Enhanced Social Group Optimization Algorithm for the Economic Dispatch Problem Including Wind Power

Abstract

The economic dispatch (ED) problem is a major challenge in power system optimization. In this article, an Enhanced Social Group Optimization (ESGO) algorithm is presented for solving the economic dispatch problem with or without wind units, considering various characteristics related to valve-point effects, ramp-rate constraints, prohibited operating zones, and transmission power losses. The Social Group Optimization (SGO) algorithm models the social dynamics of individuals within a group—through mechanisms of collective learning, behavioral adaptation, and information exchange—and leverages these interactions to guide the population efficiently towards optimal solutions. ESGO extends SGO along three complementary directions: redefining the update relations of the original SGO, introducing stochastic operators into the heuristic mechanisms, and dynamically updating the generated solutions. These modifications aim to achieve a more robust balance between exploration and exploitation, enable flexible adaptation of search steps, and rapidly integrate improved-fitness solutions into the evolutionary process. ESGO is evaluated in six distinct cases, covering systems with 6, 40, 110, and 220 units, to demonstrate its ability to produce competitive solutions as well as its performance in terms of stability, convergence, and computational efficiency. The numerical results show that, in the vast majority of the analyzed cases, ESGO outperforms SGO and other known or improved metaheuristic algorithms in terms of cost and stability. It incorporates wind generation results at an operating cost reduction of approximately 10% compared to the thermal-only system, under the adopted linear wind power model. Moreover, relative to the size of the analyzed systems, ESGO exhibits a reduced average execution time and requires a small number of function evaluations to obtain competitive solutions.

1. Introduction

The economic dispatch (ED) problem represents a central component of operational optimization mechanisms in modern power systems, being employed to determine the optimal allocation of generation among available units such that the demand is met at minimum cost, while strictly satisfying the operational constraints of the generating units and the entire electricity transmission system [1,2].

In the current context, marked by the increasing penetration of renewable resources and by the pressure on operators to maintain economic competitiveness in a volatile environment, the analysis of the ED problem has become an indispensable tool for electricity producers [3]. The outcomes of such analyses directly influence operational strategies and production planning.

In the absence of wind integration, the ED problem exhibits a deterministic structure in which the technical parameters of conventional units (cost curves, operating limits, and ramp-rate constraints) allow a well-defined non-linear mathematical formulation, suitable for solution through classical optimization methods or metaheuristics [4]. Under these conditions, generation companies can rely on predictable system behavior, facilitating the establishment of efficient production schedules and strict control of marginal operating costs.

Several recent algorithms proposed for solving the classical ED problem are summarized below. The Hybridizing Gaining–Sharing Knowledge (GSK) and Differential Evolution (DE) approach is a hybrid method combining the global search ability of DE with the solution-refinement capacity of GSK [5]. The Exponential Hybrid Mutation Differential Evolution (EHMDE) introduces two key strategies—exponential population-size reduction and a hybrid mutation operator—designed to balance exploration and exploitation during optimization [6]. The Hybrid Gray Wolf Optimization (HGWO) algorithm integrates a robust learning mechanism intended to prevent the stagnation of search agents in suboptimal regions [7]. The Growth Optimizer is a recent metaheuristic that inspires individuals to enhance their capabilities through learning and reflection as they evolve within society [8]. The Improved Pelican Optimization Algorithm introduces strategies that more accurately model more the stochastic hunting behavior of pelicans [9]. The Enhanced Social Network Search (ESNS) algorithm improves the original version by increasing search intensity around promising solutions, enhancing exploration, and avoiding entrapment in local minima [4]. The Leader White Shark Optimization algorithm incorporates a leadership mechanism aimed at intensifying exploitation and improving the efficiency of ED problem solving [10]. The Wild Horse Optimizer is a new metaheuristic designed to be simple to implement and to exhibit rapid convergence when applied to the ED problem [11]. The Enhanced Manta Ray Foraging Optimization method provides improved search mechanisms for ED systems of various sizes and characteristics [12]. Finally, the Memetic Salp Swarm Algorithm is a hybrid method that employs an adaptive β-hill climbing strategy for local search [13].

Integrating wind generation into power systems fundamentally alters the ED architecture. The variability of wind power output and the inherent uncertainty of meteorological forecasts require additional probabilistic modeling instruments capable of characterizing wind speed fluctuations in a rigorous manner [14]. For power companies, this extension introduces opportunities associated with cost reduction through the utilization of a resource with a very low marginal production cost. Mathematically, the inclusion of wind generation significantly increases the complexity of the ED problem, as the interaction between uncertainty and the optimal decision becomes a key determinant. The objective function involves multiple terms (e.g., fuel costs, penalties for forecast deviations), reflecting the operational realities of generating companies.

Several mathematical optimization models and appropriate solution algorithms have been proposed in the literature for ED problems that include wind power uncertainty. For example, Ref. [14] introduces a model that extends the classical ED formulation by integrating wind farms into the power system, where wind speed is represented by a Weibull distribution. In addition to fuel cost, the model includes specific penalties for wind power overestimation and underestimation, thus explicitly addressing the risks associated with fluctuating production. Compared to [14], the studies in [15,16] incorporate an additional objective term reflecting emission costs due to thermal unit operation. Optimization for wind–thermal power systems is performed using Quantum-Inspired Particle Swarm Optimization [15] and Gbest-Guided Artificial Bee Colony [16].

In [17], the ED problem with wind integration is solved using the Honey Bee Mating Optimization algorithm with an embedded online learning mechanism, where wind-speed uncertainty is estimated using the 2 m-point method.

The study in [18] applies the Lagrange multiplier method to several power systems (6-unit, 10-unit, and 40-unit) with various demand levels, modeling wind speed through a Weibull distribution. The Fractal Firefly Algorithm is proposed in [19] for allocating generation among thermal and wind units under constraints such as unit capacity limits, power balance, multi-fuel operation, and ramp-rate limitations.

In [20], the authors introduce a hybrid algorithm based on Particle Swarm Optimization (PSO) operating in a scenario-based framework in which wind generation uncertainty is represented through multiple probabilistic scenarios. The method combines enhanced exploration mechanisms with robust selection strategies to ensure that the obtained solutions remain effective under varying wind conditions. Numerical results show that the proposed method achieves lower operating costs and improved convergence stability compared with standard PSO and other existing methods.

An economic dispatch model focusing on the management of wind generation uncertainty is described in [21]. To ensure computational efficiency, the authors employ an Alternating Direction Method of Multipliers (ADMM) combined with a Finite-Step Consensus strategy.

In [3], the authors formulate a probabilistic-constraint ED model and propose a hybrid optimization method that integrates chaotic maps into the Artificial Bee Colony (ABC) algorithm to enhance search diversification and prevent premature convergence—key aspects when dealing with the inherent variability of wind resources. Comparative experiments on test systems (10-unit, 40-unit) demonstrate that the proposed method reduces total operating cost and improves convergence stability relative to classical ABC and other metaheuristics.

In computational intelligence, the modeling of collective and cognitive processes has proven to be a powerful paradigm for developing robust optimization techniques. This is evident across diverse domains, from structural advancements in brain–machine contextual communication [22] to the design of swarm- and social-based metaheuristics. The Social Group Optimization (SGO) algorithm [23] is part of this paradigm, being inspired by the dynamics of social interactions within human groups. It models individual behavior through mechanisms such as social learning, cooperation, and context-based adaptation, aiming to achieve an effective balance between exploration and exploitation of the search space.

The authors propose a solution-update process based on the influence of leaders and information exchange among group members. Tested on an extensive set of benchmark functions, SGO has demonstrated competitive performance relative to established metaheuristics (ABC, PSO, DE, and TLBO), highlighting its potential for global optimization. However, the standard SGO version exhibits several limitations, such as a tendency to stagnate in local optima [24], loss of population diversity [25], low sensitivity to decision-variable changes [25], and reduced stability for large-scale optimization problems [26]. These shortcomings indicate that the algorithm cannot be universally applied without adaptation, as its performance becomes limited in many contexts. The limitations highlighted for the standard version of the SGO algorithm are not isolated, but rather belong to a broader set of challenges characteristic of population-based metaheuristic algorithms. Phenomena such as early search bias toward suboptimal solutions, progressive loss of population diversity, or reduced responsiveness to variations in decision variables are frequently reported in the specialized literature for algorithms inspired by collective behaviors. In this context, improving solution-update mechanisms and search-step generation strategies represents an essential direction for increasing the efficiency of the optimization process.

To overcome these limitations, various SGO variants have been developed, targeting the following: (i) enhanced exploration–exploitation capability using a Self-Perception Probability (SAP) mechanism in the acquiring phase [27]; (ii) reduction in local stagnation by integrating chaotic maps [28]; (iii) dynamic adjustment of the exploration–exploitation balance via an adaptive inertia parameter controlled by a sigmoid function [29]; (iv) maintaining population diversity through new heuristics embedded in the improving phase [30]; and (v) achieving more robust balance through hybridizations with WOA [25] or DE [31]. In this context, the proposed ESGO variant is designed to overcome the limitations of a simple local adjustment of the SGO algorithm through an approach aimed at improving the global behavior of the search process. The introduced modifications seek to support a more effective exploration–exploitation balance, preserve population diversity, and increase the stability of the optimization process. Thus, ESGO provides a strengthened framework for addressing complex optimization problems, and its usefulness is demonstrated in this work in the context of the ED problem.

In this article, to enhance SGO’s ability to identify high-quality and stable solutions for the ED problem, we develop a new variant of the algorithm, denoted Enhanced SGO (ESGO). ESGO introduces a set of restructured heuristics relative to those in SGO, as well as new stochastic operators designed to generate more effective search steps, leading to superior solutions and lower overall costs compared to the traditional SGO.

The main contributions of this article are summarized below:

(i)

A new version of the SGO algorithm, denoted ESGO, in which the heuristics used for solution updates have been modified to enable searching both in the vicinity of the best solution (in the improving and acquiring phases) and around the current solution (in the acquiring phase). In the acquiring phase, activation of a heuristic is determined through a double conditioning based on the fitness value and a randomly generated number. Additionally, to obtain appropriate search steps, heuristics incorporate stochastic operators based on combinations of uniformly distributed and Cauchy-distributed random numbers. To the best of the authors’ knowledge, the Cauchy distribution has not previously been integrated into the SGO structure.

(ii)

Implementation of ESGO for solving the economic dispatch (ED) problem, considering various practical characteristics, including the presence of wind units.

(iii)

Evaluation of ESGO’s feasibility and efficiency for the ED problem on multiple power system configurations (6-unit, 40-unit, 110-unit, and 220-unit). Furthermore, the Wilcoxon nonparametric statistical test is used to determine whether significant differences exist between ESGO and SGO across the analyzed test cases. The results show that ESGO outperforms SGO and other benchmark algorithms reported in the literature for solving the ED problem.

The structure of the article is organized as follows. Section 2 presents the mathematical optimization model of the ED problem, with and without wind integration. Section 3 describes the structure and heuristics underpinning the SGO algorithm. Section 4 details the enhanced ESGO version. Section 5 explains how ESGO is implemented for the ED problem, with or without wind integration. Section 6 presents the analyzed cases and the numerical results obtained by SGO, ESGO, and other reference algorithms. The conclusions are summarized in Section 7.

