## 3 Methodology
### Flower Pollination Algorithm  
The fundamental concept of the Flower Pollination Algorithm 3 is inspired by the natural pollination process of flowering plants, encompassing both biotic and abiotic pollination, as well as the floral constancy exhibited by certain flower species and their pollinators. In the FPA, the position of a pollen particle is represented as a solution vector. The algorithm involves two main equations representing different pollination strategies. The global pollination mechanism, which simulates long-distance pollination, is modeled by Equation (38). Here, \(x_{i}^{t+1}\) denotes the new position of pollen particle \(i\) at iteration \(t+1\), \(x_{i}^{t}\) is the current position, \(g_{*}\) is the best solution found at iteration \(t\), \(\gamma\) is a scaling parameter, and \(L(\lambda)\) is a step size drawn from a Levy distribution characterized by the exponent \(\lambda\). Levy pseudorandom numbers can be generated using [PERSON]'s algorithm, providing a robust mechanism for global search [26].  
The local pollination mechanism, which emulates the local characteristics of pollination and floral constancy, is described by Equation (39). In this equation, \(x_{i}^{t+1}\) represents the new position of pollen particle \(i\) at iteration \(t+1\), \(x_{i}^{t}\) is its current position, \(U\) is a uniformly distributed random number, and \(x_{j}^{t}\) and \(x_{k}^{t}\) are the positions of two different pollen particles. The switching probability \(p\) is employed to decide between global and local pollination mechanisms, reflecting the natural process of abiotic and biotic pollination, as well as cross-pollination and self-pollination. Levy flights enhance the search capability in FPA by allowing for larger steps than those typically generated by Gaussian distribution, thus improving the algorithm's ability to escape local optima. Equation (40) shows the computation of Levy steps, where \(u\) and \(v\) are normally distributed random numbers, and \(\sigma\) is set to 0.7 based on an exponent \(\beta\) of \(3/2\). The parameters in FPA, such as the switching probability \(p\), population size \(n\), and scaling parameter \(\gamma\), are crucial for its performance. A typical value of \(p=0.5\) is effective for real-world pollination processes, while \(\gamma\) generally ranges from 0.01 to 0.1. For implementation in simulation software, the parameters provided in Table 9 can be applied [27].  
\[x_{i}^{t+1}=x_{i}^{t}+\gamma L(\lambda)(g_{*}-x_{i}^{t}) \tag{38}\]  
\[x_{i}^{t+1}=x_{i}^{t}+U(\lambda)(x_{j}^{t}-x_{k}^{t}) \tag{39}\]  
\[u=\text{randn}\times\sigma\] \[v=\text{randn} \tag{40}\] \[L=\gamma\times\frac{u}{|v|^{1/\beta}}\]  
The \"FOPID Tuning Flower Pollination Algorithm\" systematically optimizes Fractional Order PID controllers by mimicking the pollination process of flowering plants. Initially, the search space and algorithm parameters are declared as shown in Table 9. The flower population is initialized, with each member's position assigned to the control signal and the objective function evaluated. The algorithm iterates by comparing random values to a switching probability \(p\), generating step sizes using either Levy distribution (Equation (40)) for global pollination or uniform distribution for local pollination. New solutions are generated (Equations (38) and (39)), and the best current solution is continuously updated. This approach enhances the exploration and exploitation capabilities, ensuring optimal tuning of the FOPID controller parameters [28; 29].