A Comprehensive Approach for Transient Performance of Grounding ...

250

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 2, APRIL 2015

A Comprehensive Approach for Transient Performance of Grounding System in the Time Domain Jinpeng Wu, Bo Zhang, Jinliang He, Fellow, IEEE, and Rong Zeng, Senior Member, IEEE

Abstract—It is important to identify the impulse performance of the grounding system for lightning protection. However, the performance is nonlinear and dynamic under impulse condition. The electrical parameters of the grounding system are not only frequency dependent, but are also time dependent. Moreover, the mutual couplings between the grounding conductors cannot be ignored. Based on the hybrid electromagnetic method and vector fitting, a time-domain approach for the transient performance of the grounding system is developed. All the aforementioned phenomena can be covered in this method. Meanwhile, the sudden change of the parameters resulting from the soil ionization is also successfully dealt with in this paper. Compared with previous study results and field tests, the method proves to be reliable and effective. Index Terms—Frequency dependence, grounding, ionization, mutual coupling, time-domain analysis.

I. INTRODUCTION HE impulse performance of the grounding system plays an important role in lightning protection. The grounding system is required to have “sufficiently low impedance and currentcarrying capacity to prevent the buildup of voltages that may result in undue hazard to connected equipment and to persons” [1]. For evaluating the protection effect of the grounding system, it is essential to analyze its characteristics. In fact, unlike that under power frequency condition, the performance of the grounding system under impulse condition is nonlinear and dynamic. First, when the grounding conductor is excited by impulse currents, the phenomenon of time dependence results from the soil ionization [2], [3]. And then, the frequency dependence [4], [5] is also very important and needs to be taken into account. Moreover, the mutual couplings between the grounding conductors cannot be ignored, especially under impulse condition. Typically, there are three kinds of methods to obtain the transient performance of the grounding system under impulse condition. The first one is the circuit method such as the lumped circuit method [6]–[8] and the distributed circuit method (so-called

T

Manuscript received April 30, 2014; revised August 31, 2014 and October 2, 2014; accepted October 30, 2014. Date of publication November 14, 2014; date of current version April 13, 2015. This work was supported by the National Natural Science Foundation of China under Grant 51277107. The authors are with the State Key Lab of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]; [email protected]. cn; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2014.2366875

transmission line methods, TLM) [9]. In order to consider the soil ionization, the methods in [9] applied rational approximation to deal with the frequency-dependent parameters and solved the problem in the time domain. These circuit methods are convenient and efficient, but are not easy to consider the mutual coupling, especially for the complex structure in the multilayer medium [10], which hinders the wide application. In this situation, the electromagnetic methods are more appropriate. The second one is the finite-difference time-domain (FDTD) method [11]–[16], which is based on Maxwell’s equations, and can treat any transient excitation in the time domain. This method can deal with not only the mutual coupling and the frequency dependence, but also the soil ionization. However, FDTD requires discretization of the grounding space, which would be troublesome and complicated for a complex grounding system. The third one is the hybrid electromagnetic method (HEM) [17]–[20], as well as the partial element equivalent circuit method [21], [22]. Based on the Maxwell’s equations, this kind of methods forms an equivalent circuit to solve the electromagnetic problem. The derivation is closely related to the method of moment (MoM) [23]–[26]. The advantage of these methods is the full consideration of the mutual coupling and the frequency dependence. And also, just as the circuit method, it is convenient and efficient to calculate the transient performance of the grounding system. However, because it is a frequency-domain method, it is cumbersome to take the soil ionization into account. In order to do that, repeated Fourier transforms and multiple iterations are necessary [18], [19], [26]. Recommended by CIGRE and by IEEE Working Group [27], one method to estimate the soil ionization, using the empirical formula, was also put forward. But besides the potential rise at the injecting point, many details about the transient performance, such as the current distribution and the voltage distribution, cannot be obtained. In fact, it would be more convenient to consider the soil ionization in the time domain. The rational approximation technique was applied to solve frequency-dependent problem in the time domain [28]–[31]. Unfortunately, these papers did not tackle the soil ionization. A recently published paper [32] has presented a time-domain approach to obtain the transient performance of the grounding system with consideration of the mutual coupling, the frequency dependence and the soil ionization. In that paper, the leakage current was set at the midpoint of each segment, which was “the difference between the longitudinal current flowing into the start point and that flowing out of the end point of the segment.”