2. Statement of the ED Problem

We consider a power system in which both thermal units and wind units coexist. The economic dispatch (ED) problem consists of determining an optimal generation schedule for each generating unit such that the total operating cost is minimized, the power demand is fully supplied over the scheduling horizon, and the technical limits of the units and of the system are satisfied.

For the mathematical formulation of the optimization model, we assume that the system includes n = t + w units, where t is the number of thermal units and w is the number of wind units. The power outputs of all units are stored in the vector P = [W, T] = [P₁, P₂, …, P_j, …, P_n], which is composed of the vectors W and T, where W = [W₁, W₂, …, W_q, …, W_w] and T = [T₁, T₂, …, T_k, …, T_t] represent the power output vectors of the wind and thermal units, respectively. The total operating cost (Cost), which defines the objective function for the ED problem, is composed of the cost associated with the thermal units, (CostThermal) and the cost associated with the wind units (CostWind):

$$Cost(T,W)=CostThermal(T)+CostWind(W)$$

(1)

These two types of costs can be defined as follows [14,32]:

$$CostThermal(T)=\sum _{k=1}^t{CostT}_k\left(T_k\right)=a_k{T}_k^2+b_k{T}_k+c_k+\left|e_k\mathit{sin}(f_k(T_{min,k}-T_k))\right|$$

(2)

$$CostWind(W)=\sum _{q=1}^w{CostW}_q(W_q)=\sum _{q=1}^w\left\{ d_q\cdot W_q+o_q·M_q^{over}(W_q)+u_q·M_q^{under}(W_q) \right\}$$

(3)

where a_k, b_k, c_k, e_k, and f_k are the fuel cost coefficients of thermal unit k; T_k,min is the minimum generation capacity of thermal unit k; $M_q^{over}(W_q)$, $M_q^{under}\left(W_q\right)$ represent the average power corresponding to the overestimation and, respectively, underestimation of wind power of unit q; d_q, o_q, and u_q are the direct cost, overestimation, and underestimation coefficients for q unit.

The production of the thermal units is associated with the fuel cost for each unit k—CostT_k(T_k), modeled by the non-convex fuel cost function in (2). The wind power generation of unit q involves an aggregated cost—CostW_q(W_q) that includes the following [14]: direct operating costs—the term d_q·W_q in relation (3), overestimation penalty costs—modeled by the term $o_q·M_q^{over}(W_q)$, and underestimation penalty costs—modeled by $u_q·M_q^{under}\left(W_q\right).$ The penalty costs are average costs introduced to model the mismatch between the scheduled and the actual power output of each wind unit q. Thus, overestimation occurs when the scheduled power level for a wind unit is higher than the power that the unit can effectively deliver, given the available wind resource and the forecasted wind conditions at the site; underestimation occurs when the scheduled power is lower than the power available from the wind resource. Equations (2) and (3) are among the most widely used models for defining the costs associated with thermal and wind units and provide good accuracy.

The average powers $M_q^{over}(W_q)$ and $M_q^{under}\left(W_q\right)$ are determined considering the probability distribution of the random variable wind power (W), which depends on the random variable wind speed (V). In this work, we assume that V follows a Weibull distribution with probability density function f_V(v) given by (5), while the relationship between V and W is assumed to be linear [16]:

$$W=\left\{\begin{array}{c}0, if V<v_{in}or V>v_{out}\\ {\displaystyle \frac{(V-v_{in})Wr}{v_r-v_{in}}},\hspace{0.33em}if\hspace{0.33em}{v}_{in}\le V\le v_r\\ Wr,\hspace{0.33em} if\hspace{0.33em}{v}_r\le V\le v_{out}\end{array}\right.$$

(4)

$$f_V(v)={\displaystyle \frac{k}{c}}{\left({\displaystyle \frac{v}{c}}\right)}^{k-1}\mathit{exp}\left[-{\left({\displaystyle \frac{v}{c}}\right)}^k\right]$$

(5)

where k and c are the shape and the scale parameter, respectively, and v is a value of the random variable V.

The functional relationship W = f(V) explicitly defined in (4) requires the knowledge of the wind speed characteristics for a given location: v_in (cut-in wind speed, representing the minimum wind speed required for turbine operation), v_r (rated wind speed—speed at which turbine reaches maximum power), and v_out (cut-out wind speed—maximum safe operating speed), as well as the rated power (W_r) of the wind unit. Within the operating range v_in ≤ V ≤ v_r, a linear relationship between wind speed and generated power is assumed. This approximation is commonly adopted in economic dispatch studies to reduce model complexity and computational burden. However, since actual wind turbine power curves are inherently non-linear, this assumption may introduce estimation errors, particularly under high wind penetration levels, which can affect the accuracy of dispatch results.

Given that the wind speed at the site of wind unit q follows a Weibull distribution, the average powers $M_q^{over}(W_q)$ and $M_q^{under}\left(W_q\right)$ can be calculated as follows [14]:

$$M_q^{over}(W_q)={\int }_0^{W_q}\left(W_q-w\right)f_{W_q}(w)dw+\left(W_q-0\right)\cdot \mathrm{Prob}(W_q=0)$$

(6)

$$M_q^{under}(W_q)={\int }_{W_q}^{W{r}_q}\left(w-W_q\right)f_{W_q}(w)dw+\left(W{r}_q-W_q\right)\cdot \mathrm{Prob}(W_q=W{r}_q)$$

(7)

where w are the possible values of the random variable wind power W_q, and $f_{W_q}(w)$ is the probability density function for W_q. Since the distribution of V is known and the relation W = f(V) is given, the probability density function f_W(w) of the wind power W can be obtained as follows:

$$f_W(w)={\displaystyle \frac{k\cdot v_{in}\cdot R}{c\cdot Wr}}{\left({\displaystyle \frac{v_{in}}{c}}\cdot \left(1+{\displaystyle \frac{w\cdot R}{Wr}}\right)\right)}^{k-1}\mathit{exp}\left[-{\left({\displaystyle \frac{v_{in}}{c}}\cdot \left(1+{\displaystyle \frac{w\cdot R}{Wr}}\right)\right)}^k\right]$$

(8)

where R = (v_r − v_in)/v_in.

Prob(W_q = 0) represents the probability that the wind power W_q is equal to zero, i.e., the probability that the wind speed $V$ is either less than $v_{\mathit{in}}$ or greater than $v_{\mathit{out}}$ [14]:

Prob (W_q = 0) = Prob (V < v_in) + Prob (V > v_out) = 1 − exp[−(v_in/c)^k]+ exp[−(v_out/c)^k]

(9)

Prob(W_q = Wr_q) is the probability that the wind power of unit q equals its rated capacity W_q, i.e., the probability that the wind speed V lies between v_r and v_out [14]:

Prob(W = Wr_q) = Prob(v_r < V < v_out) = exp[−(v_r/c)^k] − exp[−(v_out/c)^k]

(10)

The set of constraints imposed on the ED problem including wind power is as following [15]:

T_k,min ≤ T_k ≤ T_k,max, k = 1, 2, …, t

(11)

W_q,min ≤ W_q ≤ W_q,max, q = 1, 2, …, w

(12)

$$Max\left(T_{k,min},T_k^0-{DR}_k\right)\le T_k\le Min\left(T_{k,max},T_k^0+{UR}_k\right), k=1, 2, \dots , t$$

(13)

$$T_{k,z}^{inf}\le T_k\le T_{k,z}^{sup}, k=1, 2, \dots ,t, z=1, 2,..., n{z}_k$$

(14)

where (11) defines the minimum (T_k,min) and maximum (T_k,max) power limits of thermal unit k. Similarly, the power outputs of the wind units W_q are bounded by a lower limit (W_q,min) and upper limit (W_q,max), as in (12). If the power output ($T_k^0$) of thermal unit k in the previous hour is imposed, then its operating limits are further restricted by (13), where DR_k and UR_k are the down-ramp and up-ramp limits of the unit k. An additional type of constraint for the generating units refers to prohibited operating zones (POZ), described mathematically by (14). POZs are power intervals in which unit operation is not allowed due to technical reasons, such as mechanical vibrations, resonance phenomena, thermal instabilities, or significant efficiency degradation. The limits P_k,z^inf and P_k,z^sup represent the lower and upper bounds of the prohibited zone z for unit t, while nz_k represents the number of prohibited zones of unit t.

Additionally, at system level, the power balance between generated power (P_G) from thermal and wind units, demanded power (P_D), and power losses (P_L) must be maintained:

$$\stackrel{P_G}{\overbrace{\sum _{k=1}^t{T}_k+\sum _{q=1}^w{W}_q}}-P_L-P_D=0$$

(15)

The transmission losses P_L can be calculated using the B coefficient loss formula (Kron’s formula):

$$P_L=\sum _{i=1}^n \sum _{j=1}^n{P}_i\cdot B_{ij}\cdot P_j+\sum _{j=1}^n{B}_{0j}\cdot P_j+B_{00}$$

(16)

where B_ij, B_0j, and B₀₀ are constant and known loss coefficients for a given load level. For a candidate solution vector [P], the satisfaction of the equality constraint (15) is achieved with less accuracy than an imposed ε:

Accuracy = |P_G − P_L − P_D| < ε

(17)

3. Social Group Optimization (SGO)

Social Group Optimization (SGO) is a metaheuristic algorithm inspired by human social behavior, proposed for solving complex optimization problems and various engineering applications by Satapathy and Naik [23]. The basic philosophy of SGO is founded on the observation that individuals within a social group tend to improve their behavior based on interactions with other members of the group.

The essential elements of the social group are the number of individuals in the group, denoted by (N), and the set of traits associated with each individual, denoted by (n). Within the social group, the following associations are established: individuals in the social group are associated with candidate solutions, X_i, i = 1, 2, …, N, the traits of the individuals correspond to the components x_ij, j = 1, 2, …, n of solution X_i = [x_1,i, x_2,i, …, x_j,i, …, x_n,i], and the knowledge accumulated by individuals is associated with the fitness function, f(X_i).

The SGO algorithm starts with an initialization phase in which the traits of the individuals x_j are generated randomly between the lower bounds (X_min = [x_1,min, x_2,min, …, x_j,min, …, x_n,min]) and upper bounds (X_max = [x_1,max, x_2,max, …, x_j,max, …, x_n,max]) imposed by the problem. The update of the individuals’ traits is carried out in two successive phases. Improving phase—the best individual in the group (X_best) attempts to transfer their knowledge to the other members (X_i, i = 1, 2, …, N) in order to help them improve their knowledge base. Acquiring phase—the traits of each individual X_i are updated to a new solution X_i,new based on the interaction among three individuals: the current individual X_i, the best individual in the group X_best o, and a randomly selected individual X_r, r ≠ i. The mathematical modeling of these concepts in the three phases is given by relations (18)–(20) [23]:

$$\begin{array}{}\text{The initialization phase:} X_i=X_{\mathit{min}}+r_1·(X_{\mathit{max}}-X_{\mathit{min}}), i=1, 2, \dots , N\end{array}$$

(18)

$$\begin{array}{}\text{The improving phase:} X_{i,\mathit{new}}=c·X_i+r_2·(X_{\mathit{best}}-X_{\mathit{i}}), i=1, 2, \dots , N\end{array}$$

(19)

$$\begin{array}{}\text{The acquiring phase: If} f\left(X_{\mathit{i}}\right)<f\left(X_{\mathit{r}}\right), \text{Then} X_{i,\mathit{new}}=X_{\mathit{i}}+r_3(X_{\mathit{i}}-X_{\mathit{r}})+r_4 (X_{\mathit{best}}-X_{\mathit{i}})\end{array}$$

(20a)

$$\begin{array}{}\text{Else} X_{i,\mathit{new}}=X_{\mathit{i}}+r_3 (X_{\mathit{r}}-X_{\mathit{i}})+r_4 (X_{\mathit{best}}-X_{\mathit{i}}), i=1, 2, \dots , N\end{array}$$

(20b)

where X_i,new is the new solution obtained by updating X_i using (19) or (20); X_best is the best solution up to that point (the individual with the best traits/knowledge); for a minimization problem, X_best corresponds to min{f_i,i = 1, 2, …, N}. X_r is a randomly selected solution from the group, r ≠ i; X_min and X_max are the imposed lower and upper bounds; c is the self-introspection factor (0 < c < 1); r₁, r₂, r₃, and r₄ are random numbers uniformly distributed in the interval (0, 1).

