## 4 GWO Algorithm Based on the Dynamic Adjustment of the Proportional Weight and Convergence Factor
### Verification of the Superiority of the Improved GWO Algorithm  
An optimal simulation was performed to test the optimal ability of the improved GWO algorithm. A single-mode function and multi-mode function were selected for the test [35; 36; 37; 38]. \(f_{1}(x)\), \(f_{2}(x)\), and are single-mode functions that represent the typical method of evaluating the global convergence of an algorithm. \(f_{4}(x)\), \(f_{5}(x)\), and \(f_{6}(x)\) are multi-mode functions, which are typically used to evaluate the development performance, and searching ability of the algorithm. This function contains additional local minimums. The solution process easily reached a local optimal solution, and the optimal result was not easily obtained. The test function dimension was set to 10, and the corresponding expression is shown in Table 2.  
A comparison experiment featuring the particle swarm optimization (PSO) algorithm, the invasive weed optimization (IWO) algorithm [39], the grey wolf optimization (GWO) algorithm [31], the grey wolf optimization/evolutionary population dynamics (GWO-EPD) algorithm [40], and the improved GWO algorithm was performed. The parameters of the PSO algorithm were as follows: the proportional weight was \(w=0.8\), the particle swarm size was \(N=100\), the learning factor was \(s_{1}=s_{2}=2\), and the maximum number of iterations was 200. The parameters of the IWO algorithm were as follows: the particle swarm size was \(N=100\), the maximum number of iterations was 200, and the maximum and minimum number of seeds were five and two, respectively. The parameters of the GWO and GWO-EPD algorithms were as follows: the grey wolf swarm size was \(N=100\), and the maximum number of iterations was 200. The parameters of the improved GWO algorithm were the same as those of the GWO algorithm, in addition to adopting the improved strategy in this study. Every test function was evaluated and solved using the above five types of algorithms. The simulation results are shown in Figure 5, where the ordinate is represented in logarithmic form.  
Figure 4: Flow chart of the bi-level optimal model of the distribution network comprising the micro energy grid, based on the improved GWO algorithm.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Test Function** & **Search Scope** \\ \hline \(f_{1}(x)=\sum\limits_{i=1}^{n}|x_{i}|+\prod\limits_{i=1}^{n}|x_{i}|\) & \([-10,10]\) \\ \(f_{2}(x)=\sum\limits_{i=1}^{n}x_{i}^{4}+random[0,1]\) & \([-1.28,1.28]\) \\ \(f_{3}(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & \([-100,100]\) \\ \(f_{4}(x)=\sum\limits_{i=1}^{n}[x_{i}^{2}-10\cos(2\pi x_{i})+10]\) & \([-5.12,5.12]\) \\ \(f_{5}(x)=-20\exp(-0.2\sqrt{\frac{1}{\frac{n}{i}}\sum\limits_{i=1}^{n}x_{i}^{2} })-\exp(\frac{1}{\frac{n}{i}}\sum\limits_{i=1}^{n}\cos(2\pi x_{i}))+20+e\) & \([-32,32]\) \\ \(f_{6}(x)=\frac{1}{4000}\sum\limits_{i=1}^{n}x_{i}^{2}-\prod\cos(x_{i}/\sqrt{i })+1\) & \([-600,600]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Test functions.  
Figure 5: _Cont._Figure 5: _Cont._  
According to Figure 5, in the process of optimizing the six test functions, the improved grey wolf optimization (GWO) algorithm, the grey wolf optimization/evolutionary population dynamics (GWO-EPD) algorithm, and the GWO algorithm exhibited better performance and a faster convergence rate than the basic particle swarm optimization (PSO) algorithm, and invasive weed (IWO) algorithm. Moreover, the improved GWO algorithm was more convergent than the GWO and GWO-EPD algorithms, and exhibited better optimization capabilities. The above conclusions verified the accuracy and superiority of the proposed algorithm.