IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 5, MAY 2013 1837

Analysis of Transient Performance of Grounding SystemConsidering Soil Ionization by Time Domain Method

Bo Zhang, Jinpeng Wu, Jinliang He, Fellow, IEEE, and Rong Zeng

State Key Lab of Power Systems, High voltage Lab., Department of Electrical Engineering, Tsinghua University,Beijing 100084, China

The grounding system is very important for lightning protection. If the lightning current is large enough, soil will be ionized, whichwill have a great effect on the transient performance of the grounding system. In this paper, an efficient time domain method is proposedto analyze the transient performance of the grounding system. The method takes account of nonlinear soil ionization, the shielding effectamong electrodes, and frequency-dependent parameters all together. First, a frequency domain circuit model is set up based on amomentmethod coupling with circuit theory. Then, the model is approximated by a frequency-independent circuit model with the help of vectorfitting method. Finally, the time-varying soil ionization is considered by making corresponding parameters time-varying when a timedomain method is used to solve the circuit model. The method is validated by field test.

Index Terms—Grounding, ionization, lightning, time domain analysis.

I. INTRODUCTION

T HE LIGHTNING impulse performance of groundingsystems plays an important role in lightning protection

[1]–[7]. Unlike that at power frequency, the performance ofgrounding systems under lightning is nonlinear and dynamicdue to soil ionization [2]–[4]. At the same time, because thelightning current has abundant high frequency components, theskin effect in the self-impedance of the ground electrode willbe significant [5]. This frequency-dependent self-impedanceis so high that most of the lightning current will leak into theearth near the current injection point, which will aggravatethe soil ionization near this part. What is more, for complexground grids, the shielding effect among electrodes will alsoaffect the current distribution in the grid. Thus, in order tothoroughly analyze the lightning impulse performance of agrounding system, the nonlinear and dynamic soil ionization,the frequency-dependent parameters, and the shielding effectamong electrodes should be taken into account all together.However, above effects have not been addressed all together

in the literature. Usually, when a grounding system is analyzed,the dynamic and nonlinear ionization phenomenon is often ig-nored [1], [7]–[10]. Because of the nature of ionization, timedomain approaches, such as transmission line method (TLM)and finite-difference time-domain (FDTD)method, were widelyused. TLM was used in [5], but shielding effect among elec-trodes was neglected. FDTD or finite element method (FEM)were used in [3], [6], and [7], but for complex grounding grid,building division meshes is nontrivial because the character-istic dimensions of the structure vary in a wide range with morethan three orders. Methods of moment (MoM), partial elementequivalent circuit (PEEC) method, and hybrid electromagneticmethod (HEM) were used in [8]–[11], respectively, which couldtake account of the frequency-dependent parameters and theshielding effect among the electrodes easily, but they did notconsider the effect of the dynamic soil ionization. MoM was

Manuscript received October 30, 2012; revised December 26, 2012, January22, 2013; accepted January 24, 2013. Date of current version May 07, 2013.Corresponding author: B. Zhang (e-mail: shizbcn@tsinghua.edu.cn).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/TMAG.2013.2243824

used in [4] with the help of fast Fourier transform, but it wasinefficient due to the iterative calculation at many frequencies.In this paper, a time domain method is developed base on

the MoM model. The method can take into account the non-linear soil ionization, the shielding effect among electrodes, andthe frequency-dependent parameters all together. Because it is atime domain method, it is more efficient than frequency domainmethod when soil ionization should be considered.

II. BASIC PRINCIPLE

In this paper, first a circuit model in frequency domain for agrounding grid is set up with the help of MoM, in which thefrequency-dependent self-impedance of the electrode and theshielding effect among electrodes are considered. Then, eachfrequency-dependent element in the circuit is approximated bya group of frequency-independent elements with the help ofvector fitting method. The frequency-independent circuit can besolved in time domain [15]. In order to take account of the time-varying soil ionization, corresponding parameters are varied ateach time step.

