp , [3] M. Rubinstein, An approximate formula for calculation of the horizontal electric field from lightning at close, intermediate, and long ranges, IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp , Aug [4] R. K. Pokharel, M. Ishii, and Y. Baba, Numerical electromagnetic analysis of lightning-induced voltage over ground of finite conductivity, IEEE Trans. Electromagn. Compat., vol. 45, no. 4, pp , Nov. 23. [5] F. H. Silveira, S. Visacro, J. Herrera, and H. Torres, Evaluation of lightning-induced voltages over a lossy ground by the hybrid electromagnetic model, IEEE Trans. Electromagn. Compat., vol. 51, no. 1, pp , Feb. 29. [6] P. Yutthagowith, A. Ametani, N. Nagaoka, and Y. Baba, Lightninginduced voltage over lossy ground by a hybrid electromagnetic circuit model method with Cooray-Rubinstein formula, IEEE Trans. Electromagn. Compat., vol. 51, no. 4, Nov. 29. [7] K. S. Yee, Numerical solution of initial boundary value problems in solving Maxwell s equations in isotropic media, IEEE Trans. Anntennas and Propat., vol. 14, no. 3, pp , Mar [8] Y. Baba, and V. A. Rakov, Voltages induced on an overhead wire by lightning strikes to a nearby tall grounded object, IEEE Trans. Electromagn. Compat., vol. 48, no. 1, pp , Feb. 26. [9] H.-M. Ren, B.-H. Zhou, V. A. Rakov, L.-H. Shi, C. Gao, and J.-H. Yang, Analysis of lightning-induced voltages on overhead lines using a 2-D FDTD method and Agrawal coupling model, IEEE Trans. Electromagn. Compat., vol. 5, no. 3, pp , Aug. 28. [1] A. K. Agrawal, H. J. Price, and H. H. Gurbaxani, Transient response of muliticonductor transmission lines excited by a nonuniform electromagnetic field, IEEE Trans. Electromagn. Compat., vol. 22, no. 2, pp , May 198. [11] P. P. Barker, T. A. Short, A. R. Eybert-Berard, and J. P. Berlandis, Induced voltage measurements on an experimental distribution line during nearby rocket triggered lightning flashes, IEEE Trans. Power Del., vol. 11, no. 2, pp , Apr [12] Z. P. Liao, H. L. Wong, B. -P. Yang, and Y. -F. Yuan, A transmitting boundary for transient wave analysis, Science Sinica, Series A, vol. 27, no. 1, pp , [13] T. Noda, and S. Yokoyama, Thin wire representation in finite difference time domain surge simulation, IEEE Trans. Power Del., vol. 17, no. 3, pp , Jul. 22. [14] C. J. Railton, D. L. Paul, and S. Dumanli, The treatment of thin wire and coaxial structures in lossless and lossy media in FDTD by the modification of assigned material parameters, IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp , Nov. 26. [15] Y. Baba, and V.A.Rakov, On the transmission line model for lightning return stroke representation, Geophys. Res. Lett, vol. 3, no. 24, 2294, doi:1.129/23gl 1847, Dec

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