The execution of the SGO algorithm stops when the maximum number of iterations, t_max, is reached. The best solution (X_best) and its associated fitness value f(X_best) are then recorded. A pseudo-code of the SGO algorithm is given below in Algorithm 1 [23].

Algorithm 1: SGO algorithm

Set: the population size (N), maximum number of iterations (tmax) {Initialization phase} Initialize solutions Xi, i = 1, 2, …, N using (18) Evaluate fitness f(Xi), then identify Xbest, and f(Xbest) while t ≤ tmax  for i = 1 to N do {Improving phase}           Update Xi,new by relation (19), then evaluate f(Xi,new)           Accept Xi,new if it has a better fitness end for Identify Xbest, and f(Xbest) in the current population  for i = 1 to N do {Acquiring phase}           Randomly select a solution Xr, r ≠ i           Update Xi,new using (20a), or (20b)           Evaluate f(Xi,new)           Accept Xi,new if it has a better fitness end for           Identify Xbest, and f(Xbest) from the population           t = t + 1 end while Store Xbest, and the fitness f(Xbest).

4. Enhanced SGO (ESGO)

The proposed Enhanced SGO (ESGO) algorithm preserves the overall structure and evolutionary steps of the original SGO algorithm; however, several internal mechanisms of SGO are modified in order to increase the global efficiency of ESGO. In its classical form, the SGO algorithm performs the solution update through search steps whose values are controlled exclusively by random variables r₁–r₄, generated uniformly in the interval (0, 1), according to relations (19) and (20). While this strategy favors diversity and exploration of the search space, its efficiency can be limited for problems with high complexity, such as those where the objective function has many local minima and/or a high dimensionality of variables. Therefore, the exploration–exploitation balance may not be adequate, potentially leading to stagnation in local minima or inconsistent solution quality.

To improve the search process in ESGO compared to the traditional SGO, three directions are followed: modification of the update relations for the solutions (heuristics); inclusion of random operators within the heuristics; and dynamic update of the obtained solutions. Each direction targets a specific objective. Thus, modifying the solution-update relations aims to achieve a better exploration–exploitation balance during the search (avoiding stagnation and obtaining more refined solutions). Including random operators in the heuristics is intended to produce search steps with appropriate amplitude and direction. Dynamically updating improved solutions aims to introduce better candidate solutions into the search process as quickly as possible. Concretely, the improvements introduced in ESGO with respect to SGO are as follows:

(i).

In the improving phase, the relation (19) of SGO is replaced by (21) in ESGO:

$$\begin{array}{}{X}_{i,\mathit{new}}=X_{\mathit{best}}+\mathit{rand}\left(1\right)·\mathit{rChy}(0, s)·(X_{\mathit{best}}-X_{\mathit{i}}), i=1, 2, \dots , N\end{array}$$

(21)

(ii).

In the acquiring phase, relations (20) are replaced by (22) and (23) in ESGO:

$$\begin{array}{}\text{If} f\left(X_{\mathit{i}}\right)<f\left(X_{\mathit{r}}\right) \text{Then}\end{array}$$

$$\begin{array}{} \text{If} \mathit{rand}\left(1\right)<0.5 \text{Then} X_{i,\mathit{new}}=X_{\mathit{best}}-\mathit{rand}\left(1\right) \mathit{cos}\left(\mathit{rand}\right(2\pi \left)\right)|X_{\mathit{best}}-X_{\mathit{i}}|\end{array}$$

(22a)

$$\begin{array}{} \text{Else} X_{i,\mathit{new}}=X_{\mathit{best}}+\mathit{sin}\left(\mathit{rand}\right(2\pi \left)\right)·(X_{\mathit{i}}-X_{\mathit{r}})+\mathit{rand}\left(1\right)·\mathit{rChy}(0, s)·(X_{\mathit{best}}-X_{\mathit{i}})\end{array}$$

(22b)

$$\begin{array}{}\text{Else} \{\text{end If} f(X_{\mathit{i}})<f(X_{\mathit{r}}\left)\right\}\end{array}$$

$$\begin{array}{} \text{If} \mathit{rand}\left(1\right)<0.5 \text{Then} X_{i,\mathit{new}}=X_{\mathit{i}}-\mathit{rand}\left(1\right) \mathit{cos}\left(\mathit{rand}\right(2\pi \left)\right)|X_{\mathit{best}}-X_{\mathit{i}}|\end{array}$$

(23a)

$$\begin{array}{} \text{Else} X_{i,\mathit{new}}=X_{\mathit{i}}+\mathit{sin}\left(\mathit{rand}\right(2\pi \left)\right)·(X_{\mathit{r}}-X_{\mathit{i}})+\mathit{rand}\left(1\right)·\mathit{rChy}(0, s)·(X_{\mathit{best}}-X_{\mathit{i}})\end{array}$$

(23b)

where rand(1) and rand(2π) are random numbers generated uniformly in (0, 1), respectively (0, 2π), and sin() and cos() are the sine and cosine trigonometric functions. rChy(0, s) is a random number generated according to the Cauchy distribution, with parameters x₀ = 0 and s.

The probability density function f(x) and the distribution function F(x) for the Cauchy distribution are defined as follows:

$$f(x;x_0,s)={\left[\pi s\left(1+{\left({\displaystyle \frac{x-x_0}{s}}\right)}^2\right)\right]}^{-1}, x\in R$$

(24)

$$F\left(x\right)={\displaystyle \frac{1}{\pi }}arctan\left({\displaystyle \frac{x-x_0}{s}}\right)+0.5$$

(25)

where x₀ is the location parameter and s is the scale parameter.

Figure 1 illustrates the shape of the Cauchy distribution for x₀ = 0 and different values of the scale parameter: very small (s = 0.01), moderately small (s = 0.1), and relatively large (s = 1). From Figure 1, it can be observed that the Cauchy distribution is symmetric and can generate positive and negative values with higher probability when the value of the parameter s increases. This behavior can be exploited to control the amplitude and direction of the computational steps in the heuristic relations (21)–(23).

Figure 1. Function f(x) for Cauchy distribution.

(iii).

In ESGO, the uniformly distributed random variables r₂–r₄, used in the original SGO algorithm, are replaced by explicitly defined random operators that are directly embedded in the solution-update relations (21)–(23). In this work, a random operator is defined as a stochastic component with a specific mathematical form that provides explicit control over the scaling and/or sign of the search step, while the geometric orientation of the step is determined by the vector differences between candidate solutions. Three types of explicitly defined random operators are employed:

(a)

Uniform random operator rand(0, 1), which introduces stochastic scaling of the update step;

(b)

Trigonometric random operator sin(rand(0, 2π)) or cos(rand(0, 2π)), whose absolute value controls the step magnitude, while its sign may reverse the orientation of the displacement along a given reference direction;

(c)

Cauchy-based random operator rChy(0, s), which generates heavy-tailed random values symmetrically distributed around zero. Due to its sign and magnitude, this operator can both amplify or attenuate the step length and invert its direction. The scale of these variations is adaptively regulated by the parameter s, which decreases linearly from b to a (b > a > 0) over the iterations, thereby progressively limiting large-amplitude steps and enabling a smooth transition from exploration to exploitation.

In the update relations (21)–(23), the vector differences, such as (X_best − X_i) and (X_i − X_r), define the reference geometric directions of movement and contribute implicitly to the step magnitude through their norms. The explicitly defined random operators act multiplicatively on these vectors and are responsible for the stochastic regulation of the effective step size and, in some cases, of the direction sign. In relation (21), a composite random operator rand(0, 1)⋅rChy(0, s) is employed to adaptively scale the displacement toward the current best solution, guiding the search process in the direction of X_best. In the acquiring phase, alternative update heuristics given by relations (22a)–(22b) and (23a)–(23b) are selected stochastically according to the condition rand(0, 1) < 0.5, enabling random switching between locally and globally oriented search behaviors. This combination of explicitly defined random operators and stochastic heuristic selection ensures a balanced and adaptive exploration–exploitation process.

Applying the inverse transformation method to relation (25), we obtain the procedure for generating a random number from a Cauchy distribution with fixed parameters x₀ and s:

$$rChy\left(x_0,s\right)=x_0+s·tan\left(\pi ·\left(u-0.5\right)\right)$$

(26)

where u is a random number uniformly distributed in the interval (0, 1), and tan(·) is the tangent trigonometric function.

In this work, the parameter x₀ is set to x₀ = 0 (to allow the generation of both positive and negative values with equal probability of 50%), while s decreases linearly from β to α (β > α > 0) depending on the current iteration index t and the maximum number of iterations t_max:

$$s=\beta -(\beta -\alpha )·{\displaystyle \frac{t-1}{t_{max}-1}}$$

(27)

According to relation (27), in the early iterations of the optimization process, the parameter s of the Cauchy distribution takes relatively high values, close to β, which favors the exploration phase of the search space. As the iterative process evolves, the value of s decreases, favoring the exploitation phase. Thus, incorporating the Cauchy distribution into the heuristics allows both exploration and exploitation phases to be carried out, as well as a gradual transition from “predominantly exploration” to “predominantly exploitation”. Figure 2 highlights the points generated by the Cauchy distribution for moderate values of the parameter s (s = 0.1, shown by the plot of rChy(0, 0.1)—green points) and for higher values (s = 1, shown by the plot of rChy(0, 1)—blue points). It can be observed that for s = 0.1, the generated values are concentrated close to the origin, x₀ = 0, while for s = 1, they exhibit a much larger spread.

Figure 2. Values generated by rChy(0, 0.1) and rChy(0, 1).

(iv).

At each update of the solutions obtained using relations (21)–(23), a greedy process is applied between X_i (the old solution) and X_i,new (the new solution) to select and retain the solution with the better fitness value. The same procedure is used when selecting the best solution (X_best) generated up to the current moment.

(v).

In ESGO, the parameter c is eliminated, and two new parameters (α, β) are introduced, which must be adjusted to increase the performance of the proposed algorithm.

The computational steps for implementing the proposed ESGO are presented in Algorithm 2.