0018-9375 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

WU et al.: COMPREHENSIVE APPROACH FOR TRANSIENT PERFORMANCE OF GROUNDING SYSTEM IN THE TIME DOMAIN

251

Due to this current arrangement, the approach in [32] neglected the mutual inductive impedance between the conductor segments into account. As a matter of fact, the mutual coupling between the conductors includes two different types: one is the mutual admittance, which results from the current leaking out of the conductor; the other is mutual inductive impedance, which results from the current along the conductor. For complex grounding device at high frequency, omitting the mutual inductive impedance may lead to some errors. Based on the previous research work, this paper introduces a new current arrangement and a corresponding equivalent circuit, which is different from those in [32] and can take the mutual inductive impedance into account. And then, this paper proposes one effective method to determine the initial state of the nonresistant elements, which is not mentioned in [32]. For validation, several examples are simulated and compared with the field test results in this paper. II. MODEL OF GROUNDING SYSTEM IN TIME DOMAIN A. General Idea The approach deals with the transient problem in the time domain by the following idea. 1) Set several pseudopoints along the conductor so that the conductor is divided into several segments. An assumption of the arrangement of the currents and potentials as unknown variables is introduced in order to simulate the real-current distribution. 2) Set up an equivalent circuit for the grounding system with MoM based on the electromagnetic quasi-static hypothesis. The equivalent circuit consists of several frequencydependent elements. 3) Deal with the frequency-dependent elements by vector fitting. Thus, the equivalent circuit converts to a frequencyindependence one, and can be solved in the time domain. 4) Calculate the current and voltage distribution by the timedomain method. 5) Judge whether the soil around the grounding conductors is ionized or not. If ionized, enlarge the radiuses of the conductors and repeat the aforementioned procedures until obtaining convergent states. The iterative scheme can be displayed in Fig. 1. In [32], the leakage currents were assumed to flow out of the midpoints of the segments. This current arrangement cut the longitudinal currents in the segments into two different ones, which makes it troublesome to take the mutual inductive impedances among the segments into account in [32]. In order to consider all the mutual coupling all together, new current arrangement should be introduced. In this paper, the leakage currents are assumed to flow out of the nodes, which are the terminal points of the segments. Then, the whole longitudinal current in each segment can be regarded as a uniform one, and it is easy to take all the mutual couple into account. A new equivalent circuit and a group of equations are set up. The nonlinear and dynamic soil ionization, the frequency-dependent parameters are considered in the model also. At the same time, how to determine the initial state of the nonresistance elements while the soil is ionized in the iterative process will be explained.

Fig. 1.

Iteration Scheme of developed program.

B. Hypothesis of Current Arrangements When an impulse current is injected into a conductor buried under the ground, the currents flow both along and out of the conductor. For convenience, the grounding conductor is first cut into K short segments and N nodes. And some hypotheses are used as follows. Regarding a segment, the current along it, or so-called longitudinal current Il , remains constant and concentrates on the axial line. Besides, the current out of it leaks from its two terminals evenly and intensively. Regarding a node, the current out of it, or so-called leakage current Ie , is half the sum of all the currents out of the segments connected to the node. Moreover, the conductor segment can be seen as a thin-wire structure. Fig. 2 shows the current arrangements based on the aforementioned hypotheses. Ilk is the current along segment k, while Ien is the current leaking out of node n, which is half of all the leakage currents of segments k, k − 1and k − 2 added together. C. Mechanism of Equivalent Circuit Considering the ununiform current distribution along the grounding conductor, the HEM is used to analyze the grounding conductor.

252

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 2, APRIL 2015

Fig. 2.

Current arrangements.

For each point on the surface of the conductor segment k, the interface condition must be satisfied as k k − Ea,e =0 Ea,i

(1)

k is the axial component of the internal electric field where Ea,i k caused by the longitudinal current of the conductor itself, Ea,e is the axial component of the external electric field impressed k can be expressed as on the conductor. Ea,i k = Zck Ilk Ea,i

where as

Zck

k Ea,e

N 

1

 ∇φ

i,k

dl =

i=1

N   i,k φ 2 − φi,k 1 i=1

=

can be expressed

K N   √ = − −1ω Aj,k ∇φi,k a −

N 

where is the axial component of the vector magnetic potential, impressed on the surface of segment k, caused by the longitudinal current of segment j; ϕi,k is the scalar electric potential, impressed on the surface of segment k, caused by the leaking current of the segment i; and ω is the angular frequency. Aj,k a can be expressed as    1 j lk μ0 Ilk dl Aj,k = (4) · k a 4π l j r l where lj (lk ) is the vector length of segment j (k); lj (lk ) is the length of segment j (k); r is the distance between dlj and lk . Thus, the following equation can be achieved by substituting (2)–(4) into (1): Zck Ilk +

√

−1ω

K 

N 

Aj,k a +

j =1

∇φi,k = 0.

