## 4 Discussion  
Let us consider examples of the application of bitwise search and dichotomy methods in the case of FL on an inhomogeneous OHPL. It was noted above that for an accurate FL on OHPL, it is necessary to take into account its inhomogeneous sections. In the case of fault on each of the OHPL sections, an individual error of the FL takes place. It depends on the ratio of the specific resistance values of individual OHPL sections and significantly complicates the procedure for calculating the distance to the fault place. In [74], the following requirements are established for determining the inspection area of the OHPL. After an emergency shutdown of the OHPL, according to the data of the FL devices, the length of the inspection zone should be no more than  
* \(\pm 15\%\) of the OHPL length with its length up to 50 km inclusive;
* \(\pm 10\%\) for OHPLs with a length of more than 50 km up to 100 km inclusive;
* \(\pm 7\%\) for OHPLs with a length from 100 to 300 km inclusive; and
* \(\pm 5\%\) for OHPLs with a length from 300 km or more. Taking the size of the inspection zone based on operating experience and statistical data from the indications of FL devices is allowed in a given electric grid company. Therefore, on an OHPL with a non-uniform distribution of resistivity along the overhead line, the inspection area should be increased by the value of the maximum error (Equation (9)). This is necessary to exclude cases when the fault place on the OHPL is outside the inspection area. For the implementation of high-speed and accurate FL on OHPLs, the most promising approach is to first calculate the distance to the fault place using the equation for homogeneous OHPLs. Next, an extended inspection area is determined, taking into account the error of the FL due to the inhomogeneous distribution of resistivity along the OHPL. Further, within the extended inspection area, the algorithm of FL is implemented for a non-uniform OHPL in order to clarify the size of the inspection area. To illustrate the application of numerical methods of one-dimensional optimization, we will use the data obtained from the operated OHPL, 220 kV and 120 km long, with a short circuit on its section. The OHPL is heterogeneous, consisting of three sections:  
\[\underline{Z}_{L}=\underline{Z}_{1L}\cdot n_{1}+(\underline{Z}_{1L}+\Delta \underline{Z}_{12})\cdot n_{2}+(\underline{Z}_{1L}+\Delta\underline{Z}_{L3}) \cdot n_{3},\]  
where \(n_{1}=n_{2}=n_{3}=1/3\); \(\Delta\underline{Z}_{12}=-0.15\cdot\underline{Z}_{1L}\); and \(\Delta\underline{Z}_{L3}=-0.3\cdot\underline{Z}_{1L}\). For simplicity of calculations, let us move from complex values to amplitude values of currents and voltages. The recorded actual values of the amplitudes of currents and voltages at the ends of a real OHPL were \(I^{\prime}=2.0\) kA, \(I^{\prime\prime}=0.56\) kA, \(U^{\prime}=40\) kV, \(U^{\prime\prime}=28\) kV. Let us implement a distance-to-fault calculation using double-ended current and voltage measurements. At the first stage, we will not take into account the inhomogeneity of the resistances in the sections of the OHPL (Figure 1). We will form the main calculation relations of the FL method based on measurements of the modules of currents and voltages at the ends of the OHPL \(I^{\prime}\), \(I^{\prime\prime}\), \(U^{\prime}\), \(U^{\prime\prime}\), as well as Equation (21):  
\[I^{\prime}\cdot n\cdot Z_{L}+U^{\prime}=U_{F};\ I^{\prime\prime}\,(1-n)\cdot Z _{L}+U^{\prime\prime}=U_{F} \tag{21}\]  
where \(U_{F}\) is the voltage at the place of the SC (fault). The calculation according to Equation (21) is performed by the components of the zero sequence. In this case, the resistivity \(Z_{0}=30\),\(426=1278\) Ohm/km, and \(\underline{Z}_{L}=L\cdot Z_{0}\).