# Modified Grey Wolf Optimizer and Application in Parameter Optimization of PI Controller
## 3 Modified Grey Wolf Optimizer  
In GWO, \(\omega\) are moved towards the \(\alpha\), \(\beta\), and \(\delta\) of the three leading wolves, which often leads to poor exploration and premature convergence. To address these limitations, we introduce an enhanced version of the Grey Wolf Optimizer, termed the Modified Grey WolfOptimizer (M-GWO). The M-GWO algorithm has been enriched by the different behaviours of \(\alpha\) wolves--alpha wolves typically isolate themselves from other wolves through specific location updates, exploring their territory and finding prey--and introducing mutation operators for the \(\omega\) wolf to enrich its search space. In the M-GWO, the search space is significantly expanded through the incorporation of sine and cosine functions specifically for the \(\alpha\) wolf. The use of sine and cosine functions greatly optimizes the search space of alpha wolves. Due to its best fitness, this scheme can enable it to search for the optimal solution faster. The specific location update formula is as follows:  
\[D_{\alpha-Best}=\left|C_{1}\times X_{\alpha}\times\sin(r_{3})-X\right|\,\ r_{4}<0.5 \tag{14}\]  
\[D_{\alpha-Best}=\left|C_{1}\times X_{\alpha}\times\cos(r_{3})-X\right|\,\ r_{4}\geq 0.5 \tag{15}\]  
\[X_{best}=D_{\alpha}-A_{1}\times D_{\alpha-Best} \tag{16}\]  
where \(r_{3}\) is a random number from 0 to 2. In the M-GWO search strategy, \(X_{best}\) plays the role of the alpha wolves with the best fitness. An alpha wolf can better explore its territory, thereby improving the exploration ability of the M-GWO algorithm. The \(r_{4}\) helps the alpha wolf in the M-GWO exhibit random behaviour, achieving a balance between exploration and development. Gaussian mutation functions as a localized search mechanism for grey wolves, with the scope of this search being regulated by the Gaussian distribution. Similarly to Gaussian mutation, Cauchy mutation operation can enrich the diversity of the population, expanding the search range of individual \(\omega\) wolves by introducing Gaussian mutation and Cauchy mutation. Gaussian mutation and Cauchy mutation make it easier for the M-GWO to escape local optimal value and find better solutions to avoid premature convergence. The specific formula for updating the position of individual \(\omega\) wolves is  
\[X_{m}(t)=(\frac{X_{1}+X_{2}+X_{3}}{3})(1+\gamma Cauchy(0,1)+(1-\gamma)Gauss( 0,1)) \tag{17}\]  
where \(X_{m}(t)\) is the position of the mutated individual of the \(\omega\) wolf; \(Cauchy(0,1)\) and \(Gauss(0,1)\) are the standard Gauss mutation and the standard Cauchy mutation, respectively; and \(\gamma=1-\left(t/Max\right)^{2}\) is the coefficient of mutation control, which is adaptively adjusted with the number of iterations to coordinate the local development and global exploration capabilities of the algorithm. When the value of \(\gamma\) is large in the early stage of iteration, the algorithm has better global exploration ability under the dominance of the Gauss mutation operator. As the iteration progresses, the local search ability of the algorithm is strengthened under the dominance of the Cauchy mutation operator. Table 1 shows the pseudo code of the M-GWO. The pseudo code can clearly demonstrate the process of algorithms, while the Cauchy mutation operator provides larger perturbations in the local search to help escape local optimal value. The introduction of Gaussian distribution further balances local and global search capabilities. The randomness of the Gaussian distribution helps maintain diversity in the global search. In conclusion, the utilization of a hybrid form of Gaussian and Cauchy distributions in the M-GWO enhances global search capabilities, increases population diversity, and improves the balance between local and global search abilities. Compared to GWO, the M-GWO features a more sophisticated position update mechanism to address more complex problems.