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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 3, AUGUST 2007

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Applications of Time-Domain Numerical Electromagnetic Code to Lightning Surge Analysis

Ramesh K. Pokharel, Member, IEEE, and Masaru Ishii, Fellow, IEEE

Abstract—Recently, applications of 3-D numerical electromagnetic analysis have been increasing either for lightning electromagnetic impulse (LEMP) studies or lightning surge analyses on transmission and distribution lines. This paper is mainly concerned with the use of time- and frequency-domain codes for electromagnetic analysis of lightning surges. The thin-wire in time-domain (TWTD) code and numerical electromagnetic code (NEC-2) in the frequency domain based on the method of moments are chosen for comparative studies. The accuracy of TWTD code in the analysis of lightning surge characteristics of a double-circuit transmission tower is ?rst investigated by comparing the computed results with the measured results on a reduced-scale tower model, computed results by NEC-2 on a full-scale tower model, and those computed by electromagnetic transients program. In the latter part of the paper, a switch model is combined with the TWTD code, and its applicability in analyzing the lightning surge characteristics of a transmission tower equipped with a surge arrester or in analyzing lightning-induced voltage on an overhead line is demonstrated. Index Terms—Lightning surge, moment method, numerical electromagnetic code (NEC-2), numerical electromagnetic analysis, nonlinearity, thin-wire in time-domain (TWTD).

I. INTRODUCTION

R

ECENTLY, applications of a 3-D numerical electromagnetic analysis have been increasing either for lightning electromagnetic impulse (LEMP) studies or lightning surge analyses on transmission and distribution lines. Among the electromagnetic codes available in the public domain, two codes, respectively, in the frequency domain called numerical electromagnetic code (NEC-2) [1] and the other referred to as thin-wire in time-domain (TWTD) code [2], both based on the method of moments, are most popular for numerical electromagnetic analysis of lightning surges either on transmission or distribution lines [3]–[13]. Both codes have their own advantages and limitations. One of the advantages of NEC-2 is that it can incorporate ?nite conductivity of ground that plays a dominant role in the evaluation of lightning-induced voltages on a distribution line [3]. TWTD code can simulate nonlinear elements such as a surge arrester or a switch. On the other hand, electromagnetic transients program (EMTP) has been extensively used to calculate lighting surge analyses of multiphase transmission lines [19], and the surge characteristics of a transmission tower is one of the key issues in the analyses. EMTP is based on the circuit theory, i.e., the

electromagnetic ?eld of the system has to be in the TEM mode to obtain correct results. Such a circuit-model approach is not appropriate to analyze the physical behavior of insulator voltages when the transmission line is struck by lightning, because the associated electromagnetic ?eld initially is far from the TEM mode. So, 3-D numerical electromagnetic ?eld analysis is effective in analyzing such problems. One of the earliest applications of TWTD code to lightning study [4] modeled a lightning channel attached to a tall structure. Another study also employs a similar time-domain code to model a lightning channel [5] for LEMP studies. Although a similar code in the time domain has been also applied to analyze lightning-induced voltages on distribution lines over perfectly conducting ground [6], the TWTD code has also been used to analyze insulator voltage waveforms on a double-circuit transmission tower when it is directly hit by lightning [9]. However, no rigorous comparisons with the measurements have been reported to verify its accuracy in lightning surge analysis on transmission lines. In this paper, accuracy of TWTD code in the analysis of lightning surge characteristics of a double-circuit transmission tower is ?rst investigated by comparing the computed results with those computed by NEC-2. The in?uence of variable radii of segments is investigated in order to obtain more accurate results. Furthermore, an arrester model represented by a nonlinear resistor is combined with TWTD code, and its applicability in modeling a transmission tower equipped with a surge arrester in an EMTP-type circuit simulator or in analyzing lightninginduced voltages on an overhead line over perfectly conducting ground is demonstrated.