(5)

i=1

By integrating (5) along the axial direction between the two ends 1 and 2 of the conductor segment k, the following expression is obtained:  2 k Ea,i dl = Zck Ilk lk 1

−1ω

K 

Aj,k a dl

=

√

−1ω

j =1

=

√

−1ω

K   j =1 K  j =1

1



2

μ0 Ilj 4π

Lj,k Ilj

 lj

L

(3)

i=1

 1 j lk dl · k dl r l

Zei,k 2 Iei −

i=1

j,k

Aj,k a

1

2

where

j =1

2√



Equivalent circuit of the grounding system.

(2)

is the internal self-impedance.

k Ea,e



Fig. 3.

μ0 = 4π

N 

Zei,k 1 Iei

(6)

i=1

   lk

lj

 1 j dl dlk r

(7)

and Zei,k 1 (Zei,k 2 ) is the leakage impedance, which represents the potential value of the node k1 (k2 ) caused by one unit current leaking out of the node i. Thus, (5) can be derived as Zck lk Ilk +

K N   √  i,k Ze 2 − Zei,k 1 Iei = 0. −1ω Lj,k Ilj + j =1

i=1

(8) Based on the aforementioned derivation, the equivalent circuit of the grounding conductor is established as Fig. 3 shows. There are four electromagnetic mechanisms, which are named as leakage self-impedance Zes , leakage mutual-impedance Zem , longitudinal self-impedance Zls (including internal one Zck lk √ k and external one −1ωL ), and longitudinal mutual-impedance √ Zlm ( − 1ωLj,k ). The leakage parameters can be computed by using the complex image methods [33], [34]. The longitudinal parameters are derived in [35]–[37], and can be conveniently calculated. It can be seen that the mutual inductive impedance is considered in this approach. Then, by vector fitting, each frequencydependent element in the equivalent circuit is transferred as a group of frequency-independent elements. Thus, the circuit can be solved in the time domain. Moreover, the cylinder model is applied to mimic soil ionization effect, which has been introduced in [32]. Further, based on the iteration process, the transient performance of the grounding system can be obtained. D. Determination of Initial State for Each Time Step It is worth noting that there are many nonresistant elements such as the inductances and capacitance in the equivalent circuit.

WU et al.: COMPREHENSIVE APPROACH FOR TRANSIENT PERFORMANCE OF GROUNDING SYSTEM IN THE TIME DOMAIN

Fig. 4. Equivalent circuits of one specified element at two time steps (a) convergent states at t (b) initial states at t + Δt.

Once the soil is ionized, the elements in the equivalent circuit will change over time. This encounters the question of how to determine the initial state for each time step in the iterating process. This is not mentioned in [32]. n , for example. By vector fitTake one specified element Zes ting, the frequency-dependent element can be equivalent to a network as Fig. 4(a) shows, where the network has a convergent n may state for the time-step t. Considering soil ionizing, Zes change at the next time-step t + Δt, and the equivalent network becomes as Fig. 4(b) shows. Herein, lays a problem to be settled regarding how to determine the initial state of t + Δt, based on the convergent state of t. In order to solve the problem, this paper proposes the following rule: It is assumed that the topology of the network keeps invariable at each time-step, which can be achieved by applying the same number of vector fitting poles P. Further, the initial states are determined by the following principle: the flux linkage of each inductance, as well as the electric charge of each capacitance, remains constant, correspondingly, and respectively. In Fig. 4, the circuit topologies of the two time steps must be kept the same. At t, assume that the convergent current of the inductance Ln (t) is Ien ([t]1 ), and the convergent voltage of capacitance C n ,i (t) is Ucn ,i ([t]1 ). While soil ionizing, the parameters in the network change, from Ln (t) and C n ,i (t) to Ln (t + Δt) and C n ,i (t + Δt), respectively. Based on the rule, the initial state of the elements at t + Δt can be obtained as Ien ([t + Δt]0 ) = Ucn ,i ([t + Δt]0 ) =

Ien

n

([t]1 ) L (t) Ln (t + Δt)

Ucn ,i ([t]1 ) C n ,i (t) (i = 1, 2, . . . , P ). C n ,i (t + Δt) (9)

III. VALIDATION AND APPLICATION In this part, the validation of the model is carried out. The model is compared with those by Grcev in [4] and Geri in [7], which are widely referred to; moreover, it is applied to compare with the result of our field test in Zhejiang, China. Before comparison, the frequency-dependent model of the soil parameter is introduced. In the simulation, the resistivity and the permittivity of the soil satisfy the following curve-fit

Fig. 5.

