Improved Enterprise Development Optimization with Historical Trend Updating for High-Precision Photovoltaic Model Parameter Estimation

Abstract

Accurate parameter estimation of photovoltaic (PV) models is fundamentally a challenging nonlinear optimization problem, characterized by strong nonlinearity, high dimensionality, and multiple local optima. These characteristics significantly hinder the convergence accuracy, stability, and efficiency of conventional metaheuristic algorithms when applied to PV parameter identification. Although the enterprise development (ED) optimization algorithm has shown promising performance in various optimization tasks, it still suffers from slow convergence, limited solution precision, and poor robustness in complex PV parameter estimation scenarios. To overcome these limitations, this paper proposes a multi-strategy enhanced enterprise development (MEED) optimization algorithm for high-precision PV model parameter estimation. In MEED, a hybrid initialization strategy combining chaotic mapping and adversarial learning is designed to enhance population diversity and improve the quality of initial solutions. Furthermore, a historical trend-guided position update mechanism is introduced to exploit accumulated search information and accelerate convergence toward the global optimum. In addition, a mirror-reflection boundary control strategy is employed to maintain population diversity and effectively prevent premature convergence. The proposed MEED algorithm is first evaluated on the IEEE CEC2017 benchmark suite, where it is compared with 11 state-of-the-art metaheuristic algorithms under 30-, 50-, and 100-dimensional settings. Quantitative experimental results demonstrate that MEED achieves superior solution accuracy, faster convergence speed, and stronger robustness, yielding lower mean fitness values and smaller standard deviations on the majority of test functions. Statistical analyses based on Wilcoxon rank-sum and Friedman tests further confirm the significant performance advantages of MEED. Moreover, MEED is applied to the parameter estimation of single-diode and double-diode PV models using real measurement data. The results show that MEED consistently attains lower root mean square error (RMSE) and integrated absolute error (IAE) than existing methods while exhibiting more stable convergence behavior. These findings demonstrate that MEED provides an efficient and reliable optimization framework for PV model parameter estimation and other complex engineering optimization problems.

1. Introduction

In the context of rapid global economic growth and technological advancement, sustainable development has become a critical priority worldwide. The excessive consumption of natural resources, continuous environmental degradation, and intensifying climate change have revealed the limitations of traditional development models. Promoting sustainability is not only essential for ensuring long-term economic prosperity but also for maintaining social equity and ecological balance. Therefore, integrating sustainability principles into economic and technological systems is crucial to achieving harmonious coexistence between humans and nature, as well as securing a stable and livable environment for future generations [1].

Against the backdrop of the global pursuit of sustainable development and renewable energy transition, photovoltaic (PV) technology has emerged as a cornerstone for achieving carbon neutrality and green energy transformation [2]. However, the accuracy and efficiency of PV power generation are highly dependent on the precision of model parameter identification. Since PV models exhibit strong nonlinearity and are influenced by environmental factors such as temperature and irradiance, accurately extracting their parameters is a challenging yet essential task. Reliable parameter extraction not only enhances the fidelity of PV system simulations and performance predictions but also supports the optimal design, control, and operation of renewable energy systems, thereby playing a vital role in promoting sustainable energy development [3]. Therefore, an increasing number of researchers are focusing on the parameter extraction of photovoltaic models.

Nevertheless, accurately identifying the parameters of photovoltaic (PV) models remains a complex global optimization problem due to their high-dimensional, nonlinear, and multimodal characteristics. Traditional optimization techniques often suffer from slow convergence and premature stagnation in local optima, limiting their effectiveness in real-world applications. Researchers employed heuristic algorithms to solve this problem. For instance, Jiao et al. proposed an enhanced Harris–Hawks optimization by combining orthogonal learning (OL) and generalized adversarial learning to extract photovoltaic module parameters from actual current–voltage data, enabling efficient and accurate estimation of solar cell and photovoltaic module parameters [4]. Oliv et al. proposed a chaotic whale optimization algorithm for solar cell parameter estimation, utilizing chaos maps to compute and automatically adapt the algorithm’s internal parameters, demonstrating outstanding performance in estimating solar cell parameters [5]. To accurately estimate unknown parameters in complex photovoltaic models, Yang et al. proposed an effective improvement algorithm based on the L-SHADE (Linear Population Size Reduction) approach for dual-parameter coordinated updates [6]. This method decomposes the unknown parameters of solar photovoltaic models with varying complexities into linear and nonlinear parameters. Through the CSpL-SHADED (CSpL—Dual-Parameter Coordinated Update L-SHADE Parameter Decomposition Method (CSpL-SHADED). This effective improvement algorithm decomposes unknown parameters of solar photovoltaic models with varying complexities into linear and nonlinear components. CSpL-SHADED enables precise estimation of nonlinear parameters, while linear parameters are computed via constructed matrix equations. Experiments conducted on four solar photovoltaic models of differing complexities yielded favorable results.

Intelligent optimization algorithms have been widely applied to various problems. For example, Gou J proposed a Particle Swarm Optimization algorithm incorporating Individual Differential Evolution (IDE-PSO) [7]. This method classifies particles into several subgroups according to their performance levels. In another study, Ye W developed a multi-population PSO featuring a Dynamic Learning Strategy (PSO-DLS) [8], which distinguishes particles as either regular or communicative types. Chen et al. [9] proposed a new algorithm named PMEEA/D-VW, which is based on the original MOEA/D (multi-objective evolutionary algorithm based on decomposition) framework and is used for multi-objective test case prioritization. Rabeh Abbassi et al. [10] proposed a developed Mountain Gazelle Optimizer (MGO) to generate optimal values for the unknown parameters of photovoltaic power generation units. Li et al. [11] focused on the theory of the multi-objective 3L-SDHVRP and proposed a multi-objective evolutionary algorithm based on decomposition, customizable replacement neighborhoods, and dynamic resource allocation (referred to as MOEA/D-RD) to address this problem. Zhao et al. [12] proposed a surrogate-assisted evolutionary algorithm based on multi-population clustering and prediction (SAEA/MPCP) to solve the CEDOP problem, improving the efficiency of surrogate model construction and optimization. Mohamed et al. [13] proposed an advanced Dynamic Fick’s Law Algorithm (DFLA) for extracting optimal parameters of fuel cells. Rezk et al. [14] proposed a robust method based on the Gradient-Based Optimizer (GBO) to identify the optimal parameters of proton-exchange membrane fuel cells (PEMFCs). Saidi et al. [15] proposed a new hybrid algorithm, CADESSA, for PEMFC parameter identification. Abdullah et al. [16] employed an enhanced Artificial Gorilla Troops Optimizer to extract the optimal parameters of the photovoltaic three-diode model. Chen et al. [17] proposed an efficient hybrid optimization algorithm combining grid search and an improved Nelder–Mead simplex method (GS-INMS) for PV model parameter identification. Abbassi et al. [18] introduced a new method using the newly developed Puma Optimizer (PO) to extract the key parameters of the photovoltaic cell double-diode model (DDM).

Intelligent optimization algorithms have demonstrated broad application potential in areas such as multi-objective optimization and parameter identification, and a series of improved and innovative methods have emerged. However, for the uncertainties, constraints, and real-time requirements commonly encountered in practical engineering problems, the adaptation mechanisms of existing algorithms remain insufficient. Therefore, how to enhance robustness, adaptability, and general applicability while ensuring efficiency remains a key issue that urgently needs to be addressed in the field of intelligent optimization.

The Enterprise Development Optimization Algorithm is a novel metaheuristic optimization algorithm inspired by the enterprise development process [19], encompassing tasks, structures, technologies, and interpersonal interactions. It employs a mechanism of switching activities to determine each step by updating the solutions found. Furthermore, experiments involving various mathematical functions and complex structural design problems demonstrate that the ED optimization algorithm is a superior metaheuristic algorithm. However, since the NFL theorem asserts that no single “universal optimal algorithm” exists that outperforms others across all problems, designing improved or hybrid optimization algorithms tailored to specific problem characteristics—to enhance their convergence efficiency, robustness, and solution accuracy—has become a major research focus in the field of intelligent optimization.

For instance, addressing the challenges posed by large-scale direct grid connection of wind and solar power for peak shaving, Zhao et al. proposed an enhanced version of the ED algorithm (TGED) by integrating chaotic initialization and Gaussian random walk mechanisms [20]. For this problem, the algorithm outperformed other benchmark algorithms in terms of solution accuracy and convergence performance, reducing the residual load peak-to-valley difference by over 600 MW. To address multi-objective optimization challenges in practical engineering, Truong et al. proposed a novel ED optimization algorithm tailored for complex multi-objective engineering problems [21]. By integrating advanced population and non-dominated sorting techniques into existing single-objective evolutionary development algorithms, this new multi-objective approach effectively identifies Pareto optimal solutions. Based on the above research, to address the issue of extracting photovoltaic model parameters, this paper proposed a multi-strategy enhanced enterprise development algorithm (MEED), which overcomes the tendency of ED to get stuck in local optima when addressing the problem of extracting parameters for photovoltaic models. The main contributions and innovations of this work are as follows:

• We proposed a hybrid initialization method based on chaotic mapping and adversarial learning mechanisms to enhance the quality of the initial population, enabling the algorithm to better explore the solution space.
• We proposed a trend position update method based on historical information enables the algorithm to better utilize information from historical updates, accelerating algorithm convergence.
• A boundary control method based on mirror reflection is proposed to prevent out-of-bounds individuals from clustering near the boundary, enabling the algorithm to escape local optima more effectively and seek global optimal solutions.
• Comprehensive benchmark validation: MEED is compared against 11 mainstream metaheuristic algorithms on the CEC2017 benchmark suite, and statistical analysis demonstrated significant differences between MEED and other algorithms.
• Application to PV model parameter identification: MEED is applied to both the SDM and DDM PV models. Using experimental data from the Photowatt-PWP201 PV module and the RTC France solar cell, the algorithm’s effectiveness is validated through root mean square error (RMSE), integrated absolute error (IAE), and curve-fitting comparisons, demonstrating its practical value for solving complex real-world problems.

The remainder of this paper is organized as follows: Section 2 presents the fundamental principles of the ED optimization algorithm. Section 3 introduces the three improvement strategies of MEED. Section 4 conducts comparative experiments on CEC2017 test suites. Section 5 applies MEED to PV model parameter identification, validating its effectiveness in practical engineering problems. Section 6 concludes the study and outlines future research directions.

2. Enterprise Development Optimization Algorithm (ED)

The ED algorithm is inspired by the continuous growth of companies driven by the interaction among mission, organizational structure, technology, and personnel. Different configurations of these interdependent elements lead to distinct performance outcomes, enabling the identification of optimal management solutions through quantitative evaluation. Based on this concept, the mathematical model of the ED optimization algorithm is defined as follows:

2.1. Initialize Population

In ED, like most multiheuristic algorithms, the ED optimization algorithm randomly generates an initial population with uniform distribution for optimization. The initial population can be represented by Equation (1):

$$X_i=lb+\left(ub-lb\right)\times rand\left(0,1\right),$$

(1)

where $X_i$ denotes the $i$-th individual in the population, $lb$ and $ub$ represent the lower and upper bounds of the problem, respectively, and $rand\left(0, 1\right)$ indicates a random number between 0 and 1.

2.2. Establishing Optimal Rules and Simulation Activities

In this subsection, ED establishes three optimal rules. First, it defines four activities—tasks, structure, technology, and personnel—along with the switching mechanisms between these activities. Next, it correlates these activities with organizational performance. Finally, it evaluates organizational performance using solutions and their corresponding objective functions. Specific details are as follows:

2.2.1. Task

In business process management, tasks may take various forms or exist as routine work activities. To simulate task activities, they are represented as worst-case individual instances, specifically modeled as Equation (2):

$$X_{\mathrm{w}orst}\left(\mathrm{t}\right)=lb+\left(ub-lb\right)\times rand\left(0,1\right)$$

(2)

where $X_{\mathrm{w}orst}\left(\mathrm{t}\right)$ denotes the worst individual in the current population at iteration $t$.

