Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow

Abstract

The conventional optimal power flow (OPF) is confronted with challenges in tackling more than three objectives and the stochastic characteristics due to the uncertainty and intermittence of the RESs. However, there are few methods available that simultaneously address high-dimensional objective optimization and uncertainty handling. This paper proposes a set-based group search optimizer (SetGSO) to tackle the stochastic many-objective optimal power flow (MaOPF) of power systems penetrated with renewable energy sources. The proposed SetGSO depicts the original stochastic variables by set-based individuals under the evolutionary strategy of the basic GSO, without using repeated sampling or probabilistic information. Consequently, two metrics, hyper-volume and average imprecision, are introduced to transform the stochastic MaOPF into a deterministic bi-objective OPF, guaranteeing a much superior Pareto-optimal front. Finally, our method was evaluated on three modified bus systems containing renewable energy sources, and compared with the basic GSO using Monte Carlo sampling (GSO-MC) and a set-based genetic algorithm (SetGA) in solving the stochastic MaOPF. The numerical results demonstrate a saving of 90% of the computation time in the proposed SetGSO method compared to sampling-based approaches and it achieves improvements in both the hyper-volume and average imprecision indicators, with a maximum enhancement of approximately 30% and 7% compared to SetGA.

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. 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. [26] propose an improved NSGA-II method that constructs directed graphs based on selected solutions to predict evolving directions. To address the problem of multi-objective microgrid dispatch, the authors of ref. [27] propose an extension to interval optimization. They employ a chaotic group search optimizer along with various methods for handling multiple producers. In a similar vein, ref. [28] introduces the micro multi-objective genetic algorithm, which incorporates interval dominance relations and crowding distance. The authors consider the interval distance between two interval numbers to define these measures. In a different approach, the authors of ref. [29] present a novel angle-based interval crowding distance. Lastly, ref. [30] presents an enhanced version of the NSGA-III algorithm. This algorithm combines the genetic algorithm, the local fruit fly optimization algorithm, and measures of hyper-volume and imprecision. The authors evaluate the algorithm to assess its suitability for implementation in the local search strategy. However, it should be noted that the optimal solution obtained through different transformation methods may vary, leading to imperfect solutions. Furthermore, the number of non-dominated solutions increases exponentially as the number of targets increases, which can result in a loss of selection pressure.

The number of non-dominated solutions exhibits exponential growth as the number of objectives increases, resulting in a decrease in selection pressure for the two aforementioned methods. Additionally, the optimal solution obtained may vary depending on the transformation methods or dominance relation employed, leading to potential imperfections in the problem resolution. Addressing the issue of processing random variables, which typically necessitates repeated sampling or the selection of probabilistic inputs, becomes increasingly burdensome as the number of objectives rises. To mitigate these challenges, ref. [31] proposes a novel approach that transforms the original optimization problem into a deterministic bi-objective problem by introducing additional objectives related to hyper-volume and imprecision. Similarly, there are some group-based, multi-resolution, and hierarchical-based types of optimization. Ref. [32] presents a hierarchical algorithm, drawing inspiration from the heap data structure. The core concept behind this algorithm revolves around the interaction among subordinates and their immediate supervisors, the collaboration among colleagues, and the individual contributions of the employees. Ref. [33] proposes an improved type of teaching–learning-based optimization, which proposes the idea of grouping to enable each learner to learn from the average value of their corresponding group during the teacher stage. The Political Optimizer algorithm, as described in ref. [34], is influenced by the multi-stage political process. It assigns dual roles to each solution by segregating the population into political parties and constituencies. Ref. [35] puts forward an optimization technique rooted in the principles of human social behavior, constituting a population-based approach. Ref. [36] unveils an optimization algorithm coined the poor and rich optimization algorithm. This innovative approach finds its roots in the aspirations of two contrasting factions: one marked by financial hardship and the other by opulence, both striving to amass wealth and enhance economic prosperity. In ref. [37], a Socio Evolution and learning optimization algorithm is introduced, wherein individuals within these family units not only learn from each other but also acquire knowledge from individuals in other families within society during the evolutionary process. The authors of ref. [38] suggest a multi-resolution-based adaptation of the original SPSA. This approach broadens the search space for the optimal solution, resulting in the faster convergence of the algorithm. However, it is worth pointing out that this hierarchical or grouping-based algorithm enhances the ability of the algorithm to search different regions of the feasible search space. For solving high-dimensional multi-objective optimization and uncertainty simultaneously, the set-based method of the conversion optimization problem, that reflects convergence and imprecision, may be more effective.

However, it should be noted that the previous SetGA utilized an internal crossover operator during the evolution process, implying a lack of information exchange among the set-based individuals. To overcome this issue, we introduce the set-based concept to the basic GSO [39] and similarly transformed the original stochastic MaOP into set-based deterministic bi-objective functions using the metrics of hyper-volume and average imprecision. We introduce the concepts of global producers and local producers to enhance information exchange among set-based individuals during the evolutionary process. The scrounger follows these producers, the Pareto solution, and the search coefficient of Rangers updates along with the number of iterations to maintain the diversity of the set solution.

