Frequency domain and time domain analysis of the transient behavior of buried grounding grids in horizontal multilayered earth model

Abstract

This study proposes a mathematical model based on the PI basic functions that uses the mutual impedance matrix between the average potential on the discrete node and the leakage current on the conductor connected around the grounding grid. By considering the effect of ionization of soil, the model can accurately calculate the distribution of lightning current and lightning response to the grounding grid buried in the horizontal multilayered soil model. To improve computational efficiency, the quasi-static complex image method and the closed form of Green’s function in the time domain are introduced to the model. Finally, certain numerical results are discussed in detail and important conclusions are derived.

1 Introduction

Accurately estimating the transient electromagnetic field around a universal grounding grid is an important issue in the design of lightning protection systems. Several methods have recently been developed to simulate the characteristics of high-frequency electromagnetic fields of grounding grids in the frequency domain, such as the method of moments [1,2,3] and hybrid methods [4,5,6,7]. Combined with appropriate fast Fourier transform methods, a number of theories based on the frequency domain have been proposed. Examples include circuit theory [8,9,10], transmission line theory [11,12,13,14], electromagnetic field theory [15,16,17,18,19,20,21,22], and hybrid methods [23,24,25,26,27,28,29,30]. Concomitantly, numerical methods based on different time domain methods have been developed, including the FDTD method [31,32,33], circuit method [34, 35], hybrid method [36,37,38,39], and transmission line theory [40, 41]. The circuit theory method [8,9,10], electromagnetic field theory method [16], and hybrid methods [28, 29] in the frequency domain, and the FDTD [32, 33] and circuit theory method [35] in the time domain considered the effect of ionization of soil in case of large pulse charging. Excluding the FDTD method, such methods consider the ionization of soil by adjusting the radius of the conductor of the grounding grid.

Hybrid methods in the frequency or the time domain can be divided into three categories: (1) PI basic function-based, (2) T basic function-based, and (3) partial-element equivalent circuit (PEEC) methods (seen in Fig. 1). The methods proposed in [4, 5, 7, 23, 24, 27, 28, 36, 39, 42] are based on PI basic functions, those developed in [6, 25, 29] are based on T basic functions, and the methods proposed in [26, 30, 37, 38] can be classified as PEEC methods.

Fig. 1

Different basic functions

The mathematical models based on PI basic functions and PEEC methods can be further divided into two types according to whether the branch potential or the node potential were used. All models based on PI basic functions in [4, 5, 7, 23, 24, 27, 28, 36, 39, 42] use the branch potential, which implies that the mutual impedance matrix between the average potential and the leakage current on segments of the discrete conductor is used. Meanwhile, the model based on PEEC methods in [26] also use the branch potential, and these models of PEEC methods in [30, 37, 38] use the node potential. Further, the methods in [4, 5, 7, 23, 24, 26,27,28, 30] use the frequency domain-based numerical method that uses the Fourier transform, whereas those in [36,37,38,39, 42] can be classified as time domain-based numerical methods. The mutual impedance matrix between the average potential on discrete nodes and the leakage current of branches connecting them can be further developed in hybrid methods based on PI basic functions. It must be pointed out that although the hybrid method with the PEEC method in [37, 38] used the node potential to determine the mutual impedance matrix. However, this method considers only the lightning impulse response of the substation grounding grid buried in uniform soil.

In fact, the determination of the earth model of a substation is mainly based on the inversion results of the earth resistivity at the measuring points. This problem is deeply discussed in Ref. [43], in which two numerical examples give a clear conclusion which means only the soil model with appropriate layers can meet the calculation error of the setting requirements. Although Ref. [43] does not clearly explain this problem in the conclusion, the two numerical examples in Ref. [43] clearly illustrate this problem. This issue is further elaborated in the appendix. Indeed, scientists and technicians have been pursuing to use the uniform medium soil model to replace the multi-layer horizontal-layered soil model, although some authors have published paper [44] to explain that they have found the analytical formula of the uniform earth model to replace the two-layer soil model and demonstrated the accuracy of the model. However, the method of how to find a uniform soil model that can represent the universal multi-layer soil model has not been realized. Moreover, with the progress of computer technology, the multi-layer soil model that is more in line with the actual situation has no technical bottleneck and is more in line with the actual engineering problems.

When a point source is assumed to be buried in a horizontal multilayered soil model, Green’s function of the point source includes the infinite integral of the Bessel function. The quasi-static complex image method (QSCIM), developed using the matrix pencil approach, has been used in this context in the frequency domain in [24,25,26] and the time domain in [36, 39].

Fig. 2

The first validation case of our model

Table 1 The CPU time used in the first two test cases

Fig. 3

The second validation case of our model

Fig. 4

The third validation case of our model

Certain errors are obtained in calculations of multiple matrix inversion—for instance in [36, 39], a time domain-based numerical method—due to the characteristics of the processing. It is difficult to accurately simulate and calculate the impulse response of a typically fast lightning current to grounding grids in the time domain, and thus the lightning current considered in [36, 39] was slow. Furthermore, the internal impedance matrix is considered to be an internal resistance matrix and the frequency characteristics of internal impedance are ignored. These problems have been solved in [42], and the time domain-based numerical method in [42] adopts branch potential. In this paper, a mathematical model for hybrid methods based on PI basic functions in both frequency and time domains have been developed, based on work in [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39, 42], in which the mutual impedance matrix between the average potential of the discrete node and leakage current on segments of the conductor connecting it was used to avoid twice matrix product operation, which means that the matrix $[\bar{\bar{\mathbf{Z }}}_{s_p}]^{-1}$ (which will be introduced later) can be used to replace the matrix $[\bar{\bar{\mathbf{K }}}]^t[\bar{\bar{\mathbf{Z }}}_{s_b}]^{-1}[\bar{\bar{\mathbf{K }}}]$ [4, 7, 24, 42]. Meanwhile, the frequency characteristics of internal impedance had been considered in the internal impedance matrix. This model can be used with arbitrary values of typically fast lightning currents along the grounding grids of substations embedded in the horizontal multilayered soil model. The model of the time domain-based hybrid method based on PI basic functions is derived in matrix form. The ionization of soil due to the impulse response of fast lightning current to the grounding grids can thus be simulated and studied. To improve computational efficiency, the QSCIM, a closed form of Green’s function in the frequency and time domains are used in the model. Analytical formulae for the coefficients of mutual inductance and mutual conductance in both the frequency and the time domains are used to avoid the numerical integral.