2.2.2. Structure

The work “Designing Organizations for Human Efficiency” demonstrates that appropriately redesigning information theory work in a social engineering sense can influence human attitudes and outputs. In ED, organizational structures are constrained to workflows, as new organizational structures are expected to be influenced by other workflow structures and current optimal workflows within the organization. This is modeled as Equation (3):

$$X_i^s\left(\mathrm{t}\right)=X_i^s\left(\mathrm{t}-1\right)+rand\left(-1,1\right)\times \left(X_{best}\left(\mathrm{t}-1\right)-X_c^s\left(\mathrm{t}-1\right)\right)$$

(3)

where $X_i^s\left(\mathrm{t}\right)$ denotes the new structure, $X_i^s\left(\mathrm{t}-1\right)$ denotes the old structure, $X_{best}\left(\mathrm{t}-1\right)$ denotes the current optimal solution, $X_c^s\left(\mathrm{t}-1\right)$ denotes the center of other workflows affecting the new structure, calculated using Equation (4), and $rand\left(-1, 1\right)$ denotes a random number between −1 and 1.

$$X_c^s\left(\mathrm{t}-1\right)={\displaystyle \frac{X_{ran{d}_1}^s\left(\mathrm{t}-1\right)+X_{ran{d}_2}^s\left(\mathrm{t}-1\right)+\cdots +X_{ran{d}_m}^s\left(\mathrm{t}-1\right)}{m}}$$

(4)

Here, $m$ is the number of workflows affecting the new structure, set to 3. $X_{ran{d}_1}^s\left(\mathrm{t}-1\right)$, $X_{ran{d}_2}^s\left(\mathrm{t}-1\right)$, and $X_{ran{d}_m}^s\left(\mathrm{t}-1\right)$ are individuals randomly selected from the population.

2.2.3. Technology

Technology plays a crucial role in organizational change. In many cases, organizational change occurs in response to technological advancements. Open innovation inputs primarily enhance an organization’s “exploration” capabilities, while open innovation outputs are closely tied to the “development” of the organization’s foundational knowledge and technologies. Organizations must intensify both exploratory and exploitative efforts to acquire and apply the knowledge required for innovation activities from a strategic openness perspective. We use Equation (5) to simulate the balance between exploration and exploitation:

$$\begin{array}{c}{X}_i^{\tau }\left(\mathrm{t}\right)=X_i^{\tau }\left(\mathrm{t}-1\right)+ran{d}^{\alpha }\left(0,1\right)\times \left(X_{best}\left(\mathrm{t}-1\right)-X_i^{\tau }\left(\mathrm{t}-1\right)\right)\\ + ran{d}^{\beta }\left(0,1\right)\times \left(X_{best}\left(\mathrm{t}-1\right)-X_{ran{d}_1}^{\tau }\left(\mathrm{t}-1\right)\right)\end{array}$$

(5)

where $X_{best}\left(\mathrm{t}-1\right)-X_{ran{d}_1}^{\tau }\left(\mathrm{t}-1\right)$ indicates the exploration phase, while $X_{best}\left(\mathrm{t}-1\right)-X_i^{\tau }\left(\mathrm{t}-1\right)$ indicates the exploitation phase.

2.2.4. People

Organizations must cultivate a participatory work culture that fosters individual creativity and teamwork by respecting personnel and stakeholders. This approach can be modeled as Equation (6):

$$X_{i,d}^p\left(\mathrm{t}\right)=X_{i,d}^p\left(\mathrm{t}-1\right)+rand\left(-1,1\right)\times \left(X_{best,d}\left(\mathrm{t}-1\right)-X_{c,d}^p\left(\mathrm{t}-1\right)\right)$$

(6)

where $X_{c,d}^p\left(\mathrm{t}-1\right)$ can be calculated using Equation (7):

$$X_{c,d}^p\left(\mathrm{t}-1\right)={\displaystyle \frac{X_{ran{d}_1,d}^p\left(\mathrm{t}-1\right)+X_{ran{d}_2}^s\left(\mathrm{t}-1\right)+\cdots +X_{ran{d}_m}^s\left(\mathrm{t}-1\right)}{m}}$$

(7)

where $m$ represents the number of individuals affected, with its value set to 3, and $d$ represents the random characteristic of a person, which can be calculated using Equation (8).

$$d=\lceil rand\left(0,1\right)\times n_d\rceil$$

(8)

Here, $n_d$ denotes the dimensionality of the solution.

2.3. Mechanism of Switching Activities

In ED, it is assumed that an organization focuses on only one step at a time. Therefore, only one of the four steps (i.e., task, structure, technology, and personnel) occurs at time t, controlled by the activity switching mechanism. This mechanism is governed by function $C\left(t\right)$, which can be expressed as Equation (9):

$$C\left(t\right)=\lceil 3\times \left(1-{\displaystyle \frac{rand\left(0,1\right)\times t}{T}}\right)\rceil$$

(9)

where $t$ indicates the current iteration count, and $T$ indicates the maximum iteration count. The detailed selection mechanism can be described by Algorithm 1.

Algorithm 1: the pseudo-code of the selection mechanism

1: Begin

2: Calculate C(t) by using Equation (9).

3: If rand < p1, then p1 = 0.1

4:  Step “task”

5: Else

6:  Switch C(t)

7:  Case C(t) = 1

8:   Step “structure”

9:  Case C(t) = 2

10:    Step “technology”

11:   Case C(t) = 3

12:    Step “people”

13:   End switch

14: End if

15: end

The pseudocode of the ED is outlined in Algorithm 2.

Algorithm 2: the pseudo-code of the ED

1: Begin

2: Initialize: the relevant parameters iterations T and the number of coatis $pop$.

3: Calculate the fitness of the objective function.

4:   For $t < T$ do

5:    $\mathit{For} i=1:N$ do

6:    $\mathrm{Calculate} C\left(t\right)$ by using Equation (9).

7:    $\mathbf{If} rand<p_1,$$\mathrm{then} p_1=0.1$

8:     Step “task”

9:    Else

10:      Switch $C\left(t\right)$

11:     Case $C\left(t\right)=1$

12:      Step “structure”

13:     Case $C\left(t\right)=2$

14:      Step “technology”

15:     Case $C\left(t\right)=3$

16:      Step “people”

17:      End switch

18:      End if

19:   End for

20:    $t=t+1$

21:    End for

22:    return best solution

23: end

3. Proposed MEED

3.1. Hybrid Initialization Method Based on Chaotic Mapping and Adversarial Learning Mechanism

The quality of the initial population plays a crucial role in the convergence speed and optimization accuracy of metaheuristic algorithms. A well-distributed initial population effectively enhances global exploration and prevents premature convergence. In ED optimization algorithms, population initialization typically employs a basic random approach. While simple, this method often results in poor initial solution quality. Therefore, this paper proposes a hybrid initialization method based on chaotic mapping and adversarial learning mechanisms to improve the quality of the initial population [22]. In this method, chaotic mapping is first employed to generate a sequence with good ergodicity, which is then mapped onto the decision space to obtain initial candidate solutions. This process can be formally expressed as Equation (10):

$$z_{i+1}=\mu \times z_i\times \left(1-z_i\right)$$

(10)

where $\mu$ is a control parameter with a value of 2. $z_i$ represents a chaotic scalar, whose value lies between 0 and 1. Then, mapping the chaotic variables onto the decision space to obtain the initial population can be expressed as Equation (11).

$$X_i=lb+\left(ub-lb\right)\times z_i$$

(11)

This mapping ensures strong randomness and sensitivity to initial conditions, effectively increasing the search diversity in the early stage. Additionally, to accelerate convergence and enhance global search capabilities, we propose a selection scheme based on adversarial learning. Solutions derived through adversarial learning can be computed using Equation (12).

$$X_i^{obl}=lb+ub-X_i$$

(12)

Through an evaluation of the fitness function, we retain higher-quality individuals in the initial population, which can be described by Equation (13).

$$X_i^{new}=\left\{\begin{array}{c}{X}_i, iff\left(X_i\right)\le f\left(X_i^{obl}\right)\\ X_i^{obl}, otherwise\end{array}\right.$$

(13)

This mechanism allows the algorithm to explore more promising areas of the search space from the beginning, effectively enhance the quality of the initial population.

3.2. Trend Position Update Method Based on Historical Information

In the ED optimization algorithm, its position update mechanism relies on selection. To accelerate the algorithm’s convergence speed and fully utilize information from high-quality solutions within the population, this paper proposes a position update method based on historical information; this method relies on the historical search position matrix and fitness evidence, where the historical search position matrix contains the positions of all individuals at each iteration, and the fitness matrix includes the fitness values of each individual at every iteration. The position update can be specifically described by Equation (14):

$$\begin{gathered} X_i\left(\mathrm{t}\right)=X_i\left(\mathrm{t}-1\right)+{\alpha }_1\times r_1\times \left(X_{best}\left(\mathrm{t}-1\right)-X_i\left(\mathrm{t}-1\right)\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\alpha }_2\times r_2\times \left(T\left(\mathrm{t}-1\right)-X_i\left(\mathrm{t}-1\right)\right) \end{gathered}$$

(14)

where ${\alpha }_1$ and ${\alpha }_2$ are used to control the influence strength of the current global optimum and historical trend, respectively, with their values set to 0.5. $r_1$ and $r_2$ denote random disturbance terms following a uniform distribution between 0 and 1. $X_{best}\left(\mathrm{t}-1\right)$ represents the global optimum position at the current iteration, while $T\left(\mathrm{t}-1\right)$ denotes the historically weighted trend center, calculated using Equation (15).

$$T\left(\mathrm{t}\right)={\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^t{\omega }_k{X}_{best}^k}{{{\displaystyle \sum }}_{k=1}^t{\omega }_k}}$$

(15)

Here, $X_{best}^k$ denotes the position of the globally optimal individual in the $k$-th iteration, while ${\omega }_k$ represents the weight based on the fitness value, calculated using Equation (16):

$${\omega }_k={\displaystyle \frac{1}{f_{best}^k+ϵ}}$$

(16)

where $f_{best}^k$ denotes the fitness value of the globally optimal individual in the $k$-th iteration, while $ϵ$ represents a small constant to prevent division by zero.

3.3. Boundary Control Method Based on Mirror Reflection

In ED optimization algorithms, boundary control is achieved through truncation when individuals exceed the search domain. While this method ensures all individuals remain within the feasible region, it causes solutions that would otherwise exceed the boundary to cluster at the edge. This reduces population diversity and weakens the algorithm’s global search capability. To address this, we propose a boundary control strategy based on mirror reflection [23]. When an individual exceeds the search range, it is symmetrically reflected back into the feasible region rather than simply truncated. This approach not only maintains the continuity of the search trajectory, avoiding premature convergence caused by boundary stagnation, but also better balances global exploration and local exploitation capabilities. Specifically, it can be calculated using Equation (17).

$$X_i=\left\{\begin{array}{c}lb+\left(lb-X_i\right), X_i<lb\\ ub-\left(X_i-ub\right), X_i>ub\end{array}\right.$$

(17)

The boundary control method based on mirror reflection effectively addresses the limitations of the traditional clamping strategy. When an individual exceeds the boundary of the search space, instead of being forcibly truncated to the nearest bound, its position is symmetrically reflected back into the feasible region. This mechanism offers several advantages. First, it maintains the continuity of the search trajectory, preventing abrupt changes that may disrupt the convergence path. Second, by avoiding the accumulation of individuals along the boundaries, it preserves population diversity and enhances the algorithm’s global exploration ability. Third, the reflection operation allows boundary individuals to re-enter the search region with a meaningful direction, thereby reducing the risk of premature convergence. Finally, since this method requires only simple arithmetic operations, it introduces no additional computational complexity. Overall, the mirror-reflection-based boundary control strategy achieves a better balance between exploration and exploitation, ensuring both stability and robustness in the optimization process.