253

Comparison with Grecv for validating.

expressions [38]:

192.2 f < 10 kHz εr (f ) = 7.6 · 103 f −0.4 + 1.3 f ≥ 10 kHz  −1

 0.65 ρ (f ) = ρ0 1 + 1.2 · 106 ρ0.73 · (f − 100) 0 (10) where f represents the frequency, and ρ0 represents the soil resistivity at 100 Hz. A. Comparison With Grcev [4] In [4], the grounding system researched by Grecv is made of copper with a diameter of 12 mm, a length of 15 m, and horizontally buried at the depth of 0.6 m. The soil resistivity is 70 Ω · m at 100 Hz. The current, illustrated in Fig. 5, is injected at one end of the copper bar. The peak value of the impulse current is about 35 A. The voltages at three points as 0 m, 3.5 m, and 7 m are concerned. It can be seen that, for the three points, the simulated results by the proposed method in this paper correlate better with the measured ones, especially at the rise time. However, it is worth noting that the impulse current is relatively small, so the soil ionization effect is ignored. B. Comparison With Visacro [5] In [5], the grounding system researched by Visacro is a copper horizontal electrode with a diameter of 14 mm and a length of 9.6 m. It is buried at the depth 0.5 m. The soil resistivity is 1400 Ω · m at 100 Hz. The peak value of the impulse current is about 0.3, which is injected at one end of the electrode. From Fig. 6, it can be seen that the simulated result by the proposed method fits the measured one by Visacro well. Moreover, the result with frequency-dependence ignored is also illustrated in the figure. The comparison shows that the frequency dependence should be considered while the grounding device is buried in the soil with high resistivity. This conclusion can be validated and explained by [39]. C. Comparison With Geri [7] In [7], the grounding system is still made of copper and buried at the depth of 0.6 m. The copper bar is with a diameter of 12 mm and a length of 5 m. The soil resistivity is 42 Ω · m at 100 Hz, and

254

Fig. 6.

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 2, APRIL 2015

Comparison with Visacro for validating. Fig. 9.

Fig. 7.

Injection impulse current in the field test.

Comparison with Geri for validating.

Fig. 10.

Corresponding impulse voltage in the field test. TABLE I RESULTS OF FIELD TEST

Peak Value of Impulse Current

Fig. 8.

Grounding model for validating in the field test.

the critical breakdown electrical field is 300 kV/m. The current, illustrated in Fig. 7, is injected at one end of the copper bar. The peak value of the impulse current reaches about 30 kA. It can be seen that the simulated result by the proposed method is closer to the measured one, compared with that of Geri. Besides, the soil is ionized at this time. While ignoring the soil ionization, the peak value of the impulse voltage will reach about 250 kV, as shown in Fig. 7. D. Comparison With Field Test The approach is also validated by the test field in Zhejiang, China. In the test, a cross horizontal conductor made of steel with a diameter of 12 mm is buried in the soil with a depth of 1 m, as shown in Fig. 8. The soil has two horizontal layers; the resistivity of the top layer with a thickness of 6.2 m is 15.8 Ω · m, and that of the bottom layer is 2.6 Ω · m. The critical electric field strength for the soil ionization is 270 kV/m, which was measured by testing on soil samples. The power frequency grounding impedance is measured as 3.14 Ω.