Fig. 5

The fourth validation case of our model

Table 2 The CPU time used in the last two test cases

2 Mathematical model of equivalent circuit of grounding grids in time domain

It is assumed that the grounding grid composed of fine interconnected conductors is completely buried in horizontal multilayered soil that consists of an $N_s$-layer medium (earth) with conductivity $\sigma _s$ and dielectric constant $\varepsilon _s$ (here, $s=1,\cdot \cdot \cdot \cdot \cdot \cdot , N_s$). The grounding grids are divided into $N_b$ segments and $N_p$ nodes.

With the above considerations, according to past studies [24,25,26,27,28,29,30], the formula for hybrid methods in the frequency domain based on PI basic functions can be given directly as Eqs. (1) (2), and a derivation is provided in the appendix:

$$\begin{aligned}&[\bar{\bar{\mathbf{A }}}]^t[\bar{\mathbf{V }}_p]+[\bar{\bar{\mathbf{Z }}}_b+j\omega \bar{\bar{\mathbf{M }}}_b][\bar{\mathbf{I }}_b]=0 \end{aligned}$$

(1)

$$\begin{aligned}&[\bar{\bar{\mathbf{Z }}}_{s_p}]^{-1}[\bar{\mathbf{V }}_p] -[\bar{\bar{\mathbf{A }}}][\bar{\mathbf{I }}_b]=[\bar{\mathbf{F }}] \end{aligned}$$

(2)

Given the above formula, and by using past work [36, 39] along with a suitable inverse Fourier transform, the equations for the frequency domain (1) and (2) can be transformed into those for the time domain and discretized using point matching. The two sides of Eqs. (3) (4) are matched at $t_k=k\cdot \Delta t$, where $k=1,2,\cdots ,N_t$, $N_t$ is the discrete time maximum and $\Delta t$ is the time step. The corresponding Eqs. (3) (4) can be obtained, and their derivation is given in the appendix.

$$\begin{aligned}&[\bar{\bar{\mathbf{A }}}]^t[\bar{\mathbf{V }}_p(t_k)] +\left[ \bar{\bar{\mathbf{Z }}}^s_b+\frac{\bar{\bar{\mathbf{M }}}^s_b}{\Delta t}\right] [\bar{\mathbf{I }}_b(t_k)]\nonumber \\&\quad =\frac{[\bar{\bar{\mathbf{M }}}^s_b]}{\Delta t}[\bar{\mathbf{I }}_b(t_{k-1})] -[\bar{\bar{\mathbf{M }}}^q_b]\frac{[\bar{\mathbf{I }}_b(t_k-t_l)]-[\bar{\mathbf{I }}_b(t_{k-1}-t_l)]}{\Delta t}\nonumber \\&\qquad -[\bar{\bar{\mathbf{Z }}}^c_b][\bar{\mathbf{I }}_b(t_k-t_m)] \end{aligned}$$

(3)

$$\begin{aligned}&[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\mathbf{V }}_p(t_k)]-[\bar{\bar{\mathbf{A }}}][\bar{\mathbf{I }}_b(t_k)]\nonumber \\&\quad =[\bar{\mathbf{F }}(t_k)] +[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\bar{\mathbf{Z }}}^q_{s_p}][\bar{\mathbf{I }}_{s_p}(t_k-t_n)] \end{aligned}$$

(4)

Fig. 6

Response caused by lightning currents at incident point

Fig. 7

Response caused by lightning currents at incident point (local amplification)

Fig. 8

Response caused by large amplitude lightning currents at incident point

Table 3 The calculated grounding impulse impedances ($\Omega$)

Table 4 The calculated grounding impulse impedances ($\Omega$)(with large amplitude)

Fig. 9

Response caused by large amplitude lightning currents at incident point (local amplification)