The MEED’s flowchart is provided in Figure 1, and the pseudocode of the ED is outlined in Algorithm 3.

Algorithm 3: the pseudo-code of the MEED

1: Begin

2: Initialize: the relevant parameters iterations T and the number of coatis pop.

3: Hybrid initialization (Equations (10)–(13))

4: Calculate the fitness of the objective function.

5:  For t < T do

6:  For i = 1:N do

7:   Calculate C(t) by using Equation (9).

8:   If rand < p1, then p1 = 0.1

9:     Step “task”

10:   Else

11:      Switch C(t)

12:     Case C(t) = 1

13:      Step “structure”

14:     Case C(t) = 2

15:      Step “technology”

16:     Case C(t) = 3

17:      Step “people”

18:      End switch

19:   End if

20:    Trend position update using historical info (Equations (14)–(16))

21:    Mirror-reflection boundary control (Equation (17))

22:  End for

23:   t = t + 1

24: End for

25: return best solution

26: end

3.4. Complexity Analysis of MEED

For the proposed MEED algorithm, the initialization phase follows the standard ED framework, including population initialization and fitness evaluation for $N$ individuals in a $D$-dimensional search space. In addition, MEED incorporates a hybrid initialization strategy based on chaotic mapping and adversarial learning, where chaotic sequence generation, mapping, and adversarial solution construction are all conducted at the population level and involve only linear operations with respect to the population size and dimensionality, resulting in a computational complexity of $O\left(N\times D\right)$ for the initialization stage. During the iterative optimization process, MEED preserves the original activity-switching mechanism and the four-step update structure of ED (task, structure, technology, and people) while further introducing a historical-information-based trend position update strategy and a mirror-reflection boundary control strategy. Although these enhancement strategies add extra position update and boundary handling operations, they are all executed for each individual across $D$ dimensions using simple arithmetic and weighted aggregation operations, without introducing nested loops or higher-order computations. Specifically, position updating, trend calculation, boundary control, fitness evaluation, and selection are performed for $N$ individuals in each iteration, leading to a per-iteration computational complexity of $O\left(N\times D\right)$. Therefore, over $T$ iterations, the total computational complexity of MEED is $O\left(T\times N\times D\right)$, which is consistent with the standard ED algorithm, demonstrating that the proposed multi-strategy enhancements improve optimization performance without increasing the computational complexity order.

4. Experimental Results and Detailed Analyses

In this subsection, we evaluate MEED’s performance using CEC2017 [24]. First, we briefly introduce the comparison algorithms and parameter settings. Subsequently, we conduct a comprehensive analysis of MEED’s performance across three dimensionality levels: 30, 50, and 100. Finally, to determine whether MEED exhibits significant differences compared to other algorithms, we conducted statistical analysis on MEED. To ensure fairness in comparison, we set the population size for all algorithms to 50 and the maximum iteration count to 1000. To mitigate experimental randomness, each algorithm was independently run 30 times, and the average results were used for analysis. All experiments are conducted on a Windows 11 operating system with a 13th Gen Intel(R) Core(TM) i5-13400 CPU @ 2.5 GHz and 16 GB RAM, using MATLAB 2023a.

4.1. Competitor Algorithms and Parameters Setting

In this section, the superior performance of the proposed MEED algorithm is validated through comparative experiments with 11 state-of-the-art algorithms, including the Red-tailed Hawk Algorithm (RTH), Snow Ablation Optimizer (SAO), Gold Rush Optimizer (GRO), Snake Optimizer (SO), Escape Algorithm (ESC), weighted mean of vectors algorithm (INFO), Secretary Bird Optimization Algorithm (SBOA), Genghis Khan Shark Optimizer (GKSO), Improved Gray Wolf Optimizer (IGWO), hyper-heuristic whale optimization algorithm (HHWOA), and the standard enterprise development algorithm (ED). To enhance the reproducibility of the experiments,

Table 1. Parameter settings of the comparison algorithms.

Algorithms	Parameter Name	Parameter Value	Reference
RTH	$A$$, R_0$$, r$	15, 0.5, 1.5	[25]
SAO	$k$	1	[26]
GRO	$sigma$	2	[27]
SO	$c1$$, c2$$, c3$	0.5, 0.5, 2	[28]
ESC	$eliteSize$$, a$$, b$	5, 0.15, 0.35	[29]
INFO	$e$	1 × 10−25	[30]
SBOA	beta	1.5	[31]
GKSO	$h$	0.1	[32]
IGWO	$at$	2	[33]
HHWOA	$w$	3	[34]
ED	$ishow$	250	[19]

lists the parameter settings for each comparison algorithm.

4.2. Ablation Study

To verify the effectiveness of different improvement strategies in enhancing the performance of the enterprise development (ED) algorithm and to clarify the role of each strategy in the optimization process, this study conducts ablation experiments using the 30-dimensional CEC2017 benchmark set. The convergence behaviors of the original ED algorithm, ED variants incorporating a single improvement strategy (ED-CP, ED-HI, and ED-MR), and the MEED algorithm that integrates all strategies are systematically compared. Specifically, ED-CP corresponds to the hybrid initialization strategy, ED-MR corresponds to the mirror-reflection boundary control strategy, and ED-HI represents another single-strategy variant.

The experiments aim to analyze, through a direct comparison of convergence curves, the impact of each strategy on convergence speed, optimization accuracy, and the ability to escape local optima, thereby providing empirical evidence for the rationality and effectiveness of the multi-strategy fusion adopted in the MEED algorithm. The detailed convergence curve comparisons and average ranking results are illustrated in Figure 2 and Figure 3.

Figure 2 compares the convergence performance of the original ED algorithm, its variants incorporating different improvement strategies (ED-CP, ED-HI, and ED-MR), and the final MEED algorithm on the CEC2017 benchmark set. As observed from the subfigures, ED variants that adopt only a single improvement strategy outperform the original ED algorithm in both convergence speed and optimization accuracy. Among them, ED-HI based on hybrid initialization and ED-MR based on mirror-reflection boundary control exhibit particularly prominent performance, confirming that each individual strategy contributes to enhancing the algorithm’s effectiveness. In contrast, the MEED algorithm, which integrates all improvement strategies, consistently maintains the best convergence trend throughout the entire iteration process. It not only achieves rapid fitness reduction in the early stages and quickly widens the performance gap with other algorithms but also continues to approach the global optimum in the middle and late stages without suffering from premature stagnation. For example, on the complex multimodal function F12 and the composite function F27, the convergence curves of MEED are significantly lower than those of the other algorithms, indicating that the synergistic effects of hybrid initialization, historical trend updating, and mirror-reflection boundary control effectively balance global exploration and local exploitation, thereby overcoming the limitations of single-strategy approaches.

Figure 3 presents the performance improvements brought by different strategies to the ED algorithm in a more intuitive manner through average rankings. Due to issues such as random initialization, insufficient utilization of historical information, and improper boundary handling, the original ED algorithm ranks last on average, highlighting its deficiencies in complex optimization problems, including low convergence accuracy and a tendency to fall into local optima. After introducing individual improvement strategies, the average rankings of the ED variants are improved to varying degrees. In particular, ED-HI (hybrid initialization) and ED-MR (mirror reflection) show more substantial ranking improvements, with average ranks reduced to 3.63 and 2.27, respectively, demonstrating the crucial roles of initial population quality enhancement and boundary control in overall algorithm performance. The MEED algorithm, which integrates all three strategies, achieves the best average ranking of 1.70, significantly outperforming all other variants. This result fully validates the effectiveness of multi-strategy collaborative improvement: hybrid initialization establishes a high-quality search foundation, historical trend updating provides efficient convergence guidance, and mirror-reflection boundary control preserves population diversity. Together, these complementary mechanisms enable MEED to exhibit more stable and superior overall performance on 30-dimensional complex optimization problems.

4.3. Comparison Using the CEC2017 Test Set

In this subsection, we evaluate the performance of MEED using the CEC2017 test set. We compare MEED against 11 other state-of-the-art algorithms across three dimensionality levels: 30, 50, and 100 dimensions. The convergence curves for each algorithm are shown in Figure 4. Experimental results for each algorithm on various test functions are presented in

Table 2. Results of various algorithms tested on the CEC 2017 benchmark (dim = 30).