Algorithm 2: ESGO algorithm

Set the parameters for ESGO: N, tmax, α, β {Initialization phase} Initialize solutions Xi, i = 1, 2, …, N using relation (18) Evaluate the fitness f(Xi), then identify Xbest, and f(Xbest) while t ≤ tmax  for i = 1 to N do {Improving phase}      Update ≤ the s parameter using relation (27)      Generate a random number, rChy(0, s), based on relation (26)      Update Xi,new using relation (21), then evaluate the fitness f(Xi,new)      Update Xi, and XBest, (and correspondingly f(Xi), and f(XBest)) if the new solution has better fitness  end for  for i = 1 to N do {Acquiring phase}      Randomly select a solution Xr, i ≠ r      Update the parameter s using relation (27)      Generate a random number, rChy(0, s), based on relation (26)      If f(Xi) < f(Xr) Then update Xi,new using relations (22a) and (22b)                               Else update Xi,new using relations (23a) and (23b) Update Xi, and XBest, (and correspondingly f(Xi), and f(XBest)) if the new solution has a better fitness  end for t = t + 1 end while Store Xbest, and the fitness f(Xbest).

5. Implementing the ESGO for the ED Problem

We consider that a solution i of the ED problem is represented by the power vector P_i = [W_i, T_i] = [[W_1i, W_2i, …, W_qi, …, W_wi] [T_1i, T_2i, …, T_ki, …, T_ti]] = [P_1,i, P_2,i, …, P_j,i, …, P_n,i], n = t + w, i = 1, 2, …, N. The vector P_i thus contains the power outputs of both wind units and thermal units (the first w components correspond to the wind units, while the remaining t components correspond to the thermal units). We use the following notation: P_i,new is the new power vector of P_i powers obtained by applying the solution-update rules (21)–(23); P_best is the best solution obtained by ESGO for the ED problem; Cost(P_best) and Cost(P_i,new) denote the values of the objective function associated with the solutions P_best. and P_i,new, respectively. The ED problem, with or without wind power integration, is solved by applying Algorithm 2 from the previous section, while considering the inequality constraints (11)–(14) and the equality constraint (15), defined in the optimization model in Section 2. The main computational steps for implementing the proposed ESGO algorithm for solving the ED problem are summarized below (Algorithm 3).

Algorithm 3: ESGO algorithm for the ED problem

Step 1.Input data for the ED problem: cost coefficients (a, b, c, e, f) of the thermal units; parameters of the Weibull distribution (c, k); characteristic wind speeds of the wind units (vin, vr, vout); rated power of the wind units (Wr); power limits for thermal and wind units (Tmin, Tmax; Wmin, Wmax); cost coefficients in the objective function for the wind units (dq, oq, uq); B loss coefficients for the transmission lines; and load demand (PD) and accuracy ε. Step 2. Set the parameters for ESGO: N, tmax, α, β. Step 3. Initialization phase: Initialize the solutions Pi, i = 1, 2, …, N using relation (18).   Evaluate the objective function Cost(Pi), and identify Pbest, and Cost(Pbest)   while t ≤ tmax Step 4. Update solutions Pi in the improving phase   for i = 1 to N do        Update the s parameter using relation (27);        Generate a random number, rChy(0, s), pe based on relation (26);        Update Pi,new using relation (21), then evaluate Cost(Pi,new);        Align Pi,new towards the closest valve points;        Apply the RHM (restrictions handling mechanism) for the inequality constraints (11)–(14);        Apply the RHM for adjusting the equality constraint (15);        Update Pi, and PBest, (respectively, Cost(Xi) and Cost(XBest)) if they have a better cost; end for Step 5. Update solutions Pi in the acquiring phase for i = 1 to N do        Randomly select a solution Pr, i ≠ r;        Update the parameter s using relation (27);        Generate a random number, rChy(0, s), based on relation (26);        If Cost(Pi) < Cost(Pr) Then update Pi,new using relations (22a) and (22b);                                           Else update Pi,new using relations (23a) and (23b);        Direct Pi,new towards the nearest valve points;        Apply the RHM for the inequality constraints (11)–(14);        Apply the RHM for adjusting the equality constraint (15);        Update Pi, and PBest, (and correspondingly Cost(Pi), and Cost(PBest)) if they have a better cost end for        t = t + 1 end while Step 6. Store the best solution: Pbest, and Cost(Pbest).

The restriction handling mechanism (RHM), for inequality and equality constraints, is used as a solution-correction stage, systematically applied after generating each candidate solution by the ESGO algorithm (in the initialization, improving, and acquiring phases) and before evaluating the objective function. The role of this mechanism is to ensure feasible solutions through direct adjustments of variables that violate operational constraints. Since the constraints in the optimization model—namely, capacity limits, ramp-rate constraints, and prohibited operating zones—apply individually to each unit and do not involve coupling relations among variables, the RHM acts independently on each component of the solution vector. Corrections are performed in a deterministic sequence as follows: first, the minimum and maximum operating limits of the thermal and wind units (if present) are checked; then, ramp-rate constraints are enforced; and finally, membership in any prohibited operating zones is verified. The RHM is described below as a set of checking and adjustment rules for variables that violate operational constraints [16,33]:

(i).

For the inequality constraints (11)–(12) related to the capacity limits of thermal and wind units: if the power of thermal unit k (T_k) or wind unit q (W_q) is outside its operating interval (T_k,min, T_k,max), or (W_q,min, W_q,max), then T_k (respectively, W_q) is set to the reached bound, T_k = T_k,min or T_k = T_k,max (respectively, W_q = W_q,min or W_q = W_q,max);

(ii).

For the inequality constraints (13) related to ramp limits: if the power of thermal unit k, T_k is outside the interval (Max(T_k,min, T⁰_k − DR_k), Min(T_k,max, T⁰_k + UR_k)), then T_k is set to the reached bound, T_k = Max(T_k,min, T⁰_k − DR_k) or T_k = Min(T_k,max, T⁰_k + UR_k);

(iii).

For the inequality constraints (14) related to POZ: if T_k lies within a prohibited zone, then T_k is set to the upper limit of that zone (T_k = T_k,z^sup);

(iv).

The RHM for the equality constraint (15) is designed as a separate procedure from the main program, aimed at correcting the power vector [P_i] (generated after each solution update in the initialization, improving, and acquiring phases), such that the power-balance condition is satisfied and the units’ operating constraints are respected. The procedure calls for a vector [P_i] that does not satisfy the power balance and includes an iterative loop that transforms it into a feasible solution. Initially, system power losses are computed using relation (16), and the existing power mismatch is computed using relation (17). Subsequently, as long as this mismatch exceeds a predefined tolerance ε, a generator is selected at random and its power is adjusted incrementally. The direction of the adjustment depends on the sign of the mismatch: The power is increased in the case of a deficit or decreased in the case of a surplus, without exceeding the admissible limits. After each modification, losses and mismatches are updated using the mentioned relations to reflect the new system state. The process is repeated until the power balance is achieved within tolerance ε. At the end, the procedure returns to the main algorithm as a feasible solution [P_i]. The pseudocode of this procedure is presented in detail in [34].

6. Numerical Results and Comparisons

The efficiency of the proposed ESGO algorithm was tested on several power systems of small (6-unit), large (40-unit), and very large (110-unit and 220-unit) sizes. Depending on the characteristics of these systems (such as transmission line losses, prohibited operating zones of generating units, valve-point effects, operating limits of units, or the inclusion of wind power sources in a thermal power system), six cases were analyzed, summarized in Table 1 and described below:

• Case 1 (C₁) investigates a power system with six thermal units, where several practical characteristics are considered, such as POZ, ramp-rate limits, and transmission losses. The power demand is 1263 MW. The cost coefficients and B loss coefficients are given in [33], and the term B₀₀ is corrected according to [35].
• Case 2 (C₂) represents a large-scale system composed of 40 thermal units, where valve-point effects are considered. The cost coefficients are taken from [32], and the total power demand is 10,500 MW.
• Case 3 (C₃): The case C₃ is a 40-unit system in which the cost coefficients are identical to those in Case C₂, but power losses are considered. The B-loss is given in [36], and the load demand is P_D = 10,500 MW.
• Case 4 (C₄): This case analyzes the 40-unit system described in Case C₃ (the cost and B-loss coefficients are identical [36]), but the first two thermal units are replaced with two wind units, each having a capacity of 550 MW and operating limits between 0 and 550 MW. The wind speed is modeled by a Weibull distribution, where the shape and scale parameters (k, c) have the values k₁ = 1.5; c₁ = 15; k₂ = 1.5; and c₂ = 15 [37]. Other characteristics associated with the two wind units are as follows [37]: v_in = 5 m/s (cut-in wind speed), v_r = 15 m/s (rated wind speed), v_out = 45 m/s (cut-out wind speed), and $c_1^u=c_2^u=5, c_1^o=c_2^o=5$ (cost coefficients associated with overestimation and underestimation). The power demand is P_D =10,500 MW.
• Case 5 (C₅): This case studies a system with 110 thermal units, where valve-point effects and transmission line losses are not considered. The power demand is 15,000 MW, and the input data for the system are taken from [38].
• Case 6 (C₆): This case is a system with 220 thermal units obtained by doubling the system in Case C₅, with total demand P_D = 30,000 MW.

Table 1. Analyzed cases and parameter settings for SGO and ESGO.

Cases	SGO and ESGO (Common Parameters)				ESGO (Specific Parameters)		PL	VPE	POZ	Wind
	n	N	tmax	NE	α	β
C1 (ED)	6	10	50	1000	0.25	0.75	√	-	√	-
C2 (ED)	40	100	50	10,000	1	3	-	√	-	-
C3 (ED)	40	50	100	10,000	0.5	5	√	√	-	-
C4 (ED-Wind)	40	50	100	10,000	0.5	6	√	√	-	√
C5 (ED)	110	50	120	12,000	0.05	0.1	-	-	-	-
C6 (ED)	220	60	200	24,000	0.05	0.1	-	-	-	-

P_L: transmission line losses; VPE: valve-point effects; POZ: prohibited operating zones; and Wind: cases that include wind power.

Table 1. Analyzed cases and parameter settings for SGO and ESGO.

Cases	SGO and ESGO (Common Parameters)				ESGO (Specific Parameters)		PL	VPE	POZ	Wind
	n	N	tmax	NE	α	β
C1 (ED)	6	10	50	1000	0.25	0.75	√	-	√	-
C2 (ED)	40	100	50	10,000	1	3	-	√	-	-
C3 (ED)	40	50	100	10,000	0.5	5	√	√	-	-
C4 (ED-Wind)	40	50	100	10,000	0.5	6	√	√	-	√
C5 (ED)	110	50	120	12,000	0.05	0.1	-	-	-	-
C6 (ED)	220	60	200	24,000	0.05	0.1	-	-	-	-

P_L: transmission line losses; VPE: valve-point effects; POZ: prohibited operating zones; and Wind: cases that include wind power.

To obtain reliable statistical estimates and reduce the risk caused by the stochastic nature of the SGO and ESGO metaheuristics, as well as to make a fair comparison between SGO and ESGO, 50 independent runs were performed for each case. Based on the objective function Cost defined in relation (1), four relevant statistical indicators were computed to provide a clear picture of the efficiency of the tested algorithms: the best cost (BestCost), the mean cost (MeanCost), the worst cost (WorstCost), and the standard deviation (SDCost). The cost-related items are complemented by the average execution time of an algorithm, denoted Time, computed based on 50 successive runs as the arithmetic mean of the recorded execution durations. The accuracy (Accuracy) in satisfying the equality constraint (15) was set to ε = 10⁻⁵ MW for cases with power losses (C₁, C₃, C₄) and ε = 10⁻¹⁰ MW for the other cases (C₂, C₅, C₆).

