## 5 Simulation Verification
### The Influence of Initial Fault Conditions  
To evaluate the proposed method's robustness and accuracy, comprehensive testing was conducted across varying parameters: initial fault phase angle \(\delta\), transition resistance \(R\), and fault distance \(d_{\text{a}}\). Here, \(d_{\text{a}}\) represents the actual distance between the fault location and the primary terminal, \(d_{\text{m}}\) denotes the calculated fault distance using our proposed method, \(\Delta\) indicates the absolute error margin, while \(t_{1}\) and \(t_{0}\) represent the arrival times of linear mode and zero-mode traveling wave heads, respectively. In order to verify the influence of the neutral point grounding method on the positioning result, the cable line \(l_{3}\) and the hybrid line \(l_{5}\) are set to have a single-phase ground fault under the neutral point ungrounded mode and the neutral point grounded through the arc suppression coil. The positioning results are shown in Table 4.  
In order to verify the influence of fault type on the positioning results, different types of faults are set in cable line \(l_{3}\) and mixed line \(l_{5}\), and the positioning results are shown in Table 5.  
To assess the impact of initial fault phase angle \(\delta\), single-phase ground faults were simulated at two locations: 1 km from section M on line \(l_{3}\) and 5 km on line \(l_{5}\). The tests were conducted at four distinct phase angles (10\({}^{\circ}\), 30\({}^{\circ}\), 60\({}^{\circ}\), and 90\({}^{\circ}\)) with a constant transition resistance of 10 \(\Omega\). The results are documented in Table 6. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(d_{\text{a}}\)/km & \(\delta\)/(\({}^{\circ}\)) & \(t_{1}\)/ms & \(t_{0}\)/ms & \(d_{\text{m}}\)/km & \(\Delta\)/m \\ \hline \(l_{3}\)-1 km & 10 & 120.569 & 120.574 & 1.036 & 36 \\ \(l_{3}\)-1 km & 30 & 121.680 & 121.685 & 1.036 & 36 \\ \(l_{3}\)-1 km & 60 & 123.347 & 123.352 & 1.036 & 36 \\ \(l_{3}\)-1 km & 90 & 125.013 & 125.018 & 1.036 & 36 \\ \(l_{5}\)-5 km & 10 & 120.583 & 120.597 & 4.989 & 11 \\ \(l_{5}\)-5 km & 30 & 121.694 & 121.708 & 4.989 & 11 \\ \(l_{5}\)-5 km & 60 & 123.361 & 123.375 & 4.989 & 11 \\ \(l_{5}\)-5 km & 90 & 125.027 & 125.041 & 4.989 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Fault location results under different fault initial phase angle \(\delta\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Neutal Point Treatment** & \(d_{\text{a}}\)/km & \(\delta\)/(\({}^{\circ}\)) & \(R\)/\(\Omega\) & \(t_{1}\)/ms & \(t_{0}\)/ms & \(d_{\text{m}}\)/km & \(\Delta\)/m \\ \hline neutral point ungrounded & \(l_{3}\)-5 km & 30 & 100 & 121.728 & 121.752 & 4.974 & 26 \\ neutral point ungrounded & \(l_{3}\)-5 km & 60 & 10 & 123.395 & 123.419 & 4.974 & 26 \\ neutral point ungrounded & \(l_{5}\)-3 km & 30 & 100 & 121.685 & 121.693 & 2.922 & 88 \\ neutral point ungrounded & \(l_{5}\)-3 km & 60 & 10 & 123.351 & 123.359 & 2.922 & 88 \\ neutral earthing via arc extinguishing coil & \(l_{3}\)-5 km & 30 & 100 & 121.728 & 121.752 & 4.974 & 26 \\ neutral earthing via arc extinguishing coil & \(l_{5}\)-3 km & 60 & 10 & 123.395 & 123.419 & 4.974 & 26 \\ neutral earthing via arc extinguishing coil & \(l_{5}\)-3 km & 30 & 100 & 121.685 & 121.693 & 2.922 & 88 \\ neutral earthing via arc extinguishing coil & \(l_{5}\)-3 km & 60 & 10 & 123.351 & 123.359 & 2.922 & 88 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fault location results under different neutral grounding modes. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Fault Type** & \(d_{\text{a}}\)/km & \(\delta\)/(\({}^{\circ}\)) & \(R\)/\(\Omega\) & \(t_{1}\)/ms & \(t_{0}\)/ms & \(d_{\text{m}}\)/km & \(\Delta\)/m \\ \hline A-G & \(l_{3}\)-5 km & 30 & 100 & 121.728 & 121.752 & 4.974 & 26 \\ A-G & \(l_{5}\)-3 km & 30 & 100 &The influence of fault transition resistance \(R\) was evaluated by introducing single-phase ground faults at 0.4 km from section M on line \(l_{5}\) and 5.4 km from line \(l_{3}\).