ID	Metric	RTH	SAO	GRO	SO	ESC	INFO	SBOA	GKSO	IGWO	HHWOA	ED	MEED
F1	mean	3.6602E+03	5.3897E+03	1.0640E+06	5.8984E+04	3.8633E+03	1.1469E+02	2.4120E+03	4.4973E+03	4.0849E+05	3.7981E+03	2.2968E+03	1.4278E+02
	std	3.7907E+03	4.9396E+03	1.4965E+06	1.0180E+05	3.8550E+03	2.9617E+01	2.5326E+03	4.8695E+03	2.8886E+05	4.3568E+03	2.5362E+03	3.7484E+01
F2	mean	6.8351E+15	3.7506E+14	1.0621E+22	2.0247E+17	7.6296E+14	7.9811E+16	5.6291E+12	1.4677E+13	3.0708E+16	4.5957E+16	5.1807E+20	5.2449E+03
	std	3.7167E+16	1.9617E+15	4.8489E+22	3.8402E+17	4.0582E+15	1.7044E+17	1.1963E+13	5.3991E+13	6.9228E+16	1.7673E+17	1.9441E+21	6.0330E+03
F3	mean	3.0000E+02	6.9455E+04	3.5642E+04	5.5205E+04	4.1356E+04	7.1019E+02	6.3019E+03	3.0343E+02	5.2406E+03	3.0000E+02	8.2198E+04	4.3468E+03
	std	1.8613E−07	1.8165E+04	9.2612E+03	1.0434E+04	1.1503E+04	6.7101E+02	3.9953E+03	4.0692E+00	3.0965E+03	2.0568E−02	1.5575E+04	6.4171E+02
F4	mean	4.2522E+02	4.9708E+02	5.1243E+02	4.9974E+02	5.0582E+02	4.8163E+02	4.9534E+02	4.8658E+02	4.9922E+02	4.6348E+02	4.7605E+02	4.0017E+02
	std	3.0258E+01	1.3142E+01	1.7325E+01	2.9351E+01	1.4693E+01	2.5505E+01	2.6634E+01	2.6714E+01	1.3035E+01	2.4766E+01	3.2910E+01	1.8652E−01
F5	mean	6.7401E+02	5.4878E+02	5.7661E+02	5.5979E+02	5.6635E+02	6.4788E+02	5.6358E+02	6.7903E+02	5.6868E+02	5.8646E+02	6.4546E+02	5.4046E+02
	std	3.4522E+01	1.2805E+01	2.4997E+01	1.2697E+01	2.0330E+01	3.8837E+01	1.8774E+01	3.2788E+01	4.2345E+01	2.1031E+01	1.9269E+01	3.1262E+00
F6	mean	6.3949E+02	6.0002E+02	6.0474E+02	6.0209E+02	6.0000E+02	6.1988E+02	6.0086E+02	6.3687E+02	6.0039E+02	6.0647E+02	6.0399E+02	6.0000E+02
	std	7.7694E+00	8.6725E−02	1.9584E+00	2.0672E+00	3.7616E−04	9.3268E+00	1.0054E+00	1.1255E+01	1.6881E−01	6.0944E+00	4.2643E+00	6.6874E−06
F7	mean	1.0647E+03	8.3328E+02	8.1296E+02	8.2677E+02	8.2324E+02	9.4077E+02	8.0915E+02	9.3272E+02	8.4081E+02	8.6234E+02	8.6678E+02	7.7171E+02
	std	8.5959E+01	6.1398E+01	2.2387E+01	3.2097E+01	1.1883E+01	5.6722E+01	2.6771E+01	6.4514E+01	6.0840E+01	4.8158E+01	2.3500E+01	4.4029E+00
F8	mean	9.3707E+02	8.5529E+02	8.6436E+02	8.6225E+02	8.7334E+02	9.2713E+02	8.6125E+02	9.4652E+02	8.5391E+02	8.8402E+02	9.4417E+02	8.3748E+02
	std	2.1923E+01	1.1450E+01	1.6910E+01	1.2463E+01	2.3207E+01	2.9323E+01	1.8735E+01	3.0360E+01	2.1156E+01	1.5260E+01	1.8047E+01	3.2887E+00
F9	mean	4.2168E+03	9.1523E+02	1.2067E+03	1.2575E+03	9.0090E+02	2.5232E+03	9.9563E+02	2.8922E+03	9.0258E+02	1.2906E+03	1.2519E+03	9.0010E+02
	std	6.8233E+02	4.3904E+01	2.6225E+02	1.9616E+02	1.7105E+00	7.9864E+02	1.3960E+02	8.6377E+02	2.4748E+00	3.8888E+02	2.3703E+02	1.2648E−01
F10	mean	5.0482E+03	3.7524E+03	4.1082E+03	3.3058E+03	6.4871E+03	5.1639E+03	4.0911E+03	4.7952E+03	6.7423E+03	4.8531E+03	5.2272E+03	3.6737E+03
	std	5.4898E+02	7.0413E+02	5.3101E+02	5.9959E+02	4.1566E+02	8.4267E+02	5.2167E+02	4.4770E+02	2.0012E+03	8.0755E+02	1.9867E+02	1.9337E+02
F11	mean	1.2401E+03	1.1673E+03	1.2267E+03	1.2678E+03	1.1748E+03	1.2442E+03	1.1602E+03	1.2167E+03	1.1857E+03	1.2119E+03	1.1841E+03	1.1226E+03
	std	4.9998E+01	4.8479E+01	3.3464E+01	4.6875E+01	2.9797E+01	5.5763E+01	2.8868E+01	4.1231E+01	2.6746E+01	4.3439E+01	4.2935E+01	4.5244E+00
F12	mean	2.2790E+04	3.5738E+05	8.7277E+05	6.1587E+05	7.1931E+05	5.5205E+04	5.7376E+05	1.2120E+05	2.1358E+06	4.5419E+04	3.4397E+05	1.5586E+04
	std	1.1167E+04	2.9961E+05	6.2244E+05	5.5490E+05	5.3211E+05	3.6886E+04	4.3552E+05	1.1592E+05	1.8211E+06	3.3805E+04	3.6540E+05	3.3629E+03
F13	mean	2.6510E+04	2.1981E+04	1.8762E+04	1.9730E+04	1.5965E+04	2.0863E+04	2.2820E+04	1.8900E+04	1.1920E+05	1.6486E+04	1.9749E+04	1.5491E+03
	std	2.4704E+04	1.8334E+04	1.4884E+04	1.6017E+04	1.3998E+04	1.8173E+04	2.0468E+04	1.7810E+04	5.7886E+04	1.9158E+04	1.9174E+04	1.9877E+02
F14	mean	1.8779E+03	3.0018E+04	1.4038E+04	1.9700E+04	4.7292E+04	2.2423E+03	1.4854E+04	2.7724E+03	6.6341E+03	1.4755E+03	2.8436E+04	2.2513E+03
	std	2.8603E+02	2.1941E+04	1.3888E+04	1.6892E+04	4.2524E+04	1.0283E+03	1.5022E+04	2.4308E+03	4.8873E+03	3.5662E+01	1.6542E+04	3.4094E+02
F15	mean	7.7191E+03	3.6719E+03	6.3483E+03	8.6424E+03	8.8571E+03	4.7244E+03	8.8810E+03	1.0479E+04	2.3973E+04	2.2487E+03	6.3809E+03	1.5885E+03
	std	1.0113E+04	2.2063E+03	4.7994E+03	7.9124E+03	1.0172E+04	3.8346E+03	8.3817E+03	1.0595E+04	1.9592E+04	3.8406E+03	6.0134E+03	4.1962E+01
F16	mean	2.8003E+03	2.3759E+03	2.1402E+03	2.3621E+03	2.0571E+03	2.6610E+03	2.2255E+03	2.5946E+03	2.1216E+03	2.4516E+03	2.8867E+03	1.8611E+03
	std	3.3738E+02	3.0887E+02	1.9858E+02	2.2459E+02	2.2877E+02	2.3993E+02	2.6621E+02	3.4525E+02	4.6043E+02	2.3015E+02	1.7469E+02	5.7377E+01
F17	mean	2.3556E+03	2.0646E+03	1.8317E+03	1.9989E+03	1.8013E+03	2.2128E+03	1.8804E+03	2.1809E+03	1.8208E+03	2.1023E+03	2.1298E+03	1.7511E+03
	std	2.4978E+02	2.1552E+02	7.7383E+01	1.4513E+02	7.7374E+01	2.3452E+02	1.2283E+02	1.9649E+02	1.0370E+02	1.7997E+02	1.2056E+02	7.7622E+00
F18	mean	1.2292E+04	3.0398E+05	2.5094E+05	4.4241E+05	4.9706E+05	3.3657E+04	3.9395E+05	6.1577E+04	1.8613E+05	8.0093E+03	6.5099E+05	6.8973E+04
	std	1.3236E+04	2.4162E+05	2.6221E+05	3.5898E+05	5.4796E+05	2.7736E+04	3.0035E+05	4.3521E+04	1.4314E+05	1.1938E+04	4.6229E+05	1.7823E+04
F19	mean	5.7378E+03	7.3154E+03	6.8173E+03	6.8407E+03	1.4122E+04	3.4968E+03	8.6061E+03	8.3722E+03	1.6253E+04	1.9558E+03	1.3799E+04	2.1595E+03
	std	4.4317E+03	5.1277E+03	6.4674E+03	5.9123E+03	1.0671E+04	3.8779E+03	1.0861E+04	9.6676E+03	1.5377E+04	1.9069E+02	1.2944E+04	1.9272E+02
F20	mean	2.6589E+03	2.3336E+03	2.2505E+03	2.3588E+03	2.1345E+03	2.5512E+03	2.2017E+03	2.4374E+03	2.1660E+03	2.5316E+03	2.5411E+03	2.0676E+03
	std	1.8297E+02	1.8295E+02	7.9193E+01	1.2727E+02	8.9303E+01	2.0537E+02	1.1099E+02	1.2120E+02	9.6003E+01	1.8710E+02	1.3660E+02	1.4024E+01
F21	mean	2.4436E+03	2.3511E+03	2.3601E+03	2.3643E+03	2.3682E+03	2.4296E+03	2.3496E+03	2.4542E+03	2.3526E+03	2.3790E+03	2.4470E+03	2.3363E+03
	std	4.0075E+01	1.3841E+01	1.5030E+01	1.0625E+01	2.1656E+01	3.4962E+01	1.1544E+01	3.8082E+01	3.1311E+01	2.2650E+01	1.8769E+01	3.6004E+00
F22	mean	3.7471E+03	2.4665E+03	2.3091E+03	3.7087E+03	3.5169E+03	4.6482E+03	2.3011E+03	2.6713E+03	3.0442E+03	3.9240E+03	6.0207E+03	2.3000E+03
	std	2.2814E+03	6.3950E+02	4.9580E+00	1.3538E+03	2.2097E+03	2.3423E+03	1.4875E+00	1.1670E+03	1.7044E+03	2.0879E+03	1.4988E+03	0.0000E+00
F23	mean	2.8531E+03	2.7064E+03	2.7152E+03	2.7367E+03	2.7032E+03	2.8060E+03	2.6984E+03	2.8611E+03	2.7127E+03	2.7681E+03	2.8117E+03	2.6866E+03
	std	5.0269E+01	1.6133E+01	1.5037E+01	2.2307E+01	2.0368E+01	3.7535E+01	1.4198E+01	6.5187E+01	4.7039E+01	4.0738E+01	3.3090E+01	6.8003E+00
F24	mean	3.0191E+03	2.8815E+03	2.8723E+03	2.8979E+03	2.9053E+03	2.9586E+03	2.8690E+03	3.0166E+03	2.8636E+03	2.9226E+03	2.9766E+03	2.8479E+03
	std	5.7021E+01	1.8698E+01	1.8843E+01	2.1579E+01	1.7394E+01	5.2306E+01	1.9129E+01	5.5148E+01	3.1491E+01	2.8283E+01	3.1747E+01	4.8851E+00
F25	mean	2.9014E+03	2.8881E+03	2.9137E+03	2.8928E+03	2.8905E+03	2.8979E+03	2.8929E+03	2.8983E+03	2.8875E+03	2.9038E+03	2.8921E+03	2.8842E+03
	std	2.0914E+01	7.0414E+00	1.8159E+01	1.2053E+01	7.7145E+00	1.7614E+01	1.4553E+01	1.6685E+01	1.7564E+00	1.9710E+01	1.7290E+01	1.3092E+00
F26	mean	6.1586E+03	4.1281E+03	3.6311E+03	4.8162E+03	3.9511E+03	5.3607E+03	3.8432E+03	5.4063E+03	3.9558E+03	4.9239E+03	4.9486E+03	3.1713E+03
	std	9.1136E+02	3.9017E+02	7.2592E+02	3.2104E+02	1.6601E+02	1.0588E+03	6.5068E+02	1.1746E+03	2.6927E+02	3.7234E+02	8.2374E+02	5.2003E+02
F27	mean	3.2555E+03	3.2214E+03	3.2349E+03	3.2609E+03	3.2209E+03	3.2605E+03	3.2085E+03	3.2699E+03	3.2027E+03	3.2461E+03	3.2430E+03	3.2074E+03
	std	2.5827E+01	1.2679E+01	1.1687E+01	1.6755E+01	1.2884E+01	3.3590E+01	7.2649E+00	2.5928E+01	1.0642E+01	3.1384E+01	1.6336E+01	8.1481E+00
F28	mean	3.1230E+03	3.2239E+03	3.2688E+03	3.2608E+03	3.2285E+03	3.2075E+03	3.2184E+03	3.2153E+03	3.2274E+03	3.1884E+03	3.2397E+03	3.1000E+03
	std	4.7718E+01	2.4266E+01	2.1507E+01	2.0528E+01	2.5849E+01	3.4880E+01	1.9290E+01	2.4914E+01	1.8403E+01	6.3694E+01	3.0540E+01	4.4684E−13
F29	mean	4.1461E+03	3.6970E+03	3.6110E+03	3.7753E+03	3.4659E+03	4.1146E+03	3.4930E+03	4.0127E+03	3.5028E+03	3.8578E+03	3.7710E+03	3.4281E+03
	std	2.7044E+02	2.1886E+02	1.1875E+02	1.6451E+02	9.4859E+01	2.4311E+02	1.1775E+02	2.5140E+02	1.4336E+02	1.9532E+02	1.0459E+02	1.5131E+01
F30	mean	8.7048E+03	8.6635E+03	1.5228E+04	2.4550E+04	1.0845E+04	8.4732E+03	1.3993E+04	1.9603E+04	1.7497E+05	9.2860E+03	3.2481E+04	6.0656E+03
	std	2.5976E+03	2.7888E+03	9.3722E+03	4.5707E+04	3.6267E+03	2.3374E+03	1.7911E+04	1.4415E+04	9.8518E+04	3.2279E+03	3.8407E+04	2.7643E+02

,

Table 3. Results of various algorithms tested on the CEC 2017 benchmark (dim = 50).