A. MoM Model in Frequency Domain

For a complex grounding grid, based on the idea of the MoM,current distribution can be arranged as follows [12].The leakage current of each segment is the difference be-

tween the longitudinal current flowing into the start point andthat flowing out of the end point of the segment. Let the longitu-dinal current from the start point to the midpoint of the segmentbe uniform and be equal to that at the start point. Similarly, letthe current from the midpoint to the end point of the segment isalso uniform and is equal to that flowing out of the end. Fig. 1(a)shows the currents on the -th segment.Because usually the size of a grounding grid is much smaller

than the wavelength associated with the maximum significantfrequency of the lightning current, quasi-static field theory canensure a satisfactory numerical accuracy. Then, a circuit modelof the grounding grid is set up in frequency domain as shown inFig. 1(b). In the figure, the potential on the outer surface of themidpoint of the segment is regarded as a voltage source such as

. are the self-imped-ances of corresponding segments which are frequency-depen-

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1838 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 5, MAY 2013

Fig. 1. Part of a grounding system and its equivalent circuit. (a) Part of agrounding system. (b) Equivalent circuit.

Fig. 2. Equivalent circuit of the internal self-impedance.

dent. The voltage sources are generated by all of the leakagecurrents and are determined by

(1)

where is the column vector of the voltage sources, isthe column vector of the leakage current, is a mutual resis-tance matrix. The capacitive reactances among segments areneglected due to their small values compared with the mutualresistances. The self-impedance can be obtained by

(2)

where is the angular frequency, is the external self in-ductance, and is the internal self-impedance which is fre-quency-dependent due to the skin effect [13]. Because the mu-tual inductances are much smaller than , they are ignored.Above model assumes that the grounding grid consists of

cylindrical conductors. For noncylindrical conductors, theycan be represented by equivalent cylindrical conductors whichare possible determined rigorously. However, in practice, theconcept of “equivalent contact area with soil” can be used toobtain an approximate equivalent cylindrical conductor whichwill give satisfactory agreement. Then, a circuit model for thegrounding grid is set up. The shielding effect is considered bythe voltage sources, and the frequency-dependent parametersare addressed by the self-impedances.

B. Frequency-Independent Model

In order to use time domain method to consider the soil ion-ization, in (2) is approximated by a group of frequency-in-dependent elements. First, is fitted by a series of rational ex-pressions in the frequency range of the lightning current

(3)

where represents the complex frequency is the numberof poles, and and are constant coefficients. Thereare several reasonable approximation methods to obtain (3). Wechoose vector fitting method because of its reliability and effi-ciency [14]. Then, the group of the frequency-independent ele-

Fig. 3. Shape of the ionized zone around an electrode [4].

ments can be obtained as Fig. 2 shows, where the elements havefollowing relationship with the parameters in (3):

C. Soil Ionization Effect

When a high magnitude current is injected into the groundingsystem, soil ionization will occur if the electric fields sur-rounding the segments exceed the critical breakdown value .The affected portion of the soil will become a good conductor.In order to take account of the effect of the soil ionization, someassumptions are adopted.First, in the ionized region, voltage drop is considered neg-

ligible. The ionized zone is assumed concentric with the con-ductor, and its radius extends up to a distance where the electricfield has decreased to as Fig. 3 shows [4]. This is equivalentto changing the radius of the conductor by

(4)

where and are the leakage current and the length of thesegment respectively, and is the soil resistivity.Second, the longitudinal current is considered only flowing

inside the conductor. Thus, the self-impedance of each segmentis not affected by the soil ionization, which means that all theequivalent circuits of the internal self-impedances in Fig. 2 donot vary with the development of soil ionization.Then, it can be seen that the soil ionization only affects the

voltage sources in Fig. 1(b). At this time, the voltage sourceis not the potential on the surface of corresponding segment,but the potential on the boundary of the ionized zone aroundthe segment. The boundary is dynamic according to (4) and thevoltage source is time-varying accordingly.

D. Solving the Model in Time Domain

The time-varying frequency-independent circuit model de-scribed above can be solved by the trapezoidal integration al-gorithm in time domain which has already been adopted by thewidely-used electromagnetic transient program (EMTP) [15].According to the algorithm, at each time step, all the inductancesand capacitances can be simulated by means of a resistance con-nected in parallel with an ideal current source. For example, aninductance can be simulated by means of a resistance con-nected in parallel with an ideal current source asFig. 4 shows [15]. In the figure, is the current flowingin the inductance from node to and are thepotentials on the nodes respectively. and can beobtained with and

(5)

ZHANG et al.: ANALYSIS OF TRANSIENT PERFORMANCE OF GROUNDING SYSTEM CONSIDERING SOIL IONIZATION 1839

Fig. 4. Equivalent circuit of an inductance.

