# Modified Grey Wolf Optimizer and Application in Parameter Optimization of PI Controller
## 1 Introduction
In [16], GWO was used to optimize the PID controller parameters for vehicle speed and yaw rate, improving the tracking speed of the tracked vehicle for vehicle speed and yaw rate under pivot steering conditions. In [17], GWO and Teaching-Learning-Based Optimization (TLBO) algorithms were jointly optimized for PID parameters of hybrid renewable energy and evaluated through different performance functions. In [18], GWO is used to find the optimal PD controller gain in drone flight control, making the time-domain results and actuator outputs smoother. Some scholars have used the improved GWO to optimize the PID controller gain for controlling the grid frequency in a single-zone power system. The experimental results show that compared to other algorithms, the improved GWO can find the optimal PID controller gain [19]. In [20], GWO is used to optimize the PID controller parameters of a laboratory scale magnetic levitation system to eliminate nonlinear problems and instability in the system. However, in the current era of computer development, the problems of premature convergence and insufficient accuracy of GWO are becoming increasingly apparent. It is very important to improve the convergence accuracy of GWO. To solve these weaknesses of GWO, a modified Grey Wolf Optimizer (M-GWO) is proposed. This research is based on proposing improvements to the position formula of individuals within the population of the existing Grey Wolf Optimizer (GWO) algorithm. The aim is to optimize and enhance the existing GWO algorithm to improve its optimization capabilities. The M-GWO introduces an innovative search strategy that integrates sine, cosine, Gaussian, and Cauchy mutation operators to enhance the search capabilities and diversity of the Grey Wolf population. This approach enriches the algorithm's ability to explore the solution space more effectively. Specifically, in the location update strategy, the sine and cosine update methods are employed to refine the position of the \(\alpha\) wolves, which are the individuals with the highest fitness, thereby facilitating more efficient local searches. Additionally, the Gaussian and Cauchy mutation operators are incorporated into the position update strategy for the \(\omega\) wolves, targeting individual mutation. The influence of these mutation operators escalates with the progression of iterations, endowing the M-GWO with an enhanced capacity for global exploration. The performance of M-GWO is developed and evaluated using the CEC2017 benchmark function [21], and the effectiveness of M-GWO is verified by comparing different algorithms. The CEC2017 benchmark functions have different types of functions, which can evaluate the exploration and development capabilities of algorithms from multiple perspectives. To better demonstrate the effectiveness of M-GWO, it is compared with some highly advanced metaheuristic algorithms currently available: Improved Grey Wolf Optimizer (IGWO) [22], GWO, Whale Optimization Algorithm (WOA) [23], Dung Beetle Optimizer (DBO) [24], and Harris Hawks Optimization (HHO) [25]. The experimental results from benchmark function tests demonstrate that the Modified Grey Wolf Optimizer (M-GWO) outperforms other algorithms in terms of effectiveness. With the exception of function 26 in 30 dimensions and function 28 in 50 dimensions, the test results of the M-GWO are superior to other algorithms. Compared to other algorithms, the M-GWO has better exploration and development capabilities. Subsequently, the designed M-GWO was used to optimize the PI controller parameters of the PMSM system current loop and compared with WOA and HHO. The experimental results demonstrate that the parameter optimization scheme of the PI controller based on the M-GWO can find the optimal PI parameters. In the rest of this paper, firstly, Section 2 introduces the Grey Wolf Optimization algorithm and mutation operator, providing a theoretical basis for improving the Grey Wolf Optimization algorithm.