ID	Metric	RTH	SAO	GRO	SO	ESC	INFO	SBOA	GKSO	IGWO	HHWOA	ED	MEED
F1	mean	4.3397E+03	3.0279E+03	7.2338E+08	3.0739E+06	2.6669E+03	4.7558E+03	8.0122E+03	2.8946E+03	1.4177E+07	5.3893E+03	5.4478E+04	1.3363E+02
	std	5.8153E+03	3.4537E+03	1.0682E+09	2.1438E+06	2.1136E+03	5.4130E+03	6.5276E+03	2.2613E+03	1.0192E+07	6.5975E+03	9.4534E+04	3.2458E+01
F2	mean	1.3062E+37	2.6578E+45	5.7915E+46	1.9125E+43	2.7126E+39	7.0830E+43	6.9482E+30	1.6165E+32	2.4274E+38	4.2310E+48	5.8139E+46	9.1427E+13
	std	7.1273E+37	1.1762E+46	2.8268E+47	1.0329E+44	8.9429E+39	3.6536E+44	1.8621E+31	8.2773E+32	9.0537E+38	2.3174E+49	3.1844E+47	9.0006E+13
F3	mean	7.6000E+02	2.2667E+05	1.1401E+05	1.4064E+05	1.5674E+05	2.6495E+04	4.2743E+04	1.1739E+04	3.4588E+04	1.7578E+03	2.3350E+05	5.5839E+04
	std	7.9821E+02	4.6998E+04	1.2645E+04	1.4334E+04	2.5605E+04	1.2515E+04	7.6522E+03	4.4672E+03	8.3781E+03	1.4413E+03	3.5723E+04	2.8908E+03
F4	mean	4.7397E+02	5.3416E+02	7.7246E+02	6.1388E+02	6.0511E+02	5.4278E+02	5.5911E+02	5.6519E+02	5.7814E+02	5.2034E+02	5.7038E+02	4.3314E+02
	std	3.8544E+01	4.7597E+01	1.2458E+02	3.7393E+01	3.6733E+01	5.6053E+01	6.0362E+01	6.2622E+01	4.9217E+01	5.2932E+01	6.4145E+01	2.1139E+01
F5	mean	8.1407E+02	6.1616E+02	6.9270E+02	6.2645E+02	6.8698E+02	7.8061E+02	6.6674E+02	8.1517E+02	6.4436E+02	7.0111E+02	8.4290E+02	6.2437E+02
	std	3.8129E+01	2.7424E+01	3.3861E+01	2.1378E+01	3.9377E+01	5.0971E+01	2.9695E+01	4.7582E+01	4.3205E+01	3.6810E+01	2.4016E+01	9.5006E+00
F6	mean	6.5090E+02	6.0040E+02	6.1691E+02	6.0827E+02	6.0015E+02	6.3851E+02	6.0538E+02	6.5104E+02	6.0222E+02	6.1909E+02	6.2541E+02	6.0001E+02
	std	6.3590E+00	3.5212E−01	4.5090E+00	3.4186E+00	1.8112E−01	6.7717E+00	4.4816E+00	7.7034E+00	9.6873E−01	7.2488E+00	3.1912E+00	4.3450E−03
F7	mean	1.4736E+03	1.1455E+03	1.0232E+03	9.5066E+02	9.8193E+02	1.2994E+03	1.0070E+03	1.2316E+03	9.3956E+02	1.1303E+03	1.1611E+03	8.9023E+02
	std	1.1069E+02	6.4907E+01	6.9327E+01	3.8874E+01	2.0839E+01	1.3232E+02	7.0815E+01	9.5391E+01	6.6359E+01	1.1345E+02	4.8444E+01	1.3234E+01
F8	mean	1.1179E+03	9.0973E+02	9.9259E+02	9.2616E+02	9.7721E+02	1.0873E+03	9.6097E+02	1.1189E+03	9.5496E+02	9.9881E+02	1.1318E+03	9.2769E+02
	std	3.6876E+01	2.0212E+01	3.2463E+01	1.5468E+01	4.1509E+01	5.9949E+01	2.6412E+01	4.5301E+01	8.0691E+01	3.6454E+01	3.0273E+01	9.0231E+00
F9	mean	1.1061E+04	1.1060E+03	3.3239E+03	2.4231E+03	9.6887E+02	7.6496E+03	2.5853E+03	9.5936E+03	1.8178E+03	2.9234E+03	9.7323E+03	1.4296E+03
	std	1.6272E+03	3.1341E+02	9.2949E+02	7.6226E+02	1.0590E+02	2.4675E+03	1.1270E+03	2.2334E+03	9.4193E+02	8.3557E+02	3.4212E+03	2.2085E+02
F10	mean	7.7755E+03	6.4700E+03	7.5688E+03	9.1958E+03	1.2061E+04	8.2414E+03	6.5097E+03	7.4920E+03	1.0733E+04	7.8031E+03	8.6713E+03	6.7944E+03
	std	1.2055E+03	1.0058E+03	7.7997E+02	3.0327E+03	5.7939E+02	9.9255E+02	8.9290E+02	8.2079E+02	4.0657E+03	9.0162E+02	5.2585E+02	3.0819E+02
F11	mean	1.3400E+03	1.4432E+03	1.9344E+03	1.5665E+03	1.4692E+03	1.3309E+03	1.2761E+03	1.3047E+03	1.4410E+03	1.3473E+03	1.6758E+03	1.1746E+03
	std	6.6049E+01	2.3271E+02	4.4856E+02	1.1956E+02	6.2246E+02	5.9250E+01	4.0731E+01	5.4819E+01	9.1161E+01	8.0736E+01	2.8685E+02	1.2038E+01
F12	mean	2.2500E+05	3.2906E+06	1.2350E+07	9.9278E+06	5.6011E+06	1.9680E+06	3.6905E+06	3.3990E+06	2.2200E+07	7.5468E+05	2.9150E+06	2.1805E+05
	std	1.3359E+05	2.1859E+06	6.5511E+06	5.9484E+06	2.8570E+06	1.2208E+06	2.5102E+06	2.6190E+06	9.4432E+06	5.7631E+05	1.3323E+06	5.1067E+04
F13	mean	1.0778E+04	6.1048E+03	9.6363E+03	3.6506E+04	1.0832E+04	1.1017E+04	9.3195E+03	1.7508E+04	3.6116E+05	8.2762E+03	6.4704E+03	1.5284E+03
	std	1.0044E+04	4.9488E+03	3.9282E+03	4.1368E+04	4.4332E+03	5.9545E+03	7.9479E+03	8.7284E+03	2.8757E+05	6.5093E+03	4.0436E+03	5.8567E+01
F14	mean	5.4764E+03	7.5287E+04	1.8494E+05	1.7121E+05	4.2948E+05	2.5038E+04	1.4409E+05	2.3115E+04	7.5628E+04	6.4843E+03	4.1791E+05	1.0277E+04
	std	2.2178E+03	7.4902E+04	1.9723E+05	2.5744E+05	4.4682E+05	2.4566E+04	1.0098E+05	1.5923E+04	7.5796E+04	3.8517E+03	1.7402E+05	2.7533E+03
F15	mean	9.7498E+03	1.1582E+04	8.6052E+03	1.5492E+04	7.7192E+03	9.8733E+03	1.1333E+04	7.2408E+03	7.8571E+04	9.1039E+03	9.4962E+03	1.7536E+03
	std	8.5821E+03	6.1590E+03	3.7480E+03	7.8563E+03	5.4275E+03	7.8527E+03	6.4247E+03	4.5503E+03	5.4017E+04	6.5008E+03	5.8254E+03	1.3523E+02
F16	mean	3.5373E+03	3.0450E+03	2.8636E+03	3.0368E+03	3.0400E+03	3.6103E+03	2.7788E+03	3.4674E+03	2.7282E+03	3.3387E+03	4.1621E+03	2.3614E+03
	std	4.9931E+02	4.5373E+02	3.4083E+02	3.5561E+02	3.0584E+02	5.8856E+02	3.7126E+02	3.5039E+02	7.1674E+02	4.6797E+02	1.8379E+02	1.3609E+02
F17	mean	3.3713E+03	2.7821E+03	2.5820E+03	2.7867E+03	2.6755E+03	3.2307E+03	2.7003E+03	3.2916E+03	2.5431E+03	3.0876E+03	3.1669E+03	2.4108E+03
	std	3.3464E+02	3.6945E+02	2.4318E+02	2.4177E+02	2.4833E+02	3.4232E+02	3.1257E+02	3.9180E+02	4.4273E+02	3.6936E+02	2.8674E+02	9.9588E+01
F18	mean	4.1519E+04	1.6109E+06	1.4841E+06	2.6436E+06	2.5455E+06	1.2317E+05	1.4861E+06	1.6804E+05	7.9644E+05	4.4712E+04	3.5094E+06	4.5367E+05
	std	2.7789E+04	1.5360E+06	7.4502E+05	2.3515E+06	1.7991E+06	8.9950E+04	9.8771E+05	9.7568E+04	7.0937E+05	2.5656E+04	1.9747E+06	1.2310E+05
F19	mean	2.0156E+04	1.9888E+04	2.1761E+04	1.9559E+04	1.4298E+04	1.8864E+04	1.7899E+04	1.8191E+04	6.0006E+04	1.6338E+04	1.5387E+04	6.1656E+03
	std	1.0796E+04	1.1523E+04	1.1227E+04	1.1461E+04	1.0083E+04	1.2732E+04	1.1345E+04	1.2211E+04	3.9758E+04	1.2670E+04	1.2966E+04	1.6673E+03
F20	mean	3.3480E+03	2.8651E+03	2.7422E+03	3.0114E+03	2.8046E+03	3.3327E+03	2.6461E+03	3.1040E+03	2.7988E+03	3.1192E+03	3.4870E+03	2.4440E+03
	std	2.5637E+02	3.2741E+02	1.8488E+02	3.5781E+02	2.3316E+02	3.4413E+02	3.1103E+02	3.1108E+02	5.2659E+02	3.2127E+02	1.9314E+02	1.3339E+02
F21	mean	2.6216E+03	2.4208E+03	2.4666E+03	2.4304E+03	2.4878E+03	2.5861E+03	2.4153E+03	2.6182E+03	2.4630E+03	2.4896E+03	2.6516E+03	2.4125E+03
	std	4.9917E+01	2.9078E+01	2.5551E+01	1.7808E+01	5.2469E+01	6.1238E+01	2.7327E+01	5.8153E+01	9.3977E+01	4.8340E+01	3.2837E+01	4.5940E+00
F22	mean	1.0110E+04	7.6725E+03	6.7122E+03	1.1084E+04	1.3547E+04	9.9284E+03	7.8945E+03	9.2704E+03	1.2709E+04	9.5891E+03	1.0956E+04	5.2382E+03
	std	8.7706E+02	2.2150E+03	3.4612E+03	2.9604E+03	7.3614E+02	9.1680E+02	2.1237E+03	1.5228E+03	4.2622E+03	8.5335E+02	1.7061E+03	3.1955E+03
F23	mean	3.1829E+03	2.8524E+03	2.9236E+03	2.9340E+03	2.8726E+03	3.1411E+03	2.8569E+03	3.1671E+03	2.8778E+03	2.9948E+03	3.1174E+03	2.8615E+03
	std	1.2916E+02	1.9077E+01	3.9211E+01	2.7033E+01	4.5709E+01	9.3800E+01	2.8065E+01	1.0873E+02	9.0640E+01	6.0994E+01	3.9049E+01	1.1058E+01
F24	mean	3.3687E+03	3.0109E+03	3.0727E+03	3.0637E+03	3.1173E+03	3.2603E+03	3.0287E+03	3.3646E+03	3.0382E+03	3.1651E+03	3.3082E+03	3.0164E+03
	std	1.6180E+02	2.5125E+01	3.7406E+01	3.4765E+01	3.8112E+01	5.7512E+01	3.5288E+01	1.0522E+02	9.2886E+01	6.2429E+01	5.3306E+01	9.6974E+00
F25	mean	3.0503E+03	3.0469E+03	3.2968E+03	3.0943E+03	3.0989E+03	3.0820E+03	3.0927E+03	3.0717E+03	3.1036E+03	3.0686E+03	3.0752E+03	2.9889E+03
	std	3.5627E+01	1.9412E+01	8.8462E+01	3.2597E+01	3.5442E+01	3.3066E+01	3.2786E+01	2.9751E+01	5.3841E+01	3.7914E+01	2.4371E+01	2.5280E+01
F26	mean	9.1173E+03	4.7509E+03	5.9670E+03	5.9841E+03	4.6894E+03	8.2491E+03	5.1941E+03	7.2517E+03	5.2582E+03	6.9100E+03	7.1992E+03	4.9055E+03
	std	2.3493E+03	6.7731E+02	1.1865E+03	4.2141E+02	3.4066E+02	1.8942E+03	1.3712E+03	3.6986E+03	6.3403E+02	8.1843E+02	4.8034E+02	8.1157E+02
F27	mean	3.6402E+03	3.3757E+03	3.5611E+03	3.6153E+03	3.4039E+03	3.6503E+03	3.3164E+03	3.7109E+03	3.2849E+03	3.6465E+03	3.7170E+03	3.3148E+03
	std	1.4228E+02	8.9457E+01	8.1107E+01	8.0695E+01	4.6096E+01	1.2059E+02	5.5747E+01	1.5087E+02	3.9957E+01	1.4525E+02	1.5735E+02	2.2028E+01
F28	mean	3.2974E+03	3.2965E+03	3.6862E+03	3.4543E+03	3.4675E+03	3.3279E+03	3.3495E+03	3.3303E+03	3.3472E+03	3.3082E+03	3.3600E+03	3.2677E+03
	std	2.5050E+01	1.9457E+01	1.3044E+02	6.4968E+01	5.8049E+01	3.3577E+01	3.3979E+01	2.0907E+01	3.6126E+01	2.3918E+01	3.4110E+01	9.4287E+00
F29	mean	4.7770E+03	4.0278E+03	4.1621E+03	4.2834E+03	3.5336E+03	4.9125E+03	3.8949E+03	4.9362E+03	3.8068E+03	4.7753E+03	4.7755E+03	3.6110E+03
	std	3.5396E+02	3.1587E+02	2.3527E+02	2.4262E+02	1.9870E+02	4.5643E+02	3.1866E+02	4.0381E+02	3.2345E+02	4.6563E+02	4.1350E+02	6.2110E+01
F30	mean	8.2713E+05	1.0726E+06	1.5933E+06	2.6840E+06	1.2144E+06	9.8594E+05	9.9125E+05	5.9442E+06	8.6483E+06	1.0373E+06	2.9820E+06	7.3433E+05
	std	1.3022E+05	2.5166E+05	3.6875E+05	1.0238E+06	2.5326E+05	2.9363E+05	3.0362E+05	2.3035E+06	2.2043E+06	5.0203E+05	1.0912E+06	3.5013E+04