(6)

Thus, if and are known, wecan set up the equivalent circuit of at time . For a capacitance,an equivalent circuit similar to Fig. 4 can also be obtained basedon the same idea. Then, the total equivalent circuit at time canbe set up which can be solved by node potential method.In order to consider the effect of soil ionization accurately, an

iterative scheme is developed at each time step.1) First, the current distribution on the grounding grid withnew lightning current at this time step is calculated basedon the radii of the segments at the last time step.

2) Then, the electric field on the surface of each segment isobtained by the leakage current

(7)

where is the original radius of the segment. If the electricfield on the surface of a segment is higher than , soilionization occurs. The equivalent radius of correspondingsegment during the ionization is calculated by (4).

3) With the new radius, current distribution on the groundinggrid is calculated again and another group of new equiva-lent radius of the ionized segments is obtained.

4) If there is almost no difference between the two groups ofequivalent radii, for example, the relative errors are smallerthan 5% for all segments, the final results at this time areobtained. Else, the analysis goes back to step 3).

In summary, the whole procedure can be displayed by Fig. 5.The nonlinear and dynamic soil ionization, the frequency-de-pendent parameters, and the shielding effect among electrodesare considered all together in the model. Because the modelis solved in time domain, it is more efficient than the tradi-tionalMoMwhich considers the soil ionization just in frequencydomain.

III. VERIFICATION AND APPLICATION

Our model was evaluated by field experiment. The test siteis located in Zhejiang Province, China. Measurement showedthat the soil can be approximated by two layers. The top layerwith a thickness of 6.2 m had a resistivity of 15.8 m, thebottom layer had a resistivity of 2.6 m. was 270 kV/m.20/40 s impulse currents with different peaks were used in thetest. All the grounding grids were made of steel electrode witha resistivity of m, a relative permeability of 636,a radius of 9 mm, and a depth of 0.8 m.First, in order to show the validity of the vector fittingmethod,

the internal self-impedance of the steel electrode with unitlength was approximated by a group of frequency-independentelements as show in Fig. 2, where is equal to 8. Calculationshowed that is 5.440 nH, is 1.552 to isnF, nF, F, F, F, mF,

mF, and F respectively, to ism m m

Fig. 5. Layout of the whole procedure.

Fig. 6. Internal self-impedance of the steel electrode with unit length. (a)magnitude, (b) angle.

m , and m respectively. Fig. 6 shows the orig-inal and the approximated . It can be seen that the two re-sults are in good agreement.Then, a 6.5 kA impulse current was injected into a 2-meter-

long horizontal electrode from one of its ends. Fig. 7 showsthe simulated and the measured ground potential rises at thecurrent injection point. It can be seen that the two results arealso in good agreement. The error is because of the variance onthe distribution of the soil resistivity and the critical breakdownvalue between our model and the real experiment.However, in Fig. 7, it is difficult to distinguish the effect of

soil ionization. In order to illustrate the effect of soil ioniza-tion, Fig. 8 is presented, which shows the variation of the im-pulse grounding resistance with the peak of the current. The im-pulse grounding resistance is defined as the ratio of the peakvoltage and the peak current at the current injection point. Sup-pose that the grounding resistance at power frequency does notvary with current, which is also shown in Fig. 8. It can be seenthat when the current is small (the peak value is smaller than

1840 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 5, MAY 2013

Fig. 7. Waveform of the ground potential rise u(t) on the horizontal electrode.

Fig. 8. Variation of the impulse grounding resistance with peak of the current.

Fig. 9. Waveform of the ground potential rise u(t) on a steel cross.

3 kA in the figure), the impulse grounding resistance is almosta constant and is somewhat higher than the grounding resis-tance at power frequency. This is because the current is so smallthat the soil ionization has not occurred and only the inductancealong the electrode takes effect. What is more, because the re-sistivity of the soil is small, the effect of the inductance becomesstrong. The inductance will increase the grounding resistance. Ifthe current waveform does not change, only the peak changes,the impulse grounding resistance will be a constant. When thecurrent increases further, the impulse grounding resistance be-gins to decrease. This means that the soil ionization appearsand becomes incrementally significant, which will decrease thegrounding resistance. If the current is large enough, the effectof the inductance will be smaller than that of the soil ioniza-tion, the impulse grounding resistance will be smaller than thegrounding resistance at power frequency. Thus, when analyzingthe transient performance of grounding system, both the induc-tance of the electrode and the soil ionization should be takeninto account.

