# 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
### Simulation Verification
#### 4.4.3 T-T Failure  
In the case of a T-T failure, the time for the line mode components to reach each endpoint is shown in Table 6, and a fault branch judgment matrix is constructed.  
The fault branch judgment matrix can be constructed according to Equation (11)  
\[\begin{bmatrix}\text{M}_{1}&\text{N}_{1}&\text{N}_{2}&\text{N}_{3}&\text{M}_{2 }\\ \text{M}_{1}&\times&0&<0&>0&>0\\ \text{N}_{1}&0&\times&<0&<0&<0\\ \text{N}_{2}&<0&<0&\times&<0&<0\\ \text{N}_{3}&<0&<0&0&\times&0\\ \text{M}_{2}&<0&<0&0&0&\times\end{bmatrix} \tag{36}\]  
From Equation (36), the elements of the rows corresponding to N2 are all less than zero. According to the T-node fault determination proposed above, only the first element of the N1 row is zero, so the fault point can be determined between T1 and T2.  
Calculate the fault distance using the N2 end as the initial end.  
\[d_{N_{2}F}=\frac{(110.012+110.074+110.087)}{3}=110.058\;km \tag{37}\]  
In order to further verify the accuracy of the distance measurement method proposed in this paper, a comparison was made using the technique in this paper and the method in [PERSON] et al. [20]. The results of the comparison are shown in Tables 7 and 8.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Measurement Point Name** & \(\mathbf{M_{1}}\) & \(\mathbf{N_{1}}\) & \(\mathbf{N_{2}}\) & \(\mathbf{N_{3}}\) & \(\mathbf{M_{2}}\) \\ \hline \(\alpha\)-mode & 864 & 956 & 1047 & 1123 & 1187 \\ \(\beta\)-mode & 793 & 902 & 1103 & 1082 & 1097 \\ \hline \end{tabular}
\end{table}
Table 6: Sampling numbers that the initial wave arrives at each measurement when the fault occurs on a T node.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fault Point**} & \multirow{2}{*}{**Fault Type**} & \multirow{2}{*}{**Beginning**} & \multicolumn{2}{c}{**Actual**} & \multicolumn{2}{c}{**Calculation**} & \multirow{2}{*}{**Positioning**} \\  & & & \multicolumn{2}{c}{**Failure**} & \multicolumn{2}{c}{**of Faults**} & \multicolumn{2}{c}{**Error/m**} \\  & & & **Distance/km** & & **Distance/km** & \\ \hline \multirow{3}{*}{N2-T2} & Bg & N2 & 40 & 40.134 & 134 \\  & ABg & N2 & 40 & 40.087 & 87 \\  & BC & N2 & 40 & 39.921 & 79 \\  & ABC & N2 & 40 & 40.104 & 104 \\ \hline \multirow{3}{*}{T2 node} & Bg & N2 & 100 & 100.094 & 94 \\  & ABg & N2 & 100 & 99.961 & 39 \\  & BC & N2 & 100 & 100.145 & 145 \\  & ABC & N2 & 100 & 100.109 & 109 \\ \hline \multirow{3}{*}{T1-T2} & Bg & N2 & 110 & 110.036 & 36 \\  & ABg & N2 & 110 & 110.127 & 127 \\ \cline{1-1}  & BC & N2 & 110 & 109.934 & 66 \\ \cline{1-1}  & ABC & N2 & 110 & 109.947 & 53 \\ \hline \hline \end{tabular}
\end{table}
Table 7: The results of the method presented in this paper are obtained under different experimental conditions.  
Due to space constraints, the results of all comparative experiments are not listed in this paper. However, the comparison results show that the method proposed in this paper is superior to the method by [PERSON] et al. (2018) in terms of distance measurement error, fault branch judgment, and calculation speed. It has a high positioning accuracy and is suitable for use in actual working conditions.