II. APPLICATION TO LIGHTNING SURGE ANALYSIS OF A TRANSMISSION TOWER A. Comparison in Reduced-Scale Model Accuracy in the analysis of tower surge response by the TWTD code is investigated by comparison with an experiment on a reduced-scale model over perfectly conducting ground as well as with the results computed by NEC-2. Similar to NEC-2, the TWTD code solves the boundary value problem of the electric ?eld integral equation by the method of moments but directly in the time domain. For the numerical analysis with the TWTD code, the entire structure needs to be modeled by a combination of straight cylindrical segments that should be short enough than the wavelengths of interest. Once the model is de?ned, excitation is imposed as a voltage source or a plane wave. It calculates current distribution on each segment of the modeled geometry, through numerically solving

Manuscript received April 2, 2006; revised December 10, 2006. R. K. Pokharel is with the Department of Electronics, Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka 819-0395, Japan (e-mail: pokharel@ed.kyushu-u.ac.jp). M. Ishii is with Institute of Industrial Science, University of Tokyo, Tokyo 153-8506, Japan (e-mail: ishii@iis.u-tokyo.ac.jp). Digital Object Identi?er 10.1109/TEMC.2007.902406

0018-9375/$25.00 ? 2007 IEEE

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 3, AUGUST 2007

Fig. 1. Tower models in the experiment. (a) Single conductor. (b) Four parallel conductors.

Fig. 3. Measured and computed tower top voltage waveforms for fourparalleled conductors. TABLE I MEASURED AND CALCULATED TOWER SURGE IMPEDANCE

Fig. 2. Measured and computed current and voltage waveforms for a single conductor. (a) Tower top voltage. (b) Current ?owing into tower.

Fig. 4. Transmission tower model for electromagnetic simulation. Crossarms and slant elements are omitted.

the electric-?eld integral equation directly in the time domain. Therefore, no Fourier transform and inverse Fourier transform are necessary to get a time-varying response unlike for NEC-2, a frequency-domain code. A good review of the basic theory of TWTD code [2] can be found in [5] and [6]. Fig. 1 illustrates the reduced-scale model towers. A steepfront current having the rise time of 5 ns was injected through a horizontal current lead wire. The current ?owing into a structure was measured through a current transformer. Voltage between the top of a structure and a horizontal voltage measuring wire, which was orthogonal to the current lead wire, was measured through a voltage probe. The rise times of the step response of the current transformer and the voltage probe were 2 and 0.7 ns, respectively [10]. The voltage at the tower top is calculated as the voltage induced on a 10-k? resistor inserted between the tower top and a horizontal voltage measuring wire. Figs. 2 and 3 show the comparison of measured and computed waveforms for the single vertical conductor [Fig. 2(a)] and for the four conductors

(Fig. 3), respectively. The computed waveforms are those calculated by NEC-2 and TWTD, respectively. The computed waveforms by TWTD are almost the same as those computed by NEC-2. Table I summarizes the tower surge impedance for model towers, de?ned as the ratio of the instantaneous values of the voltage to the current at the moment of the peak of voltage waveforms. The computed and experimental results agree well. B. Comparison in Full-Scale Tower Model In this section, the accuracy of TWTD code in analyzing a full-scale transmission tower is investigated by comparison with the results computed by NEC-2 [10]. Fig. 4 illustrates an arrangement for the analysis, which simulates a vertical lightning stroke hitting the top of an independent tower. A pulse-current generator is placed on the tower top. A voltage source of 5 kV in series to a 5-k? resistor is inserted at the connecting point of the simulated lightning channel and the tower. The tower is modeled by identical four vertical poles, which are widened at the tower base. Each leg of the tower is connected to the ground

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Fig. 6. Radii of main poles of the model tower in Fig. 4, r1 = 5 cm: r2 = 5 cm (Case I), 20 cm (Case II), and 30 cm (Case III), respectively. TABLE III COMPUTED TOWER SURGE IMPEDANCE INFLUENCED BY RADII OF MAIN POLES OF MODEL TOWER