All algorithms used in this work were implemented in MathCAD 14, and the simulations were run on a computer with the following specifications: Intel i5 processor, 4 GB of RAM, and 2.2 GHz CPU.

6.1. Setting the Parameters

Setting the parameters of metaheuristic algorithms is crucial, as it determines their efficiency, stability, quality of results, and computational cost. Without proper parameter adjustment, a potentially powerful algorithm may become inefficient or inconsistent.

The common parameters (N and t_max) and the specific parameters (α and β) of ESGO were set using experimental trials. First, several trial runs were performed to identify suitable ranges for each parameter; then, candidate values were selected from these ranges. The selection of the best values (N, t_max, α, and β) was based on the statistic MeanCost. The trial values for the ESGO parameters generally differ from case to case, depending on the specific characteristics of each system. For example, in cases C₂, C₃, and C₄, the trial values considered for parameter α, where α = 0.001 (to generate via the Cauchy distribution, small step sizes towards the end of the optimization process), α = 0.5 (for moderate values), and α = 1 (for larger step sizes in the final part of the optimization). For parameter β, which primarily influences the exploration phase, the following trial values were considered: β = {1, 2, …, 9} (for cases C₂–C₄). In contrast, for cases C₅ and C₆, which analyze systems with different characteristics compared to C₂–C₄, the trial values considered were α = 0.001, and β = {0.1, 0.2, …, 0.7}, α = 0.05, and β = {0.05, 0.1, …, 0.40}, and α = 0.1, and β = {0.1, 0.2, …, 0.7}.

Figure 3 shows the results of the tuning process for the specific parameters α and β of the ESGO algorithm for cases C₂ and C₄. Analyzing the MeanCost indicator, it can be observed that the best combinations are as follows: α = 1 and β = 3 (for C₂), and α = 0.5 and β = 6 (for C₄). In addition to the tuning plots presented in Figure 3 for cases C₂ and C₄, an extended robustness analysis was conducted for all six case studies, based on evaluating the variation in the MeanCost indicator as a function of parameters (α, β). The MeanCost analysis shows that, although high-performing (α, β) combinations can be identified for each case, the ESGO algorithm exhibits parameter-space domains where performance varies relatively little (cases C₁, C₅, and C₆, where the cost–power dependency is quadratic) or moderately (cases C₂, C₃, and C₄, where the dependency also includes a sinusoidal term). However, in general, very small values of parameter α combined with very small or very large values of parameter β represent unfavorable regions that should be avoided.

Figure 3. Selection of parameters α and β: (a) case C₂; (b) case C₄.

The selected values for the ESGO parameters, as well as for SGO, are presented in Table 1 for each studied system. For the SGO algorithm, the specific parameter c was chosen according to the indications in [23,24,27], namely, c = 0.2.

In an iterative process with t_max iterations, both SGO and ESGO perform two full update cycles over the entire population of individuals (N) per iteration. Thus, the theoretical number of objective function evaluations for both algorithms is the same, given by NE = 2N·t_max. For a fair comparison between SGO and ESGO, the number of objective evaluations (NE) and the population size (N) were set to the same values for each analyzed case (NE_SGO = NE_ESGO and N_SGO = N_ESGO), with these values being presented in Table 1. Furthermore, both algorithms start from the same initial solutions.

6.2. Case C₁ (6-Unit)

Table 2 presents the best generation schedules for the six-unit system obtained by SGO and ESGO, as well as by several representative algorithms from the literature, such as GA [33], MSSA [39], PSO [33], DHS [1], CSMA [40], and FA [41]. The best solution is obtained by SGO and ESGO, with an optimal cost of approximately BestCost = 15,449.89952487 $/h, while the other algorithms also achieve solutions close to this value. It is worth mentioning that the accuracy of the solutions obtained by SGO and ESGO is very high, with Accuracy < 4 × 10⁻¹¹ MW, which is comparable (CSMA [40], MSSA [39]) or even better than for other algorithms (GA [33], DHS [1], or FA [41]). Although the BestCost indicator has roughly the same value for all competing algorithms, the other statistics (MeanCost, WorstCost, or SDCost) obtained by ESGO are better than those of the competitors listed in Table 2. In particular, ESGO yields better statistics than SGO and shows improved stability of solutions, with SD_ESGO = 6.92 × 10⁻⁹ $/h > SD_SGO = 1.58 × 10⁻⁶ $/h.

Table 2. Optimal generation schedule obtained by SGO, ESGO, and other algorithms for Case C₁ (6-unit).

Outputs	ESGO	SGO	GA [33]	MSSA [39]	PSO [33]	DHS [1]	CSMA [40]	FA [41]
P1 (MW)	447.50377756	447.50401562	474.8066	447.5029	447.4970	447.5285	452.1021	439.293500
P2 (MW)	173.31811234	173.31842860	178.6363	173.3186	173.3221	173.2791	172.096	175.0455644
P3 (MW)	263.46280200	263.46291951	262.2089	263.463	263.4745	263.4772	251.8451	265.00000
P4 (MW)	139.06540388	139.06473599	134.2826	139.0656	139.0594	139.0291	128.9404	140.7740665
P5(MW)	165.47313121	165.47353818	151.9039	165.473	165.4761	165.4864	164.1094	167.042084
P6 (MW)	87.13500962	87.13461506	74.1812	87.1349	87.1280	87.1587	106.7149	88.78165465
PG (MW)	1275.9582366	1275.9582530	1276.03	1275.958	1276.01	1275.9590	1275.8079	1275.9367
PL (MW)	12.958236592	12.958252968	13.0217	12.958	12.9584	12.9590	12.8079	12.93686954
PD (MW)	1263	1263	1263	1263	1263	1263	1263	1263
Accuracy (MW)	<4 × 10−11	<4 × 10−11	2.21 × 10−3	1.4 × 10−11	-	8.89 × 10−9	0	1.09 × 10−8
BestCost ($/h)	15,449.89952487	15,449.89952487	15,459	15,449.8995	15,450	15,449.8996	15,451.89584	15,450.50896
MeanCost ($/h)	15,449.89952488	15,449.89952525	15,469	15,449.937	15,454	15,449.9264	-	15,452.53099
WorstCost ($/h)	15,449.89952489	15,449.89953426	15,524	15,453.545	15,492	15,449.9884	-	15,458.44268
SDCost ($/h)	6.92 × 10−9	1.58 × 10−6	-	0.3647	-	0.0204	-	2.048
Cost saving * ($/yr)	0	≈0	167,320.2	328.2	35,920.1	235.4		23,051.6
Time (s)	0.05	0.05	41.58	1.16	14.89	0.01	-	1.96
NE	1000	1000	20,000	12,000	20,000	3000	15,000	60,000

* Represents the cost saved by the ESGO algorithm compared with algorithm X: Cost saving = (MeanCost_X − MeanCost_ESGO) × 8760.

The number of objective function evaluations (NE), as well as the average execution time (Time), is smaller (1000 evaluations) for SGO and ESGO than for most of the other competitors (an exception is DHS [1], which has a shorter Time but lower-quality cost indicators).

6.3. Case C₂ (40-Unit System, Without Power Losses and Without Wind Power)

The best results obtained using the ESGO and SGO algorithms are presented in Table 3. In this case, the operating cost and the solutions identified by the two algorithms differ. The operating cost corresponding to the best solution is lower for ESGO compared to SGO, namely, BestCost_ESGO = 121,412.5355 $/h < BestCost_SGO = 121,548.4552 $/h. From Table 3, it can be observed that the accuracy ε with which the constraints are satisfied is maintained at a very high level for both ESGO and SGO (Accuracy < 10⁻¹⁰ MW).

Table 3. Generation schedule obtained by ESGO and SGO for cases C₂, C₃, and C₄ (40-unit).

P (MW)	Case C2		Case C3		Case C4
	SGO	ESGO	SGO	ESGO	SGO	ESGO
P1/W1	111.834935	110.799825	113.988149	110.799825	549.999837	550.000000
P2/W2	113.320749	110.799825	112.746971	112.538106	550.000000	550.000000
P3	97.434901	97.399913	119.998591	120.000000	119.999895	97.399913
P4	179.687845	179.733100	179.755555	179.736430	181.021287	179.733100
P5	97.000000	87.799905	87.826050	87.799905	90.273354	87.799905
P6	139.994678	140.000000	139.996736	140.000000	139.999806	105.399913
P7	299.911628	259.599650	299.997616	300.000000	299.991889	259.599650
P8	285.021570	284.599650	284.603769	284.599650	288.749159	284.599905
P9	284.912671	284.599650	296.691385	284.625366	288.445024	284.599650
P10	130.050809	130.000000	279.593375	279.599650	204.806358	204.799825
P11	94.048872	94.000000	243.608235	243.599650	94.000000	94.000000
P12	168.702143	94.000000	94.012057	94.000218	94.009197	94.000000
P13	125.009358	214.759790	484.041278	484.039160	125.031457	214.759790
P14	394.452941	394.279370	484.039443	484.039160	304.557308	394.279370
P15	304.645797	394.279370	484.037975	484.039160	394.305111	394.279370
P16	394.127450	394.279370	484.039181	484.039160	484.041217	484.039160
P17	489.345186	489.279370	489.284005	489.279370	489.303845	489.279370
P18	489.580682	489.279370	489.280874	489.279370	489.284874	489.279370
P19	511.423510	511.279370	511.308492	511.279370	511.392175	511.279370
P20	511.481012	511.279370	549.816732	511.279370	511.404565	511.279370
P21	523.956645	523.279370	523.300260	523.279370	523.421965	523.279370
P22	523.408100	523.279370	523.728818	547.018744	525.959504	523.279370
P23	523.257817	523.279370	523.280836	523.279370	523.278478	523.279370
P24	523.235375	523.279370	523.289355	523.279370	523.381482	523.279370
P25	523.291335	523.279370	523.291144	523.279370	523.383574	523.279370
P26	523.421041	523.279370	523.283877	523.279370	523.399085	523.279406
P27	10.083004	10.000000	10.005820	10.000000	10.036014	10.000000
P28	10.179693	10.000000	10.001407	10.000030	10.089374	10.000000
P29	10.142417	10.000000	10.001158	10.000000	10.005041	10.000000
P30	96.160364	87.799905	87.826857	87.799905	87.868329	87.799905
P31	189.841161	190.000000	189.990882	190.000000	189.998492	190.000000
P32	189.966836	190.000000	189.998935	190.000000	189.998537	190.000000
P33	189.995798	190.000000	189.997118	190.000000	190.000000	190.000000
P34	199.938728	164.799825	199.999515	200.000000	199.999706	169.973192
P35	199.897140	200.000000	164.830068	200.000000	165.461167	164.800904
P36	199.818717	194.397778	164.820300	164.799906	164.804244	164.799825
P37	110.000000	110.000000	110.000000	110.000000	109.997291	110.000000
P38	110.000000	110.000000	110.000000	110.000000	109.999929	110.000000
P39	110.000000	110.000000	109.999500	110.000000	109.990306	110.000000
P40	511.419091	511.279370	549.979174	550.000000	549.995348	511.279370
Accuracy (MW)	<10−10	<10−10	<2 × 10−6	<2 × 10−6	<10−5	<10−5
PL (MW)	-	-	966.291494	970.588362	951.684227	948.736492
BestCost ($/h)	121,548.45526	121,412.53551	136,504.29238	136,448.48684	123,405.49347	123,107.28172

Table 4 reports the main statistical indicators related to the operating cost, obtained by ESGO and SGO, as well as several relevant algorithms from the literature. From Table 4, it is clear that ESGO outperforms all competing algorithms mentioned—both in terms of cost (BestCost, MeanCost, WorstCost, and SDCost) and average execution time (Time), with the only exception being SGO, which achieves a slightly shorter runtime. It can be noted that ESGO surpasses not only established, well-known algorithms (such as DE [5], ABC [5], BWO [42], GWO [4], SCA [43], CSO [44], and others), but also improved or relatively recent hybrid algorithms (MINO [45], ESCSDO [46], EBWO [42], TLABC [47], Jaya-SML [48], and ESNS [4]). ESGO also exhibits good performance stability, achieving a low dispersion value SDCost_ESGO = 9.13 $/h compared to the other algorithms, which generally have higher dispersion levels, ranging from 29.56 $/h (ABC [5]) to 1391.51 $/h (FOX [10]).

