## 3 Optimization Methodology
### Mfpoa  
Adaptive orientation Gaussian (AOG) mutation is employed in the suggested MFPOA approach to optimize the controller settings. The pollen's characteristics are altered during the pollination process by using the mutation-based flower pollination method. The mutation procedure in the optimization technique often speeds up the solution by changing some of the particle properties. The AOG mutation technique is used in the FPA to accomplish this. The key benefit of the proposed MFPOA is the algorithm's quicker convergence while maintaining the features of the conventional FPA. The AOG mutation is added to the pollen produced by global pollination in the traditional FPA method following global pollination. As a result, after applying the mutation process [50], Equation (25) is changed into Equation (30):  
\[u_{n\,mut}^{i+1}=u_{n}^{i}+\delta L(x)\Big{(}u_{best}-u_{n}^{i}\Big{)}+\varphi \{AOG\} \tag{30}\]  
where \(\varphi\) is the probability factor present. Equation (31), which describes the AOG mutation function, is as follows:  
\[AOG(x)=(G_{\varphi}(x\text{sin}\,\theta)-G_{\varphi}(x\text{cos}\,\theta)) \tag{31}\]\[G_{\sigma}=\frac{1}{\sigma\sqrt{2}\pi}e^{\frac{-1}{2}\left(\frac{x-\pi}{\sigma} \right)^{2}} \tag{32}\]  
where \(m\), \(\sigma\), \(G_{\sigma}(x)\), and \(\theta\) denote the mean, variance, Gaussian function of the variable \(x\), and the rotation that must be applied to the function. The mutation is also applied to p. The one from Equation (29) is altered by the mutation process to become Equation (33).  
\[u_{n\;mut}^{i+1}=u_{n}^{i}+\in\left(u_{a}^{i}-u_{b}^{i}\right)+\varphi\{AOG\} \tag{33}\]  
As follows is the probability factor. If the pollen characteristics obtained in the most recent iteration \((i+1)\) using global or local pollination match those obtained in the most recent iteration \((i)\) using global or local pollination, the probability factor is 1; otherwise, it is zero, and the solution remains the same.  
\[\varphi=\left\{\begin{matrix}1&if\big{(}u_{n}^{i+1}-u_{n}^{i}=0\big{)}\\ 0&else\end{matrix}\right. \tag{34}\]  
The AOG-based MFPOA algorithm is provided below in Algorithm 1:  
```
1. Determine the pollen count using random solutions
2. Determine the ideal response from the starting population
3. Establish a switch probability of p
4. While \((i\;<\;max.\;No.\;of\;generation)\)
5. For \(n=1:N\)
6. If \(rand<p\)
7. Create a step vector in d dimensions that follows the Levy distribution
8. Apply Equation (25) to global pollination
9. Use Equation (34) to calculate the mutation probability factor
10. If \(\varphi=1\)
11. Apply Equation (29) to determine fresh pollen in global pollination
12. Else
13. Equation (34) is used to calculate new pollen
14. Choose from a uniform distribution in the range [0, 1]
15. Use Equation (33) to carry out local pollination
16. Use Equation (34) to calculate the mutation probability factor
17. If \(\varphi=1\)
18. Apply Equation (33) to local pollination to determine fresh pollen
19. Else
20. Equation (29) is used to calculate new pollen
21. End
22. Create fresh approaches
23. If improved solutions emerge, inform the populace of them
24. End
25. Discover the current top option
26. End
27. The fitness solution is the best option
```  
**Algorithm 1**AOG-Based MFPOA Algorithm