C. In?uence of Radii of Poles of Tower

Fig. 5. Computed waveforms for vertical injection of 0.1-?s step current. (a) Tower top voltage. (b) Injected current at the top of tower. TABLE II COMPUTED TOWER SURGE IMPEDANCE FOR THE MODEL TOWER IN FIG. 4

through a 20-? resistor to realize the footing resistance of 5 ?. The radii of all segments of the tower and the lightning channel are 5 cm in this model. At this model, the impedance of the lightning channel is close to in?nite, and the current wave propagates up the channel with the speed of light. Actual velocities of lightning current waves are slower; however, the in?uence of the propagation velocity of the lightning current wave on the insulator voltage is not signi?cant, and can be incorporated if necessary [20]. The waveforms of tower top voltage and injected current computed by TWTD code and NEC-2, respectively, are shown in Fig. 5 for vertical injection of step current having 0.1-?s rise time. The agreement in computed waveforms by the two codes is excellent such that the maximum difference is within 5% around the peak. From the comparison with the computation by NEC-2 for the model tower so far, the accuracy of the TWTD code is comparable to that of NEC-2 over perfectly conducting ground. In Table II, tower surge impedance, de?ned as the ratio of the instantaneous values of the voltage to the current at the moment of the voltage peak, is summarized. Omission of tower arms and slant elements makes the tower surge impedance higher than an actual tower by about 35% [11]. Dependence of the tower surge impedance on the method of current injection, ?rst computed by NEC-2 [14], is well reproduced by the TWTD code.

Discontinuity of the radii of segments at their junction points contributes to the error of the computed results by both the codes. This comes from the dif?culty in the treatment of the continuity of the current at such junctions between segments. To investigate the in?uence of such discontinuity, the tower surge impedance is computed for the con?guration of Fig. 4 by replacing the four main poles of the tower by the poles shown in Fig. 6. The radius of the upper part is ?xed to 5 cm, and that of the lower part is 5 cm for Case I, 20 cm for Case II, and 30 cm for Case III. The radii of the lightning channel and the orthogonal voltage measuring wire are 5 cm in all cases. Tower top voltage calculated by NEC-2 (?gure is omitted) is dependent on radii of poles of tower, and peak value decreases from Case I to Case III. While in computation by TWTD, there is almost no change in tower top voltage (not shown here). Based on these computations, Table III summarizes the in?uence of different radii of poles of tower on its surge impedance. The computed impedance by NEC-2 in Table III shows dependence on the radii of the bottom poles, whereas that computed by TWTD does not. Even this computed impedance by NEC-2 that gives tower top voltages is estimated to have errors of 4%–6% for Cases II and III due to the discontinuities of the radii of the connected poles. Errors of 10%–15% in the calculation of surge impedance of connected lines with discontinuities in radii by NEC-2, for such cases as Cases II and III, are expected [14]. It is noted that variation of the tower top voltage, in?uenced by the re?ection from the junction points of the two segments, is less than that of the surge impedance of the connected line. Therefore, it is not recommended to connect segments of different radii in the analysis by TWTD. According to the authors’ experience, an actual complex conductor system having different radii of elements can be approximated by a conductor system having an equivalent uniform radius such as Fig. 4 in TWTD calculations. Though it is desirable to con?rm the accuracy by comparing the computed result by TWTD, for a simple case,

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 3, AUGUST 2007

of (4) reduces to a simple form. In this case, the current on the loaded segment (segment k) is the algebraic expression, and is represented by

u Ik j = Ik j /[1 + RYk k ]

(5)

where Yk k is an element of matrix Y, and R is the resistive load on the segment k at time step j. The currents on the remaining segments can then be computed by

u Iij = Iij ? u Yik RIk j . 1 + RYk k

(6)

Fig. 7. Assumed voltage–current (V –I) characteristics of nonlinear resistor in TWTD analysis.

with that computed by such codes as NEC-4 [16], which is improved in handling thick elements and discontinuities in radii. III. INCORPORATION OF NONLINEAR ELEMENT A. Nonlinearity With TWTD Code In this section, a nonlinear resistor representing a surge arrester is combined with the TWTD code. The method is brie?y outlined as follows. The solution of the electric ?eld integral equation (EFIE) by the method of moments can be obtained in the following form [2], if there is no external loading in the segment: ES + EA = ZIj , j j j = 1, 2, . . . , M (1)

