# A Fault Section Location Method for Distribution Networks Based on Divide-and-Conquer
## 3 Recursive Structure of Fault Location for the Distribution Network
### Theoretical Foundation
#### 3.1.2 Approximation Gains  
Let the set of sections in the distribution network be \(\mathbf{E}\), \(\mathbf{X}\subseteq\mathbf{E}\) be the set of fault sections, and section \(x_{i}\
otin\mathbf{X}\). The approximation gain of section \(x_{i}\) to \(\mathbf{X}\) describes the contribution to reducing the objective function \(f(\mathbf{X})\) after section \(x_{i}\) is added to \(\mathbf{X}\) (that is, section \(x_{i}\) is determined to be the fault section). The approximate gain of section \(x_{i}\) to set \(\mathbf{X}\) is shown in Equation (29).  
\[\Delta_{i}^{\mathbf{X}}=f(\mathbf{X})-f(\mathbf{X}\cup x_{i}) \tag{29}\]  
By expanding \(f(\mathbf{X})\) and \(f(\mathbf{X}\cup x_{i})\) with Equation (26) and substituting Equation (19) for \(\mathbf{J}(\mathbf{X})\), the equation for \(\Delta_{i}^{\mathbf{X}}\) can be deduced:  
\[\Delta_{i}^{\mathbf{X}}=\left[(2\mathbf{I}-\mathbf{e})*\prod_{x_{k}\in\mathbf{X}}^{*}(\mathbf{e}- \mathbf{P}_{k})\right]^{\mathbf{T}}\mathbf{P}_{i} \tag{30}\]  
In particular, if \(\mathbf{X}\) is the empty set, i.e., \(\mathbf{X}=\varnothing\), then the superscript of \(\Delta_{i}^{\varnothing}\) can be omitted and abbreviated as \(\Delta_{i}\), which is referred to as the approximation gain of section \(x_{i}\), as shown in Equation (31).  
\[\Delta_{i}=f(\varnothing)-f(i) \tag{31}\]  
Substitute \(\prod\limits_{x_{k}\in\varnothing}^{*}(\mathbf{e}-\mathbf{P}_{k})=\mathbf{e}\) into Equation (30) to obtain the calculation formula of \(\Delta_{i}\).  
\[\Delta_{i}=(2\mathbf{I}-\mathbf{e})^{\mathbf{T}}\mathbf{P}_{i} \tag{32}\]  
Extending the concept of section approximation gain, the approximation gain of the set of sections is defined. Let sets of sections \(\mathbf{X},\mathbf{X}_{a}\subseteq\mathbf{E}\) be such that \(\mathbf{X}\cap\mathbf{X}_{a}=\varnothing\). The approximation gain of \(\mathbf{X}\) to \(\mathbf{X}_{a}\) describes the contribution to reducing the objective function \(f(\mathbf{X}_{a})\) after adding all sections in \(\mathbf{X}\) to \(\mathbf{X}_{a}\). The approximation gain of \(\mathbf{X}\) to set \(\mathbf{X}_{a}\) is shown in Equation (33).  
\[\Delta_{\mathbf{X}}^{\mathbf{X}_{a}}=f(\mathbf{X}_{a})-f(\mathbf{X}_{a}\cup\mathbf{X}) \tag{33}\]In particular, if \(\mathbf{X}_{a}\) is the empty set, i.e., \(\mathbf{X}_{a}=\varnothing\), then the superscript of \(\Delta_{\mathbf{X}}^{\varnothing}\) can be omitted and abbreviated as \(\Delta_{\mathbf{X}}\), which is referred to as the approximation gain of the set of sections \(\mathbf{X}\).  
The property of the approximation gains is the basis of the argument for the recursive structure. These properties are stated first, and their mathematical proofs are given in Appendix C.  
Theorem 3.2: _Let the set of sections in the distribution network be \(\mathbf{E}\). For section \(x_{i}\in\mathbf{E}\), set of sections \(\mathbf{X}\subseteq\{x_{k}\in\mathbf{E}|C_{i}\subseteq C_{k}\}\), \(\mathbf{X}\
eq\varnothing\), and set \(\mathbf{R}\subseteq C_{i}\), we have \(\Delta_{\mathbf{R}}^{\mathbf{X}}=\Delta_{\mathbf{R}}^{i}\)._