**7. Simulation result and analysis**

A typical grounding system can be seen in Fig. 4. The earth is medalled by a two-layer conductive earth model, whose conductivity and permittivity is *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> <sup>500</sup>−1*S*/*m*, *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> <sup>900</sup>−1*S*/*m*, *<sup>ε</sup>*<sup>1</sup> <sup>=</sup> <sup>10</sup>*ε*0, *<sup>ε</sup>*<sup>2</sup> <sup>=</sup> <sup>22</sup>*ε*0, respectively, the first layer height is 5 m. The material of the grounding system conductor is Cu with conductivity *<sup>σ</sup>Cu* = 5.8 × <sup>10</sup>7*S*/*m*. The conductor radii are 7 mm. The external excited lightning current is injected from the corner of the grounding system, which is described by a double-exponential function: *<sup>I</sup>*(*t*) = 1.29 × (*e*−0.019010*<sup>t</sup>* − *<sup>e</sup>*−0.292288*<sup>t</sup>* ) kA, which means that the parameters of the lightning current are *T*<sup>1</sup> = 10*µ* s, *T*<sup>2</sup> = 50*µ* s and *Im* = 1.29 kA, the lightning current has been shown in Fig. 5.

The calculated grounding impulse impedance is (14.401, *j*1.273)Ω.

A comparison between chosen total leakage currents and injecting currents of the grounding system in frequency domain is given in Table 2. It can be seen that the total leakage current of the grounding system is close to the external injected current. All this shows the accuracy of this model.


**Table 2.** Total leakage currents from the grounding grid and injecting currents in frequency domain

The quasi-static complex image in this case has two terms, the *αn* and *βn* can be seen below Table 3.


**Table 3.** Quasi-static complex image coefficients

16 Computational and Numerical Simulationse

**Figure 3.** Transient SEP at injection point

**7. Simulation result and analysis**

*<sup>I</sup>*(*t*) = 1.29 × (*e*−0.019010*<sup>t</sup>* − *<sup>e</sup>*−0.292288*<sup>t</sup>*

in Fig. 5.

of this model.

A typical grounding system can be seen in Fig. 4. The earth is medalled by a two-layer conductive earth model, whose conductivity and permittivity is *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> <sup>500</sup>−1*S*/*m*, *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> <sup>900</sup>−1*S*/*m*, *<sup>ε</sup>*<sup>1</sup> <sup>=</sup> <sup>10</sup>*ε*0, *<sup>ε</sup>*<sup>2</sup> <sup>=</sup> <sup>22</sup>*ε*0, respectively, the first layer height is 5 m. The material of the grounding system conductor is Cu with conductivity *<sup>σ</sup>Cu* = 5.8 × <sup>10</sup>7*S*/*m*. The conductor radii are 7 mm. The external excited lightning current is injected from the corner of the grounding system, which is described by a double-exponential function:

current are *T*<sup>1</sup> = 10*µ* s, *T*<sup>2</sup> = 50*µ* s and *Im* = 1.29 kA, the lightning current has been shown

A comparison between chosen total leakage currents and injecting currents of the grounding system in frequency domain is given in Table 2. It can be seen that the total leakage current of the grounding system is close to the external injected current. All this shows the accuracy

> Freq. (kHz) injecting currents (kA) Total leakage currents (kA) 16.7 (−4.077, −10.964) (−4.078, −10.959) 125.0 (−0.486, + 0.363) (−0.486, + 0.363) 250.0 (−0.289, + 0.143) (−0.289, + 0.143) 500.0 (−0.267, − 0.070) (−0.267, − 0.070) 800.0 (−4.301, + 3.428) (−4.301, + 3.428)

The calculated grounding impulse impedance is (14.401, *j*1.273)Ω.

**Table 2.** Total leakage currents from the grounding grid and injecting currents in frequency domain

) kA, which means that the parameters of the lightning

**Figure 4.** Typical grounding system

The transient SEP at the injection point is given in Fig. 5. From this figure, we can see that the maximum value of the transient SEP at the injection point disagrees with that of the lightning current, the maximum value of the transient SEP at injection point occurs at 12*µ* s, and the maximum value of the lightning current occurs at 10*µ* s.

The distribution of absolute values of grounding impedance dependance |*Z*(*jω*)| on frequency can be seen in Fig. 6. This figure shows that |*Z*(*jω*)| is independent of the frequency below 100 kHz and equal to the low frequency grounding impedance, which agrees with the viewpoint of [43].

To further discuss the electromagnetic field characteristics along the surface above the grounding grid, the distribution of the electromagnetic field along the surface with three different chosen frequencies (17 kHz, 250 kHz and 800 kHz) have been given in Figs. 7–15. Among these, Figs. 7–9 show the distribution of SEP *ϕ* along the surface, Figs. 10–12 show the distribution of the x-component of the EFI, *Ex*, along the surface, and Figs. 13–15 show the distribution of the x-component of the MFI, *Bx*, along the surface.