If the resistance of the nonlinear element is a function of current, then both that dependence and (5) must be simultaneously satis?ed, and this may require an iteration process for the correct solution. For example, if the circuit contains elements with a nonlinear V–I characteristic, each nonlinear element is assigned a value of equivalent resistance Req (tk ), and at the ?rst time step, this resistance is set equal to Req (t1 ) = ?v ?t .

v =0

where Z is a time-independent interaction matrix that relates the current Ij at the present time step to the sum of the incident ?eld and the scattered ?eld. In general, Z is not diagonal, but it will be sparse and diagonally dominant. So, the solution for unknown current in a segment is obtained by matrix inversion of (1) represented by Ij = Y ES + EA j j (2)

These values of equivalent resistance are inserted into the nodal equations and solved by an iteration process, which involves circuit analysis only and does not require any manipulation in EM solution. For simplicity, the assumed V–I characteristics of a nonlinear resistor are shown in Fig. 7 where the resistor operates with a high value (1 m?) till the breakdown voltage (Vd), and beyond that voltage, it switches into a low value (Rt). Instability in the time-domain solution of the EFIE has been a critical issue in late-time behavior, and this problem has recently been improved signi?cantly by Ji et al. [18]. B. Application in Analysis of Insulator Voltage Waveforms of a Transmission Line in the Presence of a Surge Arrester To demonstrate its application in analyzing insulator voltage waveforms, numerical analyses on a double-circuit 275-kV transmission line have been carried out, and compared with the results obtained by EMTP simulation. For TWTD analysis, two shield wires are stretched at a height of 60 m on both sides of the model tower of Fig. 4. Horizontal phase conductors are stretched at heights of 56, 48, and 40 m, respectively, 8 m apart from the tower body, though they are not shown in Fig. 4. Each phase conductor and shield wire is 420 m in length. Their ends are stretched downward vertically and connected to a perfectly conducting ground through a resistance equivalent to their surge impedance. This termination condition does not affect the surge phenomena at the tower during the ?rst 2.8 ?s. The insulator voltage is computed by inserting a 1-m? resistor in between the tower body and a phase conductor when there is no surge arrester. The computed insulator voltage waveforms for 0.1-?s rise time current injection are shown in Fig. 8(a). For EMTP analysis, the multistorey tower model [19] has been widely used for lightning surge analysis of multiphase

where Y = Z?1 , and j is the number of time steps. When a segment is loaded, the ?elds produced by the loaded segment will differ, and it will in?uence the current distribution of the whole segments. So, the correction of current distribution is necessary in the remaining segments too. So, a term ZL can be j added in (1) to include a load as in ES + EA = [Z + ZL ]Ij . j j j (3)

One of the methods to solve this equation would be to iterate directly until the solution of (3) by matrix inversion satis?es the voltage–current (V–I) characteristics of a load as shown in Fig. 7, but it costs huge computer resources. A more economical approach is employed in this paper, and is discussed brie?y. The method was previously applied to solve thin-wires antennas with a diode [17]. Multiplying both sides of (3) by Y yields Iu = U + YZL Ij j j (4)

where Iu = Y[ES + EA ], i.e., current at the present time step, j j j if the load at the present time step is zero (ZL = 0). U is the unit j matrix. If the structure is loaded at a single point, the solution

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Fig. 9

Modi?ed multistorey tower model.

TABLE IV PARAMETERS OF MODIFIED MULTISTOREY TOWER MODEL FOR THE MODEL TOWER WITH THE VALUES IN BRACKETS FOR CONVENTIONAL MULTISTOREY TOWER MODEL [19]

Fig. 8. Insulator voltage waveforms of 275-kV double-circuit transmission tower computed by TWTD or by EMTP for injection of 1-A step current of 0.1-?s rise time. (a) Computed by TWTD. (b) Computed by EMTP by using conventional multistorey tower model [19]. (c) Computed by EMTP by using modi?ed multistorey tower model.

