# Modified Grey Wolf Optimizer and Application in Parameter Optimization of PI Controller
## 4 Experimental Evaluation and Results  
### The Test Results of Benchmark Function  
In this section, the performance of M-GWO was evaluated on the CEC2017 benchmark function. All testing scenarios should be conducted in the same environment to address the different results caused by the randomness of metaheuristic algorithms. All algorithms are implemented on MATLAB 2023b which is a simulation software published by Math \begin{table}
\begin{tabular}{c c} \hline
**Algorithms** & **Parameters Used in the Algorithm** \\ \hline M-GWO & \(\mu=0\), \(\sigma=1\), \(\rho=1\), \(a=[0\), \(2]\) \\ IGWO & \(b_{1}=0.1\), \(b_{2}=0.9\), \(a=[0\), \(2]\) \\ GWO & \(a=[0\), \(2]\) \\ WOA & \(a=2\), \(b=1\) \\ DBO & \(\theta=0\) or \(\frac{\pi}{2}\) or \(\pi\), \(b=rand(0,1)\), \(\alpha=1\) or \(-1\) \\ HHO & \(beta=1.5\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters setting. Works in Natick, MA, USA. The experiment was performed on an Intel Core i7-12700 CPU (2.70 GHz) with 16 GB of main memory in Lenovo laptop (Beijing, China). The CEC2017 benchmark function set includes 29 optimization problems, including unimodal functions (F1, F3), multimodal functions (F4-F10), hybrid functions (F11-F20), and composition functions (F21-F30). The specific benchmark function expression is in Appendix A. Unimodal functions, multimodal functions, hybrid functions, and composite functions can all evaluate the development and exploration capabilities of algorithms from different aspects. The CEC 2017 benchmark function used in the experiment includes different types of functions with dimensions of 30 and 50, respectively. Each algorithm undergoes 30 independent experiments separately to ensure the fairness of the experimental results. The maximum number of iterations (\(Max\)) for the algorithm in the experiment is set to 1000. We evaluated the performance of the algorithm by comparing the average error, standard deviation, and minimum value of the global best solution obtained after 30 runs. The M-GWO is compared with other cash-element heuristic algorithms. The test results of each benchmark function provide their convergence behaviour in different dimensions. In addition, the performance of each algorithm was evaluated by comparing the minimum, average, and standard deviation of the global best solution obtained after 30 runs. The statistical analysis of minimum, mean, and standard deviation can effectively demonstrate the overall effectiveness of the compared algorithms. All the parameters of the algorithms used for comparison in this article are shown in Table 1.  
Table A1 shows the optimized numerical results of different algorithms on the unimodal function in the CEC2017 benchmark function. The experimental results demonstrate that the M-GWO has better performance in solving the problem of unimodal functions, therefore the M-GWO has better development capabilities. Table A2 shows the optimized numerical results of different algorithms on the multimodal function in the CEC2017 benchmark function. The experimental results demonstrate that the M-GWO has better development capabilities compared to other algorithms. Tables A3 and A4 show the optimized numerical results of different algorithms on the mixed and composite functions in the CEC2017 benchmark function, respectively. The experimental results demonstrate that the M-GWO has a better ability to balance development and exploration. The main reason for obtaining the experimental effect is based on the improved Grey Wolf Optimization algorithm combined with different update strategies. Firstly, the position update for alpha wolves incorporates the sine and cosine function update method. Given their superior fitness, this enhanced updated scheme allows \(\alpha\) wolves to more effectively explore the vicinity of individuals. Secondly, Gaussian mutation and Cauchy mutation were used in the position update formula for the \(\omega\) wolf.