## 4 Method Validation
### HLSSA-VMD Algorithm Validation
\begin{split} s_{1}&=1.5\sin(100\pi t)\\ s_{2}&=2\sin(300\pi t)\\ s_{3}&=\sin(500\pi t)\\ s_{4}&=2.5\sin(700\pi t)\\ s&=s_{1}+s_{2}+s_{3}+s_{4}+s_{5}\end{split}  \tag{13}\]  
where \(s_{5}\) is a Gaussian white noise signal with a mean of 0 and a standard deviation of 1. Figure 15(a) is a visual representation of each simulated signal. Signal \(s_{1}\) had an amplitude of 1.5, a frequency of 50 Hz, and a duration of 0-1 s; signal \(s_{2}\) had an amplitude of 2, a frequency of 150 Hz, and a duration of 0.35-0.65 s; signal \(s_{3}\) had an amplitude of 1.5, a frequency of 250 Hz, and a duration of 0.15-0.5 s; and signal \(s_{4}\) had an amplitude of 2.5, a frequency of 350 Hz, and a duration of 0.4-0.8 s. Figure 15(b) shows the amplitude-frequency plot obtained by performing a Fourier transform on the synthesized signal s. Figure 15(b) demonstrates that the synthesized signal \(s\) contained the frequencies corresponding to the simulated signals \(s_{1}-s_{4}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Fitness** & **[K, \(\alpha\)]** \\ \hline MVO-VMD & 8.15943 & [6; 1033] \\ PSO-VMD & 8.15474 & [6; 1097] \\ WOA-VMD & 8.15474 & [6; 1097] \\ SSA-VMD & 8.14811 & [6; 1236] \\ HLSSA-VMD & 8.14811 & [6; 1236] \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The optimization results obtained with the five algorithms. Figure 15: The iterative optimization process of the five algorithms. The optimal parameters K and \(\alpha\) for the VMD algorithm in decomposing signal \(s\) were obtained using the HLSSA. The search ranges for \(K\) and \(\alpha\) were set to [100, 6000] and [1, 10], respectively. Figure 17 illustrates the iterative optimization process for the five algorithms, and Table 8 presents the optimal solutions obtained with each algorithm. As shown in Figure 17 and Table 8, the MVO and PSO exhibited relatively poorer convergence accuracy, while the WOA, SSA, and HLSSA yielded the same optimization re  
Figure 16: (**a**) Simulated signal; (**b**) amplitude–frequency plot of the synthesized signal \(s\). Figure 17: The iterative optimization process of the five algorithms. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Fitness** & **[K, \(\alpha\)]** \\ \hline MVO-VMD & 6.4152 & [3, 5803] \\ PSO-VMD & 6.4139 & [4, 3361] \\ WOA-VMD & 6.410 & [4, 6000] \\ SSA-VMD & 6.410 & [4, 6000] \\ HLSSA-VMD & 6.410 & [4, 6000] \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The optimization results obtained with the five algorithms. sults. However, the HLSSA demonstrated a faster convergence speed, and it had the lowest initial fitness value, indicating that the HLSSA provides a higher quality initial solution. In Figure 18a, the decomposition results of MVO-VMD are displayed, revealing that the algorithm only decomposed three signals of different frequencies, while the 250 Hz signal was absent. Figure 18b shows the decomposition results of PSO-VMD, and although it successfully decomposed the signals of four different frequencies, each frequency component exhibited significant jitter, resulting in less smooth decomposition. Figure 18c-e demonstrate the decomposition results of WOA-VMD, SSA-VMD, and HLSSA-VMD, respectively. In contrast to PSO-VMD, these three algorithms not only successfully decomposed all frequencies but also provided smoother decomposition results, indicating superior performance. This comparative analysis further validates the outstanding performance of the HLSSA in optimizing VMD parameters, showcasing its faster convergence speed and higher-quality decomposition results. Figure 18: Hilbert spectra corresponding to different algorithms: (**a**) MVO-VMD; (**b**) PSO-VMD; (**c**) WOA-VMD; (**d**) SSA-VMD; (**e**) HLSSA-VMD.