## 5 Simulation Verification
### Comparison of Different Algorithms  
Taking the single-phase grounding fault at \(l_{3}\) of the cable line in Section 5.1 as an example, the proposed method is compared with the VMD-TEO calibration method [23] and the EMD_first-order differential calibration method [24]. The algorithm time is shown in Table 10, and the positioning results are as follows:  
Figure 16: Noisy zero-mode traveling wave characteristic signal after morphological gradient processing. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Noise/dB** & \(\mathbf{d_{u}}\)**/**km** & \(\mathbf{d_{m}}\)**/**km** & \(\mathbf{\Delta}\)**/m** \\ \hline \multirow{2}{*}{0} & 0.4 & 0.414 & 14 \\  & 3.2 & 3.108 & 92 \\  & 5.2 & 5.181 & 19 \\ \hline \multirow{2}{*}{40} & 0.4 & 0.414 & 14 \\  & 3.2 & 3.108 & 92 \\  & 5.2 & 5.181 & 19 \\ \hline \multirow{2}{*}{30} & 0.4 & 0.622 & 222 \\  & 3.2 & 3.108 & 92 \\  & 5.2 & 5.388 & 188 \\ \hline \multirow{2}{*}{20} & 0.4 & 0.622 & 222 \\  & 3.2 & 3.316 & 116 \\ \cline{1-1}  & 5.2 & 5.388 & 188 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Fault location results under different noise interferences. (1) VMD-TEO calibration method  
Based on the principle of two-terminal ranging, VMD is used to decompose the two-terminal line-mode traveling wave. The number of modes and the penalty factor are set to 4 and 2000 according to human experience. The arrival time of the first IMF component is calibrated by TEO. The results of the double-end calibration are shown in Figures 17 and 18. The fault distance is calculated by using the traditional double-end ranging formula. The positioning result is 5.154 km and the absolute error is 36 m.  
(2) EMD_first-order difference calibration method  
Based on the principle of single-end wave velocity difference range, EMD is used to adaptively decompose line-mode traveling wave and zero-mode traveling wave, and the first-order difference algorithm is used to calibrate the arrival time of the highest frequency IMF component. The calibration results are shown in Figures 19 and 20. The ranging \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{3}{c}{**Time/s**} \\ \cline{2-4}  & **VMD-TEO** & **EMD\_First-Order Difference** & **MVMD-Morphological Gradient** \\ \hline \multirow{2}{*}{signal preprocessing} & parameter optimization & 0 & 0 & 1.2503 \\  & mode decomposition & 3.1632 & 0.4051 & 0.9104 \\ \hline \multirow{2}{*}{traveling wave head calibration} & 0.0054 & 0.0036 & 0.0106 \\  & total time & 3.1686 & 0.4087 & 2.1713 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Times of different algorithms. Figure 17: Arrival time of M section line-mode is calibrated by VMD-TEO method. Figure 18: Arrival time of N section line-mode is calibrated by VMD-TEO method. distance is 5.388 km, and the absolute error is 188 m calculated by using the single-end mode velocity difference ranging formula. The analysis of the positioning results is as follows:  
1. VMD-TEO sets the number of decomposition modes and penalty factors according to human experience, which fails to characterize the singularity of traveling wave signals to the greatest extent. In addition, the reliability of the method of selecting the highest frequency IMF component for wave head calibration is low. When the fault occurs at the far end of the line, the attenuation of the high frequency component is serious during the propagation process, and the noise interference is easily doped into the wave head. The singularity of the wave head in the high frequency component is further weakened, which has a great influence on the result of fault location. 2. Although the EMD_first-order difference calibration method adaptively decomposes the traveling wave signal and overcomes the defect of artificially setting the decomposition parameters, the modal aliasing phenomenon is more serious, and the method of directly selecting the highest frequency IMF component for wave head calibration also has the problem of low reliability.