# A Fault Section Location Method for Distribution Networks Based on Divide-and-Conquer
## Appendix D The Proofs of Recursive Structure  
Proof of Theorem 8.: It holds that \(\forall\mathbf{X}^{\prime}\subseteq\mathbf{E}_{i}\
eq\{x_{k}\}\), \(\forall\mathbf{R}\subseteq\mathbf{E}-\mathbf{E}_{i}\), \(\mathbf{X}^{\prime}\cup\mathbf{R}\) satisfies the inequality constraint. According to Theorem 4, \(\mathbf{X}^{\prime}\), \(\mathbf{R}\) satisfy the inequality constraint. We claim that \(\mathbf{X}^{\prime}\) contains only one section. Otherwise, let \(x_{a},x_{b}\in\mathbf{X}^{\prime}\); because \(\mathbf{G}_{i}\) does not contain a T-type section, there is a path between \(x_{a}\) and \(x_{b}\), i.e., \(x_{a}\) on the road of \(x_{b}\) or \(x_{b}\) on the road of \(x_{a}\). According to Theorem 1, \(\mathbf{P}_{a}=\mathbf{P}_{a}*\mathbf{P}_{b}\) or \(\mathbf{P}_{b}=\mathbf{P}_{a}*\mathbf{P}_{b}\); thus, \(\mathbf{X}^{\prime}\) does not satisfy the inequality constraint, contradicting what is known. Let \(\mathbf{X}^{\prime}=\{x_{r}\}\), \(x_{r}\
eq x_{k}\in\mathbf{E}_{i}\). According to Theorem 4, \(\mathbf{R}\subseteq C_{r}\). Because there is no T-type section on the path between \(x_{k}\) and \(x_{r}\), according to Theorem 6, \(C_{r}=C_{k}\); thus, \(\mathbf{R}\subseteq C_{k}\). Because \(\mathbf{X}_{i}\) contains only one section, it must satisfy the inequality constraint. According to Theorem 4, \(\mathbf{X}_{i}\cup\mathbf{R}\) satisfies the inequality constraint. Applying the approximation gain definition Equations (31) and (33) to rewrite \(f(\mathbf{X}_{i}\cup\mathbf{R})\):  
\[ \begin{array}{rl}f(\mathbf{X}_{i}\cup\mathbf{R})=f(x_{k}\cup\mathbf{R})&= \left[f(x_{k}\cup\mathbf{R})-f(x_{k})\right]+\left[f(x_{k})-f(\varnothing) \right]+f(\varnothing)\\ &=-\Delta_{\mathbf{R}}^{k}-\Delta_{k}+f(\varnothing)\end{array}  \tag{11}\]  
Similarly, we obtain \(f(\mathbf{X}^{\prime}\cup\mathbf{R})=-\Delta_{\mathbf{R}}^{\prime}-\Delta_{r}+f(\varnothing)\). Because \(C_{r}=C_{k}\), \(\mathbf{R}\subseteq C_{k}\), according to Theorem 7: \(\Delta_{\mathbf{R}}^{k}=\Delta_{\mathbf{R}}^{\prime}\). Because \(\Delta_{k}>\Delta_{r}\), \(f(\mathbf{X}_{i}\cup\mathbf{R})<f(\mathbf{X}^{\prime}\cup\mathbf{R})\). Therefore, \(\mathbf{X}_{i}=\{x_{k}\}\) is the fault candidate set for \(\mathbf{G}_{i}\). \(\square\)  
Proof of Theorem 9.: It holds that \(\forall\mathbf{X}^{\prime}\subseteq\mathbf{E}_{i}\),\(\forall\mathbf{R}\subseteq\mathbf{E}-\mathbf{E}_{i}\), \(\mathbf{X}^{\prime}\cup\mathbf{R}\) satisfies the inequality constraint. The proof of Theorem 9 is divided into the following three steps. Step 1. Prove that \(\mathbf{X}_{i-t}\cup\mathbf{R}\), \(\mathbf{X}_{u}\cup\mathbf{R}\), \(\mathbf{X}_{v}\cup\mathbf{R}\), \(\mathbf{X}_{u\in\mathbf{k}v}\cup\mathbf{R}\) satisfy the inequality constraint. Because \(\mathbf{X}^{\prime}\cup\mathbf{R}\) satisfies the inequality constraint, according to Theorem 4, \(\mathbf{R}\) satisfies the inequality constraint. Therefore, it remains to be proved that (1) \(\mathbf{X}_{i-t}\), \(\mathbf{X}_{u}\), \(\mathbf{X}_{v}\), \(\mathbf{X}_{u\in\mathbf{k}v}\) satisfy the inequality constraint; (2) \(\forall x_{w}\in\mathbf{X}_{u}\), \(\forall x_{y}\in\mathbf{X}_{v}\), \(\forall x_{z}\in\mathbf{X}_{u\in\mathbf{k}v}\), there should be \(\mathbf{R}\subseteq C_{k}\), \(\mathbf{R}\subseteq C_{w}\), \(\mathbf{R}\subseteq C_{y}\), \(\mathbf{R}\subseteq C_{z}\). The proofs are given separately below. (1) Because \(\mathbf{X}_{i-t}\) contains only one element, \(\mathbf{X}_{u}\) and \(\mathbf{X}_{v}\) are fault candidate sets; thus, \(\mathbf{X}_{i-t}\), \(\mathbf{X}_{u}\), \(\mathbf{X}_{v}\) satisfy the inequality constraint. Then, \(\forall x_{w}\in\mathbf{X}_{u}\), \(\forall x_{y}\in\mathbf{X}_{v}\), we claim that \(\mathbf{P}_{w}\
eq\mathbf{P}_{w}*\mathbf{P}_{y}\). Otherwise, by \(x_{w}\in\mathbf{X}_{u}\subseteq\mathbf{E}_{u}\):\(\mathbf{P}_{w}=\mathbf{P}_{w}*\mathbf{P}_{u}\).