# A Fault Section Location Method for Distribution Networks Based on Divide-and-Conquer
## 3 Recursive Structure of Fault Location for the Distribution Network
### Recursive Structure
#### 3.2.1 Subproblem Description  
The subproblem of distribution network fault section location is the problem of solving its subnetwork fault location result. The mathematical description of the subnetwork and its fault location results are presented here, followed by the formal consistency of the original problem with the subproblem is proved. The mathematical description of the subnetwork is as follows. Let the set of sections in the distribution network be \(\mathbf{E}\). Given section \(x_{i}\in\mathbf{E}\), we define the set of sections \(\mathbf{E}_{i}\): \(\mathbf{E}_{i}=\{x_{k}\in\mathbf{E}|P_{i}=\mathbf{P}_{i}*\mathbf{P}_{k}\}\). The distribution network subgraph \(\mathbf{G}_{i}=(\mathbf{V}_{i},\mathbf{E}_{i})\) is called a subnetwork starting from the section \(x_{i}\). Among them, \(\mathbf{V}_{i}\) is the set of nodes with the same section number as \(\mathbf{E}_{i}\). According to Theorem 3.1, the essence of the subnetwork starting from section \(x_{i}\) is the set of all distribution network sections (containing \(x_{i}\)) that contain section \(x_{i}\) on all the roads. Taking the multi-branch distribution network shown in Figure 3 as an example, section \(x_{3}\) is on the road of sections \(x_{4}\sim x_{14}\). We construct the set \(\mathbf{E}_{3}=\{x_{k}\in\mathbf{E}|3\leq k\leq 14\}\), \(\forall x_{k}\in\mathbf{E}_{3}\), according to Theorem 3.1, with \(\mathbf{P}_{3}=\mathbf{P}_{3}*\mathbf{P}_{k}\). Then, we construct the set \(\mathbf{V}_{3}=\{s_{k}\in V|3\leq k\leq 14\}\), and the subgraph \(\mathbf{G}_{3}=(\mathbf{V}_{3},\mathbf{E}_{3})\) is a subnetwork starting from section \(x_{3}\). The mathematical description of the subnetwork fault location result is as follows: Let \(\mathbf{G}_{i}=(\mathbf{V}_{i},\mathbf{E}_{i})\) be a subnetwork and \(\mathbf{X}_{i}\subseteq\mathbf{E}_{i}\
eq\varnothing\) be the set of fault sections. It holds that \(\forall\mathbf{X}^{\prime}\subseteq\mathbf{E}_{i}\
eq\mathbf{X}_{i}\
eq\varnothing\), \(\forall\mathbf{R}\subseteq\mathbf{E}-\mathbf{E}_{i}\), \(\mathbf{X}^{\prime}\cup\mathbf{R}\) is a feasible solution to the inequality constraint. If \(\mathbf{X}^{\prime}\cup\mathbf{R}\) satisfies the inequality constraint and there is always \(f(\mathbf{X}_{i}\cup\mathbf{R})<f(\mathbf{X}^{\prime}\cup\mathbf{R})\), then \(\mathbf{X}_{i}\) is the fault location result of the subnetwork \(\mathbf{G}_{i}\), and \(\mathbf{X}_{i}\) is called the fault candidate set for \(\mathbf{G}_{i}\). We prove the formal consistency of the original problem with the subproblem. According to the numbering rules in this paper, section \(x_{1}\) is a feeder section surrounded by the circuit breaker on the distribution side of the main power supply and its adjacent intelligent switch; thus, \(x_{1}\) is on the road of all sections. According to Theorem 1, \(\forall x_{k}\in\mathbf{E}\), there is \(\mathbf{P}_{1}=\mathbf{P}_{1}*\mathbf{P}_{k}\). Thus, the whole distribution network can be considered a subnetwork starting from section \(x_{1}\), i.e., \(\mathbf{G}=\mathbf{G}_{1}=(\mathbf{V}_{1},\mathbf{E}_{1})\). Let \(\mathbf{X}_{1}\) be the fault candidate set for \(\mathbf{G}_{1}\). It holds that \(\forall\mathbf{X}^{\prime}\subseteq\mathbf{E}\
eq\mathbf{X}_{1}\), \(\forall\mathbf{R}=\mathbf{E}-\mathbf{E}_{1}\), \(\mathbf{X}^{\prime}\cup\mathbf{R}\) is a feasible solution to the inequality constraint. According to the definition of the fault candidate set, \(\mathbf{X}_{1}\cup\mathbf{R}\) satisfies the inequality constraint, and there is always \(f(\mathbf{X}_{1})<f(\mathbf{X}^{\prime})\). Considering \(\mathbf{E}=\mathbf{E}_{1}\), \(\mathbf{R}=\mathbf{E}-\mathbf{E}_{1}=\varnothing\), \(\mathbf{X}_{1}\cup\mathbf{R}=\mathbf{X}_{1}\), so \(\mathbf{X}_{1}\) is a feasible solution satisfying the inequality constraint. Additionally, because \(\forall\mathbf{X}^{\prime}\subseteq\mathbf{E}\
eq\mathbf{X}_{1}\), we have \(f(\mathbf{X}_{1})<f(\mathbf{X}^{\prime})\), \(\mathbf{X}_{1}\) therefore is the optimal solution of the objective function.