# A Fault Section Location Method for Distribution Networks Based on Divide-and-Conquer
## 2 Numerical Computational Model for Distribution Grid Fault Location
### Basic Principle of Distribution Network Fault Location
#### 2.2.4 Numerical Computational Expression for the Minimum Fault Set Constraint  
In Equation (6) of the objective function calculation for fault location, we introduced a second term to the right side of the equation. This is due to the possibility that the minimum value of the first term of Equation (6) corresponds to multiple assumed fault scenarios. Taking the distribution network shown in Figure 2 as an example, assuming section \(x_{3}\) faults, the fault information reported by the FTUs is \(I=[I_{1}\!-\!I_{6}]=[1\ 1\ 1\ 0\ 0\ 0]\). For the assumed fault scenarios \(\mathbf{X}_{1}=[x_{3}]\), \(\mathbf{X}_{2}=[x_{1},x_{3}]\), \(\mathbf{X}_{3}=\{x_{2},x_{3}\}\), \(\mathbf{X}_{4}=\{x_{1},x_{2},x_{3}\}\), according to Equation (19) of the switching function, we can obtain \(\mathbf{J(X_{1})}=\mathbf{J(X_{2})}=\mathbf{J(X_{3})}=\mathbf{J(X_{4})}=[1\ 1\ 1\ 0\ 0\ 0]=\mathbf{I}\) (the same results can be obtained by applying the logical expression (10) of the switching function because the two are equivalent), substituting the first term of Equation (6): \(\sum\limits_{i=1}^{n}[\mathbf{I}_{i}\mathbf{\oplus}\mathbf{J}_{i}(\mathbf{X})]=\ 0\). That is, the assumed fault scenarios \(\mathbf{X}_{1}\!\sim\!\mathbf{X}_{4}\) all lead to the minimum value of the first term of Equation (6).  
This phenomenon has been discussed in more depth in [18], and the theory of \"minimum fault set\" has been proposed to solve this problem. The meaning of the \"minimum fault set\" theory is that the fault scenario with the lowest number of fault sections has the highest probability of occurrence. The mathematical description is as follows: for the set of fault sections \(\mathbf{X}\), if there exists a proper subset \(\mathbf{X}^{\prime}\) of \(\mathbf{X}\) such that \(\mathbf{J(X)}=\mathbf{J(X^{\prime})}\), then \(\mathbf{X}\) is determined to be an infeasible solution that does not satisfy the \"minimum fault set\"  
The two preceding and following terms on the right-hand side of Equation (23) are both 0-1 vectors; thus, they are both 0 vectors by adding them to equal the 0 vector. According to Equation (14), \([e-J(X_{a})]=\prod\limits_{x_{k}\in X_{a}}^{*}(e-P_{k})\), substitute the latter term on the right-hand side of Equation (23):  
\[\begin{array}{ll}\boldsymbol{P}_{i}*[\boldsymbol{e}-\boldsymbol{J}(X_{a})]* \prod\limits_{x_{k}\in X_{c}}^{*}(\boldsymbol{e}-\boldsymbol{P}_{k})&= \boldsymbol{P}_{i}*\prod\limits_{x_{k}\in X_{a}}^{*}(\boldsymbol{e}- \boldsymbol{P}_{k})*\prod\limits_{x_{k}\in X_{c}}^{*}(\boldsymbol{e}- \boldsymbol{P}_{k})\\ &=\boldsymbol{P}_{i}*\prod\limits_{x_{k}\in X_{a}\cup X_{c}}^{*}(\boldsymbol{ e}-\boldsymbol{P}_{k})=0\end{array} \tag{24}\]  
According to Theorem 3, \(\exists x_{j}\in\boldsymbol{X}_{a}\cup X_{c}\) such that \(\boldsymbol{P}_{i}=\boldsymbol{P}_{i}*\boldsymbol{P}_{j}\), where \(x_{i}\in\boldsymbol{X}_{b}\subseteq\boldsymbol{X}\), \(x_{j}\in\boldsymbol{X}_{a}\cup\boldsymbol{X}_{c}\subseteq\boldsymbol{X}\), contradicting the assumption that \(\forall x_{i},x_{j}\in\boldsymbol{X}\), \(\boldsymbol{P}_{i}\
eq\boldsymbol{P}_{i}*\boldsymbol{P}_{j}\).  
This proves that this inference holds.