transmission lines. In the present study, the Line Constants program is used to calculate the constants of an eight-phase transmission line at a frequency of 100 kHz. The velocity of a surge wave in the tower is assumed to be 300 m/?s. The waveforms shown in Fig. 8(b) are calculated by EMTP adopting the conventional multistorey tower model [19] having the surge impedance values of Table II. According to the analysis by NEC-2, the coupling coef?cients between the shield wire and phase conductors, as well as the apparent footing impedance, are time dependent even when the real footing impedance is pure resistance [11]. The initial value of apparent footing impedance is high compared to its real footing resistance, because the re?ected wave from the tower foot observed at the tower top is initially attenuated even though

the tower is perfectly conducting. This attenuation is associated with the non-TEM propagation of current wave along a cylindrical vertical conductor [12], [13]. To cope with the initially low coupling coef?cients and initially high apparent footing impedance, it was proposed to increase the surge impedance of the multistorey tower model, and to add impedance in series to the actual footing impedance in EMTP simulation [11] as seen in the equivalent circuit of Fig. 9. In the modi?ed multistorey tower model of the present study, the time constant of added impedance at the tower foot is increased compared with that in [11] to simulate the slow decay of insulator voltage in the 3–4 τ time range, where τ is the time constant equal to the value of round-trip time of a traveling wave across the model tower (0.4 ?s). The employed parameters of the modi?ed model are listed in Table IV with the values in brackets for the conventional multistorey tower model. Fig. 8(c) shows the insulator voltage waveforms calculated by EMTP adopting the modi?ed multistorey tower model. The waveforms computed by TWTD code agree well with those computed by EMTP in Fig. 8(c) except at the initial rising part of the waveforms, while in Fig. 8(b), the computed waveforms employing the conventional multistorey tower model is lower in magnitude especially in the wavetails. Fig. 10 shows computed upper phase insulator voltage and arrester current waveforms for injection of a 100-kA ramp current having 1.0-?s rise time. The value of Vd and Rt in Fig. 7 is assumed to be 600 kV and 8 ?, respectively. The waveforms computed by TWTD [Fig. 10(a)] agree well with those computed by EMTP employing the modi?ed multistorey tower model [Fig. 10(c)]. In Table V, the peak values of arrester currents computed by TWTD and EMTP, respectively, for injection of 100-kA ramp current having different rise times are compared. The values computed by the TWTD code are almost exactly reproduced by the EMTP simulation adopting the modi?ed multistorey tower model. The amplitude of arrester currents computed by EMTP depends on the tower models postulated in the simulation. In estimating the peak arrester current by EMTP

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 49, NO. 3, AUGUST 2007

TABLE V COMPARISON OF ARRESTER CURRENTS COMPUTED BY TWTD AND EMTP EMPLOYING MULTISTOREY TOWER MODELS FOR INJECTION OF TWO TYPES OF RAMP CURRENT HAVING PEAK AMPLITUDE OF 100 kA

Fig. 11. Computed current waveforms by EM model for the simulated channel loaded with 0.6 ?/m + 6 ? H/m.

Fig. 12. Arrangement of the simulated lightning striking point and an overhead line of ?nite length.

Fig. 10. Upper phase insulator voltage waveforms including current ?owing through the arrester computed by TWTD or by EMTP for injection of 100-kA ramp current of 1.0-?s rise time, when an arrester is installed on the upper phase of a 275-kV double-circuit transmission tower. (a) Computed by TWTD. (b) Computed by EMTP by using conventional multistorey tower model [19]. (c) Computed by EMTP by using modi?ed multistorey tower model.

analysis, the behavior of the tower model in the time range of the lightning current peak is quite signi?cant. C. Application to Analysis of Lighting-Induced Voltages on an Overhead Line With Surge Arrester TWTD code has been applied to analyze lightning-induced voltages on distribution lines over perfectly conducting ground [6]; however, no studies have been reported yet on the in?uence of a surge arrester by using this code. In this section, the ability of the TWTD code is demonstrated to calculate the lightninginduced voltages on a distribution line equipped with a surge arrester over a perfectly conducting ground.