1.3 kA 2.2 kA 3.3 kA 4.4 kA 5.5 kA 6.8 kA

Peak Value of Impulse Voltage

Impulse Impedance

4.2 kV 6.9 kV 10.6 kV 14.0 kV 17.1 kV 19.8 kV

3.23 Ω 3.22 Ω 3.21 Ω 3.18 Ω 3.10 Ω 2.91 Ω

Inject a series of impulse currents with different peak values as Fig. 9 shows, and the corresponding impulse voltage is illustrated in Fig. 10. The impulse impedance can be calculated shown in Table I and Fig. 11. With the injection current increasing, the impulse impedance decreases, which is mainly caused by the soil ionization. The simulation result by the approach in this paper is compared with that of the field test, as shown in Fig. 12. There are four data curves: one is the current curve, which is in black; the other three are all voltage curves, which are in gray. Among the three voltage curves, the one with a higher peak value is the simulated one which ignored the soil ionization; the other two are the measured one and the simulated one with soil ionization considered. The latter two overlap together so that they are hard to be distinguished. It can be seen that the result by the approach in this paper agrees with the test result very well.

WU et al.: COMPREHENSIVE APPROACH FOR TRANSIENT PERFORMANCE OF GROUNDING SYSTEM IN THE TIME DOMAIN

Fig. 11.

Impulse impedance in the field test.

255

Fig. 14. Application (similar model with Grcev [4], standard current waveform of 8/20 μs).

Compared with the simulation result in the last application, the dependence has less effect on the impulse performance. This is because the front time of the impulse current of 8/20 μs is longer, which means that the high-frequency components contained in it are weaker than that in the impulse current in the last application. IV. CONCLUSION

Fig. 12.

Comparison with the field test for validating.

A numerical method is developed to analyze the transient performance of a grounding structure in the time domain. On the basis of MoM and vector fitting, an iteration scheme is established to take the mutual impedance, the frequency dependence, and the soil ionization into account, and to solve the nonlinear and dynamic problem. The method is verified by comparing with the results in the previously published papers and the results of the field test. REFERENCES

Fig. 13. Application (similar model with Grcev [4], same current waveform with [4]).

E. Applications In order to show the overall effect of frequency dependence and of soil ionization, and to distinguish the incidence of each of the two phenomena, another simulations is carried out, as an application of the proposed method in this paper. In the simulation, the grounding system is a single horizontal copper bar, the same as that in [4]. Besides, the soil resistivity is 500 Ω · m at 100 Hz, and the critical breakdown electrical field is 300 kV/m. The impulse current is with the same waveform as that in [4], but its peak value is taken up to 10 kA. The simulation result is illustrated in Fig. 13. It can be seen that the two phenomena have a large incidence on the impulse characteristic of the grounding system. Furthermore, another impulse current with a standard waveform of 8/20 μs is injected into the aforementioned grounding system. The simulation result is shown in Fig. 14.

[1] IEEE Standard Dictionary of Electrical and Electronics Terms, IEEE Std. 100-1992, Jan. 1993. [2] A. Geri, “Behavior of grounding systems excited by high impulse currents: The model and its validation,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 1008–1017, Jul. 1999. [3] A. Habjanic and M. Trlep, “The simulation of the soil ionization phenomenon around the grounding system by the finite element method,” IEEE Trans. Magn., vol. 42, no. 4, pp. 867–870, Apr. 2006. [4] L. D. Grcev and F. E. Menter, “Transient electromagnetic fields near large earthing systems,” IEEE Trans. Magn., vol. 12, no. 3, pp. 1525–1528, May 1996. [5] S. Visacro, R. Alipio, M. H. M. Vale, and C. Pereira, “The response of grounding electrodes to lightning currents: The effect of frequencydependent soil resistivity and permittivity,” IEEE Trans. Electromagn. Compat., vol. 52, no. 2, pp. 401–406, May 2011. [6] H. B. Dwight, “Calculation of the resistances to ground,” Electr. Eng., vol. 55, pp. 1319–1328, Dec. 1936. [7] A. Geri, G. M. Veca, E. Garbagnati, and G. Sartorio, “Non-linear behaviour of ground electrodes under lightning surge currents: Computer modelling and comparison with experimental results,” IEEE Trans. Magn., vol. 28, no. 2, pp. 1442–1445, Mar. 1992. [8] C. T. Mata, M. I. Fernandez, V. A. Rakov, and M. A. Uman, “EMTP modeling of a triggered-lightning strike to the phase conductor of an overhead distribution line,” IEEE Trans. Power Del., vol. 15, no. 4, pp. 1175–1181, Oct. 2000. [9] F. Menter and L. Grcev, “EMTP-based model for grounding system analysis,” IEEE Trans. Power Del., vol. 9, no. 4, pp. 1838–1849, Oct. 1994. [10] L. Grcev, “Modeling of grounding electrodes under lightning currents,” IEEE Trans. Electromagn. Compat., vol. 51, no. 3, pp. 559–571, Aug. 2009. [11] K. Tanabe, “Novel method for analyzing the transient behavior of grounding systems based on the finite-difference time-domain method,” in Proc.

