## 2 Materials and Methods  
Let us perform an analysis of the implementation FL procedure on the example of the OHPL with a double-sided power supply \(L\) km long. Figure 1 shows a simplified scheme of a single OHPL with a double-sided power supply in the case of fault. In Figure 1 OHPL has the parameters \(\underline{Z}_{1L}\), \(\underline{Z}_{2L}\), and \(\underline{Z}_{0}\), which are full impedances of the positive, negative, and zero sequence, respectively, while \(\underline{Z}_{1L}=\underline{Z}_{2L}\); \(\underline{Z}^{\prime}_{1S}\) and \(\underline{Z}^{\prime\prime}_{1S}\), \(\underline{Z}^{\prime}_{2S}\), \(\underline{Z}^{\prime\prime}_{0S}\) in \(\underline{Z}^{\prime\prime}_{0S}\) are equivalent impedances of the positive, negative, and zero sequences from system A and from system B, respectively; \(\dot{E}^{\prime}\) in \(\dot{E}^{\prime\prime}\) is equivalent EMF from system A and from system B, respectively; \(R_{trans}\) is transition resistance at the place of fault; \(L_{F}\) is distance to the place of fault from the beginning of the OHPL (system A); and \(n=L_{F}/L\). The RZA terminals installed at the ends of the OHPL record three currents and three voltages in the case of short circuit (SC). SC in the place of fault can be calculated using Equation (1):  
\[\dot{I}_{F}=\dot{I}^{\prime}+\dot{I}^{\prime\prime}. \tag{1}\]  
If the distance to the fault location \(L_{F}\) is known, then six equations can be drawn up (for three phases and three sequences):  
\[\dot{U}^{\prime}_{v}-\Delta\dot{U}^{\prime}_{v}+\Delta\dot{U}^{\prime\prime}_{ v}-\dot{U}^{\prime\prime}_{v}=0, \tag{2}\]  
where \(v\) is the phase index (A, B or C) or the sequence index (1, 2 or 0) and \(\Delta\dot{U}^{\prime}_{v}\) and \(\Delta\dot{U}^{\prime\prime}_{v}\) are voltage drops in the section to the left and to the right, respectively of the SC point. For individual sequences, they are determined by Equations (3):  
\[ \begin{split}&\Delta\dot{U}^{\prime}_{1}=n\cdot\dot{I}^{\prime}_{1 }\cdot\underline{Z}_{1L};\Delta\dot{U}^{\prime}_{2}=n\cdot\dot{I}^{\prime}_{2} \cdot\underline{Z}_{1L};\ \Delta\dot{U}^{\prime}_{0}=n\cdot\dot{I}^{\prime}_{0}\cdot \underline{Z}_{0L};\\ &\Delta\dot{U}^{\prime\prime}_{1}=(1-n)\cdot\dot{I}^{\prime\prime }_{1}\cdot\underline{Z}_{1L};\ \Delta\dot{U}^{\prime\prime}_{2}=(1-n)\cdot\dot{I}^{\prime\prime}_{2}\cdot \underline{Z}_{1L};\ \Delta\dot{U}^{\prime\prime}_{0}=(1-n)\cdot\dot{I}^{\prime\prime}_{0}\cdot \underline{Z}_{0L}.\end{split}  \tag{3}\]  
Various variants of solving Equation (2) are known. The most general solution is associated with obtaining the objective function and applying a special computational procedure [47]. At the same time, the general criterion for the FL on OHPL with a minimum error is the condition for achieving the global minimum using the objective function on a set of variable parameters. The simplest objective function can be the discrepancy of Equation (2), that is, the difference in their right side from zero. The formation of residual values \(|\,\sigma|\) is implemented by changing the distance to the fault \(n\) (for example, for \(L_{F}\in[0;L]\)). Next, the modulus of the right side is determined depending on the expected fault location. An example of four such dependences for \(R_{trans}=0\) is shown in Figure 2 for the OHPL of 220 kV and 100 km long [18]. Figure 1: Simplified scheme of single OHPL with a double-sided power supply in case of short circuit.