## 2 Materials and Methods
### Fractional-Order Particle Swarm Algorithm  
Fractional calculus is derived from traditional calculus [25]. For example, Equation (3) represents fractional differentiation based on the Grunwald-Letnikov definition, where \(D\) stands for the differential operator, \(\lambda\) denotes a fractional-order power, and \(\Gamma\) is the Euler function. According to [PERSON] et al. [26], if \(h\) is expressed in discrete terms, it can be approximated as Equation (4). \[D^{\lambda}[x(t)]=\lim_{h\to 0}\Biggl{[}\frac{1}{h^{\lambda}}\sum_{k=0}^{+ \infty}\frac{(-1)^{k}\Gamma(\lambda+1)x(t-kh)}{\Gamma(k+1)\Gamma(\lambda-k+1)} \Biggr{]} \tag{3}\]  
\[D^{\lambda}[x(t)]=\frac{1}{T^{\lambda}}\sum_{k=0}^{r}\frac{(-1)^{k}\Gamma( \lambda+1)x(t-kT)}{\Gamma(k+1)\Gamma(\lambda-k+1)} \tag{4}\]  
where \(T\) denotes the sampling period, and \(r\) represents the truncation order. In contrast to the integer-order derivative as a finite series, the fractional-order derivative requires an infinite number of terms. This implies that the information obtained using the fractional order is more global than the integer-order differentiation. Therefore, the solution space that can be explored for the fraction order is more refined than the integer order, and it is expected to obtain better solution space accuracy. Further, the fractional differentiation allows the particle swarm algorithm to memorize the position. Therefore, the velocity vector is affected by the positions of the previous and last generations. This makes the fractional-order particle swarm algorithm more conservative in the search process, making the solution space results more similar and stable every time. If \(r=4\) is used as an example, the speed and position vectors are updated with the following Equations (5) and (6). \[ \begin{split} V_{i}(k+1)=V_{i}(k)+c_{1}\cdot r_{1}\Bigl{[}P_{i}^{ l}(k)-\lambda\cdot p_{i}(k)-\frac{\lambda}{2}(1-\lambda)p_{i}(k-1)\\ -\frac{1}{6}(1-\lambda)(2-\lambda)p_{i}(k-2)-\frac{1}{24}(1- \lambda)(2-\lambda)(3-\lambda)p_{i}(k-3)\Bigr{]}+c_{2}\cdot r_{2}\\ \cdot(P^{g}(k)-p_{i}(k))\end{split}  \tag{5}\]  
\[p_{i}(k+1)=p_{i}(k)+V_{i}(k+1) \tag{6}\]  
To improve the point of the direction from which the particles are searched, [PERSON] and [PERSON] [8] added an inertia weight term \(\omega\) to the velocity \(V_{i}(k)\) to facilitate the contribution of the particle itself to the update velocity. When \(\omega\) is large, the direction of the update speed will depend on the direction of the previous generation speed. At this time, the search direction is more stable, which improves the global search ability of the particles in space. However, when \(\omega\) is considerably large, overcorrection occurs. Consequently, the particle correction speed is excessively large and deviates from the better solution, resulting in \"flying\" trajectories. When \(\omega\) is small, the update speed direction is dominated by the optimal local solution and the direction of the global optimal solution. At this time, a local search capability is provided. However, because the solution searched by the particle is not explored globally, the obtained solution may not achieve the global best solution due to its locality. Therefore, [PERSON] and [PERSON] [9] once again proposed a solution, changing the \"constant inertia weight\" to a \"linearly decreasing inertia weight.\" When \(\omega\) is set to a larger value in the initial stage, the particle swarm has a better ability to expand the search to find the best solution area in the whole domain. After the number of iterations increases, the value of \(\omega\) is gradually reduced. The particle swarm will switch from an extended search to a local search to find a better solution in the best found so far. The formula for changing the \"constant term inertia weight \(\omega\)\" to \"time-varying linear inertia weight \(\omega(k)\)\" is shown in Equation (7).