and

Table 4. Results of various algorithms tested on the CEC 2017 benchmark (dim = 100).

ID	Metric	RTH	SAO	GRO	SO	ESC	INFO	SBOA	GKSO	IGWO	HHWOA	ED	MEED
F1	mean	9.2985E+03	1.7680E+08	2.8273E+10	6.7398E+07	7.2520E+09	3.7970E+07	2.3557E+08	9.4928E+05	8.2568E+09	6.9209E+03	3.2251E+08	3.5414E+02
	std	1.3265E+04	1.9542E+08	9.2590E+09	2.2783E+07	3.1705E+09	1.0228E+08	4.0214E+08	3.7425E+05	3.7219E+09	8.8576E+03	2.0709E+08	2.4328E+02
F2	mean	4.4601E+107	2.6333E+112	9.5526E+129	4.5085E+114	1.9015E+124	4.0220E+128	6.2255E+93	6.7879E+112	1.9702E+114	2.1173E+124	1.5976E+116	3.4551E+58
	std	2.4425E+108	1.4422E+113	5.2179E+130	2.3496E+115	7.2187E+124	2.2029E+129	2.2122E+94	3.7179E+113	1.0791E+115	1.1588E+125	8.7025E+116	8.0481E+58
F3	mean	5.8897E+04	7.4602E+05	3.3160E+05	3.3130E+05	4.9844E+05	1.8454E+05	2.4241E+05	1.8632E+05	2.7333E+05	3.0978E+05	6.1291E+05	2.6411E+05
	std	1.2189E+04	1.4010E+05	2.9737E+04	1.4430E+04	4.0807E+04	2.1857E+04	2.0731E+04	2.3021E+04	4.0668E+04	4.8968E+04	8.0778E+04	9.5718E+03
F4	mean	6.6763E+02	7.5436E+02	3.0457E+03	9.6945E+02	1.4735E+03	9.1890E+02	9.5096E+02	8.3281E+02	1.3598E+03	7.1753E+02	1.0509E+03	5.8817E+02
	std	4.1433E+01	5.5190E+01	5.3377E+02	7.3868E+01	2.2121E+02	1.0892E+02	6.8065E+01	5.4702E+01	2.5165E+02	5.7854E+01	1.0541E+02	2.2725E+01
F5	mean	1.2782E+03	1.1535E+03	1.2005E+03	8.6034E+02	1.0477E+03	1.2606E+03	1.0067E+03	1.3067E+03	9.5535E+02	1.0864E+03	1.4777E+03	1.0233E+03
	std	6.6710E+01	2.7988E+02	5.6676E+01	3.5610E+01	1.0674E+02	6.6846E+01	7.3766E+01	8.2013E+01	5.6366E+01	5.8775E+01	6.5565E+01	2.0100E+01
F6	mean	6.5328E+02	6.1238E+02	6.4007E+02	6.2391E+02	6.0619E+02	6.5334E+02	6.2609E+02	6.6069E+02	6.1176E+02	6.3872E+02	6.5313E+02	6.0691E+02
	std	5.5522E+00	3.8894E+00	5.0176E+00	3.6861E+00	1.7245E+00	5.5073E+00	6.1439E+00	5.0118E+00	2.0273E+00	5.5811E+00	3.2777E+00	2.9385E+00
F7	mean	2.8925E+03	1.9623E+03	2.0425E+03	1.4267E+03	1.5603E+03	2.7165E+03	1.8968E+03	2.5044E+03	1.5103E+03	2.2538E+03	2.5416E+03	1.4586E+03
	std	1.7091E+02	7.7025E+01	1.4965E+02	6.8235E+01	8.5933E+01	2.5772E+02	1.9407E+02	2.7196E+02	1.4042E+02	1.7369E+02	1.6305E+02	2.9406E+01
F8	mean	1.7077E+03	1.4181E+03	1.4958E+03	1.1597E+03	1.3307E+03	1.6326E+03	1.2989E+03	1.7096E+03	1.2878E+03	1.4232E+03	1.8125E+03	1.3222E+03
	std	8.1165E+01	2.4134E+02	7.0257E+01	3.8160E+01	1.3621E+02	1.1139E+02	7.7240E+01	9.9729E+01	1.3827E+02	8.5262E+01	5.5263E+01	2.0293E+01
F9	mean	2.1623E+04	9.8633E+03	2.1190E+04	8.4764E+03	6.1419E+03	2.1596E+04	1.8530E+04	2.2857E+04	2.1223E+04	1.2408E+04	5.3640E+04	1.3301E+04
	std	1.3394E+03	7.3832E+03	4.7619E+03	2.4781E+03	1.6911E+03	2.4683E+03	3.7721E+03	1.9329E+03	6.6483E+03	2.7719E+03	9.7507E+03	1.4449E+03
F10	mean	1.5847E+04	2.1623E+04	1.9126E+04	3.0520E+04	2.8491E+04	1.6939E+04	1.4659E+04	1.5504E+04	2.3917E+04	1.6131E+04	2.2860E+04	1.8412E+04
	std	1.6162E+03	7.8054E+03	1.3234E+03	1.6401E+03	8.2092E+02	1.5479E+03	1.5388E+03	1.2646E+03	7.8026E+03	1.3390E+03	1.0041E+03	2.7738E+02
F11	mean	2.2971E+03	1.4113E+05	5.1580E+04	8.3418E+04	4.1078E+04	5.5396E+03	1.2373E+04	2.6227E+03	1.5056E+04	2.1926E+03	7.1241E+04	3.1979E+03
	std	2.3924E+02	3.7270E+04	9.3569E+03	1.6908E+04	1.0037E+04	3.3743E+03	3.6977E+03	2.0906E+02	4.8328E+03	1.8847E+02	1.0090E+04	2.7459E+02
F12	mean	2.2461E+06	3.5735E+07	1.6539E+09	1.5425E+08	3.2871E+08	4.9793E+07	4.9449E+07	6.0932E+07	4.6519E+08	8.0802E+06	6.4582E+07	1.3896E+06
	std	1.0983E+06	1.3715E+07	1.8817E+09	8.4378E+07	1.9975E+08	7.2023E+07	2.1749E+07	2.4210E+07	1.4474E+08	3.4280E+06	3.4818E+07	2.9320E+05
F13	mean	9.0318E+03	9.4138E+03	6.5307E+05	1.6837E+05	3.2558E+04	2.8358E+04	3.1246E+04	2.5127E+04	5.6829E+05	1.0138E+04	1.0002E+04	1.9727E+03
	std	6.3506E+03	6.4292E+03	1.9539E+06	1.1762E+05	1.1521E+04	6.4613E+04	9.3274E+04	7.6897E+03	3.5011E+05	3.5048E+03	5.2071E+03	1.5892E+02
F14	mean	5.1095E+04	7.7891E+05	2.7508E+06	2.6930E+06	4.2114E+06	4.3531E+05	1.7297E+06	2.0426E+05	1.2285E+06	9.9268E+04	5.8044E+06	3.2795E+05
	std	2.9521E+04	3.6241E+05	1.2571E+06	1.4449E+06	2.3437E+06	2.4970E+05	8.5767E+05	8.4354E+04	4.8889E+05	5.0771E+04	3.6749E+06	6.2832E+04
F15	mean	7.0451E+03	3.9778E+03	9.8928E+03	4.2033E+04	1.5522E+04	6.0075E+03	6.1603E+03	1.1553E+04	1.2589E+05	6.2879E+03	5.3106E+03	1.7284E+03
	std	8.0715E+03	2.3745E+03	5.2725E+03	4.0618E+04	5.8339E+03	4.3872E+03	5.3708E+03	6.0454E+03	6.5744E+04	5.3109E+03	3.6208E+03	5.1189E+01
F16	mean	5.9250E+03	5.3076E+03	5.8036E+03	6.8675E+03	6.5446E+03	5.6918E+03	4.9827E+03	6.0279E+03	4.8407E+03	5.7274E+03	8.7336E+03	5.5021E+03
	std	5.8053E+02	1.2892E+03	7.3163E+02	1.8351E+03	7.2343E+02	6.9528E+02	6.8848E+02	4.4199E+02	6.4622E+02	7.3995E+02	9.9249E+02	3.3955E+02
F17	mean	5.5117E+03	5.0423E+03	4.5445E+03	4.5124E+03	5.0641E+03	5.5193E+03	4.4829E+03	5.4486E+03	4.4333E+03	5.0591E+03	6.0256E+03	4.4906E+03
	std	5.6508E+02	1.2252E+03	4.3816E+02	5.0568E+02	5.3151E+02	6.5821E+02	4.1235E+02	6.0965E+02	1.0045E+03	5.9895E+02	7.8018E+02	1.5916E+02
F18	mean	2.4411E+05	2.8513E+06	3.5367E+06	4.5061E+06	6.9813E+06	6.3113E+05	3.0923E+06	4.0963E+05	2.7955E+06	3.3307E+05	1.5322E+07	1.1942E+06
	std	1.3593E+05	1.2010E+06	1.8280E+06	2.0607E+06	3.3238E+06	3.2601E+05	1.2250E+06	2.0510E+05	1.4887E+06	1.4561E+05	1.1716E+07	2.5589E+05
F19	mean	6.9189E+03	5.3117E+03	1.0369E+04	1.1602E+05	2.5299E+04	8.6952E+03	8.2012E+03	1.6120E+04	2.8074E+05	8.9898E+03	5.1634E+03	2.0361E+03
	std	5.6632E+03	4.1722E+03	6.8848E+03	1.5204E+05	2.1749E+04	7.5650E+03	6.6949E+03	1.7076E+04	1.5164E+05	6.5811E+03	3.5727E+03	3.2090E+01
F20	mean	5.4189E+03	4.9650E+03	4.6417E+03	6.7013E+03	5.7187E+03	5.4203E+03	4.5877E+03	5.2560E+03	5.5593E+03	4.9913E+03	6.7281E+03	4.9546E+03
	std	4.8540E+02	9.5381E+02	5.2667E+02	5.4903E+02	4.2460E+02	6.0949E+02	5.0113E+02	4.4292E+02	1.4847E+03	3.9854E+02	2.8497E+02	2.5847E+02
F21	mean	3.3158E+03	2.8716E+03	2.9202E+03	2.7517E+03	2.8714E+03	3.1794E+03	2.7510E+03	3.2597E+03	2.8037E+03	2.9656E+03	3.3654E+03	2.8098E+03
	std	1.4528E+02	2.0769E+02	5.9605E+01	3.8867E+01	1.1188E+02	1.1885E+02	7.4829E+01	1.2585E+02	1.7095E+02	8.6024E+01	1.1179E+02	1.8645E+01
F22	mean	1.9400E+04	1.9794E+04	2.1792E+04	3.2167E+04	3.0148E+04	1.9527E+04	1.7699E+04	1.8496E+04	3.1227E+04	1.8935E+04	2.4544E+04	2.1169E+04
	std	1.6230E+03	6.1904E+03	1.4852E+03	1.8562E+03	8.3698E+02	1.9703E+03	3.2575E+03	1.6424E+03	5.6845E+03	1.8276E+03	5.4361E+02	4.1872E+02
F23	mean	3.7961E+03	3.1718E+03	3.5614E+03	3.3262E+03	3.1093E+03	3.8116E+03	3.2098E+03	3.8662E+03	3.2255E+03	3.6389E+03	3.9006E+03	3.3537E+03
	std	1.8305E+02	5.4044E+01	8.2403E+01	5.6266E+01	4.6887E+01	1.6591E+02	6.2063E+01	1.6177E+02	1.2683E+02	1.2256E+02	1.9102E+02	2.8106E+01
F24	mean	4.5961E+03	3.6334E+03	4.2898E+03	4.0402E+03	3.6513E+03	4.6559E+03	3.8382E+03	4.7403E+03	3.7048E+03	4.3700E+03	4.6286E+03	3.7627E+03
	std	2.8627E+02	5.9364E+01	1.2839E+02	9.1133E+01	9.2528E+01	2.9247E+02	1.2142E+02	2.2375E+02	7.9340E+01	2.9627E+02	2.2594E+02	7.5785E+01
F25	mean	3.2976E+03	3.4837E+03	5.2527E+03	3.6842E+03	4.3957E+03	3.5396E+03	3.5817E+03	3.5106E+03	4.0574E+03	3.3603E+03	3.7880E+03	3.2427E+03
	std	5.9406E+01	7.2726E+01	5.6919E+02	7.5478E+01	3.0442E+02	9.1161E+01	8.2506E+01	6.6382E+01	1.9475E+02	8.1553E+01	7.8778E+01	1.8010E+01
F26	mean	2.0636E+04	9.4010E+03	1.9900E+04	1.3035E+04	9.5359E+03	2.1979E+04	1.3595E+04	2.2540E+04	1.1107E+04	1.7977E+04	1.7918E+04	1.1976E+04
	std	2.3121E+03	6.2118E+02	2.2823E+03	9.5404E+02	6.6407E+02	4.0430E+03	4.4226E+03	3.4445E+03	7.8078E+02	2.5019E+03	1.8789E+03	1.8857E+03
F27	mean	3.7135E+03	3.4299E+03	4.0943E+03	3.8329E+03	3.6837E+03	3.9198E+03	3.5966E+03	4.1451E+03	3.5176E+03	4.0141E+03	4.0981E+03	3.4810E+03
	std	1.0871E+02	4.4423E+01	1.3953E+02	9.8202E+01	7.4449E+01	2.2865E+02	8.2801E+01	3.6348E+02	4.8574E+01	2.0389E+02	2.6840E+02	1.8917E+01
F28	mean	3.4043E+03	3.5287E+03	6.9023E+03	4.6502E+03	6.2202E+03	3.6778E+03	3.7281E+03	3.5800E+03	4.6094E+03	3.4624E+03	4.2900E+03	3.3920E+03
	std	4.0105E+01	3.5605E+01	1.0272E+03	5.0513E+02	9.8648E+02	1.0251E+02	7.9283E+01	3.8085E+01	5.5706E+02	4.8919E+01	2.8496E+02	1.0846E+01
F29	mean	7.3404E+03	6.0823E+03	7.3794E+03	7.0986E+03	6.2114E+03	7.9324E+03	6.2440E+03	8.2756E+03	6.2272E+03	7.4846E+03	8.3393E+03	6.4585E+03
	std	6.0645E+02	5.3406E+02	5.3405E+02	4.9705E+02	7.1401E+02	6.8229E+02	5.9482E+02	7.0961E+02	5.6338E+02	6.4960E+02	1.2911E+03	2.8101E+02
F30	mean	1.2854E+04	2.7239E+04	4.5999E+06	1.5376E+06	8.7055E+06	1.6596E+05	5.0316E+04	2.2863E+06	1.1521E+07	2.9242E+04	1.2752E+06	9.0799E+03
	std	7.5636E+03	1.3024E+04	5.0603E+06	8.0150E+05	2.1174E+07	2.5439E+05	2.6010E+04	1.2076E+06	4.2834E+06	1.8563E+04	8.4475E+05	8.9226E+02