Fig. 10

3D distribution of positive real part of voltage in the frequency domain

Fig. 11

3D distribution of positive real part of branch current in the frequency domain

Fig. 12

3D distribution of positive real part of leakage current in the frequency domain

Fig. 13

3D distribution of positive real part of voltage in the time domain

Fig. 14

3D distribution of positive real part of branch current in the time domain

Fig. 15

3D distribution of positive real part of leakage current in the time domain

where j is an imaginary unit and $\omega$ is the angular frequency. Further, $[\bar{\mathbf{V }}_p]$ and $[\bar{\mathbf{V }}_p(t_k)]$ are the $N_p\times 1$ vectors of the scalar electrical potentials (SEP) of nodes on segments of the discrete conductor that are grounded grids in the frequency and time domains, respectively. $[\bar{\mathbf{F }}]$ and $[\bar{\mathbf{F }}(t_k)]$ are the $N_p\times 1$ vectors for lightning currents in the frequency and time domains, respectively, and $[\bar{\bar{\mathbf{Z }}}_b]$ is a $N_b\times N_b$ diagonal matrix that represents the internal impedance of the discrete conductors in the frequency domain. $[\bar{\bar{\mathbf{Z }}}^s_b]$ and $[\bar{\bar{\mathbf{Z }}}^c_b]$ are $N_b\times N_b$ diagonal matrices with and without the effects of dispersion that represent the internal impedance of discrete conductors in the time domain. $[\bar{\bar{\mathbf{M }}}_b]$ is an a $N_b\times N_b$ matrix that has zero diagonal elements, and yields the interactions of mutual induction between arbitrary pieces of discrete conductors in the frequency domain. $[\bar{\bar{\mathbf{M }}}^q_b]$ and $[\bar{\bar{\mathbf{M }}}^s_b]$ are $N_b\times N_b$ matrices with and without the effects of dispersion, respectively, that contain zero diagonal elements, and represent the interactions of mutual induction between arbitrary pieces of discrete conductors in the time domain, and $[\overline{\overline{\mathbf{Z }}}_{s_p}]$ is a $N_p\times N_p$ matrix that represents such interactions between arbitrary discrete points on discrete conductors in the frequency domain. $[\bar{\bar{\mathbf{Z }}}^q_{s_p}]$ and $[\bar{\bar{\mathbf{Z }}}^s_{s_p}]$ are $N_p\times N_p$ matrices with and without the effects of dispersion, respectively, that represent the interactions of mutual conduction between arbitrary discrete points on discrete conductors in the time domain, and $[\overline{\overline{\mathbf{A }}}]$ is a $N_p\times N_b$ incidence matrix that is used to correlate the relationship between branches and nodes of the discrete conductor, and the elements of which have been provided in [24,25,26,27]. $[\bar{\mathbf{I }}_b]$ is the branch current in the frequency domain. In Eqs. (3) (4) $1 \le l, m, n \le N_t$, $[\bar{\mathbf{I }}_b(t_k)]$, and $[\bar{\mathbf{I }}_{s_p}(t_k)]$ are the branch currents and the leakage currents at discrete point at time $t_k$, respectively.

By solving Eqs. (1) (2), the SEP of the incident point at each frequency can be obtained; then, the increase in the transient ground potential at the point of incidence of the grounding grids can be obtained by the Fourier transform.

Equations (3) (4) represent a marching-on-in-time formulation. Accordingly, they can be quickly solved through iterations.

$[\bar{\mathbf{V }}_p(t)]$ can be obtained by solving Eqs. (3) (4). The average SEP $[\bar{\mathbf{U }}_a(t)],$ branch voltage $[\bar{\mathbf{U }}_b(t)],$ leakage current at the discrete point $[\bar{\mathbf{I }}_{s_p}(t)],$ and branch current $[\bar{\mathbf{I }}_b(t)]$ can then be calculated. Furthermore, the SEP, electric field strength E, and magnetic field strength B can be obtained anywhere [24,25,26,27].

Studying the methods based on the frequency and time domain to simulate the transient lightning response of grounding grids centers on the calculation of various matrices, such as $[\bar{\bar{\mathbf{Z }}}_{s_p}]$, $[\bar{\bar{\mathbf{Z }}}_b]$, $[\bar{\bar{\mathbf{M }}}_b]$ (frequency domain), $[\bar{\bar{\mathbf{Z }}}^s_{s_p}]$, $[\bar{\bar{\mathbf{Z }}}^q_{s_p}]$, $[\bar{\bar{\mathbf{Z }}}^s_b]$, $[\bar{\bar{\mathbf{M }}}^s_b],$ and $[\bar{\bar{\mathbf{M }}}^q_b]$ (time domain). Methods to calculate elements of the matrix in the frequency domain ($[\bar{\bar{\mathbf{Z }}}_b]$ and $[\bar{\bar{\mathbf{M }}}_b]$) and the time domain ($[\bar{\bar{\mathbf{Z }}}^s_b]$, $[\bar{\bar{\mathbf{M }}}^s_b]$, and $[\bar{\bar{\mathbf{M }}}^q_b]$) have been discussed (in [24,25,26] for the frequency domain and [36, 39] for the time domain). We discuss only the method to calculate the mutual impedance matrix in the frequency ($[\bar{\bar{\mathbf{Z }}}_{s_p}]$) and time domains ($[\bar{\bar{\mathbf{Z }}}^s_{s_p}]$ and $[\bar{\bar{\mathbf{Z }}}^q_{s_p}]$).

The expressions for elements of the mutual impedance matrix in the frequency and time domains are

$$\begin{aligned} Z^{^f_{u,v}}_{s_p}= & {} \frac{1}{N^v_\theta }\sum ^{N^v_\theta }_{\theta =1} \frac{1}{l^\theta _v}\frac{1}{N^u_\vartheta }\sum ^{N^u_\vartheta }_{\vartheta =1} \frac{1}{l^\vartheta _u}\int _{l^\theta _v}\int _{l^\vartheta _u}G({\bar{r}}, \bar{r'}~~)d\tau ^\vartheta _ud\tau ^\theta _v \nonumber \\\end{aligned}$$

(5)

$$\begin{aligned} Z^{^t_{u,v}}_{s_p}= & {} \frac{1}{N^v_\theta }\sum ^{N^v_\theta }_{\theta =1} \frac{1}{l^\theta _v}\frac{1}{N^u_\vartheta }\sum ^{N^u_\vartheta }_{\vartheta =1} \frac{1}{l^\vartheta _u}\int _{l^\theta _v}\int _{l^\vartheta _u}G({\bar{r}}, \bar{r'},t)d\tau ^\vartheta _ud\tau ^\theta _v\nonumber \\ \end{aligned}$$

(6)

where $Z^{^f_{u,v}}_{s_p}$ and $Z^{^t_{u,v}}_{s_p}$ represent elements of the mutual impedance matrix in frequency and time domains, $G({\bar{r}},\bar{r'})$ and $G({\bar{r}},\bar{r'},t)$, denote Green’s functions in these domains, respectively, u and v represent the nodes of the source point and the field point, respectively, $u, v = 1, \cdots , N_p$; $N^u_\vartheta$, and $N^v_\theta$ represent the total number of connected segments of the conductor around nodes u and v, respectively, and $\vartheta$ and $\theta$ represent the source point and the field point, respectively. $\vartheta =1,\cdots ,N^u_\vartheta$, $\theta =1,\cdots ,N^v_\theta$. $l^\vartheta _u$ and $l^\theta _v$ denote the lengths of the $\vartheta$th and $\theta$th segments of the conductor connected at the source point u and the field point v, respectively, and $\tau ^o_u$ and $\tau ^p_v$ are integral variables on the segments $\vartheta$ and $\theta$ of the conductor at the source point u and the field point v, respectively.

