# Modified Grey Wolf Optimizer and Application in Parameter Optimization of PI Controller
## 5 Conclusions  
To address the challenges of inadequate exploration, reduced diversity, and the imbalance between exploration and exploitation inherent in the Grey Wolf Optimizer (GWO), this paper introduces an enhanced version of the algorithm, known as the Modified Grey Wolf Optimizer (M-GWO). The M-GWO incorporates sine and cosine search strategies, along with Gaussian and Cauchy mutation operators. This enhanced strategy empowers the algorithm to augment both the search capability and the diversity within the population. The performance of the M-GWO was rigorously assessed and statistically analyzed through the execution of the CEC 2017 benchmark functions. In the 30 dimensions, except for F26, the M-GWO has stronger performance and better average fitness values in 30 independent solutions. In 50 dimensions, the M-GWO can find better fitness values in all test functions. The experimental results demonstrate that the M-GWO algorithm has better performance compared to other advanced metaheuristic algorithms. In addition, through testing of the motor current, the control effect of the M-GWO-PI on the motor resulted in a stable q-axis current of around \(-\)2\(\sim\)1 A, which is superior to WOA-PI and HHO-PI. The control effect of the M-GWO-PI on the motor results in smaller fluctuations in the d-axis current of the motor, with a stable value of around \(-\)2\(\sim\)2 A. The experimental results of PI controller parameter optimizations prove that the designed M-GWO has a good application prospect in engineering problems. Conceptualization, L.S. and S.W. ; methodology, L.S. and S.W. ; software, S.W. ; validation, S.W. and Z.L. ; formal analysis, S.W. and Z.L. ; data curation, S.W. ; writing--original draft preparation, L.S. and S.W. ; writing--review and editing, L.S. and S.W. ; funding acquisition, L.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science Foundation of Xinjiang Uygur Autonomous Region Prefecture under Grant 2022D01F35. Not applicable. Due to technical reasons, no public link has been established. If you need any data, please contact the second author via email. Thanks again to all authors for their contributions in this article, including algorithm selection and simulation. The authors declare no conflicts of interest. The expression for basic functions is as follows:  
1. Bent Cigar Function \[f_{1}(x)={x_{1}}^{2}+10^{6}{\sum\limits_{i=2}^{D}{x_{i}}^{2}}\]
2. Sum of Different Power Function \[f_{2}(x)={\sum\limits_{i=1}^{D}{\mid x_{i}\mid}^{i+1}}\]2. Zakharov Function \[f_{3}(x)=\sum_{i=1}^{D}x_{i}^{2}+\left(\sum_{i=1}^{D}0.5x_{i}^{2}\right)^{2}+ \left(\sum_{i=1}^{D}0.5x_{i}^{2}\right)^{4}\] (4) Rosenbrock's Function \[f_{4}(x)=\sum_{i=1}^{D-1}\left(100(x_{i}-x_{i+1})^{2}+(x_{i}-1)^{2}\right)\] (5) Rastrigin's Function \[f_{5}(x)=\sum_{i=1}^{D}\left(x_{i}^{2}-10\cos(2\pi x_{i})+10\right)\] (6) Expanded Schaffer's F6 Function \[f_{6}(x)=g(x_{1},x_{2})+\cdots+g(x_{D-1},x_{D})+g(x_{D},x_{1})\] where \(g(x,y)=0.5+\frac{\left(sin^{2}\left(\sqrt{x^{2}+y^{2}}\right)-0.5\right)}{ \left(1+0.001(x^{2}+y^{2})\right)^{2}}\). (7) Lunacek bi-Rastrigin Function \[f_{7}(x)=\min\left(\sum_{i=1}^{D}(\hat{x}_{i}-\mu_{0})^{2},dD+s\sum_{i=1}^{D}( \hat{x}_{i}-\mu_{1})^{2}+10\left(D-\sum_{i=1}^{D}\cos(2\pi\hat{z}_{i})\right)\right)\] where \(\mu_{0}=2.5\), \(\mu_{1}=-\sqrt{\frac{b\mu^{2}-d}{s}}\), \(s=1-\frac{1}{2\sqrt{D+20}-8\hat{2}}\), \(d=1\) \(y=\frac{10(x-o)}{100}\), \(\hat{x}_{i}=2 sign(x_{i}{}^{*})y_{i}+\mu_{0}\), for \(i=1,2,\ldots,D\), \(z=\Lambda^{100}(\hat{x}-\mu_{0})\) (8) Non-continuous Rotated Rastrigin's Function \[f_{8}(x)=\sum_{i=1}^{D}\left(z_{i}^{2}-10\cos(2\pi z_{i})+10\right)\] where \(\hat{x}=M_{1}\frac{5.12(x-o)}{100}\) \(y_{i}=\left\{ \begin{array}{l}\hat{x}_{i}\,if|\hat{x}_{i}|\leq 0.5\\ \frac{round(2\hat{x}_{i})}{2}\,if|\hat{x}_{i}|\leq 0.5\end{array}