## 2 Modeling Technique  
Computer technology for determining EMF [34] was intended to calculate the strengths of plane-parallel fields of objects whose cross-sectional dimensions are significantly smaller than their length. This approach did not allow analysis of the EMF of current-carrying parts of limited length, which include, for example, most components of substations. In addition, this method did not have the ability to take into account edge effects. In the presence of limited length conductors, the EMF becomes three-dimensional, and the modeling becomes significantly more complicated. The method proposed in [33] to resolve this issue is based on the following basic principles:  
* The current-carrying parts are represented in the form of straight segments, which are located in space according to the design of the object; some components, in particular, cables and parts of grounding systems, may be buried. To use the equations of a quasi-stationary zone, the size of the set of objects should not exceed several hundred meters. * The analyzed network may include power lines and traction networks, transformers, loads, and sets of short conductors, which are modeled in the same way as power transmission line wires for calculation of operating parameters. The resistances of the wires are small; hence, this approach does not distort the network operating parameters. In addition, short-length grounded conductors can be used, specifically to model supports of power transmission lines and traction networks, substation portals, lightning rods, and others. * The potentials and currents of short wires are determined by calculating the operating parameters in phase coordinates [32]. * The electric field is determined by the method of equivalent charges; the Bio-Savart formulas are used to calculate the magnetic field induction. * Plain X0Z of the Cartesian coordinate system corresponds to the surface of the earth, the X axis is perpendicular to the railway route, and the Y axis is directed vertically upward. The coordinate system and the short wire (SW) are shown in Figure 1. In the general case, the presence of \(N_{w}\) SW is assumed. Currents \(\hat{\mathbf{i}}_{ij}\) of normal or emergency modes can flow through them. Some grounded objects have \(\hat{\mathbf{j}}_{ij}=0\). Each SW \(i\) of length \(L_{i}\) is represented as \(n_{i}\) elementary segments (ES) with lengths \(\Delta l_{i}=L_{i}/n_{i}\) (Figure 2). For a more precise indication of the location of ES on wire \(i\), the notation \(\Delta l_{i\ j}\) can be used. The numbering of ES starts with one. The potential created by wire \(i\) at observation point \(M\) with coordinates (\(x\), \(y\), \(z\)) is defined as follows [33]:  
\[\hat{\varphi}_{i}=\frac{1}{4\pi\epsilon_{0}}\int\limits_{0}^{L_{i}}\!\!\!\!\! \int\limits_{0}^{L_{i}}\!\!\!\!\!\int\limits\left[\left(\frac{1}{r}-\frac{1}{r }\right)\hat{\tau}_{i}dl_{i\ j}\right]\approx\frac{1}{4\pi\epsilon_{0}}\sum \limits_{j=1}^{n_{i}}\left(\frac{1}{r_{i\ j}}-\frac{1}{r_{i\ j}}\right)\!\! \hat{q}_{i\ j} \tag{1}\]where \(r_{i\,j}\)--distance from \(M\) to the midpoint of the segment \(\Delta l_{i\,j}\); \(r_{i\,j}\)--similar parameter for mirror image of ES; \(\hat{q}_{i\,j}=\tilde{\alpha}_{i\,j}\Delta\,l_{i\,j}\)--charge of ES \(j\) of wire \(i\); and \(n_{i}\)--number of ES. Distances \(r_{i\,j}\) and \(r_{i\,j}\)\({}^{\prime}\) are calculated using the coordinates of the beginning \(x_{i1},\ y_{i1},\ z_{i1}\) and end \(x_{i2},\ y_{i2},\ z_{i2}\) of the short wire  
\[r_{i\,j}=\sqrt{\left[x-x_{i1}-(j-0.5)\Delta x_{i}\right]^{2}+\left[y-y_{i1}-(j -0.5)\Delta y_{i}\right]^{2}+\left[z-z_{i1}-(j-0.5)\Delta z_{i}\right]^{2}};\]  
\[r_{i\,j}\,^{\prime}=\sqrt{\left[x-x_{i1}-(j-0.5)\Delta x_{i}\right]^{2}+\left[ y+y_{i1}\,^{\prime}+(j-0.5)\Delta y_{i}\right]^{2}+\left[z-z_{i1}-(j-0.5)\Delta z _{i}\right]^{2}};\]  
where \(\Delta x_{i}=(x_{i2}-x_{i1})/n_{i}\); \(\Delta y_{i}=(y_{i2}-y_{i1})/n_{i}\); \(\Delta z_{i}=(z_{i2}-z_{i1})/n_{i}\).