# Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow
## 3 For the Stochastic MaOPF
### Set-Based Evolutionary Strategy  
Figure 1 illustrates the evolutionary strategy employed by SetGSO. Initially, a set of individuals is selected based on pairwise comparisons of their set-based metrics. These selected individuals may contain the same ensemble. The offspring generated through the evolutionary process are not identical due to randomness, and it is also appropriate to select the same set-based individual. At each iteration, the search members in the \(s_{th}\)\((s=1,\ldots,S)\) set compute all objective functions, and the fitness values for each local producer are determined by sorting the objective function values of all search members. A global producer is chosen for each objective from the set-based individuals. The scrounger imitates both the local and global producers, and is also influenced by Pareto solutions, which can expedite the approximation of the entire Pareto front. The Pareto solutions discovered by comparing the objective function values of the search members are stored in their archives at each iteration. The parent and child set-based individuals are then ordered based on the set-based dominance relation, resulting in the acquisition of new \(S\) set-based individuals. Figure 1: The set-based evolutionary strategy of SetGSO. Due to the presence of stochastic uncertainty, the objective function value is no longer a single value but rather an interval value. This deviates from the conventional practice of directly selecting the optimal value for each goal as a producer. Consequently, a novel dominance relation is necessary to compare these interval values. We let two interval values be represented by midpoint of interval \(A=[a_{L},a_{R}]=[f_{\underline{m}}(\mathbf{x_{1}},\mathbf{u}),\overline{f_{m}}( \mathbf{x_{1}},\mathbf{u})]\) and \(B=[b_{L},b_{R}]=[\underline{f_{\underline{m}}}(\mathbf{x_{2}},\mathbf{u}), \overline{f_{\underline{m}}}(\mathbf{x_{2}},\mathbf{u})]\). As shown in Figure 2, when intervals are separate or intersect, determining the dominant relation is effortless; for instance, when \(a_{R}<b_{L}\), \(A\) dominates \(B\), when the interval is containment, we use the interval credibility strategy [42] to determine the dominant relation. The interval credibility strategy is defined as follows:  
\[ \begin{split}& P_{\text{rel}}(A<B)=\frac{\text{Dis}(B,D)}{\text{ Dis}(A,D)+\text{Dis}(B,D)}\\ &\text{Dis}(A,D)=\max\{|a_{\text{L}}-D_{\text{L}}|,|a_{\text{R}}- D_{\text{R}}|\}\\ &\text{Dis}(B,D)=\max\{|b_{\text{L}}-D_{\text{L}}|,|b_{\text{R}}- D_{\text{R}}|\}\\ & D_{\text{L}}=\min\{a_{\text{L}},b_{\text{L}}\}\\ & D_{\text{R}}=\min\{\{a_{\text{L}},b_{\text{L}},a_{\text{R}},b_{ \text{R}}\}-\{D_{\text{L}}\}\}\end{split}  \tag{16}\]  
where \(P_{\text{rel}}(A<B)\) represented the reliability of the interval \(A\) being less than the interval \(B\). We assume \(P_{\text{rel}}(A<B)>0.5\) denotes that \(A\) dominates \(B\), otherwise, \(B\) dominates \(A\). The interval dominant relation is based on the definition of domination possibility. We choose \(m\) producers with the best value for each objective as local producers of that set-based individual based on the aforementioned dominating relation, where \(m\) is the quantity of objective functions. Additionally, we identify \(m\) global producers with the best value throughout the whole solution space using the interval reliability technique in order to continue communication across set-based individuals. As for the scroungers of the SetGSO, the updated formulation is shown as follows:  
\[\mathbf{X}_{i}^{k+1}=\mathbf{X}_{i}^{k}+r_{1}\circ(\mathbf{X}_{g}^{k}-\mathbf{ X}_{i}^{k})+r_{2}\circ(\mathbf{X}_{i}^{k}-\mathbf{X}_{i}^{k})+r_{3}\circ( \mathbf{X}_{p}^{k}-\mathbf{X}_{i}^{k}) \tag{17}\]  
where \(r_{1},r_{2},r_{3}\in R^{1}\) are constant, and \(\mathbf{X}_{g}^{k},\mathbf{X}_{i}^{k},\mathbf{X}_{p}^{k}\) represent the globally optimal solution, locally optimal solution, and Pareto optimal solution. The concept of a locally optimal solution allows the scrounger to efficiently approach the optimal solution within a given set.