The contributions of this paper can be summarized as follows:

(1)

A set-based group search optimizer (SetGSO), introducing the set-based concept into the basic GSO, is proposed to transform the stochastic many-objective optimal power flow (MaOPF) into a set-based deterministic bi-objective function, utilizing metrics of hyper-volume and average imprecision. This transformation effectively addresses the issue of decreased selection pressure on Pareto optimal solutions that arises as the number of objectives increases.

(2)

The proposed SetGSO, operating under the evolutionary strategy of the fundamental GSO, delineates the underlying stochastic variables using the average imprecision metric, devoid of repeated sampling or probabilistic information. Subsequently, the set-based evolutionary strategy is embedded into the GSO, updating the set-based individuals and appointing the global and local producers using the interval credility strategy, which enhances the exchange of evolutionary information among the set-based individuals.

(3)

In terms of performance comparison, the simulation results based on three distinct node systems incorporating varying proportions of renewable energy sources indicate that the proposed algorithm yields greater computational time savings compared to the sampling method. Furthermore, in comparison to the existing set-based algorithm, SetGA, SetGSO attains superior Pareto frontier solutions.

The rest of the paper is arranged as follows. Section 2 introduces an optimal power flow problem model. Section 3 presents the stochastic many-objective optimization algorithm, SetGSO. In Section 4, simulation studies on a modified three difference bus system are discussed and Section 5 concludes the paper.

2. Modelling of Stochastic Many-Objective Optimal Power Flow Problem

The stochastic many-objective optimal power flow problem involves adjusting the control variables in a given power grid structure with uncertain output variables to find a power flow distribution that satisfies all operating constraints and optimizes system performance indicators. This study focuses on optimizing the power generation dispatch of distributed generators, minimizing actual losses in transmission lines, enhancing voltage stability at each node, and ensuring the safety and power quality of each bus bar. In addition to the equation constraints, there are also inequality constraints for distributed generators, transformers, parallel compensators, and transmission lines. These constraints include limits on generator active power, reactive power, and voltage, as well as constraints on transformers, parallel compensators, and transmission lines. Further details on these constraints are provided below.

2.1. Problem Formulation

The efficient operation of the power grid necessitates the simultaneous consideration of multiple objectives, including economic efficiency, power quality, and operational safety [27]. The specific details regarding this process are elaborated upon in a subsequent section. The stochastic OPF problem is a many-objective optimization problem that must comply with specific constraints. The subsequent formulation provides a general overview of this problem.

$$\begin{array}{cc}\hfill min& \mathbf{F}(\mathbf{X},\mathbf{U})={\left(f_1\left(\mathbf{x},\mathbf{u}\right),f_2\left(\mathbf{x},\mathbf{u}\right),\cdots ,f_M\left(\mathbf{x},\mathbf{u}\right)\right)}^T\hfill \\ \hfill \mathrm{subjected}\mathrm{to}& \mathbf{g}(\mathbf{x},\mathbf{u})=0\hfill \\ \hfill & \mathbf{h}(\mathbf{x},\mathbf{u})\le 0\hfill \\ \hfill & \mathbf{x}\in \mathbf{D}\subseteq {\mathbf{R}}^N\hfill \end{array}$$

(1)

where ${\left(f_1\left(\mathbf{x},\mathbf{u}\right),f_2\left(\mathbf{x},\mathbf{u}\right),\cdots ,f_M\left(\mathbf{x},\mathbf{u}\right)\right)}^T$ is the vector of optimization objectives, i is the $i_{th}$ objective function with interval parameters, and g and h represent the equality and inequality constraints. Due to the presence of stochastic variables, the objective function can be expressed as $f_i(\mathbf{x},\mathbf{u})\triangleq [\underline{f_i}(\mathbf{x},\mathbf{u}),\overline{f_i}(\mathbf{x},\mathbf{u})]$, $i=1,2,\cdots ,M$; M is the number of objectives. The upper and lower bounds of the objective function can be obtained from the following formulations [27]:

$$\begin{array}{cc}\hfill \underline{f_i}(\mathbf{x},\mathbf{u})& =\underset{\mathbf{u}}{min}{f}_i(\mathbf{x},\mathbf{u})\hfill \\ \hfill \overline{f_i}(\mathbf{x},\mathbf{u})& =\underset{\mathbf{u}}{max}{f}_i(\mathbf{x},\mathbf{u}).\hfill \end{array}$$

(2)

$\mathbf{x}$ is the set of decision variables, including the real power of each generator $P_G$, the voltages of generator bus $V_G$, the tap ratios of transformers $T_m$, and the reactive power generation of volt-ampere reactive sources $Q_{C_n}$. Therefore, x can be presented as follows:

$$\begin{array}{cc}\hfill & {\mathbf{x}}^{\mathbf{T}}=[P_{G_1},\dots ,P_{G_{N_G}},V_{G_1},\dots ,V_{G_{N_G}},T_1,\dots ,T_{N_T},Q_{C_1},\dots ,Q_{C_{N_C}}]\hfill \end{array}$$

(3)

where $N_G,N_T,N_C$ are the numbers of controllable generators, transformer branches, and reactive power compensation sources, respectively.