By introducing the frequency and time domains in the context of Green’s function of a point source in an infinitely conducting medium [24,25,26, 36, 39], expressions for elements of the mutual impedance matrix in the frequency and time domains can be obtained as follows, and the derivation is provided in the appendix:

$$\begin{aligned} Z^{^f_{u,v}}_{s_p}= & {} \frac{1}{N^v_\theta }\sum ^{N^v_\theta }_{\theta =1} \frac{1}{l^\theta _v}\frac{1}{N_\vartheta }\sum ^{N^u_\vartheta }_{\vartheta =1} \frac{1}{l^\vartheta _u}\frac{1}{4\pi {\tilde{\sigma }}_1}\int _{l^\theta _v}\int _{l^\vartheta _u}\frac{1}{R}d\tau ^\vartheta _ud\tau ^\theta _v \end{aligned}$$

(7)

$$\begin{aligned} Z^{^{t_s}_{u,v}}_{s_p}= & {} \frac{1}{N^v_\theta }\sum ^{N^v_\theta }_{\theta =1} \frac{1}{l^\theta _v}\frac{1}{N^u_\vartheta }\sum ^{N^u_\vartheta }_{\vartheta =1} \frac{1}{l^\vartheta _u}\frac{1}{4\pi \sigma _1}\int _{l^\theta _v}\int _{l^\vartheta _u} \frac{1}{R}d\tau ^\vartheta _ud\tau ^\theta _v \end{aligned}$$

(8)

$$\begin{aligned} Z^{^{t_q}_{u,v}}_{s_p}= & {} \frac{1}{N^v_\theta }\sum ^{N^v_\theta }_{\theta =1} \frac{1}{l^\theta _v}\frac{1}{N^u_\vartheta }\sum ^{N^u_\vartheta }_{\vartheta =1} \frac{1}{l^\vartheta _u}\sum ^{M_{q_0}}_{q_0=1}\nonumber \\&\qquad \alpha _{q_0}\frac{\delta (t-t_{\beta })}{4\pi \sigma _1}\int _{l^\theta _v}\int _{l^\vartheta _u}\frac{1}{R}d\tau ^\vartheta _ud \tau ^\theta _v \end{aligned}$$

(9)

where R is the distance between the field point and the source point, and $\delta (t)$ is the delta function. $Z^{^f_{u,v}}_{s_p}$ represents the element of the mutual impedance matrix in the frequency domain, and ${\tilde{\sigma }}_1=\sigma _1+j\omega \varepsilon _1$. $Z^{^{t_q}_{u,v}}_{s_p}$ and $Z^{^{t_s}_{u,v}}_{s_p}$ represent the elements of the mutual impedance matrix in the time domain with and without the effects of dispersion, respectively, $t_\beta =j\beta _{q_0}$, $M_{q_0}$ represents the total number of complex images, and $\alpha _{q_0}$ and $\beta _{q_0}$ are the coefficients of a complex image.

By employing Eqs. (7) to (9) with reference to the literature [24,25,26, 36, 39], three two-dimensional (2D) line integral formulae can be estimated analytically. The medium surrounding the source of the point current is assumed to be homogeneous and infinite. However, a horizontal multilayered earth model is more generally used in practice. A closed form of Green’s function with quasi-static complex images in the frequency [24,25,26] and time domains [36, 39] has been developed to deal with the earth model, and has been used in the mathematic model. Because this algorithm avoids the numerical integration, especially the numerical calculation of Bessel infinite integral by quasi-static complex image method, this algorithm has very good computational efficiency.

3 Results of simulation and analysis

3.1 Effects of soil ionization

A mathematical method for calculating the radius of the virtual equivalent conductor, generated by the phenomenon of soil ionization at each time step, is presented here. It simulates the ionization of soil and has been discussed in [39].

3.2 Verification of model algorithm

In the following, the transient response of lightning current to the grounding grid is obtained by a mathematical algorithm (in the time and frequency domains), and its performance is compared with the results of experiments and simulations from the literature.

The first example is taken from Ref. [45], and the configuration of the grounding grid for it is shown in Fig. 2. The grounding grid was composed of round copper conductors with a cross-sectional area of $50\,\mathrm{mm}^2$ buried at a depth of $0.5\,\mathrm{m}$. The parameters of the injected lightning current were set to $T_1=65 \,\upmu \mathrm{s}$, $T_2=185 \,\upmu \mathrm{s}$, and $I_m=6.86\,A$, and the feed point was set at the corner of the grid. The conductivities and permittivities of the two-layer earth model were set to $\frac{\sigma _1=50 \,\Omega \mathrm{m}}{\sigma _2=20 \,\Omega \mathrm{m}}$, and $\frac{\varepsilon _1=10\varepsilon _0}{\varepsilon _2=10\varepsilon _0}$ and $h=0.6\,\mathrm{m}$. Figure 2 shows curves of the transient voltage in the time and frequency domains using hybrid methods based on the PI basic function with branch or node potential used. The curves of the hybrid methods were based on the PEEC [26] and T [25] basic functions in the frequency domain. It is clear that curves of the simulation of the three algorithms were completely consistent, and those of the calculated voltage agreed well with the measured curves given in Ref. [45].

