## 2 Method
### Improved Black-Winged Kite Algorithm (IBKA)  
Considering that the BKA algorithm lacks dynamic adaptation, it tends to lose the diversity of the population, and the parameters present in it are relatively fixed and lack the ability of hierarchical search. Therefore, the study proposes an IBKA algorithm, which can dramatically improve the algorithm's global search ability, local optimization ability, and stability of the results through the dynamic adaptive mechanism, the population clustering strategy, and the dynamic migration strategy.  
In comparison with traditional optimization algorithms such as the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), the IBKA offers several notable advantages. On the one hand, GA operates based on evolutionary principles, using crossover and mutation operations, which can result in a slower convergence rate and a higher chance of becoming stuck in local optima due to its reliance on predefined genetic operators.  
PSO, on the other hand, relies on a swarm of particles exploring the search space, but this method can suffer from premature convergence if the swarm lacks diversity, especially in complex, high-dimensional problems. In contrast, IBKA incorporates dynamic adaptation mechanisms that allow it to adjust its search process over time, enhancing its ability to explore the search space more thoroughly. The introduction of population clustering and dynamic migration strategies in IBKA further improves its robustness and avoids the pitfalls of local optima. These enhancements lead to faster convergence, higher-quality solutions, and better adaptability in complex, high-dimensional optimization problems, such as those encountered in our study. The IBKA's superior ability to maintain diversity within the population and adaptively fine-tune the optimization process provides a clear advantage over GA and PSO in the context of solving intricate optimization problems like the one discussed in this research.  
Initializing the Population  
Similar to the original BKA, the population initialization equation is  
\[P_{i}=L_{b}+\text{rand}\cdot(U_{b}-L_{b}) \tag{28}\]  
where \(L_{b}\) and \(U_{b}\) are the lower and upper bounds of the search range, respectively; rand is a random number within [0,1].  
Adaptive Trapping Behavior  
A dynamic learning rate \(\eta\) is introduced to dynamically adjust the trapping step size according to the number of iterations:  
\[\eta=\eta_{\text{max}}\cdot(1-\frac{n}{N})+\eta_{\text{min}}\cdot\frac{n}{N} \tag{29}\]  
The trapping behavior equation is  
\[P_{k,m}^{n+1}=\begin{cases}P_{k,m}^{nt}+\eta\cdot(1+\sin(q))\cdot P_{k,m}^{nt},&\text{Condition A}:c<q\\ P_{k,m}^{nt}+\eta\cdot(2q-1)\cdot P_{k,m}^{nt},&\text{Condition B}:c\geq q \end{cases} \tag{30}\]  
where \(\eta_{\text{max}}\) and \(\eta_{\text{min}}\) are the initial and minimum values of the learning rate, \(n\) is the current number of iterations, and \(N\) is the maximum number of iterations; the rest of the symbols have the same meaning as in the original BKA.  
Population Clustering Strategy  
The population is divided into \(G\) clusters, and individuals within each cluster use independent hunting strategies. The cluster division formula is as follows:  
\[G_{j}=\left\{P_{i}|\ \ \|P_{i}-P_{\text{leader}}\|<\tau_{j}\right\} \tag{31}\]  
where \(\tau_{j}\) is the radius of the cluster; the center of the cluster is the current leader \(P_{\text{leader}}\).  
Individuals in each cluster interact with the following information after hunting:  
\[P_{j,i}^{n+1}=\lambda\cdot P_{j,i}^{n}+(1-\lambda)\cdot P_{\text{leader}}^{nt} \tag{32}\]  
where \(\lambda\) is the fusion coefficient, which usually takes values in the range of \([0.5,0.9]\).