# A Fault Section Location Method for Distribution Networks Based on Divide-and-Conquer
## Appendix D The Proofs of Recursive Structure
Substitute \(\mathbf{P}_{w}=\mathbf{P}_{w}*\mathbf{P}_{y}\) for \(\mathbf{P}_{u}=\mathbf{P}_{w}*\mathbf{P}_{u}=(\mathbf{P}_{w}*\mathbf{P}_{y})*\mathbf{P}_{u}=\mathbf{P}_{y}*( \mathbf{P}_{w}*\mathbf{P}_{u})=\mathbf{P}_{y}*\mathbf{P}_{u}\); that is, \(x_{y}\in\mathbf{E}_{u}\). According to Figure 4, \(\mathbf{E}_{u}\cap\mathbf{E}_{v}=\varnothing\); thus, \(x_{y}\
otin\mathbf{E}_{v}\), contradicting what is known, \(x_{y}\in\mathbf{X}_{v}\subseteq\mathbf{E}_{v}\). Similarly, \(\mathbf{P}_{z}\
eq\mathbf{P}_{w}*\mathbf{P}_{z}\). Therefore, \(x_{y}\in C_{w}\), \(\mathbf{X}_{v}\subseteq\mathbf{C}_{w}\). According to Theorem 4, \(\mathbf{X}_{u\in\mathbf{k}v}=\mathbf{X}_{u}\cup\mathbf{X}_{v}\) satisfies the inequality constraint. (2) Let section \(x_{i}\) be the starting section of \(\mathbf{G}_{i}\). Then, \(\forall x_{r}\in\mathbf{R}\), \(\forall x_{q}\in\mathbf{X}^{\prime}\), according to Theorem 4, \(x_{r}\in C_{q}\). We claim that \(\mathbf{P}_{i}\
eq\mathbf{P}_{i}*\mathbf{P}_{r}\) and \(\mathbf{P}_{r}\
eq\mathbf{P}_{i}*\mathbf{P}_{r}\). If \(\mathbf{P}_{i}=\mathbf{P}_{i}*\mathbf{P}_{r}\), then \(x_{r}\in\mathbf{E}_{i}\), contradicting what is known, \(x_{r}\in\mathbf{R}\subseteq\mathbf{E}-\mathbf{E}_{i}\). If \(\mathbf{P}_{r}=\mathbf{P}_{i}*\mathbf{P}_{r}\), by \(x_{q}\in\mathbf{E}_{i}\),\(\mathbf{P}_{i}=\mathbf{P}_{i}*\mathbf{P}_{q}\), then \(\mathbf{P}_{r}=\mathbf{P}_{i}*\mathbf{P}_{r}=(\mathbf{P}_{i}*\mathbf{P}_{q})*\mathbf{P}_{r}=\mathbf{P}_{q} *(\mathbf{P}_{i}*\mathbf{P}_{r})=\mathbf{P}_{q}*\mathbf{P}_{r}\);thus, \(x_{r}\
otin C_{q}\), contradicting what is known, \(x_{r}\in C_{q}\). Thus, \(\mathbf{P}_{i}\
eq\mathbf{P}_{i}\ast\mathbf{P}_{r}\) and \(\mathbf{P}_{r}\
eq\mathbf{P}_{i}\ast\mathbf{P}_{r}\) hold, i.e., \(x_{r}\in C_{i}\), \(\mathbf{R}\subseteq C_{i}\). Due to \(x_{k},x_{w},x_{y},x_{z}\in\mathbf{E}_{i}\): \(\mathbf{P}_{i}=\mathbf{P}_{i}\ast\mathbf{P}_{k}\), \(\mathbf{P}_{i}=\mathbf{P}_{i}\ast\mathbf{P}_{w}\), \(\mathbf{P}_{i}=\mathbf{P}_{i}\ast\mathbf{P}_{y}\), \(\mathbf{P}_{i}=\mathbf{P}_{i}\ast\mathbf{P}_{z}\), according to Theorem 1, section \(x_{i}\) is on the road of \(x_{k},x_{w},x_{y},x_{z}\). According to Theorem 5, \(C_{i}\subseteq C_{k}\), \(C_{i}\subseteq C_{w}\), \(C_{i}\subseteq C_{z}\). Combined with \(\mathbf{R}\subseteq C_{i}\): \(\mathbf{R}\subseteq C_{k}\), \(\mathbf{R}\subseteq C_{w}\), \(\mathbf{R}\subseteq C_{y}\), \(\mathbf{R}\subseteq C_{z}\). Step 2. Prove that the fault candidate set of \(\mathbf{G}_{i}\) is only one of \(\mathbf{X}_{i-t}\), \(\mathbf{X}_{u}\), \(\mathbf{X}_{v}\), \(\mathbf{X}_{u\mathbf{k}\mathbf{v}}\). Combined with the conclusion of Step 1, the proposition is transformed: if \(\mathbf{X}^{\prime}\
eq\mathbf{X}_{i-t}\), \(\mathbf{X}_{u},\mathbf{X}_{v},\mathbf{X}_{u\mathbf{k}v}\), then one of the following equations must hold: \(f(\mathbf{X}_{i-t}\cup\mathbf{R})<f(\mathbf{X}^{\prime}\cup\mathbf{R})\), \(f(\mathbf{X}_{u}\cup\mathbf{R})<f(\mathbf{X}^{\prime}\cup\mathbf{R})\), \(f(\mathbf{X}_{v}\cup\mathbf{R})<f(\mathbf{X}^{\prime}\cup\mathbf{R})\), or \(f(\mathbf{X}_{u\mathbf{k}\mathbf{v}}\cup\mathbf{R})<f(\mathbf{X}^{\prime}\cup\mathbf{R})\). Because all sections in \(\mathbf{E}_{i-t}\) are on the road of sections in \(\mathbf{E}_{u}\) and \(\mathbf{E}_{v}\), there are four cases of \(\mathbf{X}^{\prime}\) that meet the inequality constraint: \(\mathbf{X}^{\prime}\subseteq\mathbf{E}_{i-t}\), \(\mathbf{X}^{\prime}\subseteq\mathbf{E}_{u}\), \(\mathbf{X}^{\prime}\subseteq\mathbf{E}_{v}\), and \(\mathbf{X}^{\prime}\subseteq\mathbf{E}_{u}\cup\mathbf{E}_{v}\). The proofs are given separately below. 1. \(\mathbf{X}^{\prime}\subseteq\mathbf{E}_{i-t}\). Same procedure as the proof of Theorem 8: the \(\mathbf{X}^{\prime}\) satisfying the inequality constraint contains only one section, and let \(\mathbf{X}^{\prime}=\{x_{q}\}\). Apply Equations (31) and (33) to rewrite \(f(\mathbf{X}_{i-t}\cup\mathbf{R})\): \(f(\mathbf{X}_{i-t}\cup\mathbf{R})=-\Delta_{\mathbf{R}}^{k}-\Delta_{k}+f(\varnothing)\) (the derivation process is similar to Equation (A8)).