## 5 Simulation Verification
### The Influence of Line Type  
(1) A test scenario was configured with a single-phase ground fault occurring 5.2 km from section M on cable line \(l_{3}\) at 0.135 s, using parameters of 10 \(\Omega\) grounding resistance and 270\({}^{\circ}\) initial fault phase angle.  
Initially, MPA was employed to optimize MVMD core parameters using Equation (16), guided by the minimum comprehensive entropy kurtosis ratio principle for both wave modes. To validate MPA's effectiveness, PSO and the Sparrow Search Algorithm (SSA) are used for comparative analysis, averaging results across 10 experimental iterations. To ensure fair comparison, consistent parameters were applied across all three algorithms: population size of 30, maximum iterations of 15, mode number (\(K\)) search range [2; 10], and quadratic penalty factor (\(\alpha\)) range [50; 10,000]. Algorithm-specific parameters were configured as follows: for PSO, individual and social learning factors were set to 1.5; for SSA, safety threshold (ST) = 0.7, with discoverers comprising 30% of population and alerters 50%; for MPA, the fish aggregation device effect coefficient was set to 0.2. The comparative results are illustrated in Figure 6.  
From Figure 6, SSA demonstrated rapid initial convergence but showed susceptibility to local optima, reaching its optimal solution with a comprehensive entropy kurtosis ratio of 0.022424 by the seventh generation. PSO exhibited robust global search capabilities but slower convergence, achieving optimal solution with a comprehensive entropy kurtosis ratio of 0.022426 at the tenth generation. MPA combined quick convergence with effective global search, reaching optimal solution with a comprehensive entropy kurtosis ratio of 0.022410 by the fourth generation, surpassing both alternatives in optimization performance. In summary, MPA demonstrated superior optimization capabilities compared to PSO and  
Figure 6: Comparison of results of different optimization algorithms.  
The line-mode signal's morphological gradient reaches its peak at sampling point 1066, indicating maximum local amplitude variation. Given that sampling point 1000 corresponds to the fault initiation time of 0.135 s, the line-mode wave head arrives at the first measuring device at 0.135066 s.  
For the zero-mode signal, the wave head reaches the first-end measuring device at 0.135091 s, establishing a 25 ms temporal difference between the line-mode and zero-mode arrival times. Applying Formula (22) yields a fault distance of 5.181 km with an absolute error of 19 m.  
(2) A single-phase ground fault was simulated on hybrid line \(l_{5}\), 4.2 km from section M, occurring at 0.1225 s with 100 \(\Omega\) ground resistance and a 45\({}^{\circ}\) initial fault phase angle. The morphological gradient calibration results for both the line-mode and zero-mode traveling waves are presented in Figures 11 and 12.  
It can be seen from Figures 11 and 12 that the line-mode traveling wave characteristic signal calibrated by the morphological gradient has a sudden change at sampling point 1021 and 0.122521 s, and the amplitude is the largest. The zero-mode traveling wave characteristic signal has a sudden change at sampling point 1032 and 0.122532 s, and the amplitude is the largest. The time difference between the arrival time of the line-mode  
Figure 11: Line-mode traveling wave characteristic signal after morphological gradient processing in the second fault condition.  
Figure 12: Zero-mode traveling wave characteristic signal after morphological gradient processing in the second fault condition.  
traveling wave and the zero-mode traveling wave is 11 ms. The fault distance is 4.223 km and the absolute error is 23 m, as calculated by Formula (25).