Because the lightning wave considered in the first example was not sufficiently steep, the second example considered the shock produced by a steeper lightning wave. Using the work in Ref. [19], the length of the horizontal conductor was set to $15\,\mathrm{m},$ and it was buried at a depth of $0.6\,\mathrm{m}$. The parameters of the soil, viz., conductivity and relative permittivity, were $70 \,\Omega \mathrm{m}$ and 15,  respectively. The radius of the conductor was $12\,\mathrm{mm}$, and the parameters of the injected lightning current were set to $T_1=0.8 \,\upmu \mathrm{s}$, $T_2=12.5 \,\upmu \mathrm{s},$ and $I_m=36.5\,A$. The feed point was set at the end of the conductor. Figure 3 shows curves of the transient voltage in the time and frequency domains obtained by using the hybrid methods based on the PI basic function with branch or node potential used, and the curves of the simulation were the results of hybrid methods based on the PEEC [26] and T [25] basic functions in frequency domain. The curves obtained from the simulation using the four algorithms were identical and agreed well with the measured curve in Ref. [19]. In order to show the calculation efficiency of this algorithm, the calculation time of different algorithms in the first two examples is compared, as shown in Table 1.

The third examples used for verification are from Ref. [9], where the length of the horizontal conductor was 5  m, and it was buried at a depth of 0.6 m. The conductivity and relative permittivity of soil were $43 \Omega$ m and 10, respectively. The radius of the conductor was 4 mm. Figure 4 shows the injected current; its amplitude was 25kA and induced the ionization of soil. A comparison of the results of the simulation, depicted in Fig. 4, with the measured values in Ref. [9] clearly reflects the excellent agreement obtained by the method presented here. In this example, 350 kV/m was set as the threshold for the ionization of soil, as in [9].

The last case for verification is also from Ref. [9]. Here, the length of the vertical bars was 1 m, and they were on the surface of the ground. The conductivity and relative permittivity of the half-homogenous soil were $43 \Omega$ m and 10, respectively, and the radius of the vertical conducting bar was 25 mm. Figure 5 shows an injected current with an amplitude of 30 kA to provoke the ionization of soil. The results of the simulation are also presented in Fig. 5, where we compare the model with the measured values from Ref. [9]. It is clear that a basic agreement was obtained between the results of the proposed method and observational data. In this example, the criterion of soil ionization was 350 kV/m [9]. The calculation time of different algorithms in the last two examples is compared, as shown in Table 2.

3.3 Validation of the model algorithm

The maximum effective applicable frequency of this method is limited by the quasi-stationary approximation of electromagnetic field, which means that the propagation effect of electromagnetic field around the grounding system can be ignored, so $e^{-\gamma _eR}\approx 1$, where $\gamma ^2_e=j\omega \mu (j\omega \varepsilon _e+\sigma _e)$, $e=1,\cdots , N_e$. $N_e$ is the number of layers of the horizontal layered earth model [21].

Fig. 16

3D distribution of virtual radius of the grid in the time domain

3.4 Results and analysis

The typical configuration of the grounding grid analyzed in this study is shown in Fig. 6. A four-layer earth model was used, the conductivities and permittivities of which were $\sigma _1=100^{-1}\mathrm{S/m}$, $\sigma _2=300^{-1} \mathrm{S/m}$, $\sigma _3=200^{-1} \mathrm{S/m}$, $\sigma _4=100^{-1} \mathrm{S/m}$, $\varepsilon _1=10\varepsilon _0$, $\varepsilon _2=10 \varepsilon _0$, $\varepsilon _3=10 \varepsilon _0$, and $\varepsilon _4=10 \varepsilon _0$. Further, the heights of the first, second, and third layers were 1.2 m, 2.6 m, and 2.2 m, respectively. The grounding grid was made of copper and had a radius of 7 mm. It was buried at a depth of 0.5 m. The two types of lightning currents injected at the corner of the grounding grid were described by two double-exponential functions as $I(t)=43.5\times (e^{-0.032t}-e^{-0.308t})\,\mathrm{kA}$ and $I(t)=12.54\times (e^{-0.01t}-e^{-1.728t})\,\mathrm{kA}$. The parameters of the lightning currents were $T_1=8\,\,\upmu \mathrm{s}$, $T_2=12\,\,\upmu \mathrm{s}$, $I_m=30\,\mathrm{kA}$, $T_1=3.5\, \,\upmu \mathrm{s}$, $T_2=73 \,\,\upmu \mathrm{s}$, and $I_m=12\,\mathrm{kA}$, as shown in Figs. 6 and 8. To ensure the ionization of soil, lightning current with a large amplitude was used, which means $I(t)=83.5\times (e^{-0.032t}-e^{-0.308t})$ kA and $I(t)=66.54\times (e^{-0.01t}-e^{-1.728t})$ kA, and the critical value of electric field intensity set in Ref. [46] was employed.

The calculated grounding impulse impedances are given in Table 3 and 4 for the configuration in the frequency and time domains based on the two types of lightning currents with and without the effects of ionization considered and with large amplitude or not. The time domain $^2$ or $^1$ indicated whether the effects of ionization were considered or not, respectively. It can be seen from the two tables that a large lightning current will cause soil ionization and greatly reduce the impact impedance.