$\mathbf{U}$ is a set of stochastic variables, that represents uncertain power outputs of the WT and PV units. An interval method is used to describe uncertainty variables by using interval endpoints or midpoint and radius [40]; its form is shown in (4).

$$\begin{array}{cc}\hfill & {\mathbf{U}}^{\mathbf{T}}=[P_{W{G}_1},\dots ,P_{W{G}_{N_{WT}}},P_{V{G}_1},\dots ,P_{V{G}_{N_{PV}}}]\hfill \\ \hfill & P_{W{G}_i}=\left[\underline{P_{W{G}_i}},\overline{P_{W{G}_i}}\right],i=1,2,\cdots ,N_{WT}\hfill \\ \hfill & P_{V{G}_i}=\left[\underline{P_{V{G}_i}},\overline{P_{V{G}_i}}\right],i=1,2,\cdots ,N_{PV}\hfill \end{array}$$

(4)

where $N_{WT},N_{PV}$ are the numbers of wind and photovoltaic farms, $\underline{P_{W{G}_i}},\overline{P_{W{G}_i}},\underline{P_{V{G}_i}},\overline{P_{V{G}_i}}$, and the upper and lower bounds of wind and photovoltaic farms, respectively.

2.2. Objectives

To optimize the power flow problem, several key factors are taken into account, including the overall fuel cost of distributed generators, power loss, voltage deviation, and voltage stability index. These considerations are essential for meeting the economic, power quality, and safety requirements.

2.2.1. Fuel Cost

Economic dispatching is a crucial concern in the operation of power systems, as it aims to minimize the fuel cost of generator units while adhering to various constraints. The relationship between the output power of the distributed generator and the total fuel cost can be mathematically expressed as a quadratic function [41].

$$F_1\left({\mathit{P}}_{\mathrm{G}}\right)=\sum _{i=1}^{N_{\mathrm{G}}}{F}_i=\sum _{i=1}^{N_{\mathrm{G}}}\left(a_i{\left(P_{{\mathrm{G}}_i}\right)}^2+b_i{P}_{{\mathrm{G}}_i}+c_i\right)$$

(5)

where $F_i$ is the fuel cost of generator, and i, $a_i,b_i$ and $c_i$ are the corresponding coefficients.

2.2.2. Power Loss

The power grid operators have a strong dedication to reducing the overall power loss, which is determined by both the magnitude and phase angle of the bus voltage [39]. The formula used to calculate the total power loss is as follows:

$$F_2\left(\mathit{V},\theta \right)=\sum _{k=1}^{N_{\mathrm{E}}}{g}_k[{\left((V_i\right)}^2-{\left(V_j\right)}^2+2V_i{V}_{j}cos({\theta }_i-{\theta }_j))]$$

(6)

where $g_k$ is the conductance of a transmission line k connected between the $i_{\mathrm{th}}$ and $j_{\mathrm{th}}$ bus, and $V_i,V_j,{\theta }_i,{\theta }_j$ are the voltage magnitudes and phase angles of the $i_{\mathrm{th}}$ and $j_{\mathrm{th}}$ bus.

2.2.3. Voltage Deviation

The measurement of bus voltage deviation is a significant parameter in assessing the safety and power quality of power grid operations. It pertains to the disparity between the voltage of individual load buses and the reference voltage amplitude. Consequently, the cumulative voltage deviation across all buses is regarded as a key objective.

$$F_3\left(\mathit{V}\right)=\sum _j^{N_{\mathrm{B}}}\left|V_i-V_{\mathrm{ref}}\right|$$

(7)

where the $V_{\mathrm{ref}}$ is the referenced voltage magnitude.

2.2.4. Constraints

The efficient functioning of the power grid necessitates the fulfillment of various equality constraints $\mathbf{g}(\mathbf{x},\mathbf{u})$ and inequality constraints $\mathbf{h}(\mathbf{x},\mathbf{u})$ to maintain power balance. The power balance, which involves the equilibrium between active and reactive power, is an example of an equality constraint. Specifically, the active and reactive power generated by the generator must align with the corresponding load consumption. Consequently, a collection of nonlinear power flow constraints can be employed to represent the equality constraints $\mathbf{g}(\mathbf{x},\mathbf{u})$ within the following expression: $\mathbf{g}(\mathbf{x},\mathbf{u})$:

$$\begin{array}{cc}\hfill 0=& P_{G_i}-P_{D_i}-V_i\sum _{j=1}^{N_i}{V}_j\left(G_{ij}cos{\theta }_{ij}+B_{ij}sin{\theta }_{ij}\right)\hfill \\ \hfill & 1\le i\le N_0\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill 0=& Q_{G_i}-Q_{D_i}-V_i\sum _{j=1}^{N_i}{V}_j\left(G_{ij}sin{\theta }_{ij}+B_{ij}cos{\theta }_{ij}\right)\hfill \\ \hfill & 1\le i\le N_{PQ}\hfill \end{array}$$

