## 3 Methodology
### Grey Wolf Optimizer  
Grey Wolf Optimizer (GWO) emulates the social leadership hierarchy and hunting strategy of grey wolves in the wild. Wolves in a pack have a strict social hierarchy, and GWO leverages this to solve optimization problems. GWO does not require prior initialization of control parameters other than population size, dimension of control variables, and maximum iterations. This method offers advantages such as avoiding local optima, requiring fewer function evaluations, and being free from the need for extensive parameter tuning. In GWO, the population is divided into four categories (alpha (\(\kappa\)), beta (\(\beta\)), delta (\(\delta\)), and omega (\(\omega\))), reflecting the hierarchical structure of a wolf pack. Alpha wolves lead the pack, beta wolves assist in decision-making, delta wolves follow alpha, beta wolves lead omega wolves, and omega wolves follow the other wolves. The hunting strategy involves monitoring, pursuing, and attacking the prey, which in this context translates to selecting the best parameters for FOPID tuning [25]. The circling behavior of wolves around prey is modeled using Equations (33) and (34). The vectors \(\vec{A}\) and \(\vec{C}\) are computed as \(\vec{A}=2\times\text{rand}\times\vec{\gamma}-\vec{\gamma}\) and \(\vec{C}=2\times\text{rand}\), where rand is a random vector between 0 and 1, and \(\vec{\gamma}\) decreases linearly from 2 to 0 over iterations. The position vector \(\vec{X}(t+1)\) is updated based on the prey's position \(\vec{X}_{p}(t)\) and the computed distance \(\vec{\delta}=|\vec{C}\cdot\vec{X}_{p}(t)-\vec{X}(t)|\). During hunting, alpha, beta, and delta wolves guide the search. Their positions are updated using Equation (35), where \(\vec{\delta}_{a}\), \(\vec{\delta}_{\beta}\), and \(\vec{\delta}_{\delta}\) represent distances from alpha, beta, and delta wolves to the prey, respectively. The new positions \(\vec{X}_{1}\), \(\vec{X}_{2}\), and \(\vec{X}_{3}\) are calculated using Equation (36), and the average position \(\vec{X}(t+1)\) is obtained from Equation (37). GWO's exploration and exploitation balance is controlled by the values of \(\vec{A}\) and \(\vec{C}\). When \(\vec{A}>1\), exploration is favored; when \(\vec{A}<1\), exploitation is emphasized. This mechanism ensures that the GWO can effectively search the space and avoid local optima, with \(\vec{A}\) and \(\vec{C}\) dynamically adjusting to maintain a balance between diversification and intensification of the search process. \[\vec{A}=(2\times\text{rand}\times\vec{\gamma})-\vec{\gamma} \tag{33}\] \[\vec{C}=2\times\text{rand}\]  
\[\vec{\delta}=\left|\vec{C}\cdot\vec{X}_{p}(t)-\vec{X}(t)\right| \tag{34}\] \[\vec{X}(t+1)=\vec{X}_{p}(t)-(\vec{A}\times\vec{\delta})\]  
\[\vec{\delta}_{\alpha}=\left|\vec{C}_{1}\cdot\vec{X}_{\alpha}-\vec{X}\right|\]  
\[\vec{\delta}_{\beta}=\left|\vec{C}_{2}\cdot\vec{X}_{\beta}-\vec{X}\right| \tag{35}\] \[\vec{\delta}_{\delta}=\left|\vec{C}_{3}\cdot\vec{X}_{\delta}-\vec {X}\right|\]  
\[\vec{X}_{1}=\vec{X}_{\alpha}-(\vec{A}_{1}\times\vec{\delta}_{\alpha})\]  
\[\vec{X}_{2}=\vec{X}_{\beta}-(\vec{A}_{2}\times\vec{\delta}_{\beta}) \tag{36}\] \[\vec{X}_{3}=\vec{X}_{\delta}-(\vec{A}_{3}\times\vec{\delta}_{ \delta})\]  
\[\vec{C}=2\times\text{rand}\]0  
The proposed \"FOPID Tuning Grey Wolf Optimizer\" Algorithm 2 systematically optimizes Fractional Order PID controllers by emulating the social hierarchy and hunting strategy of grey wolves. Initially, the search space and algorithm parameters are declared as shown in \[\vec{C}=2\times\text{rand}\]1 The population is initialized, with each member's position assigned to the control signal, and the objective function evaluated. The algorithm iterates through several steps: identifying the three best solutions (\(X_{\alpha}\), \(X_{\beta}\), \(X_{\delta}\)), establishing coefficients \(\vec{A}\) and \(\vec{C}\) (Equation (33)), and calculating distances \(\vec{\delta}_{\alpha}\), \(\vec{\delta}_{\beta}\), \(\vec{\delta}_{\delta}\) (Equation (35)).