Figures 6 and 8 show two groups of six curves of the transient voltage at the point of injection of the typical grounding grid under two kinds of lightning currents, with and without the effect of ionization and with large amplitude or not. The corresponding local large curves (Figs. 7 and 9) were evident, and it is clear that the peak time of the lightning current was not consistent with that of its responses in methods based on the frequency and time domains. In the latter cases, the maximum values of the transient voltage at the injection point appeared to be nearly identical. Figure 7 shows that the lightning impulse response with the effect of ionization of soil was not obvious compared with that without the effect of ionization; meanwhile, Fig. 8 shows that the lightning impulse response with the effect of ionization of soil was obvious compared with that without the effect of ionization.

Three-dimensional (3D) pictures of the spatial distribution of the positive real parts of the voltage, branch, and leakage currents along the grounding grids at different frequencies, due to the first kind of lightning current, are shown in Figs. 10, 11 and 12. 3D photos of the temporal and spatial distributions of these values at different times are shown in Figs. 13, 14 and 15.

Figures 10, 11 and 12 show that at a zero frequency, values of the voltage were evenly distributed on the grounding grid, branch current, and leakage current with thee clear rules of distribution evident in the discrete grounding grids. Further, with an increase in frequency, the voltage, branch, and leakage currents obtained their largest values at the incident point (0.0, 0.0, 0.0) and their minimum values at the diagonal point (50.0, 50.0, 0.0). For each frequency, values of the voltage, branch, and leakage currents distributed on the discrete grounding grids between the incident point and the diagonal point gradually decreased. With the further increase of frequency, their distribution in the whole discrete grounding grid gradually decreases from the incident point to both sides until it disappears, and is the smallest at the set maximum calculation frequency.

Figure 13 contains four small graphs that reflect the spatiotemporal evolution of the transient voltage from weak to strong, following which it disappeared. The first sub-figure shows the result early on in time ($0.25 \,\upmu \mathrm{s}$), and subsequent sub-figures show those obtained later on ($2.0 \,\upmu \mathrm{s}$, $8.0 \,\upmu \mathrm{s}$, and $127.25 \,\upmu \mathrm{s}$). The maximum value of the overall distribution of voltage appeared at $2.0 \,\upmu \mathrm{s}$, and this situation is reflected in the curve of the transient voltage in Fig. 7.

Figure 14 also shows four sub-figures. The first one shows the distribution of branch current early on ($0.25 \,\upmu \mathrm{s}$) along the grounding grids, and the three subsequent sub-figures show the temporal and spatial evolutions of the branch current at the given times ($2.0 \,\upmu \mathrm{s}$, $8.0 \,\upmu \mathrm{s}$, and $127.25 \,\upmu \mathrm{s}$). It is clear that the distributed branch current was very weak early on and was present in all parts except for that passing through the incident point. Similarly, the maximum value of the distribution of the overall branch current appeared at $8.0 \,\upmu \mathrm{s}$, and this situation is reflected in the curve of the first kind of lightning current in Fig. 7.

Figure 15 shows graphs reflecting the evolution of leakage current from weak to strong, and then to zero. Like the branch current, the leakage current was weak early on, and appeared only near the incident point. With time, the overall distribution of the leakage current reached its maximum value at $2.0 \,\upmu \mathrm{s}$. At this time, the voltage distributed on the grounding grids reached its maximum value as well (see the third sub-figure in Fig. 13, and then gradually weakened and disappeared completely.

Regardless of whether large lightning current is considered or not, the temporal and spatial distribution of grounding grid transient voltage, branch current and leakage current under the action of the second type of lightning current is basically consistent with that of the first type of lightning current. As the second kind of lightning current was steeper than the first kind, the maximum values and vanishing times of the corresponding voltage, branch current, and leakage current changed. At the same time, when soil ionization occurs due to large lightning current, in addition to the changes of voltage, branch and leakage current on the grounding grid, the virtual radius of grounding grid conductor also changes accordingly. Figure 16 shows the three-dimensional change diagram of virtual radius of grounding grid conductor caused by soil ionization.

4 Conclusions

This paper established a hybrid mathematical model in the time and frequency domains based on PI basic functions by using the mutual impedance matrix between the average potential at the discrete node and leakage current on the conductor around it. This model can accurately determine the lightning response of a substation grounding grid embedded into a horizontal multilayer soil model by considering the effect of ionization of soil. The closed form of Green’s function was used in the QSCIM and its time domain. Finally, certain numerical results were discussed, and the following conclusions can be drawn:

• 1. The proposed hybrid mathematical model can adequately simulate the impulse response of any typical lightning impulse current.

• 2. The hybrid mathematical models in the frequency and time domains can be verified through numerical calculations of the lightning impulse response of substation grounding grids.

• 3. Sufficient lightning current will cause soil ionization around the grounding grid conductor, so as to reduce the lightning impulse impedance of the grounding grid.

Appendix

1.1 Derivation for Eqs. (1) (2) (3) and (4)

For an electrical circuit,

$$\begin{aligned}{}[\bar{\mathbf{U }}_d]+[\bar{\bar{\mathbf{Z }}}_b+j\omega \bar{\bar{\mathbf{M }}}_b][\bar{\mathbf{I }}_b]=0 \end{aligned}$$

(10)

Here, $[\bar{\mathbf{U }}_d]$ is the voltage (potential difference) between the end points of a segment of the conductor. Considering that $[\bar{\mathbf{U }}_d]=[\bar{\bar{\mathbf{A }}}]^t[\bar{\mathbf{V }}_p]$,

$$\begin{aligned}{}[\bar{\bar{\mathbf{A }}}]^t[\bar{\mathbf{V }}_p]+[\bar{\bar{\mathbf{Z }}}_b+j\omega \bar{\bar{\mathbf{M }}}_b][\bar{\mathbf{I }}_b]=0 \end{aligned}$$

(11)

By considering the circuit model of the grounding grid, according to Kirchhoff’s law of current,

$$\begin{aligned}{}[\bar{\mathbf{I }}_{s_p}]=[\bar{\bar{\mathbf{A }}}][\bar{\mathbf{I }}_b]+[\bar{\mathbf{F }}] \end{aligned}$$

(12)

where $[\bar{\mathbf{I }}_{s_p}]$ is an $1\times N_p$ vector of leakage currents.

