# Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow
## 1 Introduction
Commonly used probability density functions, such as Weibull and Beta, are employed to describe uncertainties in variables like wind speed and solar irradiance [9]. However, the applicability of these standard probability density functions in practical scheduling remains uncertain [10]. Furthermore, the calculation of uncertainty in stochastic optimization methods can be costly due to the need for generating a large number of scenes using specific sampling techniques [11]. In recent years, Two-Stage Robust Optimization, as an effective uncertainty-handling method, has inspired extensive research and application in the field of power system scheduling [12; 13]. This method involves representing uncertainty variables through uncertainty sets and optimizing all uncertainty parameters across the entire uncertainty set. The authors of ref. [14] propose a robust optimization technique to handle uncertainty in wind power, where the wind power output is characterized as an uncertainty set and the optimal solution is obtained by minimizing the reactive power loss under the worst-case scenario of wind power. The authors of ref. [15] propose a novel robust multi-objective optimization model based on flexible uncertainty sets to handle uncertainties, which aims to obtain Pareto optimal solutions that are less conservative. It is worth noting that robust optimization is typically considered overly cautious as it focuses on the most unfavorable scenario, which has a very low probability of occurrence [16]. Consequently, an alternative method known as interval optimization has been introduced to tackle the uncertainty problem. The interval optimization method will be elaborated upon in the subsequent discussion. Apart from robust optimization, interval optimization methods can be categorized into two main approaches. The first approach involves transforming interval optimization problems into deterministic multi-objective optimization problems. This can be achieved by converting uncertain problems into deterministic ones using the endpoints, width, and midpoint of the interval objectives. For example, a previous study [17] developed an evolutionary optimization algorithm that weighs the interval endpoints and widths to handle optimization problems with both explicit indices and implicit indices. Another study [18] converts a multi-objective interval problem into a deterministic one based on the midpoint and radius of an interval, and utilized order relations to compare interval numbers. Similarly, the authors of ref. [19] propose a distance assessment approach to address the optimization dispatch of a wind-thermal power system by splitting a single objective uncertain problem into two deterministic problems. The midpoint and width of the interval were weighted to convert each uncertain objective function into a deterministic single-objective optimization problem [20]. To address the various uncertainties present in demand response and renewable energy generation, the authors of ref. [21] convert a single objective, the operational cost of an integrated energy system, into a deterministic bi-objective function by midpoint and radius. The second approach in interval optimization involves the comparison of interval aims using alternative dominance relations or measures of potential domination, similarity, or closeness. For example, a stochastic Pareto genetic algorithm was proposed to address issues of stochastic uncertainty [22]. Another study introduced a modified NSGA-II method that defined dominance relations and crowding distance suitable for interval objectives [23]. The interval many-objective quantum-inspired cultural algorithm [24] introduces the concept of potential dominance and presents three different crowding operators. Similarly, the authors of ref. [25] enhanced the interval artificial bee colony method by incorporating credibility dominance and fresh computations for spectator bees. Additionally, the authors of ref.