Table 4. Statistical item values obtained by various algorithms for case C₂ (40-units, P_D = 10,500 MW).

Algorithm	BestCost ($/h)	MeanCost ($/h)	WorstCost ($/h)	SDCost ($/h)	Time (s)	NE	Cost Saving * ($/yr)
ABC [5]	121,418.1073	121,457.1952	121,516.3408	29.56	4.08	400,000	328,915.2
MIMO [45]	122,758.7	124,621.8	126,059.2	866.20	−	400,000	28,050,853.3
DE [5]	121,420.8933	121,479.7156	121,587.3117	42.88	4.03	400,000	526,193.9
SOA [42]	125,704.4	128,066.9	127,019.5	667.16	110.32	1,500,000	58,229,929.3
SMA [49]	121,658.66	−	−	−	−	25,000	-
BWO [42]	122,875.8	123,858.9	123,398.6	240.70	45.88	1,500,000	21,367,849.3
WSO [10]	122,488.5	123,286.7	123,982.5	490.72	-	-	16,355,377.3
GWO [4]	121,951.3	122,254.1	122,965.9	299.97	37.25	500,000	6,589,729.3
SDO [46]	121,750.2	122,460.1	123,222.7	405.01	−	4,000,000	9,114,361.3
ESCSDO [46]	121,626.97	122,351.7	123,128.9	412.29	−	4,000,000	8,164,777.3
EBWO [42]	121,600.9	122,180.9	122,012.6	163.42	98.69	1,500,000	6,668,569.3
SCA [43]	121,506.58	121,857.90	122,056.15	347.26	−	30,000	3,839,089.3
TLABC [47]	121,468.38	121,739.44	122,192.32	160.88	−	200,000	2,801,379.7
FOX [10]	123,745.7	125,954.3	129,811.1	1391.51	-	-	39,723,553.3
CSO [44]	121,465.99	121,988.48	122,781.75	275.92	19.77	1,000,000	4,982,970.1
Jaya-SML [48]	121,476.3977	121,689.0773	122,039.8731	147.8901	12.89	90,000	2,360,202.4
SNS [4]	121,465.7	121,640.8	122,267.9	194.86	52.05	500,000	1,937,293.3
MSSA [13]	121,415.45	121,468.18	-	-	-	-	425,142.1
GSK [5]	121,421.1734	121,481.4039	121,605.7015	53.81	4.64	400,000	540,983.4
ESNS [4]	121,415.6	121,742.2	122,155.2	206.70	61.44	500,000	2,825,557.3
SGO	121,548.4553	121,936.4643	122,418.6658	215.56	1.65	10,000	4,527,312.5
ESGO	121,412.5355	121,419.6478	121,443.4261	9.13	2.23	10,000	0.0

* Cost saving = (MeanCost_X − MeanCost_ESGO) × 8760.

An additional observation relevant for case C₂ is that, for most of the compared algorithms, the range of cost values obtained by ESGO does not overlap with those of competing methods. More specifically, the maximum cost produced by ESGO is lower than the minimum cost reported by most algorithms in the literature, except for a few methods (ABC [5], DE [5], MSSA [13], GSK [5], and ESNS [4]). This clear separation of performance is even more significant given that ESGO achieves these results using a considerably smaller number of objective-function evaluations (NE = 10,000), compared with the 400,000–500,000 evaluations generally required by competing algorithms. Therefore, the results indicate that ESGO combines superior solution quality with high computational efficiency, which increases its potential for practical medium- and large-scale applications.

The average execution time is very good for both SGO (Time = 1.65 s) and ESGO (Time = 2.23 s), considering the characteristics and size of the analyzed system (40-unit). In addition, ESGO obtains a faster average runtime than all the other algorithms listed in Table 4 (except for SGO). This behavior is partly explained by the reduced number of function evaluations required by ESGO to achieve statistically consistent results.

6.4. Case C₃ (40-Unit, with Power Losses and Without Wind Power)

This case analyzes a system with 40 thermal units, no wind units, and explicit consideration of power losses. The optimal unit commitment obtained by the ESGO and SGO algorithms is presented in Table 3. To cover the load demand (10,500 MW) and the power losses (approximately 971 MW for ESGO and 967 MW for SGO), the vast majority of generating units are loaded near their maximum capacity; the exceptions are units P₂₇, P₂₈, and P₂₉, which are kept at the minimum (10 MW) due to their high operating costs. The optimal solutions found by ESGO and SGO satisfy all constraints, with high accuracy (below 2 × 10⁻⁶ MW). The best solutions identified are BestCost_ESGO = 136,448.4868 $/h and BestCost_SGO = 136,504.2924 $/h. Table 5 reports the set of standard statistical indicators used in the literature to assess the quality, stability, and robustness of metaheuristic algorithms for the C₃ system. The results in Table 5 show that the proposed ESGO algorithm achieves higher performance, in terms of cost (BestCost, MeanCost, WorstCost, and SDCost), than a suite of metaheuristics, such as ACS [50], BBO [51], HGWO [52], SCA [53], SGO, etc. The stability of ESGO, evaluated via SDCost = 42.25 $/h, is moderate and better than that of other algorithms, which range from 49.65 $/h (for SCA [53]) to 752.74 $/h (for MIMO [45]). Additionally, ESGO exhibits a very good average runtime, being shorter than for other algorithms (with the exceptions of HGWO [52] and SGO, which achieve lower Time values but poorer cost–quality metrics).

Table 5. Statistical item values obtained by various algorithms for case C₃ (40-units, with loss of power, P_D = 10,500 MW).

Algorithm	BestCost ($/h)	MeanCost ($/h)	WorstCost ($/h)	SDCost ($/h)	Time (s)	Cost Saving * ($/yr)
GAAPI [36]	139,864.96	-	-	-	-	-
ACS [50]	137,413.73	-	-	-	7.32	-
BBO [51]	137,026.82	137,116.58	137,587.8200	-	40.00	5,409,865.9
DE/BBO [51]	136,950.77	136,966.77	137,150.7700	-	32.00	4,097,530.3
ORCCRO [51]	136,855.1900	136,855.1900	136,855.1900	-	14.00	3,120,089.5
HPSO-DE [54]	136,835.0021	-	-	-	27.00	-
HGWO [52]	136,681.0000	136,684.000	-	-	1.46	1,620,465.1
KHA-IV [55]	136,670.3701	136,671.2293	136,671.8648	-	-	1,508,593.8
OKHA [56]	136,575.9680	136,576.1500	136,576.6400	-	-	675,699.1
MIMO [45]	137,034.2000	138,472.9000	140,124.3000	752.74	10.45	17,291,229.1
SCA [53]	136,775.4102	136,894.5140	137,045.0486	49.65	140.51	3,464,567.7
FPA [53]	136,454.3766	136,511.1059	136,787.6439	61.28	139.30	105,912.8
SGO	136,504.2924	137,144.3464	138,447.9214	423.63	4.26	5,653,099.6
ESGO	136,448.4868	136,499.0154	136,598.5730	42.25	4.60	0.0

* Cost saving = (MeanCost_X − MeanCost_ESGO) × 8760.

6.5. Case C₄ (40-Unit, with Power Losses and with Wind Power)

Case C₄ is a power system containing 38 thermal units and 2 wind units of 550 MW each, with power losses taken into account. The optimal generator loading identified by SGO and ESGO is presented in Table 3. The presence of the two wind units, loaded at the rated power (550 MW), causes the thermal units to operate at lower power, and, therefore, at lower cost. Operating the wind units at their rated power (550 MW) places both the linear and non-linear wind–power curve representations in a saturated regime. As a result, the dispatch solution in case C4 is unaffected by the choice of curve model. However, a linear approximation may influence the probabilistic estimation of available power and the exact magnitude of cost reduction in scenarios with sub-rated wind speeds.

From Table 3, it can be seen that the BestCost indicator decreases significantly in Case C₄ compared to C₃ (by approximately 10% for both SGO and ESGO) as a result of including the wind units in the power system. The best operating costs obtained by ESGO and SGO are BestCost_ESGO = 123,107.28172 $/h and BestCost_SGO = 123,405.49347 $/h, respectively. To study the efficiency of ESGO, it is compared with other well-known algorithms (DE, PSO, SCA, and SGO) using the standard statistical set (BestCost, MeanCost, WorstCost, SDCost, and Time). The values of these statistical items are presented in Table 6. It can be observed that ESGO performs better than all the mentioned algorithms in terms of cost and average execution time (except for SGO, which has a slightly better Time than ESGO).

Table 6. Statistical item values obtained by various algorithms for case C₄ (40-units, with loss of power and wind, P_D = 10,500 MW).

Algorithm	BestCost ($/h)	MeanCost ($/h)	WorstCost ($/h)	SDCost ($/h)	Time (s)	NE	Cost Saving * ($/yr)
DE	124,742.5544	125,182.4747	125,437.5446	143.89	4.40	10,000	17,838,766.8
PSO	123,806.6709	124,588.4918	125,586.2667	442.84	4.03	10,000	12,635,476.6
SCA	124,801.1087	125,316.4121	125,658.6213	193.75	5.17	10,000	19,012,058.4
SGO	123,405.4935	124,134.6517	125,379.7301	492.16	4.31	10,000	8,659,837.3
ESGO	123,107.2817	123,146.0858	123,294.5221	47.13	4.83	10,000	0.0

* Cost saving = (MeanCost_X − MeanCost_ESGO) × 8760.

The stability of the solutions obtained by ESGO is good, SDCost_ESGO = 47.13 $/h, compared with the other algorithms (which are greater than 143 $/h). It should be noted that, to compare ESGO correctly and rigorously with the competing algorithms (DE, PSO, SCA, and SGO), a set of necessary conditions was imposed: the same initial solutions, the same number of runs (50 runs), the same population size (N = 50), and the same number of objective-function evaluations (NE = 10,000). In addition, the parameters of the competing algorithms were set to the following values: N = 50, t_max = 200, c₁ = c₂ = 2, w_min = 0.3, and w_max = 0.9, where c₁ and c₂ are acceleration coefficients, and w_min and w_max are the initial and final inertial weights (for the PSO algorithm); N = 50, t_max = 200, CR = 0.2, F = 1; where CR is the crossover rate, and F is the scaling factor (for the DE); N = 50, t_max = 200, and a = 0.1 (for the SCA).

