## 3 Results  
### Influence of the Heterogeneity at Overhead Transmission Line on the Accuracy of the Fault Location  
According to the requirements of the standard [71], the inhomogeneity of the resistivity distribution along the OHPL can be neglected if the error of the FL increases by no more than 2%. Sections of extended OHPLs operated by electric grid companies may include wires of different brands suspended on towers of various types. Additionally, the line route often passes through terrain with different soil characteristics [72; 73]. The equivalent circuit of a single-circuit OHPL, consisting of two heterogeneous sections of different lengths \(n_{1}\cdot L\) and \(n_{2}\cdot L\) with resistances \(\underline{Z}_{1L}\cdot n_{1}\cdot L\) and \((\underline{Z}_{1L}+\Delta\underline{Z}_{L2})\cdot n_{2}\cdot L\), is shown in Figure 3.  
Taking into account the metallic SC (\(R_{trans}=0\)), as well as the uniform resistivity of the OHPL sections, the error in calculating the distance to the fault place in the section \(n_{1}\cdot L\) will be determined by Equation (4):  
\[\Delta n_{1}=\left[\left(\frac{\underline{Z}_{1L}}{\underline{Z}_{L}}\right) \ -1\right]\cdot n, \tag{4}\]  
where \(\underline{Z}_{L}=\underline{Z}_{1L}n_{1}+(\underline{Z}_{1L}+\Delta\underline {Z}_{L2})\cdot n_{2}\). At the same time, the maximum value of the FL error of OHPL is achieved when the SC occurs at the end of the section \(n_{1}\cdot L\) and corresponds to Equation (5):  
\[ \begin{split}\Delta n_{\text{1 max}}=[(\underline{Z}_{1L}-( \underline{Z}_{1L}+\Delta\underline{Z}_{L2}))\cdot n_{1}+(\underline{Z}_{1L }+\Delta\underline{Z}_{L2})\cdot n_{2}]/\underline{Z}_{L}=\\ =[(\underline{Z}_{1L}+\Delta\underline{Z}_{L2})\cdot n_{2}-\Delta \underline{Z}_{L2}\cdot n_{1}]/\underline{Z}_{L}.\end{split}  \tag{5}\]  
Figure 3: Equivalent circuit of a heterogeneous single-circuit OHPL with two sections. In the event of a SC in the second OHPL section, the error in calculating the distance to the fault place should be determined by Equation (6):  
\[\Delta n_{2}=[(\underline{Z}_{1L}-(\underline{Z}_{1L}+\Delta\underline{Z}_{L2})) \cdot n_{1}+((\underline{Z}_{1L}+\Delta\underline{Z}_{L2})-\underline{Z}_{L}) \cdot n]/\underline{Z}_{L}, \tag{6}\]  
which takes the maximum value at \(n=n_{2}\). \[\Delta n_{2\max}=[(\underline{Z}_{1L}-\underline{Z}_{L})\cdot n_{1}+(( \underline{Z}_{1L}+\Delta\underline{Z}_{L2})-\underline{Z}_{L})\cdot n_{2}]/ \underline{Z}_{L} \tag{7}\]  
An analysis of Equations (4) and (6) shows that the error value changes according to a linear law, and \(\Delta\underline{Z}_{L2}\) can have both a positive and a negative sign. Figure 4 shows the dependence of the calculation error in the FL on OHPL for cases where \(\Delta\underline{Z}_{L2}<0\) and \(\Delta\underline{Z}_{L2}>0\), and the maximum error value is reached at the end of the first section of the OHPL. Generalizing the approaches to calculating the error of the FL on OHPL, including \(N\) inhomogeneous sections, let us move on to Equation (8) for calculating the error during an SC in the \(N\)th section of the OHPL:  
\[ \begin{array}{c}\Delta n_{N}=[(\underline{Z}_{1L}-(\underline{Z}_{1L}+ \Delta\underline{Z}_{LN}))\cdot n_{1}\\ +\sum_{i=2}^{N-1}\left\{(\underline{Z}_{1L}+\Delta\underline{Z}_{Li})-( \underline{Z}_{1L}+\Delta\underline{Z}_{LN})\right\}\cdot n_{i}+((\underline{ Z}_{1L}+\Delta\underline{Z}_{LN})-\underline{Z}_{L})\cdot n]/\underline{Z}_{L}\\ =[\sum_{i=2}^{N-1}\left(\Delta\underline{Z}_{Li}-\Delta\underline{Z}_{LN} \right)\cdot n_{i}-\Delta\underline{Z}_{LN}\cdot n_{1}+((\underline{Z}_{1L}+ \Delta\underline{Z}_{LN})-\underline{Z}_{L})\cdot n_{N}]/\underline{Z}_{L}. \end{array}  \tag{8}\]  
Based on the requirements of [71], when estimating the magnitude of the maximum error, it is necessary to consider SC at the end of each section of the OHPL and choose the largest of them, which determines the admissibility of neglecting the inhomogeneity of the OHPL. The value of the maximum error in the \(N\)th section should be determined using Equation (9):  
\[