(9)

where $G_{ij},B_{ij}$ are the transfer conductance and transfer susceptance between bus i and j, and $N_0,N_{PQ}$ are the number of buses excluding the slack bus and total number of PQ buses, respectively. In addition to the presence of equality constraints, the inclusion of inequality constraints $\mathbf{h}(\mathbf{x},\mathbf{u})$ primarily represents the physical limitations imposed on power equipment. These limitations primarily pertain to the active and reactive power output of distributed generators, as well as the terminal voltage of the generator. Furthermore, certain constraints related to power transformers, shunt compensator capacity, and transmission lines are also considered. These constraints are outlined as follows:

$$\begin{array}{cc}\hfill P_{G_i}^{min}& \le P_{G_i}\le P_{G_i}^{max}1\le i\le N_G\hfill \\ \hfill Q_{G_i}^{min}& \le Q_{G_i}\le Q_{G_i}^{max}1\le i\le N_G\hfill \\ \hfill V_{G_i}^{min}& \le V_{G_i}\le V_{G_i}^{max}1\le i\le N_G\hfill \\ \hfill Q_{C_i}^{min}& \le Q_{C_i}\le Q_{C_i}^{max}1\le i\le N_C\hfill \\ \hfill T_k^{min}& \le T_k\le T_k^{max}1\le k\le N_T\hfill \\ \hfill V_i^{min}& \le V_i\le V_i^{max}1\le i\le N_B\hfill \\ \hfill \left|S_k\right|& \le S_k^{max}1\le k\le N_E\hfill \end{array}$$

(10)

where $N_B,N_E$ are the total number of buses and the number of network branches. The parameters utilized in this paper can be referred to in the nomenclature section.

3. For the Stochastic MaOPF

3.1. 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 $I_{\mathrm{ave}}\left(X\right)$, 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).

$$\begin{array}{cc}\hfill & H\left(X\right)=[\underline{H\left(X\right)},\overline{H\left(X\right)}]=\hfill \\ \hfill & \wedge \left(\bigcup _{\mathbf{x}\in X}\left\{\mathbf{y}\in R^N\mid {\mathbf{x}}_{\mathrm{ref}}{\prec }_{IP}{\mathbf{y}}_{IP}\prec \mathbf{x}\right\}\right)\hfill \end{array}$$

(11)

$$I_{\mathrm{ave}}\left(X\right)=\frac{{\sum }_{\mathbf{x}\in X}{\sum }_{i=1}^m\left({\overline{f}}_i\left(\mathbf{x},{\mathbf{c}}_i\right)-{\underline{f}}_i\left(\mathbf{x},{\mathbf{c}}_i\right)\right)}{num\_A}$$

(12)

where X is an approximate Pareto-optimal solution, $\overline{H\left(X\right)}$ is the best-case hyper-volume, $\underline{H\left(X\right)}$ is the worst-case hyper-volume, ${\mathbf{x}}_{\mathrm{ref}}$ is the reference point, ⋀ 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.

$$\begin{array}{cc}\hfill & X_1{\succ }_{SetP}{X}_2\iff \hfill \\ \hfill & \left(\underline{H(X_1}\right)>\underline{H(X_2})\bigcap I_{\mathrm{ave}}\left(X_1\right)<I_{\mathrm{ave}}\left(X_2\right))\hfill \end{array}$$

(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:

$$Spread\left(X\right)=\sqrt{\sum _{i=1}^m{(max{\overline{f}}_i(\mathbf{x},c_i)-min{\underline{f}}_i(\mathbf{x},c_i))}^2}$$

(14)

where $\mathbf{x}$ is a solution in X. The better the $Spread\left(X\right)$, the broader the distribution of its corresponding front.

Therefore, the original stochastic MaOPF is transformed into a bi-objective problem, shown as follows:

$$\begin{array}{cc}\hfill max& F\left(X\right)\triangleq (F_1\left(X\right),F_2\left(X\right))=(\underline{H\left(X\right)},-I_{\mathrm{ave}}\left(X\right))\hfill \\ \hfill \mathrm{subjected}\mathrm{to}& X\in 2^D\hfill \end{array}$$

(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.

3.2. 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,\dots ,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.

