## 3 Three-Phase Unbalanced Distribution Network Fault Location Algorithm
### Fault Location Algorithm  
(1) _Fault location start criterion_  
In this paper, the mutation detection algorithm [30] will be used to determine the moment when the fault occurs. For the voltage \(U_{M}\) collected by the intelligent terminal on the low-voltage side of the distribution transformer and at the key nodes of the system, when the voltage sampling value change \(\Delta U_{M}\) in the adjacent cycle is set to the limit three times in a row, it is considered that the previous sampling that exceeded the limit for the first time is the time of failure. The out-of-limit expression is:  
\[|\Delta U_{M}(k+N)-\Delta U_{M}(k)|\geq k_{f}U_{dz} \tag{10}\]  
where \(k_{f}\) is the constant value adjustment coefficient of mutation (usually \(k_{f}\) = 1); \(U_{dz}\) is the starting value of mutation. (2) _Fault location and fault phase identification criteria_  
In case the line parameters and topology of the distribution network are known, the fault state is used as the variable to be estimated to construct the measurement equation:  
\[z^{\prime}=H(x)+v \tag{11}\]  
where \(z^{\prime}\) is the fault measurement vector and \(x\) is the fault state vector. When a short-circuit fault occurs in the distribution network, it can be seen from the three-phase unbalanced distribution network operation model that the changes in node voltage and current due to the fault satisfy the following equation:  
\[\Delta I^{\varphi}=Y^{\varphi}\Delta U^{\varphi} \tag{12}\]  
where \(\Delta I^{\varphi}\) is the node equivalent three-phase fault current and \(\Delta U^{\varphi}\) is the node voltage variation. Reversing Equation (11), take the fault equivalent current as the state quantity, and take the node voltage measurement as the quantity measurement:  
\[ \begin{bmatrix}\Delta U^{\varphi}\\ \Delta U^{\varphi}_{2}\\ \vdots\\ \Delta U^{\varphi}_{m}\end{bmatrix} =H\cdot\Delta I^{\varphi} \tag{13}\]  
where \(m\) is the number of intelligent terminal devices in the distribution network. Due to the existence of DG in the distribution network, considering that it is difficult to estimate its effect on the fault current during the fault period, the voltage change value of \begin{table}
\begin{tabular}{c c c} \hline \hline
**MV Side Fault Type** & \begin{tabular}{c} **MV Side Voltage** \\ **Characteristics** \\ \end{tabular} & \begin{tabular}{c} **Whether LV Side Has Same Fault** \\ **Characteristics as MV Side** \\ \end{tabular} \\ \hline Single-phase-to-ground & The voltage of fault phase is lowered and that of the & No \\  & \begin{tabular}{c} non-fault phase is raised \\ \end{tabular} & No \\  & \
\begin{tabular}{c} The voltage of fault phase is lowered and that of the \\ non-fault phase is raised \\ \end{tabular} & No \\  &
\begin{tabular}{c} The voltages of all the phases are raised \\ \end{tabular} & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Propagation effect of the transformer on fault characteristics. the node where it is located is taken as the fault state quantity, and Equation (13) is further transformed into:  
\[ \begin{bmatrix}\Delta U_{1}^{\theta}\\ \Delta U_{2}^{\theta}\\ \vdots\\ \Delta U_{m}^{\theta}\end{bmatrix} =H^{\prime}\cdot \begin{bmatrix}\Delta U_{DG1}^ {\theta}\\ \Delta U_{DG2}^{\theta}\\ \vdots\\ \Delta U_{DG\ s}^{\theta}\\ \Delta I^{\varphi}\end{bmatrix}  \tag{14}\]  
where \(s\) is the number of DGs in the distribution network and \(\Delta I^{\theta^{\prime}}\) is the fault injection current of \(n\)-\(s\) nodes.