# Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow
## 4 Case Study
### Algorithm Comparisons  
The Monte Carlo methodology [43] was employed to assess the effectiveness of the algorithm in comparison to the proposed algorithm. By utilizing the Monte Carlo method, a random sample of 100 wind speed data points from the fan was obtained to determine the optimal value of the objective function. To assess the effects of different numbers of objectives, simulation studies have been conducted on the IEEE-30 bus system. The optimal Pareto front for the three objectives is illustrated in Figure 6 using three different approaches. It is evident that the SetGSO methods exhibit superior performance compared to the sampling method, as they converge towards a more evenly distributed set of non-dominated solutions. The performance metrics commonly regarded as important indicators for assessing algorithm performance include: Hyper-volume (HV) [44], Mean Euclidean Distance (MED) [45], Spacing Metric (S) [46], and Run Time (RT). The calculation Formula (11) for hyper-volume has been mentioned earlier, and the calculation formulas for the MED and S Metric are as follows:  
\[\text{MED}=\frac{\sum_{i=1}^{n}dis(x_{i},y)}{n} \tag{19}\]  
where \(n\) is the number of true Pareto-optimal solutions, \(dis\) dictates the Euclidean distance, and \(y\) is the reference point. The MED metric reflects the convergence of the Pareto front solutions, where a smaller value indicates a better convergence of the algorithm. Figure 5: Influence of different number of set-based individuals on the performance of SetGSO. Figure 6: The Pareto-optimal front obtained using SetGSO, SetGA, and GSO-MC under three objectives. \[\text{S}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\left(d-d_{i}\right)^{2}} \tag{20}\]  
where \(d_{i}\) represents the Euclidean distance between two neighboring individuals within the non-dominated solution set, \(\bar{d}\) is the average of all \(d_{i}\), and the S metric reflects the uniformity of the distribution of individuals within the solution set, with a smaller value indicating a better uniformity of the solution set. The algorithm introduced in this paper builds upon the original GSO framework. Therefore, our initial step involves a performance comparison between the set-based SetGSO and SetGA algorithms, as well as the GSO-MC. This comparison is based on the four metrics mentioned earlier, and the results are presented in Table 1. Finally, for a more comprehensive assessment of the relative performance between the two set-based algorithms, we conduct a comparative analysis of systems with varying proportions of renewable energy sources. The details of this analysis are elaborated in the subsequent section. According to the findings presented in Table 1, it is evident that set-based methods exhibit significantly higher computational efficiency compared to the sampling method. Additionally, the HV, MED, and S metrics also demonstrate superior performance in comparison to the GSO-MC. Moreover, SetGSO outperforms SetGA in terms of Pareto frontier solution convergence, as evidenced by the HV and MED metrics. This improvement is likely attributed to the incorporation of global and local leaders, thereby enhancing the exchange of evolutionary information among ensemble-based individuals, whereas SetGA only employs crossover evolution within the set. As the number of objectives increases, the performance of the algorithm is minimally affected. This can be attributed to the fact that multiple objectives ultimately converge into a bi-objective. To overcome the stochastic nature of algorithm results, we conducted ten independent runs of SetGA and SetGSO algorithms on an IEEE-30 bus system featuring three objectives. The statistical outcomes are presented in Table 2. It is pertinent to note that GSO-MC, being an algorithm that relies on repetitive sampling, inherently exhibits lower stochasticity and, consequently, obviates the need for repeated executions.