# Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow
## 1 Introduction  
Optimal power flow (OPF) is a crucial aspect of power system operation and planning. Typically, OPF problems optimize various objective functions by scheduling a set of control variables, while simultaneously satisfying system constraints on generator capacity, line capacity, bus voltage, and power flow balance [1]. More and more renewable energy sources (RES), including wind and solar facilities, are being incorporated into the power grid owing to the environmental deterioration and climate change brought on by the excessive use of fossil fuels [2, 3]. The stochastic many-objective optimization problem (MaOP), considering economic, reliability, and environmental indexes, and the intermittency and randomness of RESs, are crucial to the stable and safe operation of power systems. The stochastic MaOP is a challenging and trending issue, attracting much attention in the area of evolutionary computation. For one, the computation burden and the selection of non-dominated solutions become heavy, along with the increasing number of objective functions. Many researchers have focused on addressing high-dimensional multi-objective problems through dimensionality reduction methods. However, it is important to acknowledge that the reduced objectives are derived from the original objectives, and their optimization performance still requires improvement. For another, stochastic multi-objective problems necessitate the consideration of the instability of uncontrollable sources,such as wind farms and photovoltaic power plants. Failure to account for this uncertainty makes it difficult to effectively manage the output of each unit. Consequently, the handling of stochastic variables always requires repeated sampling or probability input, consuming lots of computation resources and running time. To address the first concern, the authors of ref. [4] view the weighted average of various objectives, such as economy, dependability, and safety, as a single objective. The authors of ref. [5] propose a novel aggregation function, in which they consider both convergence and diversity by using the weighted sum of horizontal and vertical distances between the subproblem and the relevant weight factor. The concept of dimensionality reduction is introduced in ref. [6], where the authors explore the relationship between multiple objectives. The study in ref. [7] employs the principal component analysis technique to investigate the correlation and significance level of the objective function. The primary aim is to retain as many of the original data as possible while eliminating redundant objectives from the complementary aims of the two pairs. The weighted method, although useful in reducing the number of objectives, is limited by its subjectivity, which hinders the attainment of genuine Pareto frontier solutions. Even when fewer objectives are considered, their independence can be compromised when highly relevant objectives are combined. Additionally, the approach of reducing objectives based on correlation tends to prioritize objectives with higher correlation for the weighted sum, thereby sacrificing the independence of the target and leading to the problem being trapped in a local optimum. In the realm of addressing uncertain multi-objective optimization problems involving stochastic variables, there exist four main approaches: fuzzy optimization, stochastic optimization, robust optimization, and interval optimization methods. The fuzzy optimization approach entails representing uncertain renewable energy as a fuzzy variable, determining a fuzzy membership degree, and constructing a fuzzy scheduling model. Nevertheless, it is crucial to acknowledge that the resultant optimal solution may be susceptible to considerable subjectivity [8]. In stochastic optimization methods, uncertainty is regarded as a random variable and is typically assumed to follow a standard probability density function.