256

[12] [13] [14]

[15]

[16] [17] [18] [19] [20] [21]

[22] [23] [24] [25] [26]

[27] [28]

[29] [30]

[31]

[32] [33] [34]

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 2, APRIL 2015

IEEE Power Eng. Soc. Winter Meet., Columbus, OH, USA, 2001, vol. 3, pp. 1128–1132. G. Ala and F. Viola, “Soil ionization in earth electrodes by a finite difference time domain scheme,” in Proc. IEEE Int. Symp. Electromagn. Compat., Aug. 2004, vol. 3, pp. 850–855. Y. Baba, N. Nagaoka, and A. Ametani, “Modeling of thin wires in a lossy medium for FDTD simulations,” IEEE Trans. Electromagn. Compat., vol. 47, no. 1, pp. 54–60, Feb. 2005. M. Tsumura, Y. Baba, N. Nagaoka, and A. Ametani, “FDTD simulation of a horizontal grounding electrode and modeling of its equivalent circuit,” IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp. 817–825, Nov. 2006. 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. 1262–1263, Apr. 2008. G. Ala, P. L. Buccheri, P. Romano, and F. Viola, “Finite difference time domain simulation of earth electrodes soil ionization under lightning surge condition,” IET Sci. Meas. Technol., vol. 2, no. 3, pp. 134–145, 2008. L. Grcev and F. Dawalibi, “An electromagnetic model for transients in grounding systems,” IEEE Trans. Power Del., vol. 5, no. 4, pp. 1773– 1781, Nov. 1990. A. F. Otero, J. Cidras, and J. L. del Alamo, “Frequency-dependent grounding system calculation by means of a conventional nodal analysis technique,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 873–878, Jul. 1999. J. Cidr´as, A. F. Otero, and C. Garrido, “Nodal frequency analysis of grounding systems considering the soil ionization effect,” IEEE Trans. Power Del., vol. 15, no. 1, pp. 103–107, Jan. 2000. S. Visacro and A. Soares, “HEM: A model for simulation of lightningrelated engineering problems,” IEEE Trans. Power Del., vol. 20, no. 2, pp. 1206–1208, Apr. 2005. P. Yutthagowith, A. Ametani, N. Nagaoka, and Y. Baba, “Application of the partial element equivalent circuit method to analysis of transient potential rises in grounding systems,” IEEE Trans. Electromagn. Compat., vol. 53, no. 3, pp. 726–736, Aug. 2011. P. Yutthagowith, A. Ametani, F. Rachidi, N. Nagaoka, and Y. Baba, “Application of a partial element equivalent circuit method to lightning surge analyses,” Elect. Power Syst. Res., vol. 94, no. 1, pp. 30–37, 2013. F. Dawalibi and A. Selbi, “Electromagnetic fields of energized conductors,” IEEE Trans. Power Del., vol. 8, no. 3, pp. 1275–1284, Jul. 1993. R. Andolfato, L. Bernardi, and L. Fellin, “Aerial and grounding system analysis by the shifting complex images method,” IEEE Trans. Power Del., vol. 15, no. 3, pp. 1001–1009, Jul. 2000. B. Zhang, Z. Zhao, X. Cui, and L. Li, “Diagnosis of breaks in substation’s grounding grid by using electromagnetic method,” IEEE Trans. Magn., vol. 38, no. 2, pp. 473–476, Mar. 2002. B. Zhang, J. He, J.-B. Lee, X. Cui, Z. Zhao, J. Zou, and S.-H. Chang, “Numerical analysis of transient performance of grounding systems considering soil ionization by coupling moment method with circuit theory,” IEEE Trans. Magn., vol. 41, no. 5, pp. 1440–1443, May 2005. L. Grcev, “Time- and frequency-dependent lightning surge characteristics of grounding electrodes,” IEEE Trans. Power Del., vol. 24, no. 4, pp. 2186–2196, Oct. 2009. M. Vargas, D. Rondon, J. Herrera, J. Monta˜na, D. Jimenez, M. Camargo, and H. Torres, “Influence of grounding system modeling on transient analysis of transmission lines due to direct lightning strike,” in Proc. Power Tec, St. Petersburg, Russia, Jun. 2005, pp. 1–6. B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del., vol. 21, no. 3, pp. 1587–1592, 2006. N. Theethayi, R. Thottappillil, M. Paolone, C. A. Nucci, and F. Rachidi, “External impedance and admittance of buried horizontal wires for transient studies using transmission line analysis,” IEEE Trans. Dielectr. Electr. Insul., vol. 14, no. 3, pp. 751–761, Jun. 2007. D. Jim´enez, M. Camargo, J. Herrera, M. Vargas, and H. Torres, “Application of modified vector fitting to grounding system modeling,” presented at the 9th Int. Symp. Lightning Protection, Foz do Iguacu, Brazil, Nov., 2007. B. Zhang, J. Wu, J. He, and R. Zeng, “Analysis of transient performance of grounding system considering soil ionization by time domain method,” IEEE Trans. Magn., vol. 49, no. 5, pp. 1837–1840, May 2013. Y. L. Chow, J. J. Yang, and G. E. Howard, “Complex images for electrostatic field computation in multilayered media,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1120–1125, Jul. 1991. B. Zhang, X. Cui, L. Li, and J. He, “Parameter estimation of horizontal multilayer earth by complex image method,” IEEE Trans. Power Del., vol. 20, no. 2, pp. 1394–1401, Apr. 2005.