In evaluation of lightning-induced voltages or LEMP analyses by the TWTD code, the electromagnetic model (EM) is employed as a return-stroke model [20]. In this model, a lightning channel is modeled by a vertical thin conductor, and its bottom is connected to the ground through a voltage source [3]. To reproduce the reduced speed of a current wave or return-stroke channel, the model channel is loaded with series resistance and inductance uniformly. Distortion and the propagation speed of the current wave on the simulated lightning channel affects the upward traveling current wave. Fig. 11 shows variation of computed current waveforms on the vertical channel dependent on the height for series loading of 0.6 ?/m + 6 ?H/m, which are similar to those computed by NEC-2 [21]. The propagation speed is about 145 m/?s, and the rise time of the wave front increases with the height. The impedance of the channel at its bottom is about 1.3 k?, increased from almost 400 ? when the model channel is not loaded [3]. Fig. 12 illustrates the arrangement of the model subject to analysis; it comprises a simulated lightning striking point and a distribution line at a height of 10 m from the ground plane. The line is 500 m in length and terminated with a 540-? resistor at its both ends. The lightning channel is at a distance of 100 m from the middle of the line. Though the ?nite conductivity of

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Fig. 13. Computed induced voltage and arrester current waveforms for lightning current of 10 kA having the rise time of 1 ?s.

Fig. 14. Computed induced voltage and arrester current waveforms under similar simulated conditions reproduced from [22].

the ground plays a dominant role in the evaluation of lightninginduced voltage [3], the existing version of the TWTD code cannot simulate the system over ?nitely conducting ground. So, a perfectly conducting ground is assumed throughout the present analysis. A surge arrestor having V–I characteristic as in Fig. 7 (Vd = 30 kV, Rt = 1 ?) is connected at the middle of the line to the ground through a grounding resistance (10 ?). The computed induced voltages and arrester current waveforms at the middle and at the ends of the line are shown in Fig. 13 for lightning current having a rise time of 1.0 ?s and a peak amplitude of about 10 kA (see Fig. 11). The thin line shows the discharge current of the arrester. The voltage waveform at the middle shows that when the instantaneous induced voltage exceeds 30 kV at around 1.1 ?s, the arrester turns on, and it turns off at around 2.1 ?s. These results are very close to those computed by a different method, which computes lightning-induced voltage in the frequency domain, on almost the same model with an arrester [22]. For comparison purposes, those results are reproduced in Fig. 14. In the model employed to calculate the results in Fig. 14 [22], the lightning channel was an ideal transmission line, which produced slightly different EM ?elds from those produced by the EM model. This is the reason for the slight differences between the waveforms in Figs. 13 and 14. In Fig. 15, voltage waveforms at the ends of the line are shown where peaks of voltage waveforms in?uenced by the arrester depend heavily on the rise time of the lightning current. The geometry subject to numerical analysis producing Fig. 15

Fig. 15. Computed induced voltage waveforms at ends of line dependent on rise time of lightning current. Dotted lines show voltage waveforms in?uenced by an arrester.

is the same as that for Fig. 13(b), which is included in Fig. 15. The voltage waveform induced at the end of the line by a lightning current of 2.0-?s rise time is not affected by an arrester because the voltage across the arrester never exceeds 30 kV for a lightning current of 10 kA for the simulated conditions. Fig. 16 shows discharge current of an arrester dependent on the rise time of the lightning current in simulated conditions. The discharge current heavily depends on the rise time of the lightning current. This also shows that the arrester does not operate if the rise time of a lightning current exceeds 1.5 ?s in the simulated condition. Furthermore, the model employed in this paper can incorporate lumped inductance as demonstrated

