# 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
#### 3.2.3 Recursive Case  
For a subnetwork containing a T-type section (e.g., subnetwork \(\mathbf{G}_{3}\) shown in Figure 3), a recursive solution is required. Its fault candidate set is constructed from the fault candidate set of a subnetwork smaller than its size, as described in Theorem 9.  
**Theorem 9**.: _Let the subnetwork \(\mathbf{G}_{i}=(\mathbf{V}_{i},\mathbf{E}_{i})\) contain T-type sections, and let section \(x_{t}\) be the T-type section in \(\mathbf{G}_{i}\) that is closest to the main power supply, assuming that \(xu\) and \(xv\) are the downstream sections of \(x_{t}\), as shown in Figure 4. Let \(\mathbf{G}_{u}=(\mathbf{V}_{u},\mathbf{E}_{u})\)and \(\mathbf{G}_{v}=(\mathbf{V}_{v},\mathbf{E}_{v})\) be subnetworks and set \(\mathbf{E}_{i-t}=\mathbf{E}_{i}-\mathbf{E}_{u}-\mathbf{E}_{v}\). Suppose that for section \(x_{k}\in\mathbf{E}_{i-t}\), and \(\forall x_{q}\
eq x_{k}\in\mathbf{E}_{i-t}\), there is always \(\Delta_{k}>\Delta_{q}\). Let \(\mathbf{X}_{i-t}=\{x_{k}\}\), and let its corresponding approximation gain be \(\Delta_{\mathbf{X}_{i-t}}=\Delta_{k}\). Suppose that \(\mathbf{X}_{u}\) and \(\mathbf{X}_{v}\) are the fault candidate sets for subnetworks \(\mathbf{G}_{u}\) and \(\mathbf{G}_{v}\) respectively, and construct the set \(\mathbf{X}_{u\mathbf{k}v}=\mathbf{X}_{u}\cup\mathbf{X}_{v}\). Let \(\Delta_{\mathbf{X}_{i-t}}\), \(\Delta_{\mathbf{X}_{u}}\), \(\Delta_{\mathbf{X}_{v}}\), and \(\Delta_{\mathbf{X}_{u\mathbf{k}v}}\) be the approximation gains of sets \(\mathbf{X}_{i-t}\), \(\mathbf{X}_{u}\),\(\mathbf{X}_{v}\), and \(\mathbf{X}_{u\mathbf{k}v}\) respectively. Then, the set corresponding to the largest of \(\Delta_{\mathbf{X}_{i-t}}\), \(\Delta_{\mathbf{X}_{u}}\), \(\Delta_{\mathbf{X}_{v}}\), and \(\Delta_{\mathbf{X}_{u\mathbf{k}v}}\) is the fault candidate set \(\mathbf{X}_{i}\) of \(\mathbf{G}_{i}\)._  
The distribution network shown in Figure 3 is taken as an example to illustrate the meaning of Theorem 9. Subnetwork \(\mathbf{G}_{3}\) contains T-type section \(x_{6}\). The fault location result of this network is constructed from the fault candidate sets of subnetworks \(\mathbf{G}_{7}\) and \(\mathbf{G}_{12}\), which are smaller than its size. Let the fault candidate sets of \(\mathbf{G}_{7}\) and \(\mathbf{G}_{12}\) be \(\mathbf{X}_{7}\) and \(\mathbf{X}_{12}\); their corresponding approximation gains are \(\Delta_{\mathbf{X}_{7}}\) and \(\Delta_{\mathbf{X}_{12}}\). Assume that the maximum approximation gain section found in the set of sections \(\mathbf{E}_{3-6}=\mathbf{E}_{3}-\mathbf{E}_{7}-\mathbf{E}_{12}=\{x_{3},x_{4},x_{5},x_{6}\}\) is \(x_{k}\); construct the set \(\mathbf{X}_{3-6}=\{x_{k}\}\), and its corresponding approximation gain is \(\Delta_{\mathbf{X}_{3-6}}=\Delta_{k}\). The set \(\mathbf{X}_{7k12}=\mathbf{X}_{7}\cup\mathbf{X}_{12}\) is constructed, and its corresponding approximation gain is \(\Delta_{\mathbf{X}_{7k12}}\). Then, the set corresponding to the largest of \(\Delta_{\mathbf{X}_{3-6}}\), \(\Delta_{\mathbf{X}_{7}}\), \(\Delta_{\mathbf{X}_{12}}\), and \(\Delta_{\mathbf{X}_{7k12}}\) is the fault location result of \(\mathbf{G}_{3}\).  
The proof of Theorem 9 is given in Appendix D.