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 $A=[a_L,a_R]=[\underline{f_m}({\mathbf{x}}_{\mathbf{1}},\mathbf{u}),\overline{f_m}({\mathbf{x}}_{\mathbf{1}},\mathbf{u})]$ and $B=[b_L,b_R]=[\underline{f_m}({\mathbf{x}}_{\mathbf{2}},\mathbf{u}),\overline{f_m}({\mathbf{x}}_{\mathbf{2}},\mathbf{u})]$. 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:

$$\begin{array}{cc}\hfill & P_{\mathrm{rel}}(A<B)=\frac{\mathrm{Dis}(B,D)}{\mathrm{Dis}(A,D)+\mathrm{Dis}(B,D)}\hfill \\ \hfill & \mathrm{Dis}(A,D)=max\left\{\right|a_{\mathrm{L}}-D_{\mathrm{L}}|,|a_{\mathrm{R}}-D_{\mathrm{R}}\left|\right\}\hfill \\ \hfill & \mathrm{Dis}(B,D)=max\left\{\right|b_{\mathrm{L}}-D_{\mathrm{L}}|,|b_{\mathrm{R}}-D_{\mathrm{R}}\left|\right\}\hfill \\ \hfill & D_{\mathrm{L}}=min\{a_{\mathrm{L}},b_{\mathrm{L}}\}\hfill \\ \hfill & D_{\mathrm{R}}=min\{\{a_{\mathrm{L}},b_{\mathrm{L}},a_{\mathrm{R}},b_{\mathrm{R}}\}-\left\{D_L\right\}\}\hfill \end{array}$$

(16)

where $P_{\mathrm{rel}}(A<B)$ represented the reliability of the interval A being less than the interval B. We assume $P_{\mathrm{rel}}(A<B)>0.5$ 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:

$${\mathbf{X}}_i^{k+1}={\mathbf{X}}_i^k+r_1\circ ({\mathbf{X}}_g^k-{\mathbf{X}}_i^k)+r_2\circ ({\mathbf{X}}_l^k-{\mathbf{X}}_i^k)+r_3\circ ({\mathbf{X}}_p^k-{\mathbf{X}}_i^k)$$

(17)

where $r_1,r_2,r_3\in R^1$ are constant, and ${\mathbf{X}}_g^k,{\mathbf{X}}_l^k,{\mathbf{X}}_p^k$ 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. On the other hand, the Pareto optimal solution acts as a safeguard against particles moving toward undesirable positions. Lastly, the globally optimal solution guarantees that, as one set discovers a superior solution, other sets will also converge toward the optimal solution.

For the rangers of the SetGSO, the search coefficient for updating the formula may help the algorithm escape from the local optimum [27], the formula of which is defined as follows in (18):

$$\begin{array}{cc}\hfill & {\mathbf{X}}_i^k={\mathbf{X}}_i^k+r\left(n\right)\circ ({\mathbf{X}}_p^k-{\mathbf{X}}_i^k)\hfill \\ \hfill & r\left(n\right)={\mu }_1\times r(n-1)\times (1-r(n-1))\hfill \end{array}$$

(18)

where ${\mu }_1$ is the parameters of the logistic map and n is the number of iterations.

3.3. 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 intialized. 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.

4. Case Study

In this section, the performance of the proposed multi-objective optimization algorithm is validated through two main comparisons. Firstly, a comparison is conducted with the Monte Carlo sampling method. This simulation is based on the IEEE 30-bus system, which comprises generators, power loads, wind farms, and photovoltaic plants. The topology diagram is presented in Figure 4. Secondly, a comparison is made with existing algorithms rooted in the idea of ensembles. The simulation results are based on IEEE 57-bus and IEEE 118-bus systems, which share structural similarities with the IEEE 30-bus system.

4.1. Parameter Settings

This particular case has been resolved through the utilization of three methods: SetGSO, SetGA, and GSO-MC. The parameters employed in SetGSO include a population size of 3330 solutions within a set-based individual, and 100 iterations. Additionally, the coefficient r in (18) has been set to 0.7. In SetGA, the crossover and mutation probabilities have been set to 0.9 and 0.1, respectively, while the remaining parameters remain consistent with SetGSO. The parameter settings for the GSO-MC method are identical to those of the SetGSO method.

4.2. Parameter Effect

In this section, we will investigate the influence of different quantities of set-based individuals on the search effectiveness of the SetGSO algorithm. To illustrate this, we will employ a modified version of the IEEE 30-bus system, keeping all parameters constant except for the number of set-based individuals, which will vary from 2 to 8. This study focuses on optimizing fuel cost, reducing power loss, and minimizing voltage deviation as the three main objectives. The research utilizes SetGSO to identify optimal solutions and records the average imprecision and hyper-volume of each group. Figure 4 illustrates the hyper-volume and average imprecision curves, demonstrating the impact of increasing the number of set-based individuals.