6.6. Case C₅ (110-Unit)

The best-performing solutions obtained by the SGO and ESGO algorithms are summarized in Table 7. The experimental results show that ESGO is more efficient than SGO in obtaining the optimal solution, BestCost_ESGO = 197,988.1775 $/h < BestCost_SGO = 197,998.0824 $/h. For the 110-unit system to cover the required power (15,000 MW), 52 units are loaded at minimum capacity (they have higher operating costs), 40 units at maximum capacity (they have relatively low operating costs), and 18 units at intermediate capacity. The accuracy of the optimal solution identified by SGO or ESGO is very high, Accuracy < 2 × 10⁻¹¹ MW.

Table 7. Generation schedule obtained by SGO and ESGO for case C₅ (110-unit, P_D = 15,000 MW).

P (MW)	SGO	ESGO	P (MW)	SGO	ESGO	P (MW)	SGO	ESGO
P1	2.40091	2.40000	P38	69.766436	69.999999	P75	89.87014	90.00000
P2	2.40428	2.40000	P39	99.891629	100	P76	49.91488	50.00000
P3	2.40626	2.40000	P40	116.36312	120	P77	160.11553	160.00000
P4	2.40694	2.40000	P41	155.79066	157.1805	P78	311.57157	295.75330
P5	2.40134	2.40000	P42	219.98698	220	P79	175.39153	175.06004
P6	4.00567	4.00000	P43	439.99644	440	P80	96.25830	98.01123
P7	4.00504	4.00000	P44	559.92203	560	P81	10.00221	10.00000
P8	4.00260	4.00000	P45	659.95383	660	P82	12.83785	12.00000
P9	4.01074	4.00000	P46	617.69517	616.45095	P83	20.78197	20.00000
P10	64.22005	64.40766	P47	5.4006051	5.4	P84	199.28246	200.00000
P11	60.86614	62.16581	P48	5.4065787	5.4	P85	323.92980	325.00000
P12	32.40132	36.29050	P49	8.4011838	8.4	P86	439.96649	440.00000
P13	54.48428	56.64826	P50	8.4118577	8.4	P87	21.19490	14.39906
P14	25.00659	25.00000	P51	8.4016544	8.4	P88	20.00000	24.32084
P15	25.11744	25.00000	P52	12.012191	12	P89	81.63090	82.41757
P16	25.00095	25.00000	P53	12.00007	12	P90	87.78758	89.26627
P17	154.86054	155.00000	P54	12	12	P91	57.32497	57.61011
P18	154.96585	155.00000	P55	12.00002	12	P92	98.80979	99.99890
P19	154.97970	155.00000	P56	26.924038	25.2	P93	439.95058	440.00000
P20	154.93202	155.00000	P57	25.500516	25.200001	P94	499.81957	500.00000
P21	68.90597	68.90000	P58	35.086748	35	P95	599.99000	600.00000
P22	68.90000	68.90000	P59	35.012709	35	P96	472.51608	471.45390
P23	68.90084	68.90000	P60	45.000745	45	P97	3.62369	3.60000
P24	349.96768	350.00000	P61	45.005063	45	P98	3.60001	3.60000
P25	399.99378	400.00000	P62	45.002462	45	P99	4.41376	4.40000
P26	399.97066	400.00000	P63	184.78732	185	P100	4.40076	4.40000
P27	499.88484	500.00000	P64	184.9757	185	P101	10.00106	10.00000
P28	499.99256	500.00000	P65	184.88997	185	P102	10.00054	10.00000
P29	199.72061	200.00000	P66	185	185	P103	20.00700	20.00000
P30	99.94403	100.00000	P67	70.000405	70	P104	20.00120	20.00000
P31	10.00024	10.00000	P68	70.00047	70	P105	40.00307	40.00000
P32	19.91851	20.00000	P69	70.000105	70	P106	40.00022	40.00000
P33	77.40539	80.00000	P70	359.95548	360	P107	50.00069	50.00000
P34	249.91225	250.00000	P71	399.96043	400	P108	30.00273	30.00000
P35	359.92432	360.00000	P72	399.99977	400	P109	40.00409	40.00000
P36	399.80109	400.00000	P73	107.27423	104.96143	P110	20.00559	20.00000
P37	39.66947	40.00000	P74	189.52096	191.50367

Table 8 summarizes the statistical indicators obtained by various algorithms applied recently or relatively recently to solve Case C₅. Based on these statistics, ESGO outperforms all competitors (HHO [38], IS [38], SNS [4], DiDE [6], GBO [4], ESNS [4], and others) in terms of BestCost, MeanCost, WorstCost, SDCost, and Time. An exception is PPSO [6], which has slightly better stability than ESGO (WorstCost_ESGO = 197,988.1778 $/h > WorstCost_PPSO [6] = 197,998.1776 $/h, and SDCost_ESGO = 6.22 × 10⁻⁵ $/h > SDCost_PPSO [6] = 1.88 × 10⁻⁵ $/h). Also, SGO has a slightly shorter Time than ESGO. It should be noted that ESGO achieves these results using only 12,000 evaluations, whereas PPSO [6] uses 50,000 objective-function evaluations. ESGO shows low dispersion (SDCost_ESGO = 6.22 × 10⁻⁵ $/h), indicating that the algorithm is stable and predictable, with consistent convergence even for large-scale systems (110 units).

Table 8. Statistical item values obtained by various algorithms for case C₅ (110-units, P_D = 15,000 MW).

Algorithm	BestCost ($/h)	MeanCost ($/h)	WorstCost ($/h)	SDCost ($/h)	Time (s)	NE	Cost Saving ($/yr)
HHO [38]	219,950.8	222,320.1	225,741.9	2593.77	1106.04	5,000,000	213,147,640
JS [38]	213,120.4	213,389.5	213,982.6	356.98	479.18	5,000,000	134,915,584
SNS [4]	207,194.8	209,977.1	211,106.9	1063.51	201.80	7,500,000	105,022,960
DiDE [6]	201,332.1755	202,188.9128	203,240.5577	544.55	-	50,000	36,798,440
GBO [4]	202,888.6	207,338.9	215,710.8	3248.17	267.08	7,500,000	81,912,328
PSO [57]	200,272.8409	201,483.4217	203,491.7762	662.23	35.69	1,100,000	30,618,338
ESNS [4]	200,266.5	202,925.5	204,908	1669.81	254.54	7,500,000	43,250,944
GWO [4]	201,505.7	202,480.1	203,869.7	598.09	144.26	7,500,000	39,349,240
SOS [58]	201,034.1008	-	-	-	-	5,000,000	-
JADE [6]	198,011.7169	198,019.5193	198,028.4368	3.85	-	50,000	274,553
DSOS [58]	198,007.60256	-	-	-	-	5,000,000	-
ISMA [38]	198,565.9	198,782.1	198,949.1	153.46	899.75	5,000,000	6,954,760
CSO [57]	197,993.3466	197,997.0751	198,001.5873	1.80	28.61	1,100,000	77,942
QILDE [6]	197,990.6250	197,991.8523	197,993.2696	0.62	-	50,000	32,190
OLCSO [57]	197,988.8576	197,989.5832	197,990.4551	0.36	33.55	1,100,000	12,313
MPSO [6]	197,988.2292	197,988.3453	197,988.6536	0.10	-	50,000	1469
SGO-L [59]	197,988.1782	197,988.1840	197,988.1950	4.80 × 10−3	5.21	21,000	56
TFWO [60]	197,988.1790	197,988.1823	197,988.1904	6.80 × 10−3	46.22	320,000	41
DSPSO [6]	197,988.2133	197,988.2457	197,988.2999	2.30 × 10−2	-	50,000	596
PPSO [6]	197,988.1775	197,988.1776	197,988.1776	1.88 × 10−5	-	50,000	0
EHMDE [6]	197,988.1778	197,988.1784	197,988.1795	3.75 × 10−4	-	50,000	7
SGO	197,998.0824	198,010.4653	198,022.0416	5.13	3.32	12,000	195,240
ESGO	197,988.1775	197,988.1776	197,988.1778	6.22 × 10−5	4.21	12,000	0

The number of evaluations required by ESGO to reach these performances is small (NE = 12,000), considering the system size (110 units) and compared with other algorithms (which range between 21,000 and 7,500,000 evaluations).

6.7. Case C₆ (220-Unit)

To demonstrate the ability of the ESGO algorithm to maintain its performance as the problem size increases, ESGO is applied to a 220-unit thermal power system. The statistical indicators obtained with ESGO and with several well-known algorithms (SGO, PSO, DE, and SCA) are shown in Table 9. From this Table, it can be observed that ESGO is more effective in extracting a better solution than the competitors mentioned. Moreover, the quality of the solutions (measured by BestCost, MeanCost, WorstCost, and SDCost) and their stability (SDCost) is better for ESGO than for SGO, PSO, DE, and SCA. Considering the size of the system, the average execution time of ESGO is very good, being relatively close to that obtained by the other algorithms. An objective comparison of the algorithms was carried out under identical experimental conditions: the same initial solutions, the same number of runs (50 runs), the same population size (N = 60), and the same number of objective function evaluations (NE = 24,000). The parameters of the competing algorithms were set as follows: N = 60, t_max = 400, c₁ = c₂ = 2, w_min = 0.3, and w_max = 0.9 (for the PSO); N = 60, t_max = 400, CR = 0.1, and F = 0.4 (for the DE); and N = 60, t_max = 400, and a = 0.1 (for the SCA).

Table 9. Statistical item values obtained by various algorithms for case C₆ (220-units, P_D = 30,000 MW).

Algorithm	BestCost ($/h)	MeanCost ($/h)	WorstCost ($/h)	SDCost ($/h)	Time (s)	NE	Cost Saving ($/yr)
PSO	395,970.8063	395,972.9673	395,976.5272	1.32	10.02	24,000	30,090
DE	407,164.0918	410,149.2913	411,833.6176	892.41	9.95	24,000	124,214,688
SCA	404,563.7960	406,959.4278	408,080.6368	609.28	9.75	24,000	96,271,484
SGO	396,012.8182	396,032.8183	396,061.5487	11.60	10.45	24,000	554,385
ESGO	395,969.5321	395,969.5323	395,969.5329	1.57 × 10−4	15.07	24,000	0

6.8. Comparison of ESGO with SGO

Statistical testing: The sets of statistical items (BestCost, MeanCost, WorstCost, and SDCost) describing the behavior of the SGO and ESGO algorithms on the different test system configurations are given in Table 2, Table 4, Table 5, Table 6, Table 8 and Table 9 for the experimental cases C1–C6. In all cases, these indicators are better for ESGO than for SGO. However, these statistics provide only a descriptive analysis of the results (best, worst, and average behavior of an algorithm or its stability) and do not explicitly account for the variability and distribution of the samples. To demonstrate that there are statistically significant differences between the SGO and ESGO algorithms, and to avoid conclusions based merely on coincidences due to randomness, it is necessary to apply an inferential statistical test. In this paper, the Wilcoxon signed-rank test for paired data is used, with a significance level of 1%. The number of paired data points is 50, because both SGO and ESGO perform 50 independent runs starting from the same initial solutions. The statistical processing of the results obtained by SGO and ESGO is performed using the Wilcoxon test implemented in the Statistical Package for the Social Sciences (SPSS^® version 16). Table 10 reports the p-values obtained by applying the Wilcoxon test to compare the performance of SGO and ESGO. For each dataset considered, the p-value is below the 0.01 threshold, which indicates statistically significant differences and confirms the superiority of the solutions generated by ESGO over SGO.

