## 3 Results
### Analysis of the Application of Iterative Procedures in Case of Fault Location on OHPLs  
Let us consider the application of numerical methods of one-dimensional optimization for FL on OHPLs. The problem of the FL on OHPLs can be reduced to finding the smallest  
Figure 4: Dependence of the calculation error in the case of FL on OHPL: 1 corresponds to \(\Delta\underline{Z}_{L2}<0\); 2 corresponds to \(\Delta\underline{Z}_{L2}>0\). value of some function, which is chosen as the target one. This approach is universal and can be applied to both homogeneous and heterogeneous OHPLs. It can be assumed that the objective functions characterizing the distribution of the OHPL parameters of the recorded currents and voltages are continuous and differentiable. Let \(f(x)\) be a continuous function on the interval \([a,b]\). The task of finding the minimum of a function can be written as  
\[f(x)\to min;x\in[a,b]. \tag{10}\]  
The solution of task (10) will be denoted by \(x_{\text{opt}}\):  
\[f(x_{\text{opt}})=\underset{[a,b]}{min}f(x). \tag{11}\]  
Note that maximizing the objective function (\(f(x)\to max\)) is equivalent to minimizing the opposite value (\(-f(x)\to min\)); therefore, without loss of generality, we will consider only minimization tasks. Due to the specific features of the FL on OHPLs, when solving the optimization task, as a rule, unimodal functions take place, in which the local minimum is simultaneously global. Therefore, \(f(x)\) is a convex function (the second derivative \(f^{\prime\prime}(x)\geq 0\) on the section \([a,b]\)). The use of some one-dimensional optimization methods is possible only when the rate of change of the objective function \(f(x)\) in any parts of the segment \([a,b]\) is limited by a certain number, the same for all sections. Then, we say that \(f(x)\) satisfies the Lipschitz condition. It is observed if there exists a number \(\Theta>0\) (Lipschitz constant) such that  
\[\left|f(x^{\prime})-f(x^{\prime\prime})\right|<\Theta\cdot\left|x^{\prime}-x^ {\prime\prime}\right| \tag{12}\]  
for all \(x^{\prime}\) and \(x^{\prime\prime}\) belonging to \([a,b]\). Methods that use only the values of the function and do not require the calculation of derivatives are called direct optimization methods. In this case, the function \(f(x)\) may not be specified in an analytical form, but it is important that it is possible to determine the function \(f(x)\) at given points. The classification of search algorithms for unconditional one-dimensional optimization based on various mathematical methods is shown in Figure 5.  
The simplest way to solve the problem is the method of sequential enumeration (Figure 5). A typical example of the use of sequential enumeration is the iterative method of FL by EMP for single-ended measurements on OHPLs [49]. This method is applicable not only to homogeneous OHPLs but also, with some modernizatio  
Figure 5: Classification of search algorithms for unconditional one-dimensional optimization. heterogeneous sections. Let us explain the implementation of the iterative FL using the example of a single homogeneous OHPL (Figure 1). In the event of a SC on the OHPL, the currents \(\dot{I}^{\prime}\) (from the side of System A) and \(\dot{I}^{\prime\prime}\) (from the side of System B) and voltages \(\dot{\dot{U}}^{\prime}\) (on the buses of System A) and \(\dot{\dot{U}}^{\prime\prime}\) (on the buses of System B). The  
3. At the time of fault on the OHPL, based on the entered settings, the distance to the fault place \(n_{(i)}\) is determined as a first approximation using the assumption that the current distribution coefficient \(\underline{\zeta^{\prime}}_{1(i)}\) is a real number (for \(i=0\)). 4. An iterative process is launched, and in its course, the current distribution coefficient \(\underline{\zeta^{\prime}}_{1(i+1)}\) is specified, as well as the distance to the fault place \(n_{(i+1)}\), where \(i\) is the iteration number. 5.