Figure 5 illustrates the relationship between the hyper-volume generated by SetGSO and the number of set-based individuals. It is observed that, as the number of set-based individuals increases, the search members also increase, leading to the discovery of more Pareto-front solutions. However, the average imprecision remains relatively stable, suggesting that it is not significantly affected by this shift. Consequently, there is a conflict between the hyper-volume and average imprecision, resulting in instances where the hyper-volume is smaller when a high number of sets is present compared to when the number of sets is small. Ultimately, the computational time will experience a substantial increase as more set-based individuals are incorporated. To summarize, based on the research objectives outlined in this study, it is determined that employing four set-based individuals is satisfactory for the forthcoming experiment.

4.3. Algorithm Comparisons

The Monte Carlo methodology [43] was employed to assess the effectiveness of the algorithm in comparison to the proposed algorithm. By utilizing the Monte Carlo method, a random sample of 100 wind speed data points from the fan was obtained to determine the optimal value of the objective function. To assess the effects of different numbers of objectives, simulation studies have been conducted on the IEEE-30 bus system.

The optimal Pareto front for the three objectives is illustrated in Figure 6 using three different approaches. It is evident that the SetGSO methods exhibit superior performance compared to the sampling method, as they converge towards a more evenly distributed set of non-dominated solutions.

The performance metrics commonly regarded as important indicators for assessing algorithm performance include: Hyper-volume (HV) [44], Mean Euclidean Distance (MED) [45], Spacing Metric (S) [46], and Run Time (RT). The calculation Formula (11) for hyper-volume has been mentioned earlier, and the calculation formulas for the MED and S Metric are as follows:

$$\mathrm{MED}=\frac{{\sum }_{i=1}^{n}dis(x_i,y)}{n}$$

(19)

where n is the number of true Pareto-optimal solutions, $dis$ dictates the Euclidean distance, and y is the reference point. The MED metric reflects the convergence of the Pareto front solutions, where a smaller value indicates a better convergence of the algorithm.

$$\mathrm{S}=\sqrt{\frac{1}{n-1}\sum _{i=1}^n{\left(\overline{d}-d_i\right)}^2}$$

(20)

where $d_i$ represents the Euclidean distance between two neighboring individuals within the non-dominated solution set, $\overline{d}$ is the average of all $d_i$, and the S metric reflects the uniformity of the distribution of individuals within the solution set, with a smaller value indicating a better uniformity of the solution set.

The algorithm introduced in this paper builds upon the original GSO framework. Therefore, our initial step involves a performance comparison between the set-based SetGSO and SetGA algorithms, as well as the GSO-MC. This comparison is based on the four metrics mentioned earlier, and the results are presented in

Table 1. Three objectives of results obtained using three methods on quantitative metrics.

Algorithm	HV	MED	S	RT (s)
GSO-MC	273,530	79.51	2.39	11,484.0
SetGA	287,230	54.52	0.80	1599.0
SetGSO	305,480	44.03	1.71	1605.5

Bold text represents superior values.

. Finally, for a more comprehensive assessment of the relative performance between the two set-based algorithms, we conduct a comparative analysis of systems with varying proportions of renewable energy sources. The details of this analysis are elaborated in the subsequent section.

According to the findings presented in Table 1, it is evident that set-based methods exhibit significantly higher computational efficiency compared to the sampling method. Additionally, the HV, MED, and S metrics also demonstrate superior performance in comparison to the GSO-MC. Moreover, SetGSO outperforms SetGA in terms of Pareto frontier solution convergence, as evidenced by the HV and MED metrics. This improvement is likely attributed to the incorporation of global and local leaders, thereby enhancing the exchange of evolutionary information among ensemble-based individuals, whereas SetGA only employs crossover evolution within the set. As the number of objectives increases, the performance of the algorithm is minimally affected. This can be attributed to the fact that multiple objectives ultimately converge into a bi-objective.

To overcome the stochastic nature of algorithm results, we conducted ten independent runs of SetGA and SetGSO algorithms on an IEEE-30 bus system featuring three objectives. The statistical outcomes are presented in

Table 2. Statistical results for three metrics in the modified IEEE-30 Bus system.

	HV		MED		S
	SetGA	SetGSO	SetGA	SetGSO	SetGA	SetGSO
Average	296,060	299,970	52.8	49.1	2.49	2.35
Median	295,810	300,090	53.2	48.9	1.43	2.02
STD.Dev	1486.2	355.9	2.3	1.2	1.42	1.47
Worst	294,560	299,570	53.4	49.5	4.73	5.10
Best	298,870	300,510	46.6	44.7	1.27	0.73

Bold text represents superior values.

. It is pertinent to note that GSO-MC, being an algorithm that relies on repetitive sampling, inherently exhibits lower stochasticity and, consequently, obviates the need for repeated executions. Based on the statistical results of the MED and HV indicators presented in Table 2, it is evident that SetGSO exhibits the best average convergence in comparison to SetGA. This further underscores the exceptional advantage of SetGSO in terms of solution convergence. Furthermore, a statistical analysis of the S metric reveals that the SetGA algorithm yields a more evenly distributed Pareto front, while the SetGSO algorithm exhibits denser clustering.

