## 4 Discussion
Let us calculate the distance to the fault place on the OHPL by jointly solving Equation (21), with the substitution \(Z_{0}\), which corresponds to the calculation equation:  
\[n = \left[U^{\prime\prime}-U^{\prime}+I^{\prime\prime}\cdot Z_{0} \right]/\left[(I^{\prime}+I^{\prime\prime})\cdot Z_{0}\right] \tag{22}\] \[= \left[28\text{--}40+0.56\cdot 1.278\right]/\left[(2.0+0.56) \cdot 1.278\right]=0.188;\] \[L_{F} = n\cdot L=22,582\text{km}.\]  
Let us set the permissible error in calculating the distance to the FL on OHPLs with a nonuniform distribution of resistivity based on the operating experience of an OHPL. The operational and technological personnel of electric grid companies consider the error in calculating the distance to the FL on OHPL to be low if it does not exceed the length of one OHPL span. The length of one span of 220 kV OHPL is 250-350 m, so we will set the allowable error in calculating the distance to the fault place in the form \(\delta=0.3\) km. Let us assume that according to the results of preliminary calculations of the FL on OHPL, taking into account the heterogeneity of the sections of the OHPL, the average distance to the fault place was \(L^{\prime}_{F}=19.2\) km. The simulation may include multiple random factors to obtain the average distance to a fault, and the results may differ from the actual values due to the current combination of random factors. Therefore, the error in the implementation of the FL on OHPL due to the non-uniform distribution of resistivity along the overhead line will be equal to  
\[\Delta L_{F}=L_{F}-L^{\prime}{}_{F}=22.582-19.2=3.382\text{ km}.\]  
Let us form an extended inspection area of OHPLs by a line crew for the example under consideration (\(\pm 7\%\)) for OHPLs with a length from 100 to 300 km inclusive (\(\pm 120\cdot 0.07=8.4\) km), taking into account the FL error with the value \(\pm\Delta L_{F}=3.382\) km. The extended inspection area of the OHPL will be \(\pm 12.382\) km, and relative to the fault place \(L_{\text{K}}=22.582\) km, it will correspond to the section of the OHPL from 10.2 km to 34.964 km. When implementing the developed method of FL on OHPLs, we will use an iterative algorithm based on the bitwise search method, which provides a calculation of the distance to the fault place on the OHPL with a non-uniform distribution of resistivity along the OHPL with a given allowable error \(\delta\) = 0.3 km. The computational algorithm of the iterative bitwise search procedure (Figure 7a) is implemented within the extended inspection area of the OHPL within \(a\) = 10.2 km and \(b\) = 34.964 km. The application of the criterion of equality of voltages calculated from two ends of the OHPL makes it possible to form the required objective function of the iterative algorithm \(f(\mathrm{x})=\Delta U\), which provides a minimum of the module of the voltage difference \(\Delta U\) at the fault point (\(U_{F}\)) (Equation (21)), calculated from different ends of power lines:  
\[\Delta U=\big{|}\big{[}I^{\prime}n\cdot\underline{Z}_{L}+U^{\prime}\big{]}-[I ^{\prime\prime}\left(1-n\right)\cdot\underline{Z}_{L}+U^{\prime\prime}]\big{|}. \tag{23}\]  
Equation (23) is valid for a homogeneous OHPL; therefore, in the presence of sections with non-uniform resistance, it is necessary to specifically take into account the voltage drop in these sections. FL devices that involve the use of search algorithms are made by both European and North American manufacturers, which use different processors and controllers. Therefore, the operating time is estimated by the number of steps of the iterative procedure in order not to be tied to a particular manufacturer and to ensure the comparability of the effect for different devices. Let us apply a step-by-step calculation of the distance to the fault place for an inhomogeneous OHPL using the bitwise search method with the chosen objective function (23). In the first step:  
\[