## 5 Example Analysis
### Optimization Performance Tests of IBWO  
To compare the optimization performance of meta-heuristic algorithms, we use CEC benchmarking functions for comparison. In this paper, three kinds of single-peak, three kinds of multi-peak, and three kinds of fixed-dimensional multi-peak benchmark test functions are selected to test the performance and make a comparative analysis of PSO, Grey Wolf Optimization Algorithm (GWO), Improved GWO (IGWO), BWO, and the proposed IBWO. The parameter set is shown in Table 2, the benchmark test functions are shown in Table 3, the curve of results is shown in Figures 7 and 8, respectively; and the results of the function simulation test and the Wilcoxon rank-sum test are shown in Tables 4 and 5, respectively. Figure 6: Meteorological and historical data. As illustrated in Table 4, it can be clearly seen that IBWO achieves the optimal value of the minimum among the nine test functions of \(F_{1}\) to \(F_{9}\) in comparison with other algorithms. Regarding the average and STD, it can be observed that IBWO consistently identifies the minimum value of the function, demonstrating the stable optimization search performance of the proposed benchmark functions convincingly. Meanwhile, as shown in Figure 8 and Table 4, the IBWO can not only find the optimal fitness value but also exhibits a faster convergence speed than the remaining four algorithms. Specifically, the convergence curves, such as \(F_{1}\), \(F_{3}\), \(F_{4}\), etc., demonstrate the efficiency and effectiveness of the two improvement methods of nonlinear balancing factor and elite learning strategy. Moreover, the optimal value is already found at the early stage of iterations in the results of \(F_{4}\), \(F_{5}\), \(F_{6}\), etc. The convergence speed is fast, while the optimization accuracy is high. Combined with the above analysis, it is not difficult to conclude that compared with PSO, GWO, IGWO, and BWO, the IBWO algorithm proposed in this paper has superior optimization performance and high stability. In addition, Table 4 presents the Wilcoxon rank-sum test results for the nine types of tests mentioned above, aimed at assessing whether IBWO is significantly different from the other four algorithms. The \(p\)-value is calculated for this purpose. Under the standard significance level of \(p\) = 0.05, if it is greater than 0.05, there is no significant difference between the above algorithms and IBWO; otherwise, the difference with IBWO is significant. According to Table 5, the \(p\)-values calculated for the four algorithms are far \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **Parameters** & **Values** \\ \hline All Algorithms & \begin{tabular}{c} Population size \\ \(T_{\text{max}}\) \\ \end{tabular} & \begin{tabular}{c} 30 \\ 500 \\ \end{tabular} \\ \hline PSO & \begin{tabular}{c} \(V_{\text{max}}\) \\ \(c_{1},c_{2}\) \\ \end{tabular} & \begin{tabular}{c} 6 \\ 2 \\ \end{tabular} \\ \hline GWO and IGWO & \(a\) & \begin{tabular}{c} \([0,2]\) \\ \end{tabular} \\ \hline SOA & \begin{tabular}{c} \(f_{c}\) \\ \(A\) \\ \end{tabular} & \begin{tabular}{c} 2 \\ [0,2] \\ \end{tabular} \\ \hline BWO and IBWO & \(W_{f}\) & \begin{tabular}{c} [0.05, 0.1] \\ \end{tabular} \\ \hline Search Scope & \
\begin{tabular}{c} \(c\) \\ \end{tabular} &
\begin{tabular}{c} [0.1, 100] \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of algorithms. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Benchmark Function** & **Dim** & **Rang** & _fmin_ \\ \hline \(F_{1}(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & 30 & \begin{tabular}{c} \([-100,100]\) \\ \end{tabular}  & 0 \\ \(F_{2}(x)=\sum\limits_{i=1}^{n-1}\left[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}\right]\) & 30 & \begin{tabular}{c} \([-30,30]\) \\ \end{tabular}  & 0 \\ \(F_{3}(x)=\sum\limits_{i=1}^{n}x_{i}^{4}+random[0,1)\) & 30 & \begin{tabular}{c} \([-1.28,1.28]\) \\ \end{tabular}  & 0 \\ \(F_{4}=\sum\limits_{i=1}^{n}-x_{i}\sin(\sqrt{|x_{i}|})\) & 30 & \begin{tabular}{c} \([-500,500]\) \\ \end{tabular}  &