4.4. Simulation Results on the Different Proportion of REs

In this part, a simulation study is conducted using a modified IEEE 57-bus and 118-bus system. Meanwhile, three types of renewable energy with various ratios are set in this research in order to examine the effects of such renewable energy with respect to the search performance of SetGSO: (1) 30% REs; (2) 40% REs; and (3) 50% REs.

Table 3. The uncertain renewable power position of WTs and PVs in the modified IEEE-57 Bus system.

30% REs		40% REs		50% REs
Bus NO.	Type of Stochastic Variables	Bus NO.	Type of Stochastic Variables	Bus NO.	Type of Stochastic Variables
21	WT	21	WT	21	WT
22	WT	22	WT	22	WT
48	PV	48	PV	48	PV
-	-	36	WT	36	PV
-	-	37	PV	37	PV
-	-	-	-	45	WT
-	-	-	-	46	WT

and

Table 4. The uncertain renewable power position of WTs and PVs in the modified IEEE 118-Bus system.

30% REs		40% REs		50% REs
Bus NO.	Type of Stochastic Variables	Bus NO.	Type of Stochastic Variables	Bus NO.	Type of Stochastic Variables
15	WT	15	WT	15	WT
18	WT	18	WT	18	WT
19	PV	19	PV	19	PV
55	PV	24	WT	24	PV
56	PV	27	PV	27	PV
74	WT	32	WT	32	WT
76	PV	34	WT	34	WT
77	PV	36	WT	36	WT
112	WT	40	WT	40	WT
113	WT	55	PV	55	PV
-	-	56	PV	56	PV
-	-	74	-	62	WT
-	-	76	PV	70	WT
-	-	77	PV	74	WT
-	-	92	-	76	PV
-	-	99	PV	77	PV
-	-	104	-	94	WT
-	-	112	PV	95	PV
-	-	113	PV	96	PV
-	-	116	-	92	WT
-	-	-	-	99	PV
-	-	-	-	104	WT
-	-	-	-	107	WT
-	-	-	-	110	WT
-	-	-	-	112	PV
-	-	-	-	113	PV
-	-	-	-	116	WT
-	-	-	-	84	PV
-	-	-	-	82	PV
-	-	-	-	83	PV

show the specific locations of wind farms and photovoltaic power generation for the above three schemes.

In the case of four set-based individuals, the novel stochastic many-objective optimization technique, SetGSO, and the conventional approach, SetGA, function autonomously. The outcomes are presented and compared in

Table 5. Results obtained by SetGA and SetGSO on different proportions of REs in the modified IEEE-57 Bus system.

Scenario	SetGSO			SetGA
	HV	${\mathit{I}}_{\mathrm{ave}}\left(\mathit{X}\right)$	MED	HV	${\mathit{I}}_{\mathrm{ave}}\left(\mathit{X}\right)$	MED
30% REs	215,720	0.45	2485.2	173,910	0.56	2586.1
40% REs	201,360	0.61	2341.8	160,850	1.74	2477.3
50% REs	254,180	1.78	2142.4	202,080	1.37	2280.0

Bold text represents superior values.

and

Table 6. Results obtained by SetGA and SetGSO on different proportions of REs in the modified IEEE-118 Bus system.

Scenario	SetGSO			SetGA
	HV	${\mathit{I}}_{\mathrm{ave}}\left(\mathit{X}\right)$	MED	HV	${\mathit{I}}_{\mathrm{ave}}\left(\mathit{X}\right)$	MED
30% REs	6,248,200	0.13	2268.2	4,587,000	0.14	3055.5
40% REs	6,983,400	0.30	2374.0	5,028,900	0.31	3079.5
50% REs	6,969,600	0.42	1298.2	6,848,600	0.38	1586.4

Bold text represents superior values.

.

The metrics of mean Euclidean distance and hyper-volume indicate that the SetGSO approach has the potential to yield a superior Pareto front or Pareto optimum solution under similar circumstances. Furthermore, the SetGSO approach demonstrates superiority over the original SetGA method in terms of the average imprecision index, suggesting a higher level of solution stability. Furthermore, the results from Table 5 and Table 6 also demonstrate that there was limited variability in the imprecision measure, while the hyper-volume measure displayed a significant rise with an increase in the number of wind and photovoltaic farms. This implies that the associated uncertainty increases significantly as the proportion of renewable energy sources in the system continues to rise. The quest for a stable optimal solution becomes challenging. Consequently, when balancing the convergence of the Pareto frontier solutions and addressing uncertainty, the algorithm tends to prioritize the pursuit of solutions with superior convergence rather than aiming solely for minimized imprecision.

