## 3 Results
### Analysis of the Application of Iterative Procedures in Case of Fault Location on OHPLs
At the end of each iteration, the following condition is checked: \[\left|n_{(i)}\ -\ n_{(i+1)}\right|\leq\delta.\] (15) If the inequality condition (15) is not met, then the process goes to the next iteration and repeats step 3. If the inequality condition (15) is met, then the iterative process ends and the distance to the fault place in relative units (r.u.) is determined by Equation (16): \[L_{F}=n_{(i+1)}\cdot L.\] (16) The high accuracy of the iterative method of FL on OHPLs is confirmed by the results of simulation [49]. Figure 6: Block diagram of the FL algorithm based on the impedance method for a single-circuit OHPL with two-way power supply. An improved method of sequential enumeration is called the bitwise search method and is characterized by a reduced number of \(f(x)\) values that must be found in the process of searching for the extremum of the function. If \(f(x_{i})\leq f(x_{i+1})\), then there is no need to calculate \(f(x)\) at the points \(x_{i+2}\), \(x_{i+3}\), etc. since the unimodality of the function implies that \(x_{opt}\leq x_{i+1}\). It is advisable to first determine the section of the OHPLs containing the point \(x_{opt}\) with a small accuracy and then search for it in this section with a smaller step. It is this logic that is implemented in the bitwise search method. The enumeration of the section points occurs first with a step \(\Delta x=x_{i+1}-x_{i}>\delta\) until the condition \(f(x_{i})\leq f(x_{i+1})\) is satisfied or until the next of these points coincides with the end of the OHPL section. After that, the step decreases (usually four times) and the enumeration of points with a new step is carried out in the opposite direction until the values of \(f(x)\) stop decreasing or the next point coincides with the end of the OHPL section. The algorithm for the FL on OHPLs based on the bitwise search method is shown in Figure 7a. The process of calculating the distance to the fault place is completed when the enumeration in the given direction is completed. In this case, the discretization step does not exceed the error \(\delta\). Let us consider the issue of applying the methods of excluding sections for FL on OHPLs. One of the most efficient ways to search for \(x_{opt}\) follows from the definition of unimodal functions when organizing the FL procedure. Let \(a<x_{1}<x_{2}<b\); then, by comparing the values of \(f(x_{1})\) and \(f(x_{2})\), we can shorten the search area \(x_{opt}\) by going to the area \([a,x_{1}]\), if \(f(x_{1})<f(x_{2})\) or to the area \([x_{1},b]\), if \(f(x_{1})>f(x_{2})\)  
Figure 7: Block diagrams of the FL algorithm for OHPLs based on the following methods: (**a**) bitwise search; (**b**) dichotomies. The above procedure can be repeated several times, successively reducing the section of the OHPL containing the minimum point. There are various methods (Figure 5) for excluding segments depending on the method of choosing the analyzed points (dichotomy, golden section, Fibonacci numbers, etc.). Let us consider the application of methods for eliminating segments using the example of a dichotomy. In this case, a small value \(E\) is chosen, which belongs to the interval \(\mathrm{E}\in[0;2\delta]\), where \(\delta\) is the FL error (Figure 7b). Relative to the middle of the segment [\(a\), \(b\)], the points are determined according to Equation (17):  
\[f(x)\to min;x\in[a,b]. \tag{10}\]0  
as well as the values of the function \(f(x)\) at these points \(f_{1}=f(x_{1}),f_{2}=f(x_{2})\). Let us verify the condition \(f_{1}\leq f_{2}\) (Figure 7b) and select points for forming a segment for subsequent analysis depending on the ratio of \(f_{1}\) and \(f_{2}\). The ratio of the new segment length to the length of the previous one will be as follows:  
\[f(x)\to min;x\in[a,b]. \tag{10}\]1  
The iterative process ends when the length of the next segment is (\(b-a\)) < \(2\delta\). At the last step of the iterative process, the optimal value of the function \(f(x)\) argument is determined, which is \(x_{opt}=(a+b)/2\).