# A Fault Section Location Method for Distribution Networks Based on Divide-and-Conquer
## 4 Fault Section Location Method for Distribution Networks Based on Divide-and-Conquer
### Algorithm Design  
The fault location method closely follows the divide-and-conquer paradigm. Intuitively, it operates as follows. * Divide. Divide the distribution network into smaller subnetworks using the T-type section closest to the main power source as the boundary (the decomposition effect is shown in Figure 4). * Conquer. Recursively invoke the algorithm to solve the fault candidate set of the subnetwork. If the subnetwork no longer contains a T-type section, the recursion is stopped, and the fault candidate set is constructed directly according to Theorem 8. Figure 4: Subnetwork with T-type section. * Combine. Combine the fault candidate set of subnetworks to create the fault candidate set of the original network according to Theorem 9. The topological information of the network is the basis of the fault location for the distribution network, and the method of determining the topological information required by the algorithm is described below. * Node number and section number. By applying the depth-first search algorithm, the section is traversed with the main power supply as the source node, and the sections are numbered in the order they are accessed so that the section numbers of any subnetwork in the network are consecutive and conform to the numbering rules in this paper. * Section type vector \(\mathbf{T}\). Construct the section type vector \(\mathbf{T}\) to determine the type of the section. When \(x_{k}\) is a T-type section, let \(\mathbf{T}(k)=1\); otherwise, \(\mathbf{T}(k)=0\). * Adjacency list(\(Adj\)). Construct the adjacency list of the section to record the downstream section of each section for a fast search of the adjacency section of the T-type section, which helps implement the \"divide\" step in the recursive case. When the distribution network topology changes, this operation is re-executed to make the algorithm flexible to adapt to the topology change. The pseudocode DACFL gives the implementation of applying DAC to solve the subnetwork \(\mathbf{G}_{i}\) fault candidate set. Because the subnetwork section numbers are consecutive, the diagnostic range is determined by the starting section number \(i\) and the maximum section number \(j\). DACFL accepts \(i\), \(j\), network topology information Graph (containing adjacency list and section type vector \(\mathbf{T}\)), and the fault information vector \(\mathbf{I}\) reported by FTUs as input. The fault candidate set \(\mathbf{X}\) and its approximation gain \(\Delta_{\mathbf{X}}\) are returned. As described in Section 3.2.1, the distribution network fault location problem can be solved by solving the fault candidate set \(\mathbf{X}_{1}\) of its subnetwork \(\mathbf{G}_{1}\). Therefore, the algorithm initially set \(i=1\) and \(j\) to take the maximum number of the distribution network sections. The pseudocode for the algorithm is shown as Algorithm 1. In detail, the DACFL procedure works as follows. Lines 1-2 of DACFL set the initial pointer position and calculate the approximation gain of the starting section, where the value of D is 0 when the algorithm is started, and the approximation gain of the upstream T-type section of the starting section xi is taken at each recursion thereafter (the calculation of the approximation gain of section \(x_{k}\) in lines 2 and 5 is described in the last paragraph of this section). Lines 3-6 traverse the section from the starting section \(x_{i}\), determine whether there is a T-type section in \(\mathbf{G}_{i}\), and complete the calculation of the approximation gain of the T-type section and its upstream section. If \(\mathbf{G}_{i}\) does not contain a T-type section, the traversal process is not interrupted and ends with \(k=j\). Otherwise, the process ends with \(k\) recording the T-type section number in \(\mathbf{G}_{i}\) that has the shortest distance from the main power supply. Lines 7-9 correspond to the base case, where \(\mathbf{G}_{i}\) does not contain a T-type section.