For an infinitely homogeneous conductive medium, the average SEP at the discrete nodes can be calculated by the leakage currents along segments of the conductor connecting these nodes:

$$\begin{aligned} \frac{1}{N^u_\vartheta }\sum ^{N^u_\vartheta }_{\vartheta =1}\frac{1}{4\pi {\tilde{\sigma }}_1}\int _{l^\vartheta _u}\frac{1}{R}\frac{I^\vartheta _u}{l^\vartheta _u}d\tau ^\vartheta _u=\varphi _v \end{aligned}$$

(13)

where $u, v=1,\cdots ,N_p$. $I^\vartheta _u$ is the leakage current along the $\vartheta$th link of the discrete segment of the uth discrete node, and $\varphi _v$ is the SEP at the vth discrete node.

Considering Galerkin’s method of moment,

$$\begin{aligned}&\frac{1}{N^v_\theta }\sum ^{N^v_\theta }_{\theta =1}\frac{1}{l^\theta _v}\int _{l^\theta _v}\frac{1}{N^u_\vartheta }\sum ^{N^u_\vartheta }_{\vartheta =1}\frac{1}{4\pi {\tilde{\sigma }}_1}\int _{l^\vartheta _u}\frac{1}{R}\frac{I^\vartheta _u}{l^\vartheta _u}d\tau ^\vartheta _ud\tau ^\theta _v\nonumber \\&\quad =\frac{1}{N^v_\theta }\sum ^{N^v_\theta }_{\theta =1}\frac{1}{l^\theta _v}\int _{l^\theta _v}\varphi _vd\tau ^\theta _v \end{aligned}$$

(14)

Equation (14) can be written in matrix form as

$$\begin{aligned}{}[\bar{\bar{\mathbf{Z }}}_{s_p}][\bar{\mathbf{I }}_{s_p}]=[\bar{\mathbf{V }}_p] \end{aligned}$$

(15)

Considering Eqs. (12) and (15),

$$\begin{aligned}{}[\bar{\bar{\mathbf{Z }}}_{s_p}]^{-1}[\bar{\mathbf{V }}_p]-[\bar{\bar{\mathbf{A }}}][\bar{\mathbf{I }}_b]=[\bar{\mathbf{F }}] \end{aligned}$$

(16)

Thus, Eqs. (11) and (16) can reveal the origin of Eqs. (1) (2).

Equations (11) and (16) can be further developed through the inverse Fourier transform as:

$$\begin{aligned}&[\bar{\bar{\mathbf{A }}}]^t[\bar{\mathbf{V }}_p(t)]+[\bar{\bar{\mathbf{Z }}}^s_b] [\bar{\mathbf{I }}_b(t)]+[\bar{\bar{\mathbf{M }}}^s_b] \frac{d[\bar{\mathbf{I }}_b(t)]}{dt}\nonumber \\&\quad +[\bar{\bar{\mathbf{M }}}^q_b] \frac{d[\bar{\mathbf{I }}_b(t-t_m)]}{dt}=0 \end{aligned}$$

(17)

$$\begin{aligned}&[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\mathbf{V }}_p(t)] -[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\bar{\mathbf{Z }}}^q_{s_p}] [\bar{\mathbf{I }}_p(t-t_n)]\nonumber \\&\quad -[\bar{\bar{\mathbf{A }}}][\bar{\mathbf{I }}_b(t)] =[\bar{\mathbf{F }}(t)] \end{aligned}$$

(18)

By using difference approximation to replace the differential operation, time t is treated as discrete time $t_k$. Hence,

$$\begin{aligned}&[\bar{\bar{\mathbf{A }}}]^t[\bar{\mathbf{V }}_p(t_k)] +[\bar{\bar{\mathbf{Z }}}^s_b][\bar{\mathbf{I }}_b(t_k)] +[\bar{\bar{\mathbf{M }}}^s_b]\frac{[\bar{\mathbf{I }}_b(t_k)] -[\bar{\mathbf{I }}_b(t_{k-1})]}{\Delta t}\nonumber \\&\quad +[\bar{\bar{\mathbf{M }}}^q_b]\frac{[\bar{\mathbf{I }}_b(t_k-t_m)] -[\bar{\mathbf{I }}_b(t_{k-1}-t_m)]}{\Delta t}=0 \end{aligned}$$

(19)

$$\begin{aligned}&[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\mathbf{V }}_p(t_k)] -[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\bar{\mathbf{Z }}}^q_{s_p}] [\bar{\mathbf{I }}_p(t_k-t_n)]\nonumber \\&\quad -[\bar{\bar{\mathbf{A }}}][\bar{\mathbf{I }}_b(t_k)] =[\bar{\mathbf{F }}(t_k)] \end{aligned}$$

(20)

Hence, the following can be obtained:

$$\begin{aligned}&[\bar{\bar{\mathbf{A }}}]^t[\bar{\mathbf{V }}_p(t_k)] +\left[ \bar{\bar{\mathbf{Z }}}^s_b+\frac{\bar{\bar{\mathbf{M }}}^s_b}{\Delta t}\right] [\bar{\mathbf{I }}_b(t_k)] =[\bar{\bar{\mathbf{M }}}^s_b]\frac{[\bar{\mathbf{I }}_b(t_{k-1})]}{\Delta t}\nonumber \\&\quad -[\bar{\bar{\mathbf{M }}}^q_b]\frac{[\bar{\mathbf{I }}_b(t_k-t_m)]-[\bar{\mathbf{I }}_b(t_{k-1}-t_m)]}{\Delta t} \end{aligned}$$

(21)

$$\begin{aligned}&[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\mathbf{V }}_p(t_k)] -[\bar{\bar{\mathbf{A }}}][\bar{\mathbf{I }}_b(t_k)]=[\bar{\mathbf{F }}(t_k)]\nonumber \\&\quad +[\bar{\bar{\mathbf{Z }}}^s_{s_p}]^{-1}[\bar{\bar{\mathbf{Z }}}^q_{s_p}] [\bar{\mathbf{I }}_p(t_k-t_n)] \end{aligned}$$

(22)

1.2 Derivation of Eqs. (5) (6) (7) (8) and (9)

Equation (5) can be obtained from Eqs. (14), and (6) can be obtained from Eq. (5), through the inverse Fourier transform.