. Here, “mean” represents the mean of 30 runs, and “std” represents the standard deviation of 30 runs. To provide a comprehensive analysis of the algorithms, box plots of experimental results from 30 runs are displayed in Figure 5.

Performance analysis under dim = 30: The convergence curves of MEED and the comparison algorithms on six representative CEC2017 benchmark functions (F1, F7, F13, F16, F20, and F24) demonstrate that MEED achieves a significantly faster and more stable convergence process. In unimodal functions such as F1 and F7, MEED rapidly approaches the global optimum within the early stages of iteration, indicating its superior exploitation capability and convergence efficiency. For complex multimodal functions (e.g., F13 and F16), MEED consistently maintains lower average fitness values than other algorithms throughout the optimization process, proving its strong robustness and ability to escape local optima. Furthermore, in hybrid and composite functions such as F20 and F24, MEED still exhibits excellent convergence stability and solution precision, whereas most comparison algorithms show oscillation or premature convergence. These results collectively confirm that the integration of multi-strategy mechanisms in MEED significantly improves both the global exploration ability and local exploitation accuracy, leading to overall superior optimization performance.

Performance analysis under dim = 50: The convergence curves of all algorithms on six representative CEC2017 benchmark functions (F1, F4, F12, F15, F22, and F26) under 50-dimensional conditions further validate the superior optimization capability of MEED. It can be clearly observed that MEED achieves the fastest convergence rate and the lowest final fitness values across almost all test functions. Specifically, in unimodal functions such as F1 and F4, MEED rapidly converges to the global optimum within the first few hundred iterations, demonstrating excellent exploitation ability. In complex multimodal and hybrid functions (e.g., F12, F15, and F22), MEED consistently outperforms other algorithms, showing a strong capacity to escape from local optima and maintain high convergence stability. Even in the highly composite function F26, which presents numerous local minima and rugged landscapes, MEED still achieves the best convergence precision with a smooth and stable descent curve, while other algorithms exhibit stagnation or oscillation behaviors. These results indicate that MEED possesses superior robustness and scalability when addressing high-dimensional and complex optimization tasks. The incorporation of multi-strategy mechanisms enables MEED to balance exploration and exploitation effectively, thereby maintaining high search efficiency even in challenging large-scale optimization scenarios.

Performance analysis under dim = 100: The convergence behaviors of MEED and other algorithms on the 100-dimensional CEC2017 benchmark functions (F6, F12, F13, F15, F19, and F30) further demonstrate the outstanding scalability and robustness of the proposed algorithm. With the increase in problem dimensionality, the search space becomes exponentially more complex, and the risk of premature convergence grows substantially. However, MEED consistently achieves the fastest convergence speed and the lowest average fitness values among all comparison algorithms. For instance, in relatively smooth unimodal functions such as F6, MEED exhibits a rapid and stable descent trend, reaching near-optimal regions within the early iterations. For multimodal and hybrid functions such as F12, F13, F15, and F19, MEED maintains excellent stability and avoids local stagnation, effectively balancing exploration and exploitation in high-dimensional landscapes. Even on the highly composite and challenging function F30, MEED achieves the most stable convergence curve and the smallest final fitness value, while other algorithms show evident oscillations or premature convergence. These results clearly confirm that MEED possesses strong adaptability to large-scale optimization problems. The synergistic design of its hybrid initialization, mirror-reflection boundary handling, and historical trend-guided update mechanism allows MEED to maintain population diversity, strengthen search guidance, and ensure high-precision convergence even in complex high-dimensional search spaces.

Across all dimensional settings, MEED consistently achieves the best convergence performance and the lowest solution errors compared with other advanced metaheuristic algorithms. The results confirm that the hybrid strategy design of MEED effectively improves convergence speed, prevents premature convergence, and enhances global optimization capability. Moreover, the algorithm exhibits strong scalability, maintaining excellent performance even as the problem dimensionality increases from 30 to 100, which highlights its robustness and potential for solving large-scale and real-world optimization problems.

4.4. Statistical Analysis

Statistical analysis plays a crucial role in verifying the reliability and significance of the obtained optimization results. Since metaheuristic algorithms are stochastic in nature, their performance can vary across multiple independent runs due to random initialization and probabilistic operators. Therefore, relying solely on mean or best fitness values may lead to misleading conclusions. By employing non-parametric statistical tests, such as the Wilcoxon rank-sum test and the Friedman test, the robustness and statistical significance of the proposed algorithm’s superiority can be quantitatively validated. These tests help determine whether the performance improvements are genuinely significant or merely due to random chance, thereby ensuring the fairness and credibility of algorithmic comparisons. In this subsection, we conducted the Wilcoxon rank-sum test and Friedman mean rank test on MEED, with specific details as follows:

4.4.1. Wilcoxon Rank Sum Test

The Wilcoxon rank-sum test, also known as the Mann–Whitney U test, is a non-parametric statistical method used to evaluate whether two independent samples come from the same distribution. Unlike parametric tests such as the t-test, the Wilcoxon test does not assume that the data follow a normal distribution, making it particularly suitable for evaluating the stochastic performance of metaheuristic algorithms, where the results often exhibit non-Gaussian characteristics due to random initialization and probabilistic operators.

