# Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow
## 3 For the Stochastic MaOPF
### Procedure of SetGSO  
Based on the above design, the procedure of the algorithm of SetGSO is described as follows:  
Setp 1: A population \(S\) with \(P\) individuals randomly selected from a searching group is initialized. Compute the objective function values of the OPF problem according to Equations (5)-(7), and obtain the boundary values for each function using Equation (2). Then, simultaneously calculating the hyper-volume and average imprecision indicators for each set-based individual, use Equations (11) and (12) for transforming the MOPF problem into a bi-objective optimization problem. Then, sort the non-dominated sets based on the hyper-volume, average imprecision, and spread of the Pareto optimum solution set. The set iteration is \(k=0\).  
Step 2: The SetGSO archive is updated by performing several iterations. In each iteration, calculate the upper and lower bounds of each objective function presented in Equation (2). Subsequently, the midpoint value of each objective function is determined and compared with all solutions in all sets. The non-dominant sorting method is employed to identify the optimal Pareto solution, which is then stored in the external archive.  
Step 3: Select the top \(m\) set-based individuals for parent generation. To choose \(m\) set-based individuals, we use the set-based dominate relation \(\succ_{SetP}\). This randomly chooses two set-based individuals and uses the dominance relationship to find the best-performing set-based individuals. This process is repeated \(m\) times to choose \(m\) set-based individuals.  
Step 4: Set-based individuals update and create offspring individuals. The local producer is chosen from each set-based individual using the interval credibility approach illustrated in Equation (16), while the global producer is chosen from all set-based individuals via all local producers. Local and global producers, as well as the Pareto front solution, define the position of the scroungers using Equation (17). For the rest of the members, rangers adopt ranging behavior to update themselves using Equation (18).  
Step 5: Use set-based dominate relations \(\succ_{SetP}\) to produce the \(m\) set-based individuals.  
Step 6: Judge whether the termination criterion is met or not. If yes, output the optimal set-based individual; otherwise, set \(n=n+1\), and go to Step 2.  
The computing procedure is illustrated in Figure 3.  
It is worth noting that applying SetGSO to solve the OPF problem requires the transformation of the original problem. To transform the OPF problem with multiple objectives and uncertainties into a deterministic bi-objective optimization problem, firstly, We calculate the objective function values of the OPF problem represented by Equations (5)-(7). Then, obtain the upper and lower bounds of the objective function using Equation (2) and compare the midpoint values of each objective function for each member to obtain the Pareto front. Finally, compute the hyper-volume and average imprecision metrics expressed by Equations (11) and (12) for all sets. Therefore, the evolving individuals are all members of the entire set rather than individual members.