## 5 MPA-Optimized MVMD and Improved Wavelet Threshold Noise Reduction Method  
### MPA for Parameter Optimization of MVMD  
The second noise reduction uses MPA-optimized MVMD to decompose the noisy PD signal, filtering out narrowband interference and combining the 3\(\sigma\) criterion to obtain the reconstructed PD signal. The following is the MPA parameter setting:  
(1) Selection of population size: The population size directly affects the search ability and computational efficiency of the MPA algorithm. The larger population size can increase the coverage of the search space, but it will also reduce the computational efficiency. After many experiments, the influence of different population sizes on the comprehensive performance of the optimization algorithm is compared. In this paper, the population size is 15. (2) Determination of the search space boundary: The mode number \(k\) and the penalty factor \(\alpha\) determine the accuracy of the decomposition and the bandwidth of the IMF component. The definition of the search space boundary is also extremely important. Paper [30] determined the value of \(k\) by observing the center frequency of each IMF decomposed of the cable partial discharge signal at different values (\(k=2,3,4,5,6,7\)). The results show that when the value of \(k\) is 6, the VMD algorithm effectively decomposes the frequency component of the PD signal and avoids decomposing too many high-frequency components. Considering the difference between the partial discharge signal model and its noise characteristics, in order to realize the full decomposition of the noisy partial discharge signal as much as possible, this paper sets the search space boundary of \(k\) as [2,15]. In paper [31], the sum of the kurtosis values of each IMF is used as the objective function, and the Beetle Antennae Search optimization algorithm is used to optimize the selection of the penalty factor. The results show that when \(k\) is 5, the penalty factor of the fifth-layer IMF component is 3850, which can decompose the PD characteristic signal most effectively. Considering the \(k\) value search space set in this paper, the search space boundary of the penalty factor is set as [100, 15,000]. (3) The determination of the number of modes \(k\) and the penalty factor value \(\alpha\): From Figure 4, it can be seen that with the increase in SNR, the average envelope entropy of the noisy PD signal gradually decreases. Therefore, with the minimum average envelope entropy value as the objective function, the noisy PD signal can decompose the PD characteristic signal with the greatest degree of certainty. Therefore, taking (15) as the objective \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Multiples of the Amplitude** & **SNR (dB)** & **NCC** & **RMSE** \\ \hline \multirow{4}{*}{SVD} & 1.0 & 20.1378 & 0.9952 & 0.0220 \\  & 0.8 & 19.8896 & 0.9948 & 0.0223 \\  & 0.5 & 12.1543 & 0.9696 & 0.0539 \\  & 0.3 & 9.3233 & 0.9414 & 0.0762 \\  & 1.0 & 21.1683 & 0.9957 & 0.0203 \\ \multirow{2}{*}{ISVD} & 0.8 & 20.4748 & 0.9955 & 0.0210 \\  & 0.5 & 20.3421 & 0.9954 & 0.0212 \\ \multirow{2}{*}{} & 0.3 & 19.1884 & 0.9939 & 0.0245 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Noise reduction performance evaluation of different methods for different narrowband interference amplitude signals. function and using MPA to optimize the noisy PD signal, the optimal values of the modal number \(k\) and the penalty factor can be obtained. The optimization results of MPA are shown in Figure 6. At the same time, the parameters of MVMD are optimized by particle swarm optimization (PSO) and the sparrow search algorithm (SSA), respectively. To guarantee the fairness of the experiment, the population size of the three algorithms is set to 15, the maximum number of iterations is set to 20, the search space of the modal number \(k\) is [2; 15] and the search space of the quadratic penalty factor \(\alpha\) is [100; 15,000].