## 3 Method
### Improved Parameter Estimation for Particle Swarm Hybridization  
In the PSO algorithm, the inertia weight factor is a key parameter that affects the search capability, velocity, and search range of each particle in the solution space. In this section, we introduce a non-linearly decreasing inertia weight that varies with the number of iterations. The non-linearly decreasing inertia weight allows particles to have higher inertia in the initial stage, enhancing their global search capability. As particles approach the optimal position, the inertia weight decreases, restricting their movement to a smaller local range and improving search precision. The updated formula for the inertia weight is given as follows:  
\[\omega(k)=\omega_{start}-(\omega_{start}-\omega_{end})\left(\dfrac{k}{N_{\max }}\right)^{2} \tag{20}\]  
where \(\omega_{start}\) represents the initial inertia weight, and \(\omega_{end}\) represents the inertia weight at the maximum number of iterations. \(k\) represents the current iteration number, and \(N_{max}\) represents the maximum number of iterations. The PSO algorithm suffers from the problem of premature convergence, where particles becomes trapped in local optima, thereby affecting the convergence accuracy of the algorithm. To address this issue, we introduce the selection and recombination operations from genetic algorithms into the PSO algorithm, referred to as the HPSO algorithm. This approach significantly improves the problem of premature convergence in the population. After updating the positions and velocities of all particles, a certain number of particles are selected for recombination based on the hybrid pool size ratio \(P_{s}\). With a given recombination probability \(P_{c}\), two particles are randomly selected and undergo recombination, and the resulting offspring particles replace the parent particles. This process helps particles escape from local optima. The recombination formula for parent particles is as follows:\[ \begin{split} child(x)&=P_{c}*parent_{1}(x)+(1-P_{c})*parent_{2} (x)\\ child(v)&=\frac{parent_{1}(v)+parent_{2}(v)}{|parent_{1}(v)+ parent_{2}(v)|}|parent_{1}(v)|\end{split}  \tag{21}\]  
where parent (\(x\)) and parent (\(v\)) represent the position and velocity of the hybrid particles, while child (\(x\)) and child (\(v\)) represent the position and velocity of the offspring particles resulting from the hybridization process. When two particles trapped in different local optima undergo hybridization, they can escape from their respective local optima and improve the global search capability of the population. Additionally, if the hybrid particle group adopts a fixed hybridization probability \(P_{c}\), it may result in repetitive and ineffective hybridization operations within the local range during the later iterations, leading to decreased algorithm efficiency and the loss of the advantages of hybridization. To address this issue, a nonlinearly decreasing hybridization probability is employed, where the hybridization probability of the parent particles decreases nonlinearly with the increase in iteration count. The updated formula for the hybridization probability \(P_{c}\), is as follows:  
\[P_{c}=(P_{c1}-P_{c2})\bigg{(}1-\frac{k}{N_{\text{max}}}\bigg{)}^{2}+P_{c2} \tag{22}\]  
where \(P_{c1}\) represents the initial crossover probability and \(P_{c2}\) represents the crossover probability at the maximum number of iterations. The introduction of a nonlinearly decreasing crossover probability enhances the global search capability of particles in the early iterations while addressing the issue of local search efficiency in the later iterations. The basic steps of the proposed HPSO algorithm in this study are as follows:  
**Step 1. ** Initialize the particle swarm, including the swarm size \(N\), the inertia weight \(\omega\) for particle updates, the learning factors \(c_{1}\) and \(c_{2}\), as well as parameters for crossover operations such as the crossover pool size ratio \(P_{s}\) and the crossover probability \(P_{c}\).