For an infinitely homogeneous conductive medium, Green’s function of the point source in the frequency and time domains can be obtained as follows [24,25,26, 36, 39]:

$$\begin{aligned} G_\varphi ({\bar{r}},\bar{r'})= & {} \frac{1}{4\pi {\tilde{\sigma }}_1}\frac{1}{R} \end{aligned}$$

(23)

$$\begin{aligned} G_\varphi ({\bar{r}},\bar{r'},t)= & {} \frac{1}{4\pi \sigma _1}\frac{1}{R}+ \frac{1}{4\pi \sigma _1}\sum ^{M_{q_0}}_{q_0=1}\alpha _{q_0}\frac{\delta (t-t_{\beta })}{R} \end{aligned}$$

(24)

Equation (21) can be further set as

$$\begin{aligned}&G_\varphi ({\bar{r}},\bar{r'},t)=G^s_\varphi ({\bar{r}},\bar{r'},t)+G^q_\varphi ({\bar{r}},\bar{r'},t) \end{aligned}$$

(25)

$$\begin{aligned}&G^s_\varphi ({\bar{r}},\bar{r'},t)=\frac{1}{4\pi \sigma _1}\frac{1}{R} \end{aligned}$$

(26)

$$\begin{aligned}&G^q_\varphi ({\bar{r}},\bar{r'},t)=\frac{1}{4\pi \sigma _1}\sum ^{M_{q_0}}_{q_0=1}\alpha _{q_0}\frac{\delta (t-t_{\beta })}{R} \end{aligned}$$

(27)

Table 5 The estimated model parameters of different layered model and the model parameters of paper [47]

Fig. 17

The measure results and theoretical results of different layered models of case A

Table 6 The estimated model parameters of different layered model and the model parameters of paper [48]

Fig. 18

The measure results and theoretical results of different layered models of case B

Considering Eqs. (5) and (6), the elements of the mutual impedance matrix in the frequency and time domains can be obtained as

$$\begin{aligned} Z^{^f_{u,v}}_{s_p}= & {} \frac{1}{N^v_p}\sum ^{N^v_p}_{p=1}\frac{1}{l^p_v}\frac{1}{N_o}\sum ^{N^u_o}_{o=1}\frac{1}{l^o_u}\frac{1}{4\pi {\tilde{\sigma }}_1}\int _{l^p_v}\int _{l^o_u}\frac{1}{R}d\tau ^o_ud\tau ^p_v \end{aligned}$$

(28)

$$\begin{aligned}&Z^{^{t_s}_{u,v}}_{s_p}=\frac{1}{N^v_p}\sum ^{N^v_p}_{p=1}\frac{1}{l^p_v}\frac{1}{N^u_o}\sum ^{N^u_o}_{o=1}\frac{1}{l^o_u}\frac{1}{4\pi \sigma _1}\int _{l^p_v}\int _{l^o_u}\frac{1}{R}d\tau ^o_ud\tau ^p_v \nonumber \\\end{aligned}$$

(29)

$$\begin{aligned}&Z^{^{t_q}_{u,v}}_{s_p}=\frac{1}{N^v_p}\sum ^{N^v_p}_{p=1}\frac{1}{l^p_v}\frac{1}{N^u_o}\sum ^{N^u_o}_{o=1}\frac{1}{l^o_u}\sum ^{M_{q_0}}_{q_0=1}\nonumber \\&\alpha _{q_0}\frac{\delta (t-t_\beta )}{4\pi \sigma _1}\int _{l^p_v}\int _{l^o_u}\frac{1}{R}d\tau ^o_ud\tau ^p_v \end{aligned}$$

(30)

1.3 Rationality demonstration of the multilayered soil model

This section mainly uses the content of previously published Ref. [43] to prove the argument of this paper. It is mainly based on the global inversion method, i.e., genetic algorithm, to invert the horizontal multilayered soil model. Two numerical examples from [43] are given directly in this paper to demonstrate the appropriate layered soil model, which is mainly based on the inversion results.

• Case A Two-layer, three-layer and four-layer models are used to interpret the resistivity sounding data [47]. Soil parameters of different layered model optimized by this method and Ref. [47] are shown in Table 5. Theoretical apparent resistivity curves of different layered models are plotted in Fig. 17. By observing Fig. 17, it is obvious that two-layer and four-layer models are not enough to simulate soil electrical properties. So the three-layer model is adopted, which is consistent with that in Ref. [47].

• Case B Two-layer, three-layer and four-layer models are used to interpret the resistivity sounding data [48]. The model parameters of different layered models obtained by this inversion algorithm and Ref. [48] are listed in Table 6. Figure 18 shows the measured apparent resistivity and the theoretical apparent resistivity curves. It is found from Fig. 18 that the four-layer model can describe the soil inhomogeneity. So the number of soil layers is fixed as four in frequency domain, this choice is consistent with that in Ref. [48].

These two examples demonstrate that the number of soil layers should be determined on the basis of inversion results of different layered models and the appropriate layered soil model was of course mainly based on the inversion results.