At last, in order to evaluate the method for complexgrounding grid, experiment was also done on a steel cross witha leg-length of 5 m. A 7 kA 20/40 s impulse current wasinjected at the center of the cross. Fig. 9 shows the simulatedresult and the test result. It can be seen that the two results arealso in good agreement.

IV. CONCLUSION

In this paper, a time domain method is developed, which ad-dresses the nonlinear soil ionization, the shielding effect, and thefrequency-dependent parameters all together. Field test showsthat the method is effective and efficient.

ACKNOWLEDGMENT

This work was supported by the National Natural ScienceFoundations of China under Grant No. 51277107.

REFERENCES

[1] B. Zhang et al., “Effect of grounding system on electromagnetic fieldsaround building struck by lightning,” IEEE Trans. Magn., vol. 46, no.8, pp. 2955–2958, Aug. 2010.

[2] A. Geri et al., “Non-linear behaviour of ground electrodes under light-ning surge currents: Computer modelling and comparison with exper-imental results,” IEEE Trans. Magn., vol. 28, no. 2, pp. 1442–1445,Mar. 1992.

[3] A. Habjanic and M. Trlep, “The simulation of the soil ionization phe-nomenon around the grounding system by the finite element method,”IEEE Trans. Magn., vol. 42, no. 4, pp. 867–870, Apr. 2006.

[4] B. Zhang et al., “Numerical analysis of transient performance ofgrounding systems considering soil ionization by coupling momentmethod with circuit theory,” IEEE Trans. Magn., vol. 41, no. 5, pp.1440–1443, May 2005.

[5] Y. Liu, N. Theetahyi, and R. Thottappillil, “An engineering model fortransient analysis of grounding system under lightning strikes: Nonuni-form transmision-line approach,” IEEE Trans. Power Del., vol. 20, no.2, pp. 722–730, Apr. 2005.

[6] M. Tsumura, Y. Baba, N. Nagaoka, andA.Ametani, “FDTD simulationof a horizontal grounding electrode and modeling of its equivalent cir-cuit,” IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp. 817–825,Nov. 2006.

[7] J. Li et al., “Numerical and experimental investigation of groundingelectrode impulse-current dispersal regularity considering the transientionization phenomenon,” IEEE Trans. Power Del., vol. 26, no. 4, pp.2647–2658, Oct. 2011.

[8] L. D. Grcev and F. E. Menter, “Transient electromagnetic fieldsnear large earthing systems,” IEEE Trans. Magn., vol. 12, no. 3, pp.1525–1528, May 1996.

[9] P. Yutthagowith et al., “Application of the partial element equivalentcircuit method to analysis of transient potential rises in grounding sys-tems,” IEEE Trans. EMC, vol. 53, no. 3, pp. 726–736, Aug. 2011.

[10] S. Visacro and A. Soares, “HEM: A model for simulation of light-ningrelated engineering problems,” IEEE Trans. Power Del., vol. 20,no. 2, pp. 1206–1208, Apr. 2005.

[11] B. Zhang, Z. Zhao, X. Cui, and L. Li, “Diagnosis of breaks in substa-tion’s grounding grid by using electromagnetic method,” IEEE Trans.Magn., vol. 38, no. 2, pp. 473–476, Mar. 2002.

[12] R. F. Harrington, Field Computation by Moment Methods. NewYork: Macmillan, 1968.

[13] R. L. Stoll, The Analysis of Eddy Currents. Oxford, U.K.: OxfordUniv. Press, 1974.

[14] B. Gustavsen, “Improving the pole relocating properties of vector fit-ting,” IEEE Trans. Power Del., vol. 21, no. 3, pp. 1587–1592, 2006.

[15] H. W. Dommel, “Digital computer solution of electromagnetic tran-sient in single and multiple networks,” IEEE Trans. Power App. Syst.,vol. 88, no. 4, pp. 388–399, Apr. 1969.