# 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
### Active Distribution Network Application Note  
For active distribution networks containing distributed generations (DGs), the short-circuit current after a fault will flow in both directions, and then, the encoding of fault information described in Equation (1) would no longer applicable. In this paper, we refer to the sections that reach the substation power supply through node \(s_{i}\) as the downstream sections of node \(s_{i}\) and the remaining sections in the distribution network as the upstream sections of node \(s_{i}\). The fault information of node \(s_{i}\) is determined by Equation (35) [33]. \[I_{i}(\mathbf{X})=\left\{\prod_{\mathbf{G}_{u}}\left[K_{\mathbf{G}_{u}}\left(1-\prod_{x_{ \mathbf{G}_{u}}\in\mathbf{E}_{i,\mathbf{G}_{u}}}y_{\mathbf{G}_{u}}\right)\right]\right\}\prod_{ x_{\mathbf{G}}\in\mathbf{E}_{i,\mathbf{G}_{u}}}y_{\mathbf{d}}-\left\{\prod_{\mathbf{G}_{u}} \left[K_{\mathbf{G}_{d}}\left(1-\prod_{x_{\mathbf{G}_{d}}\in\mathbf{E}_{i,\mathbf{G}_{d}}}y_{ \mathbf{G}_{d}}\right)\right]\right\}\prod_{y_{\mathbf{x}}\in\mathbf{E}_{i,\mathbf{G}_{u}}}y_{ \mathbf{u}} \tag{35}\]  
where \(G_{u}\) denotes the power source connected to the upstream section of node \(s_{i}\), which is referred to as the upstream power source of node \(s_{i}\). \(K_{\mathbf{G}_{u}}\) denotes the throw-off factor of \(G_{u}\), when \(G_{u}\) is put into operation \(K_{\mathbf{G}_{u}}=1\); otherwise, \(K_{\mathbf{G}_{u}}=0\). \(y_{\mathbf{G}u}\) denotes the operating state of the feeder section \(x_{\mathbf{G}u}\) between \(s_{i}\) and \(G_{u}\). In contrast, \(G_{d}\), \(K_{\mathbf{G}_{d}}\), and \(y_{\mathbf{G}d}\) denote the variables associated with the downstream power supply of node \(s_{i}\). \(G_{d}\) denotes the downstream power supply of \(s_{i}\), \(K_{\mathbf{G}_{d}}\) denotes the throw-off factor of \(G_{d}\), and \(y_{\mathbf{G}d}\) denotes the operating state of the feeder section \(x_{\mathbf{G}d}\) between \(s_{i}\) and \(G_{d}\). In addition, \(y_{d}\) and \(y_{u}\) indicate the operation status of the downstream section of and the upstream section of \(s_{i}\), respectively, and \"\(\prod\)\" indicates the logical \"OR\" superposition operation. In Equation (35), it can be seen that each grid-connected power source will only provide fault current in a single direction to the node after a distribution network fault, and the fault state of the node can be regarded as the superposition of the states under the action of each power source individually. Therefore, the coding rule of node \(s_{i}\) to grid-connected power sources \(S\) is proposed as  
\[\mathbf{I}_{i,S}= \begin{cases}1&\text{Short circuit current is detected in the same direction}\\ &\text{as the short circuit current supplied by power supply S}\\ 0&\text{Others}\end{cases}  \tag{36}\]  
Meanwhile, because the road from the section to each power source is unique, the distribution network with each power source as the main power source has a radial structure. In view of this, the following fault section location process for an active distribution network based on DAC is proposed. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Steps** & **Number of Executions** & **Operation Cost** & \(\mathbf{f}\)**(X)** \\ \hline Lines 1–2 & \(l\) & \(\Theta(1)\) & \(\Theta(l)\) \\ Lines 3–5 & \(n\) & \(\Theta(1)\) & \(\Theta(n)\) \\ Line 6–7 & \(l\) & \(\Theta(1)\) & \(\Theta(l)\) \\ Lines 8, 21 & \(<\)\(l\) & \(\Theta(j\)-\(i)\) or \(\Theta(k\)-\(i)\) & \(\Theta(n)\) \\ Line 9 & \(l\) & \(\Theta(1)\) & \(\Theta(l)\) \\ Lines 10–20 & \(<\)\(l\) & \(\Theta(1)\) & \(\Theta(l)\) \\ Lines 22–32 & \(<\)\(l\) & \(\Theta(1)\) & \(\Theta(l)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Efficiency analysis. Step 1: Store the network topology information (including node section number, adjacency list, and section type vector \(T\)) of the distribution network with each power source as the main power source.