# Fault Location Method of Multi-Terminal Transmission Line Based on Fault Branch Judgment Matrix
## 4 Research on Multi-Branch Line Fault Ranging Technology Based on Fault Branch Judgment Matrix
### T-Type Transmission Line Fault Branch Judgment Method
#### 4.1.1 Fault Occurred on The Branch Line  
As shown in Figure 4, the fault occurs on M1-M2 and the fault distances of branches M1-M2 and M1-N1 with M1 as the starting point are calculated according to (11).  
\[d_{M_{1}FM_{2}}=\frac{t_{M11}-t_{M12}}{t_{M11}-t_{M12}+t_{M21}-t_{M22}}L_{M_{1} M_{2}}=L_{M_{1}\bar{T}_{1}}+d_{T_{1}\bar{F}} \tag{24}\]  
\[d_{M_{1}FN_{1}}=\frac{t_{M_{11}}-t_{M_{12}}}{t_{M_{11}}-t_{M_{12}}+t_{N_{11}}-t _{N_{12}}}L_{M_{1}N_{1}}=L_{M_{1}\bar{T}_{1}} \tag{25}\]  
Similarly, we can calculate the fault distance starting from M2 as \(d_{M_{1}FM_{2}}\) and \(d_{M_{1}FN_{1}}\), and starting from N1 as \(d_{N_{1}FM_{1}}\) and \(d_{N_{1}FM_{2}}\).  
To further determine the location of the fault on the branch line, the calculated fault distance and the distance from the beginning of the branch line to the T node are used to construct a fault branch line judgment matrix, where \(\times\) represents elements that are not considered.  
The fault branch judgment matrix constructed by modeling the T-shaped transmission line in Figure 4 is given in Equation (26).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Fault Type**} & \multicolumn{2}{c}{**In This**} & \multirow{2}{*}{**Positioning**} & \multirow{2}{*}{**Relative**} & \multirow{2}{*}{**Residence**[19]**} & \multirow{2}{*}{**Positioning**} & \multirow{2}{*}{**Relative**} \\  & **Positioning** & & & & & **Positioning** \\  & **Results/km** & & & & & **Error/\%** \\ \hline Ag & 80.124 & 124 & 0.155 & 79.703 & 297 & 0.371 \\ AB & 79.824 & 176 & 0.220 & 80.367 & 367 & 0.459 \\ BC & 80.192 & 192 & 0.240 & 80.344 & 344 & 0.430 \\ ABg & 80.137 & 137 & 0.171 & 79.641 & 359 & 0.449 \\ BCg & 79.845 & 155 & 0.194 & 79.639 & 361 & 0.451 \\ ABC & 80.131 & 131 & 0.163 & 80.369 & 369 & 0.461 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Location error under different fault types.  
Figure 4: T transmission line.  
\[\begin{bmatrix}&M_{1}&N_{1}&M_{2}\\ M_{1}&\times&d_{M_{1}FN_{1}}-L_{M_{1}T_{1}}&d_{M_{1}FM_{2}}-L_{M_{1}T_{1}}\\ N_{1}&d_{N_{1}FM_{1}}-L_{N_{1}T_{1}}&\times&d_{N_{1}FM_{2}}-L_{N_{1}T_{1}}\\ M_{2}&d_{M_{2}FM_{1}}-L_{M_{2}T_{1}}&d_{M_{2}FN_{1}}-L_{M_{2}T_{1}}&\times\end{bmatrix} \tag{26}\]  
when the fault occurs on \(M_{1}-M_{2}\), \(d_{M_{1}FN_{1}}-L_{M_{1}T_{1}}=0\), \(d_{M_{1}FM_{2}}-L_{M_{1}T_{1}}>0\), \(d_{N_{1}FM_{1}}-L_{N_{1}T_{1}}=0\), \(d_{N_{1}FM_{2}}-L_{N_{1}T_{1}}>0\), \(d_{M_{2}FM_{1}}-L_{M_{2}T_{1}}<0\), \(d_{M_{2}FN_{1}}-L_{M_{2}T_{1}}<0\), therefore, the fault branch judgment matrix is  
\[\begin{bmatrix}&M_{1}&N_{1}&M_{2}\\ M_{1}&\times&0&>0\\ N_{1}&0&\times&>0\\ M_{2}&<0&<0&\times\end{bmatrix} \tag{27}\]  
As can be seen from the fault branch judgment matrix (27), the elements in the row corresponding to the M2 endpoint are all \(<\)0, while the elements in the column corresponding to the M2 endpoint are all \(>\)0. Therefore, the principle for determining the fault branch on a T-type transmission line is as follows: when the elements in the row of a fault branch judgment matrix corresponding to an endpoint are all \(<\)0 and the elements in the column corresponding to the same endpoint are all \(>\)0, then the endpoint corresponds to the fault branch.