Table 10. Comparison of SGO and ESGO by the Wilcoxon test.

Cases	R−	R+	z-Values *	p-Values	Winner
C1	28.13	17.17	−4.167	0.000	ESGO
C2–C6	25.50	0	−6.154	0.000	ESGO

R⁻ and R⁺ represent the mean of the negative and positive ranks, respectively. * Based on positive ranks.

The test results confirm that the modifications introduced to strengthen the ESGO version (described in Section 4, points (i)–(v)) lead to an improvement in the efficiency of this algorithm compared with the traditional SGO.

Convergence process: Figure 4a–f illustrate the convergence curves, associated with the best solutions, of the SGO and ESGO algorithms for each analyzed case, C₁–C₆. By examining these curves, we can observe differences in the evolution of the iterative process towards the optimum. In all cases, the ESGO curve decreases more steeply than the SGO curve in the early iterations, which shows that ESGO has better exploration capability than SGO. Furthermore, ESGO stabilizes faster than SGO and then remains stable in the final iterations. This behavior reflects a successful transition from exploration to exploitation, followed by a strong ability to refine solutions. In the detailed plots of Figure 4e,f, corresponding to cases C₅ and C₆, it is clearly visible that SGO requires more iterations to reach the performance achieved by ESGO in 120 and 200 iterations, respectively. Thus, the ESGO version, equipped with the new solution-update relations and random operators, exhibits better local and global search behavior than the original SGO algorithm.

Figure 4. Convergence characteristics obtained by SGO and ESGO: (a) case C₁; (b) case C₂; (c) case C₃; (d) case C₄; (e) case C₅; and (f) case C₆.

The annual Cost saving is computed for each case by taking the difference between MeanCost_X and MeanCost_ESGO, where X denotes a competitor algorithm to ESGO. The Cost saving values are reported in Table 2, Table 4, Table 5, Table 6, Table 8 and Table 9 for each case studied.

For Case C₁, the 6-unit small-scale system, the saved cost is negligible, with the exception of the cost corresponding to the GA algorithm [33]. For large-scale systems (40-unit, 110-unit, or 220-unit), the savings are significant, ranging in most situations from hundreds of thousands of $/yr (e.g., ABC [5], DE [5], and MSSA [13] in Case C₂; FPA [53] in Case C₃; and JDE [6] and SGO in Case C₅) to tens of millions of $/yr (for most algorithms applied in Cases C₂–C₆), and even hundreds of millions of $/yr (HHO [38], JS [38], SNS [4] in Case C₅, and DE in Case C₆).

6.9. Analysis of the Exploration–Exploitation Dynamics

To explicitly characterize the exploration–exploitation dynamics, this section analyzes the algorithmic behavior of ESGO and SGO, with emphasis on the internal evolution of the search process throughout optimization. The analysis is based on three complementary perspectives: population diversity, search trajectories, and the correlation of these dynamics with the achieved numerical performance. For this purpose, two dynamic metrics are employed: the average population diversity, denoted Div(t), and the average search-step magnitude, denoted Step(t), evaluated at each iteration and subsequently averaged over 50 independent runs. The mathematical definitions of these metrics are provided in Appendix A.

Population diversity: The evolution of Div(t), comparatively illustrated for ESGO and SGO in Figure 5a,b for cases C₂ and C₄, highlights consistent differences in the way population dispersion is managed. Although both algorithms start from the same initial diversity level, ESGO systematically maintains higher values of this metric during the early and intermediate stages of the optimization process. The ESGO curves indicate a progressive and smooth reduction in diversity, suggesting a controlled narrowing of the search space as solutions concentrate in promising regions. By comparison, SGO exhibits a faster and nearly monotonic decrease in Div(t), reaching low diversity levels within a smaller number of iterations. This behavior indicates a less gradual transition toward an exploitation-dominated regime compared with ESGO. Maintaining diversity over a longer interval enables ESGO to delay pronounced population contraction and to preserve exploration capability during the intermediate stages of the search.

Figure 5. Evolution of population diversity Div(t): (a) for cases C₂, and (b) for cases C₄.

Search trajectories and step adaptation: The evolution of the average step magnitude Step(t), presented in Figure 6a,b for cases C₂ and C₄, provides additional insight into population mobility and into the adaptation of displacement amplitudes during optimization. In both analyzed cases, ESGO is characterized by higher values of the average step in the early and intermediate stages, reflecting broader exploration of the solution space. As iterations progress, Step(t) in ESGO decreases gradually and continuously, indicating a progressive adjustment of displacement amplitudes as a function of the search stage. In contrast, SGO shows a more abrupt reduction in step magnitude, associated with a faster decrease in population mobility and an earlier intensification of exploitation. The observed differences between the Step(t) curves suggest that ESGO implements a more flexible adaptation of search steps over the iterative process.

Figure 6. Evolution of the average step size Step(t): (a) for cases C₂, and (b) for cases C₄.

Correlation between dynamics and numerical performance: The dynamic behaviors evidenced by the evolution of Div(t) and Step(t) are fully consistent with the numerical results reported previously. For both cases, C₂ and C₄, ESGO attains superior values of the minimum, mean, and maximum cost, as well as significantly lower standard deviations compared with SGO, indicating more stable and robust convergence across the 50 independent runs. The convergence curves confirm that ESGO reaches promising regions of the search space more rapidly and stabilizes earlier in the neighborhood of the best identified solution.

The combined analysis of population diversity and search trajectories provides direct quantitative evidence regarding differences in internal dynamics between ESGO and SGO. Maintaining higher diversity in the initial stages, together with a gradual reduction in search-step magnitude, indicates a progressive and controlled transition from exploration to exploitation in the case of ESGO. This dynamic is coherent with the superior numerical performance and robustness observed experimentally and is compatible with the introduced heuristic mechanisms, including the use of Cauchy-based perturbations with an adaptive scale parameter.

7. Conclusions

This article presents a new variant of the SGO algorithm, called ESGO, in which the set of heuristics for updating solutions is modified in both the improving phase and the acquiring phase. The new heuristics are also endowed with stochastic operators based on the uniform and Cauchy distributions. In the acquiring phase, a heuristic is activated by the simultaneous fulfillment of two conditions: the first compares the fitness values of two competing solutions (f(X_i) < f(X_r)), and the second is based on a random condition of the form rnd(1) < 0.5. ESGO was validated on power systems of varying complexity—small (6-unit), large (40-unit, with or without wind integration), and very large (110-unit and 220-unit). For all six analyzed cases, ESGO outperforms SGO in terms of cost, as confirmed by the Wilcoxon test. ESGO is also more cost-efficient than a range of established algorithms (e.g., PSO, DE, TLBO, GWO, and SCA) or hybrid/enhanced algorithms (e.g., Jaya-SML, MSSA, MIMO in Case C₂; DE/BBO, ORCCRO, HPSO-DE in C₃; DSOS, ISMA, and MPSO in C₅). The average runtime of ESGO is very good across all cases (0.05 s for C₁; 2.23 s for C₂; 4.60 s for C₃; 4.83 s for C₄; 4.21 s for C₅; and 15.07 s for C₆), though slightly higher than SGO’s, due to the more complex heuristics.

A key benefit of ESGO is the reduced number of objective-function evaluations—10,000 for the 40-unit configurations, 12,000 for the 110-unit configurations, and 24,000 for the 220-unit configurations—to achieve the performances reported in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. This computational load is substantially lower than that of the competing algorithms.

In terms of stability, the standard deviations of cost (SDCost) are very low in Cases C₁ (6.92 × 10⁻⁹ $/h), C₅ (6.22 × 10⁻⁵ $/h), and C₆ (1.57 × 10⁻⁴ $/h), and moderate in C₂ (9.13 $/h), C₃ (42.25 $/h), and C₄ (47.13 $/h), reflecting good consistency of results. The annual savings resulting from using ESGO in large-scale systems (40-unit, 110-unit, or 220-unit) are considerable, reaching the order of millions, tens of millions, or even hundreds of millions of dollars, compared to the costs generated by other optimization algorithms. The integration of wind capacity in case C₄ reduces total operating costs by approximately 10% compared to the thermal-only case C₃. This result confirms that ES-GO effectively solves wind-integrated dispatch problems under a linear wind power model.

The results indicate that the modifications introduced in ESGO improve the quality and stability of solutions as well as convergence compared to SGO. However, for the 40-unit system (C₂–C₄), where non-convex functions appear, solution stability remains moderate, suggesting the need for further improvements to ESGO. In addition, the two ESGO-specific parameters (α and β) must be selected based on the characteristics of the analyzed system. Although this requirement represents a practical limitation, the robustness analysis performed shows that ESGO exhibits relatively stable performance within the investigated ranges for these parameters, and only extreme combinations of them should be avoided.

Given the results obtained with the ESGO algorithm, a natural direction for future research concerns algorithmic enhancements, such as introducing parallel computing to increase execution speed, which is particularly relevant for real-world medium- and large-scale problems. Another important direction is the development and validation of a fully automatic strategy for setting the bounds of parameters α and β. The applicability of ESGO can be extended to related economic dispatch problems, including dynamic economic dispatch with or without renewable sources (wind, solar) and static or dynamic multi-area economic dispatch, as well as hydro-thermal systems. In this context, incorporating non-linear models for wind power generation could lead to a more faithful representation of real-system behavior. Moreover, testing ESGO on complex mathematical benchmark functions, with and without constraints, and conducting rigorous statistical comparisons with the original SGO and other well-established algorithms from the literature, are relevant directions for further evaluating the method’s performance.

Appendix A.1. Average Population Diversity

The average population diversity at iteration t, Div(t), is defined as follows:

$$Div(t)={\displaystyle \frac{1}{N}}\sum _{i=1}^N‖X_i\left(t\right)-\overline{X}(t)‖$$

where X_i(t) is the solution vector of individual i at iteration t; N is the population size; $\overline{X}\left(t\right)={\sum }_{i=1}^N{X}_i(t)/N$ is the population center (centroid); and ‖·‖ denotes the Euclidean norm. This metric quantifies the degree of population dispersion in the search space and is used to characterize the exploration capability of the algorithm.

Appendix A.2. Average Search-Step Magnitude

The average search-step magnitude at iteration t, Step(t), is defined as follows:

$$Step(t)={\displaystyle \frac{1}{N}}\sum _{i=1}^N‖X_i\left(t+1\right)-X(t)‖$$

where X_i(t) and X_i(t + 1) are successive positions of individual i.

This metric provides a direct measure of the amplitude of the movements generated by the algorithm and enables analysis of search-step adaptation during the optimization process.

The metrics Div(t) and Step(t) are computed for each independent run and then averaged over 50 runs to obtain a robust estimate of the dynamic behavior of the analyzed algorithms.