In a parallel vein, to fortify the robustness of our results, we executed ten independent runs of both the SetGA and SetGSO, employing the IEEE-57 bus and IEEE-118 bus systems as illustrative benchmarks. Based on the statistical results across three distinct test scenarios presented in

Table 7. Statistical results for the IEEE-57 bus system with three different scenarios.

Scenario		HV		${\mathit{I}}_{\mathbf{ave}}\left(\mathit{X}\right)$		MED
		SetGA	SetGSO	SetGA	SetGSO	SetGA	SetGSO
30% REs	Average	165,510	214,100	0.51	0.51	2651.6	2515.2
	Median	168,910	213,470	0.51	0.50	2648.9	2497.8
	STD.Dev	9307.4	7088.3	0.034	0.016	63.9	26.8
	Worst	148, 230	205,930	0.57	0.53	2745.6	2553.9
	Best	175,780	227,130	0.47	0.49	2578.3	2480.7
40% REs	Average	162,080	225,960	1.78	1.65	2515.6	2347.9
	Median	168,400	226,420	1.81	1.66	2489.0	2351.2
	STD.Dev	19,609	2395.6	0.14	0.016	116.7	30.54
	Worst	129,010	223,210	1.96	1.67	2717.4	2397.6
	Best	183,500	229,050	1.55	1.62	2357.2	2304.6
50% REs	Average	184,660	267,880	1.80	1.72	2351.4	2129.1
	Median	178,290	266,690	1.82	1.73	2427.4	2137.2
	STD.Dev	28,001	4327.9	0.13	0.018	107.3	21.5
	Worst	155,310	263,470	2.00	1.73	2435.1	2154.5
	Best	234,970	274,890	1.64	1.68	2160.6	2091.6

Bold text represents superior values.

and

Table 8. Statistical results for the IEEE-118 bus system with three different scenarios.

Scenario		HV		${\mathit{I}}_{\mathbf{ave}}\left(\mathit{X}\right)$		MED
		SetGA	SetGSO	SetGA	SetGSO	SetGA	SetGSO
30% REs	Average	4,411,300	6,285,160	0.13	0.13	3090.6	2305.8
	Median	4,435,700	6,284,700	0.13	0.12	3084.3	2310.3
	STD.Dev	321,175.1	46,912.2	0.038	0.017	109.1	37.7
	Worst	3,965,800	6,220,500	0.23	0.17	3248.6	2352.6
	Best	4,673,500	6,347,300	0.11	0.12	2980.8	2258.0
40% REs	Average	4,963,667	6,975,260	0.35	0.31	3047.5	2378.4
	Median	4,890,600	6,992,700	0.34	0.30	3054.2	2380.2
	STD.Dev	436,916.7	16,691.6	0.076	0.016	77.7	14.0
	Worst	4,325,400	6,954,200	0.47	0.34	3159.0	2395.9
	Best	5,579,500	7,002,500	0.25	0.29	2920.3	2354.3
50% REs	Average	6,565,667	7,058,280	0.40	0.38	1628.9	1284.0
	Median	6,578,500	7,069,600	0.38	0.38	1624.2	1283.4
	STD.Dev	298,562.7	41,298.0	0.031	0.024	68.98	18.08
	Worst	6,247,500	6,979,400	0.40	0.41	1694.0	1307.3
	Best	6,879,500	7,114,500	0.36	0.35	1523.5	1255.4

Bold text represents superior values.

, encompassing MED, HV, and $I_{\mathrm{ave}}\left(X\right)$ indicators, it can be concluded that SetGSO exhibits the optimal convergence average. This signifies that the proposed algorithm notably ameliorates imprecision whilst concurrently upholding the convergence and diversity of hyper-volume. While SetGA achieves the best outcome in terms of the $I_{\mathrm{ave}}\left(X\right)$ metric, SetGSO demonstrates relatively low result fluctuations, with even its worst outcomes surpassing those of SetGA.

5. Conclusions

This paper has proposed SetGSO to handle the stochastic many-objective optimal power flow of power systems penetrated with renewable energy sources.

The following conclusions have been drawn:

(1)

A modified IEEE-30 bus system is used to conduct numerical case studies to illustrate that the proposed solution yields more significant time savings compared to the sampling methods. The obtained Pareto fronts and multiple performance metrics have shown that the SetGSO obtains better convergence solutions. Compared with the basic GSO-MC, the SetGSO could solve the problem with much less running time.

(2)

Numerical case studies are conducted using enhanced IEEE 57 and IEEE 118 node systems. The results indicate that, in terms of performance metrics, the algorithm introduced in this study yields superior Pareto optimal solutions compared to the existing SetGA algorithm. Moreover, it produces solutions that are more stable in terms of average imprecision.

(3)

As the proportion of renewable energy sources progressively increases, the solutions demonstrate relatively stable convergence, albeit with a modest increase in average imprecision. In summary, the SetGSO method outperforms the SetGA approach in terms of the convergence and stability of the obtained frontier solution.