[35] J. Van Bladel, Electromagnetic Field. Bristol, PA, USA: Hemisphere, 1985. [36] R. L. Stoll, The Analysis of Eddy Currents. London, U.K.: Oxford Univ. Press, 1974. [37] J. R. Wait, “Electromagnetic wave propagation along a buried insulated wire,” Can. J. Phys., vol. 50, pp. 2402–2409, 1972. [38] S. Visacro and R. Alipio, “Frequency dependence of soil parameters: Experimental results, predicting formula and influence on the lightning response of grounding electrodes,” IEEE Trans. Power Del., vol. 27, no. 2, pp. 927–935, Apr. 2012. [39] R. Alipio and S. Visacro, “Impulse efficiency of grounding electrodes: Effect of frequency-dependent soil parameters,” IEEE Trans. Power Del., vol. 29, no. 2, pp. 716–723, Apr. 2014.

Jinpeng Wu was born in Hebei, China, in 1987. He received the B.Sc. degree in electrical engineering from the Department of Electrical Engineering, Tsinghua University, Beijing, China, in 2010, where he is currently working toward the Ph. D. degree in the same Department. His research interests include overvoltage analysis in power system, grounding technology, and electromagnetic compatibility.

Bo Zhang was born in Datong, China, in 1976. He received the B.Sc. and Ph.D. degrees in theoretical electrical engineering from the North China Electric Power University, Baoding, China, in 1998 and 2003, respectively. He had been a Postdoctoral Researcher in the Department of Electrical Engineering, Tsinghua University, Beijing, China, from September 2003 to June 2005, where he is currently an Associate Professor in the same department. His research interests include computational electromagnetics, grounding technology and EMC in power system.

Jinliang He (F’07) was born in Changsha, China, in 1966. He received the B.Sc. degree from Wuhan University of Hydraulic and Electrical Engineering, Wuhan, China, the M.Sc. degree from Chongqing University, Chongqing, China, and the Ph.D. degree from Tsinghua University, Beijing, China, all in electrical engineering, in 1988, 1991, and 1994, respectively. He became a Lecturer in 1994 and an Associate professor in 1996, in the Department of Electrical Engineering, Tsinghua University, where he was promoted to a Professor in 2001. Since 2009, he has been evaluated as Changjiang Scholar Professor of China Education Ministry. Currently, he is the Chair of High Voltage Research Institute, Tsinghua University. His research interests include overvoltage and EMC in power systems and electronic systems, lightning protection, grounding technology, power apparatus, and dielectric material. He is the author of five books and 240 technical papers.

Rong Zeng (SM’06) was born in Shaanxi, China, in 1971. He received the B.Sc., M.Eng., and Ph.D. degrees in electrical engineering from Tsinghua University, Beijing, China, in 1995, 1997, and 1999, respectively. He became a Lecturer in the Department of Electrical Engineering and an Associate Professor at Tsinghua University in 1999 and 2002, respectively, where he is currently the Vice Dean of the same department. His research interests include high-voltage technology, grounding technology, power electronics, and distribution system automation.