From Figs. 7–9, we know that the ground SEP rise is dependent on the magnitude of the injecting current, the ground SEP rise at 17 kHz is generated by injecting current with (-4.077,-10.964) kA, the ground SEP rise at 250 kHz is generated by injecting current with (-0.289,+ 0.143) kA, and the ground SEP rise at 800 kHz is generated by injecting current

**Figure 7.** The distribution of the SEP *ϕ* on the ground surface (f=16.7 kHz)

**Figure 8.** The distribution of the SEP *ϕ* on the ground surface (f=250 kHz)

at the opposite corner.

represents the case of a corner current injection. Earth potentials near the injection corner present a very sharp peak, while they are quite low and flat everywhere else, with a minimum

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From Figs. 10–12, we can see that as in the ground SEP rise case, the maximum value of the x-componential of the EFI *Ex* is dependent on the magnitude of the injecting current. The distribution of the x-componential of the EFI *Ex* is along the x-direction. Meanwhile, the distribution of the electrical field is not dependent on the current injection location at low frequencies. This conclusion no longer holds at higher frequencies, as shown in Fig. 12,

**Figure 5.** Lightning current

**Figure 6.** Lightning current

with (-4.301,+ 3.428) kA. So the ground SEP rise at 17 kHz is the maximum, and the ground SEP rise at 250 kHz is the minimum. Meanwhile, we can see that almost an equipotential surface occurs for the low frequency case (17 kHz and 250 kHz), and at higher frequencies, the impedances of the grid conductors are no longer negligible and most of the earth currents dissipate close to the injection point. This phenomenon is well illustrated in Fig. 9 which <sup>410</sup> Computational and Numerical Simulations Numerical calculation for lightning response to grounding systems buried in horizontal multilayered earth model based on quasi-static Numerical Calculation for Lightning Response to Grounding Systems Buried in Horizontal Multilayered Earth Model Based on Quasi-Static Complex Image Method http://dx.doi.org/10.5772/57049 411

**Figure 7.** The distribution of the SEP *ϕ* on the ground surface (f=16.7 kHz)

18 Computational and Numerical Simulationse

**Figure 5.** Lightning current

**Figure 6.** Lightning current

with (-4.301,+ 3.428) kA. So the ground SEP rise at 17 kHz is the maximum, and the ground SEP rise at 250 kHz is the minimum. Meanwhile, we can see that almost an equipotential surface occurs for the low frequency case (17 kHz and 250 kHz), and at higher frequencies, the impedances of the grid conductors are no longer negligible and most of the earth currents dissipate close to the injection point. This phenomenon is well illustrated in Fig. 9 which

**Figure 8.** The distribution of the SEP *ϕ* on the ground surface (f=250 kHz)

represents the case of a corner current injection. Earth potentials near the injection corner present a very sharp peak, while they are quite low and flat everywhere else, with a minimum at the opposite corner.

From Figs. 10–12, we can see that as in the ground SEP rise case, the maximum value of the x-componential of the EFI *Ex* is dependent on the magnitude of the injecting current. The distribution of the x-componential of the EFI *Ex* is along the x-direction. Meanwhile, the distribution of the electrical field is not dependent on the current injection location at low frequencies. This conclusion no longer holds at higher frequencies, as shown in Fig. 12,

**Figure 11.** The distribution of the EFI *Ex* on the ground surface (f=250 kHz)

**Figure 12.** The distribution of the EFI *Ex* on the ground surface (f=800 kHz)

different from the electrical field case.

**8. Conclusion**

field is dependent on the current injection location from low to higher frequencies, which is

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Based on Quasi-Static Complex Image Method

http://dx.doi.org/10.5772/57049

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A novel mathematical model for accurately computing the lightning currents flowing in the grounding system of a high voltage a.c. substation, buried in multilayered earth, has been developed in this paper. Together with the FFT, not only the conducting effect of harmonic wave components of these currents, but also capacitive and inductive effects from the interface between different soil layers have been analyzed in the frequency domain. To

**Figure 9.** The distribution of the SEP *ϕ* on the ground surface (f=800 kHz)

**Figure 10.** The distribution of the EFI *Ex* on the ground surface (f=16.7 kHz)

which clearly shows that the region of maximum electrical field shifts from the edge of the grid to the injection point location as the frequency increases from low to high.

The distribution of the x-componential of MFI *Bx* is given in Figs. 13–15. We can also see that as in the ground SEP rise case, the maximum value of the x-componential of the MFI *Bx* is dependent on the magnitude of the injecting current. Unlike the distribution of the x-componential of the EFI *Ex*, the distribution of the x-componential of the MFI *Bx* is along the y-direction, which can be easily explained in that the distribution of the electrical field is perpendicular to that of the magnetic field. Meanwhile, the distribution of the electrical <sup>412</sup> Computational and Numerical Simulations Numerical calculation for lightning response to grounding systems buried in horizontal multilayered earth model based on quasi-static Numerical Calculation for Lightning Response to Grounding Systems Buried in Horizontal Multilayered Earth Model Based on Quasi-Static Complex Image Method http://dx.doi.org/10.5772/57049 413

**Figure 11.** The distribution of the EFI *Ex* on the ground surface (f=250 kHz)

**Figure 12.** The distribution of the EFI *Ex* on the ground surface (f=800 kHz)

field is dependent on the current injection location from low to higher frequencies, which is different from the electrical field case.
