### Model for geoelectric field

In order to model the GICs in the power grid, we need to determine the geoelectric field first. Various approaches were applied to determine the geoelectric field from variations of the geomagnetic field, for example, a plane-wave method or complex image method (Pirjola and Viljanen 1998). For the purpose of our study, we decided to use the plane-wave model. The disturbances caused by magnetospheric–ionospheric currents propagate vertically downwards as a plane wave, and disturbances caused by geoelectromagnetic variations are described as a wave in this approach. Our model showed good consistency with Hejda and Bochníček (2005) where they modelled geomagnetically induced pipe-to-soil voltages in the Czech oil pipelines during the Halloween storms.

A common assumption in similar studies of the geoelectric field is that the conductivity of the Earth only varies with depth (1D structure of the Earth). The Earth is replaced by a half-space with a flat surface, which is an acceptable approximation since GIC is a regional phenomenon. We are using a uniform ground resistivity model, which means that the Earth structure is regionally homogeneous with a constant conductivity \(\sigma\). For the purpose of this study, we use the value of conductivity \(\sigma =10^{-3}\) \(\Omega ^{-1}\) \(\hbox {m}^{-1}\) which is a typical value for Czech territory, and it was used also by Hejda and Bochníček (2005). Assumption of the sufficiently homogeneous geomagnetic field in the whole region of the Czech Republic let us to compute the geoelectric field only from its variations, and so we were not forced to consider ionospheric currents.

The plane-wave model yields the integrodifferential equation coupling electric and magnetic field,

$$\begin{aligned} E_y(t) = -\frac{1}{\sqrt{\pi \mu _0 \sigma }} \int _{0}^{\infty } \frac{g(t - u)}{\sqrt{u}} \mathrm{d}u \end{aligned}$$

(1)

in the time domain, where \(g(t)= \mathrm{d} B_x(t)/\mathrm{d}t\). It is in agreement with causality, which means that at the time *t*, \(E_y(t)\) depends only on the previous values of *g*(*t*). The weight of affection by past values decreases with time. A stable solution can be achieved by integration over several hours. The geomagnetic coordinates are defined by the *x* axis in the south to north direction and *y* coordinates in the west to east direction. We note to the reader that compared to the S-JTSK coordinate system (*X*, *Y*) of the tension towers and substations of the transmission grid, we have \(x \parallel X\) and \(y \parallel Y\), where each of them have the opposite senses. A similar equation to (1) can be written for \(E_x\).

The square root in the denominator of (1) has a singularity at \(t=0\). This problem is caused due to quasi-static approximation, where displacement currents are ignored. The solution to the problem exists; the details can be found in Love and Swidinsky (2014).

Since we approximated the Earth’s surface as an infinite half-space, we are interested only in *x* and *y* components of the measured geomagnetic field. The time derivative of \({\varvec{B}}=(B_x,B_y)\) can be discretised as:

$$\begin{aligned} \frac{\Delta {\varvec{B}}(t_j)}{\tau }=\frac{\Delta {\varvec{B}}(t_{j+1})-{\varvec{B}}(t_j)}{\tau }, \end{aligned}$$

(2)

where \(\tau\) represents the data sampling (\(\tau = 60\) s in our case). From the discrete values of \({\varvec{B}}\) the induced electric field can be obtained by (numerical) convolution with the transfer function \(\chi _R(t;\tau )\),

$$\begin{aligned} \chi _R(t;\tau )=\frac{2}{\sqrt{\pi }}[\sqrt{t}H(t)-\sqrt{t-\tau }H(t-\tau )], \end{aligned}$$

(3)

where *H*(*t*) is the Heaviside function.

Then

$$\begin{aligned} {\varvec{C}} {\varvec{E}} (t_i;\sigma )=\frac{1}{\sqrt{\mu \sigma }} \sum _{j=1}^{i}\chi _R(t_i-t_j)\frac{\Delta {\varvec{B}}}{\tau }(t_j), \end{aligned}$$

(4)

where

$$\begin{aligned} \mathbf{C } = \begin{bmatrix} 0 &{} -1\\ 1 &{} 0\\ \end{bmatrix} \end{aligned}$$

is a spin matrix coming from the curl operator. Due to the plane-wave approximation, \({\varvec{C}}\) has only two dimensions.