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REFERENCES

[1] G. J. Burke and A. J. Poggio, “Numerical electromagnetic code (NEC)— Method of moments,” Naval Ocean Systems Center, San Diego, CA, Tech. Document 116, Jan. 1980. [2] E. K. Miller, A. J. Poggio, and G. J. Burke, “An integro-differential technique for time-domain analysis of thin wire structures,” J. Comput. Phys., vol. 12, pp. 24–48, 1973. [3] R. K. Pokharel, M. Ishii, and Y. Baba, “Numerical electromagnetic analysis of lightning-induced voltage over ground of ?nite conductivity,” IEEE Trans. Electromagn. Compat., vol. 45, no. 4, pp. 651–656, Nov. 2003. [4] A. S. Podgorski and J. A. Landt, “Three dimensional time domain modelling of lightning,” IEEE Trans. Power Del., vol. 2, no. 3, pp. 931–937, Jul. 1987. [5] R. Moini, B. Kordi, G. Z. Ra?, and V. A. Rakov, “A new lightning return stroke model based on antenna theory,” J. Geophys. Res., vol. 105, no. D24, pp. 29 693–29 702, 2000. [6] R. Moini, B. Kordi, and M. Abedi, “Evaluation of LEMP effects on complex wire structures located above a perfectly conducting ground using electric ?eld integral equation in time domain,” IEEE Trans. Electromagn. Compat., vol. 40, no. 2, pp. 154–162, May 1998. [7] F. Heidler and K. Muller, “LEMP calculations with the traveling current source model,” presented at the 1989 Int. Conf. Lightning Static Electr., Bath, U.K., Sep., Paper 7B. 2.1. [8] S. Cristina and A. Orlandi, “Lightning channel’sin?uence on currents and electromagnetic ?elds in a building struck by lightning,” in Proc. 1990 IEEE Int. Symp. Electromagn. Compat., Washington, DC, Aug., pp. 338– 342. [9] T. Mozumi, Y. Baba, M. Ishii, N. Nagaoka, and A. Ametani, “Numerical electromagnetic ?eld analysis of archorn voltages during a back-?ashover on a 500 kV twin-circuit line,” IEEE Trans. Power Del., vol. 18, no. 1, pp. 207–213, Jan. 2003. [10] M. Ishii and Y. Baba, “Numerical electromagnetic ?eld analysis of tower surge response,” IEEE Trans. Power Del., vol. 12, no. 1, pp. 483–488, Jan. 1997. [11] Y. Baba and M. Ishii, “Numerical electromagnetic ?eld analysis on lightning surge response of tower with shield wire,” IEEE Trans. Power Del., vol. 15, no. 3, pp. 1010–1016, Jul. 2000. [12] Y. Baba and V. A. Rakov, “On the mechanism of attenuation of current waves propagating along a vertical perfectly conducting wire above ground: Application to lightning,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 521–532, Aug. 2005. [13] Y. Baba and V. A. Rakov, “On the interpretation of ground re?ections observed in small-scale experiments simulating lightning strikes to towers,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 533–542, Aug. 2005. [14] Y. Baba and M. Ishii, “Numerical electromagnetic ?eld analysis on measuring methods of tower surge impedance,” IEEE Trans. Power Del., vol. 14, no. 2, pp. 630–635, Apr. 1999. [15] Y. Baba and M. Ishii, “Numerical electromagnetic ?eld analysis of unit step response characteristics of impulse voltage measuring systems,” IEEE Trans. Power Del., vol. 19, no. 1, pp. 21–27, Jan. 2004. [16] G. J. Burke, “Numerical electromagnetics code—NEC-4. Method of moments, part II: Program description—Theory,” Lawrence Livermore Nat. Lab., Livermore, CA. [17] J. A. Landt, “Network loading of thin-wire antennas and scatters in the time domain,” Radio Sci., vol. 16, no. 6, pp. 1241–1247, Dec. 1981. [18] Z. Ji, T. K. Sarkar, B. H. Jung, M. Yuan, and M. Salzar-Palma, “Solving time domain electric ?eld integral equation without the time variable,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 258–262, Jan. 2006. [19] M. Ishii, T. Kawamura, T. Kouno, E. Ohsaki, K. Shiokawa, K. Murotani, and T. Higuchi, “Multistory transmission tower model for lightning surge analysis,” IEEE Trans. Power Del., vol. 6, no. 3, pp. 1327–1335, Jul. 1991. [20] Y. Baba and M. Ishii, “Numerical electromagnetic ?eld analysis of lightning current in tall structures,” IEEE Trans. Power Del., vol. 16, no. 2, pp. 324–328, Apr. 2001. [21] Y. Baba and M. Ishii, “Characteristics of electromagnetic return-stroke models,” IEEE Trans. Electromagn. Compat., vol. 45, no. 1, pp. 129– 134, Feb. 2003. [22] K. Michishita, “A method of analysis of lightning-induced voltages taking account of surge arrester in frequency domain,” Trans. Inst. Electr. Eng. Jpn., vol. 120-B, no. 6, pp. 893–898, Jun. 2000.

