## 6 Tree Flash Fault Location Simulation Analysis
Figure 15: The VMD-Hilbert transform results of the zero-mode component at measurement terminal M.  
Similarly, it can be inferred that the zero-mode component at measurement terminal N and its VMD decomposition results are shown in Figure 16. The VMD-Hilbert transform results are depicted in Figure 17, indicating that the time for the zero-mode component to reach terminal N is 255.9 \(\upmu\)s.  
The line-mode component at measurement terminal M and its VMD decomposition results are shown in Figure 18. The VMD-Hilbert transform results are illustrated in Figure 19, indicating that the time for the line-mode component to reach terminal M is 163.4 \(\upmu\)s.  
The line-mode component at measurement terminal N and its VMD decomposition results are shown in Figure 20. The VMD-Hilbert transform results are depicted in Figure 21, indicating that the time for the line-mode component to reach terminal N is 202.3 \(\upmu\)s.  
Figure 16: The zero-mode signal at measurement terminal N and its VMD decomposition results. Figure 17: The VMD-Hilbert transform results of the zero-mode component at measurement terminal N.  
Figure 19: The VMD-Hilbert transform results of the line-mode component at measurement terminal M.  
Figure 14.0 The line-mode component at measurement terminal N and its VMD decomposition results. Substituting the four time data into Equation (11):  
Figure 14.1  
It can be observed that the deviation from the actual tree flash fault location, which is 38 km away from terminal \(M\), is 97 m. This meets the requirement for precise localization of the tree flash fault point. Based on the calculation formula of the propagation speed of tree flash fault traveling waves on overhead lines and cables derived from the hybrid line model's parameters of inductance, capacitance, and the wave velocity of transient faults, the propagation speed of the tree flash fault traveling wave on the overhead line is denoted as \(V_{J}=2.92128046\times 10^{8}\) m/s, and on the cable as \(V_{D}=1.47046104\times 10^{8}\) m/s. Since the lengths of the overhead line and cable of the hybrid line and the values of the wave velocity are known, according to the formula \(t=l/v\), the travel times of the tree flash fault traveling wave on the overhead line segments ML\({}_{1}\) and L\({}_{2}\)N can be, respectively, calculated as \(t_{a}=6.846312867\times 10^{-5}\) s, and on the cable segment L\({}_{1}\)L\({}_{2}\), they can be calculated as \(t_{b}=6.800588202\times 10^{-5}\) s.  
Using Equation (12), the distance of the tree flash fault point from terminal \(M\) is calculated to be 37.867 km, with an error of 133 m. By comparing the two methods, it can be concluded that the wave-based positioning method, independent of wave speed, exhibits higher positioning accuracy and better meets the requirement for precise localization of tree flash faults. Figure 14.2  
At positions 20 km, 25 km, 30 km, 35 km, and 45 km from terminal M, simulation experiments were conducted to introduce tree flash faults. The localization of the tree flash fault was performed using both the method combining VMD decomposition and Hilbert transform and the formula-based wave-speed-based double-end positioning method. The calculated results of the tree flash fault point positions are presented in Tables 4 and 5. By analyzing the data in the table, it can be concluded that the tree flash fault localization method utilizing VMD decomposition and Hilbert transform is effective in pinpointing tree flash fault points. Moreover, as the occurrence position of the tree flash fault point moves closer to terminals M and N, the localization error shows a trend of gradual increase. Figure 14.3 The VMD-Hilbert transform results of the line-mode component at measurement terminal N.  
Capturing phase currents may be more practically feasible in real-world applications, and it is a more common practice to collect phase currents for fault extraction, especially in the presence of loaded distributed transformers.