### Pirjola–Lehtinen method of GIC modelling

In the ideal case, the GICs would be measured with sensors intended for this purpose. Unfortunately, these sensors are rare and expensive, so they were installed only at a few points in the power grids with higher risks of GICs-related damages. In the Czech Republic, only one sensor (Ripka et al. 2019) was installed recently in substation Mírovka, unfortunately, the interpretation of the measurements is not straightforward (Hejda et al. 2019) and the measurements are not publicly available.

Several studies from several countries showed that the GICs may successfully be modelled. The modelled values were satisfactorily compared to the direct measurements. The advantage of the GIC modelling is that it is very cheap compared to the development, installation, and maintenance of the GIC sensor and that the modelling may be performed for any situation for which the measurements of the geomagnetic field are available.

The commonly used method stems from the direct application of Kirchhoff’s and Ohm’s laws when the power-distribution network is virtually replaced by an electric circuit (see Fig. 2). In this approach (Lehtinen and Pirjola 1985; Pirjola 2012), the network consists of *N* grounded nodes that are connected by the power lines with a known resistivity. The lines are subjects to the induction by the external electric field. The induced voltages are computed from the known geoelectric field \({\varvec{E}}\) by

$$\begin{aligned} V_{im}=\int _{i}^{m}{\varvec{E}}\cdot \mathrm{d}{\varvec{s}}, \end{aligned}$$

(5)

where the voltage contributions \({\varvec{E}}\cdot \mathrm{d}{\varvec{s}}\) are integrated along the curve representing the respective line between nodes *i* and *m*. \(V_{im}\) is anti-symmetric to the swapping of *i* and *m* due to the opposite orientation of integration paths.

From the known induced voltages, we may compute the ideal-earthing current \(J_{e,m}\) in node *m* from the equation

$$\begin{aligned} J_{e,m}=\sum _{i=1,i\ne m}^{N}\frac{V_{im}}{R_{im}}, \end{aligned}$$

(6)

where \(V_{im}\) is the geomagnetic voltage calculated from equation (5) and *N* is the number of earthed nodes which represent transformer substations of the considered network. \(R_{im}\) is the resistance of wires between the nodes *i* and *m*. Formally, the set of nodes \(m=1,\dots ,N\) builds a vector of ideal-earthing currents \({\varvec{J}}_e\), and it represents the currents flowing through the lines under the assumption of the perfect grounding, which will close the circuit.

Finally, the GICs in the nodes are at once in the vector form given by

$$\begin{aligned} {\varvec{I}}_e=({\varvec{1}}+{\varvec{Y}} \cdot {\varvec{Z}}_e)^{-1}\cdot {\varvec{J}}_e, \end{aligned}$$

(7)

where \({\varvec{Y}}\) is the admittance matrix and \({\varvec{Z}}_e\) is the impedance matrix of the whole transmission system. The matrix \({\varvec{1}}\) indicates an identity matrix \(N\times N\). The resulting currents \({\varvec{I}}_e\) are the sought GICs flowing through the grounding line of the respective node. The convention is such that the positive \(I_{e,m}\) indicates the current flowing from the network to the Earth in node *m*, negative \(I_{e,m}\) occurs when the current flows from bedrock to the network. We note that usually more than one transformer share the grounding point of the substation. The current flowing through the substation’s grounding point is then split between the neutrals of these transformers according to the electrical parameters of the connected transformers.

To close the set, we give the definitions of the admittance matrix \({\varvec{Y}}\):

$$\begin{aligned} Y_{im}=-\frac{1}{R_{im}}\ \mathrm{for}\ i \ne m\ \mathrm{and}\ Y_{im}=\sum _{k=1,k\ne i}^{N}\frac{1}{R_{ik}}\ \mathrm{for}\ i = m, \end{aligned}$$

(8)

where \(R_{n,im}\) is the line resistance between two grounded *i* and *m* nodes.

The impedance matrix \({\varvec{Z}}_e\) is defined by the relation

$$\begin{aligned} Z_{e,im}=R_{i}\ \mathrm{for}\ i = m\ \mathrm{and}\ Z_{e,im}=0\ \mathrm{for} \ i \ne m, \end{aligned}$$

(9)

where \(R_{i}\) is the grounding resistance of the appropriate node. In the case when the nodes are spatially separated (which is what we assume in our case), the impedance matrix is diagonal.