Fig. 16. Computed arrester current waveforms dependent on rise time of lightning current of 10 kA.

in Fig. 11 where the computed waveforms are almost the same as computed by NEC-2, another method-of-moments code in the frequency domain. Therefore, the model developed in this paper can easily take account the frequency-dependent characteristics of an arrester.

IV. CONCLUSION A time-domain code for numerical electromagnetic analysis, i.e., TWTD, is applied to analyze the surge response of transmission towers. TWTD code is more advantageous than NEC-2 in incorporating the nonlinear elements in the system, though it is dif?cult to analyze a system including a ?nitely conducting ground. The accuracy of the TWTD code is investigated by comparison with the experimental results in a reduced-scale model and those computed by the frequency-domain code (NEC-2). Especially, in analyzing lightning surges in complex models that incorporates segments of different radii, discontinuity is introduced in both the codes. However, this disadvantage can be overcome by carefully constructing the model composed of thin-wire segments of equal radii. An arrester model represented by a nonlinear resistor is combined with the TWTD code, and is applied to simulate arrester currents when the arrester is installed on the upper phase of a transmission tower. The arrester current computed by the TWTD code is well reproduced by EMTP analysis postulating a modi?ed multistorey tower model, while computation adopting the conventional multistorey model produces lower current amplitudes. The improved multistorey tower model takes account of the slow decay of the apparent footing impedance in the 1–2 ?s time range, which can never be predicted by an equivalentcircuit approach. Furthermore, the handling of a nonlinear element by the TWTD code is demonstrated in analyzing the lightning-induced voltage on a distribution line over a perfectly conducting ground equipped with a surge arrester. Its accuracy is veri?ed by comparison with previously calculated results [22]. It has been shown that operation of an arrester is heavily in?uenced by rise time of the lightning current.

POKHAREL AND ISHII: APPLICATIONS OF TIME-DOMAIN NUMERICAL ELECTROMAGNETIC CODE TO LIGHTNING SURGE ANALYSIS

631

Ramesh K. Pokharel (M’03) received the B.S. degree from Aligarh Muslim University, Aligarh, India, in 1994 and the M.E. and Ph.D. degrees in electrical engineering from the University of Tokyo, Tokyo, Japan, in 2000 and 2003, respectively. In 1994, he joined the Institute of Engineering, Tribhuwan University, Lalitpur, Nepal, as a Research Associate. In 1995, he moved to Nepal Telecommunications Corporation, Chauni, Kathmandu. From April 2003 to March 2005, he was a Postdoctoral Research Fellow with the Department of Electrical Engineering and Electronics, Aoyama Gakuin University, Tokyo. Since April 2005, he has been a Research Associate with the Department of Electronics, Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan. His current research interests include the application of numerical electromagnetic analysis to EMC problems. Dr. Pokharel is a member of the Institute of Electronics, Information and Communication Engineering and the Institute of Electrical Engineers in Japan. He was a recipient of the Monbukagusho Scholarship of the Japanese Government from 1997–2003 and an excellent COE Research Presentation Award from the University of Tokyo in 2003.

Masaru Ishii (SM’87–F’04) was born in Tokyo, Japan. He received the B.S., M.S., and Dr.Eng. degrees in electrical engineering from the University of Tokyo, Tokyo, Japan in 1971, 1973, and 1976, respectively. He joined the Institute of Industrial Science, University of Tokyo in 1976, where he has been a Professor since 1992. Dr. Ishii is a member of the American Geophysical Union and the International Conference on Large Electric High-Tension Systems. From 2004 to 2006, he was the President of the Power and Energy Society, Institute of Electrical Engineers of Japan.

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