# Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow
## 3 For the Stochastic MaOPF  
### Stochastic MaOPF to Set-Based Bi-Objective OPF  
This section aims to transform the original stochastic MaOPF problem, as presented in (1), into a deterministic bi-objective problem by utilizing the worst-case hyper-volume and average imprecision measures. The hyper-volume measure captures the distribution of representative points along the Pareto-optimal front and the convergence efficiency of the genuine Pareto-optimal front. On the other hand, the average imprecision measure quantifies the fluctuation of the Pareto-optimal solution due to the stochastic variables. The quality of an approximation front is enhanced with a higher hyper-volume value, indicating a better performance. Additionally, a lower value of the average imprecision measure, denoted as \(I_{\text{ave}}(X)\), signifies reduced uncertainty in the solution set. The definitions of the average imprecision and hyper-volume [31] measures can be found in (11) and (12).  
\[\begin{split}& H(X)=[\underline{H(X)},\overline{H(X)}]=\\ &\wedge\left(\bigcup_{\mathbf{x}\in\mathcal{X}}\left\{\mathbf{y} \in R^{N}\mid\mathbf{x}_{\text{ref}}\ \prec_{IP}\mathbf{y}_{IP}\prec\mathbf{x}\right\}\right)\\ & I_{\text{ave}}(X)=\frac{\sum_{\mathbf{x}\in X}\sum_{i=1}^{m} \left(\underline{f_{i}(\mathbf{x},\mathbf{c}_{i})}-\underline{f_{i}(\mathbf{ x},\mathbf{c}_{i})}\right)}{num\_A}\\ \end{split} \tag{11}\]  
where \(X\) is an approximate Pareto-optimal solution, \(\overline{H(X)}\) is the best-case hyper-volume, \(\underline{H(X)}\) is the worst-case hyper-volume, \(\mathbf{x}_{\text{ref}}\) is the reference point, \(\bigwedge\) is the Lebesgue measure, and \(num\_A\) is the size of Pareto-optimal solution.  
For any two solutions, a set-based dominate relation \(\succ_{SetP}\) is defined to select a better solution.  
\[\begin{split}& X_{1}\succ_{SetP}X_{2}\Leftrightarrow\\ &(\underline{H(X_{1})}>\underline{H(X_{2})}\bigcap I_{\text{ave} }(X_{1})<I_{\text{ave}}(X_{2}))\\ \end{split} \tag{13}\]  
At least one greater-than relation is held; \(X_{1}\) dominates \(X_{2}\) and is denoted as, \(X_{1}\succ_{SetP}X_{2}\). Otherwise, when the two solution sets do not dominate each other, then those sets are in descending order according to their spread metrics, which measure the extension of an approximate front. The spread metrics [31] are defined as follows:  
\[Spread(X)=\sqrt{\sum_{i=1}^{m}(\max\overline{f}_{i}(\mathbf{x},c_{i})-\min\underline {f}_{i}(\mathbf{x},c_{i}))^{2}} \tag{14}\]  
where \(\mathbf{x}\) is a solution in \(X\). The better the \(Spread(X)\), the broader the distribution of its corresponding front.  
Therefore, the original stochastic MaOPF is transformed into a bi-objective problem, shown as follows:  
\[\begin{array}{ll}\max&F(X)\triangleq(F_{1}(X),F_{2}(X))=(\underline{H(X)},-I_{\text{ave}}(X))\\ \text{subjected to}&X\in 2^{D}\end{array} \tag{15}\]  
where \(X\) is the solution set of problem and \(X\in 2^{D}\) is the power set of \(D\), which includes all subsets of \(D\).  
The aforementioned dominant relation based on sets involves comparing solutions from distinct set-based individuals, thereby intensifying the selection pressure as the number of objectives grows.