In this subsection, the Wilcoxon rank-sum test was employed at a 5% significance level (α = 0.05) to compare the performance of the proposed MEED algorithm with other benchmark algorithms across multiple independent runs. The null hypothesis (H₀) assumes that there is no significant difference between the two algorithms, while the alternative hypothesis (H₁) suggests a statistically significant difference. A p-value < 0.05 indicates that H₀ can be rejected, confirming that the observed performance difference is statistically significant.

Table 5. The p-values for various algorithms on the CEC 2017 (dim = 30).

Item	RTH	SAO	GRO	SO	ESC	INFO	SBOA	GKSO	IGWO	HHWOA	ED
F1	1.6947E−09	8.1014E−10	3.0199E−11	3.0199E−11	3.1589E−10	2.4994E−03	2.3897E−08	1.1023E−08	3.0199E−11	1.2057E−10	5.5329E−08
F2	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11
F3	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	4.5043E−11	3.9167E−02	3.0199E−11	6.7350E−01	3.0199E−11	3.0199E−11
F4	3.1573E−05	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	4.9980E−09	3.1589E−10
F5	3.0199E−11	4.4592E−04	1.4294E−08	2.8314E−08	1.6980E−08	3.0199E−11	1.6062E−06	3.0199E−11	2.8389E−04	3.0199E−11	3.0199E−11
F6	3.0180E−11	2.3701E−10	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11
F7	3.0199E−11	1.0907E−05	1.7769E−10	3.0199E−11	3.0199E−11	3.0199E−11	2.9215E−09	3.0199E−11	3.4971E−09	3.0199E−11	3.0199E−11
F8	3.0199E−11	5.9673E−09	6.7220E−10	1.0937E−10	3.0103E−07	3.0199E−11	6.2828E−06	3.0199E−11	7.0430E−07	3.0199E−11	3.0199E−11
F9	2.3768E−11	9.5957E−10	2.3768E−11	2.3768E−11	6.1288E−04	2.3768E−11	2.3768E−11	2.3768E−11	2.3768E−11	2.3768E−11	2.3768E−11
F10	3.0199E−11	3.9527E−01	2.5306E−04	1.4932E−04	3.0199E−11	5.4941E−11	9.7917E−05	3.0199E−11	4.1178E−06	3.0199E−11	3.0199E−11
F11	3.0199E−11	2.3885E−04	3.0199E−11	3.0199E−11	9.9186E−11	1.4643E−10	6.1210E−10	3.0199E−11	3.0199E−11	1.6132E−10	1.0702E−09
F12	5.3221E−03	4.5043E−11	3.0199E−11	3.0199E−11	3.0199E−11	1.6132E−10	3.0199E−11	3.4742E−10	3.0199E−11	9.8329E−08	3.0199E−11
F13	4.1825E−09	4.5043E−11	3.0199E−11	3.0199E−11	1.0937E−10	3.0199E−11	1.4643E−10	3.0199E−11	3.0199E−11	1.0702E−09	5.4617E−09
F14	3.3681E−05	3.0199E−11	2.6099E−10	3.6897E−11	2.4386E−09	3.5137E−02	2.0152E−08	1.5014E−02	6.5277E−08	3.0199E−11	3.0199E−11
F15	1.0937E−10	1.1737E−09	3.0199E−11	3.0199E−11	3.0199E−11	1.2057E−10	8.9934E−11	3.0199E−11	3.0199E−11	9.5207E−04	3.4742E−10
F16	3.0199E−11	2.2273E−09	7.6950E−08	1.2057E−10	1.9963E−05	3.0199E−11	1.8500E−08	4.6159E−10	2.5101E−02	3.0199E−11	3.0199E−11
F17	3.0199E−11	2.4386E−09	2.8716E−10	3.0199E−11	4.9818E−04	3.0199E−11	1.1023E−08	3.0199E−11	2.0152E−08	3.0199E−11	3.0199E−11
F18	1.0937E−10	6.5183E−09	1.8567E−09	5.4941E−11	1.2870E−09	2.0023E−06	4.9980E−09	3.0317E−02	1.0188E−05	6.6955E−11	3.0199E−11
F19	9.2603E−09	3.0103E−07	5.9673E−09	1.1077E−06	3.8249E−09	3.5137E−02	1.8500E−08	7.7387E−06	3.0199E−11	2.6015E−08	1.0666E−07
F20	3.0199E−11	5.0723E−10	1.0937E−10	4.1997E−10	3.2651E−02	3.0199E−11	4.6856E−08	3.0199E−11	1.5846E−04	3.0199E−11	3.0199E−11
F21	3.0199E−11	4.4205E−06	1.6132E−10	3.0199E−11	4.1127E−07	3.0199E−11	4.1127E−07	3.0199E−11	1.1058E−04	8.9934E−11	3.0199E−11
F22	1.1902E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2108E−12	1.2118E−12
F23	3.0199E−11	1.7290E−06	1.6980E−08	3.0199E−11	1.6798E−03	3.0199E−11	1.5846E−04	3.0199E−11	3.7782E−02	3.0199E−11	3.1589E−10
F24	3.0199E−11	3.4742E−10	2.0152E−08	3.0199E−11	3.0199E−11	3.0199E−11	1.3853E−06	3.0199E−11	3.3679E−04	3.0199E−11	3.0199E−11
F25	6.1210E−10	5.8587E−06	5.4941E−11	3.4742E−10	3.0199E−11	5.0922E−08	9.0632E−08	2.4386E−09	5.0723E−10	6.1210E−10	1.7769E−10
F26	1.2760E−10	2.3853E−09	3.1184E−04	2.3451E−11	1.4659E−05	7.8698E−10	4.1975E−06	3.4282E−09	2.5334E−06	2.3451E−11	7.8698E−10
F27	3.0199E−11	5.4620E−06	4.1997E−10	3.0199E−11	1.9963E−05	1.9568E−10	9.1171E−01	3.0199E−11	1.6955E−02	1.6132E−10	2.3715E−10
F28	2.2668E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12
F29	3.0199E−11	2.1540E−06	3.0199E−11	4.9752E−11	8.7663E−01	3.0199E−11	8.5000E−02	3.0199E−11	1.3017E−03	3.0199E−11	3.0199E−11
F30	1.7479E−05	1.2860E−06	3.0199E−11	2.9215E−09	3.0199E−11	7.1186E−09	1.0937E−10	3.0199E−11	3.0199E−11	1.1077E−06	3.0199E−11

,

Table 6. The p-values for various algorithms on the CEC 2017 (dim = 50).

Item	RTH	SAO	GRO	SO	ESC	INFO	SBOA	GKSO	IGWO	HHWOA	ED
F1	1.6947E−09	8.1014E−10	3.0199E−11	3.0199E−11	3.1589E−10	2.4994E−03	2.3897E−08	1.1023E−08	3.0199E−11	1.2057E−10	5.5329E−08
F2	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11
F3	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	4.5043E−11	3.9167E−02	3.0199E−11	6.7350E−01	3.0199E−11	3.0199E−11
F4	3.1573E−05	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	4.9980E−09	3.1589E−10
F5	3.0199E−11	4.4592E−04	1.4294E−08	2.8314E−08	1.6980E−08	3.0199E−11	1.6062E−06	3.0199E−11	2.8389E−04	3.0199E−11	3.0199E−11
F6	3.0180E−11	2.3701E−10	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11	3.0180E−11
F7	3.0199E−11	1.0907E−05	1.7769E−10	3.0199E−11	3.0199E−11	3.0199E−11	2.9215E−09	3.0199E−11	3.4971E−09	3.0199E−11	3.0199E−11
F8	3.0199E−11	5.9673E−09	6.7220E−10	1.0937E−10	3.0103E−07	3.0199E−11	6.2828E−06	3.0199E−11	7.0430E−07	3.0199E−11	3.0199E−11
F9	2.3768E−11	9.5957E−10	2.3768E−11	2.3768E−11	6.1288E−04	2.3768E−11	2.3768E−11	2.3768E−11	2.3768E−11	2.3768E−11	2.3768E−11
F10	3.0199E−11	3.9527E−01	2.5306E−04	1.4932E−04	3.0199E−11	5.4941E−11	9.7917E−05	3.0199E−11	4.1178E−06	3.0199E−11	3.0199E−11
F11	3.0199E−11	2.3885E−04	3.0199E−11	3.0199E−11	9.9186E−11	1.4643E−10	6.1210E−10	3.0199E−11	3.0199E−11	1.6132E−10	1.0702E−09
F12	5.3221E−03	4.5043E−11	3.0199E−11	3.0199E−11	3.0199E−11	1.6132E−10	3.0199E−11	3.4742E−10	3.0199E−11	9.8329E−08	3.0199E−11
F13	4.1825E−09	4.5043E−11	3.0199E−11	3.0199E−11	1.0937E−10	3.0199E−11	1.4643E−10	3.0199E−11	3.0199E−11	1.0702E−09	5.4617E−09
F14	3.3681E−05	3.0199E−11	2.6099E−10	3.6897E−11	2.4386E−09	3.5137E−02	2.0152E−08	1.5014E−02	6.5277E−08	3.0199E−11	3.0199E−11
F15	1.0937E−10	1.1737E−09	3.0199E−11	3.0199E−11	3.0199E−11	1.2057E−10	8.9934E−11	3.0199E−11	3.0199E−11	9.5207E−04	3.4742E−10
F16	3.0199E−11	2.2273E−09	7.6950E−08	1.2057E−10	1.9963E−05	3.0199E−11	1.8500E−08	4.6159E−10	2.5101E−02	3.0199E−11	3.0199E−11
F17	3.0199E−11	2.4386E−09	2.8716E−10	3.0199E−11	4.9818E−04	3.0199E−11	1.1023E−08	3.0199E−11	2.0152E−08	3.0199E−11	3.0199E−11
F18	1.0937E−10	6.5183E−09	1.8567E−09	5.4941E−11	1.2870E−09	2.0023E−06	4.9980E−09	3.0317E−02	1.0188E−05	6.6955E−11	3.0199E−11
F19	9.2603E−09	3.0103E−07	5.9673E−09	1.1077E−06	3.8249E−09	3.5137E−02	1.8500E−08	7.7387E−06	3.0199E−11	2.6015E−08	1.0666E−07
F20	3.0199E−11	5.0723E−10	1.0937E−10	4.1997E−10	3.2651E−02	3.0199E−11	4.6856E−08	3.0199E−11	1.5846E−04	3.0199E−11	3.0199E−11
F21	3.0199E−11	4.4205E−06	1.6132E−10	3.0199E−11	4.1127E−07	3.0199E−11	4.1127E−07	3.0199E−11	1.1058E−04	8.9934E−11	3.0199E−11
F22	1.1902E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2118E−12	1.2108E−12	1.2118E−12
F23	3.0199E−11	1.7290E−06	1.6980E−08	3.0199E−11	1.6798E−03	3.0199E−11	1.5846E−04	3.0199E−11	3.7782E−02	3.0199E−11	3.1589E−10
F24	3.0199E−11	3.4742E−10	2.0152E−08	3.0199E−11	3.0199E−11	3.0199E−11	1.3853E−06	3.0199E−11	3.3679E−04	3.0199E−11	3.0199E−11
F25	6.1210E−10	5.8587E−06	5.4941E−11	3.4742E−10	3.0199E−11	5.0922E−08	9.0632E−08	2.4386E−09	5.0723E−10	6.1210E−10	1.7769E−10
F26	1.2760E−10	2.3853E−09	3.1184E−04	2.3451E−11	1.4659E−05	7.8698E−10	4.1975E−06	3.4282E−09	2.5334E−06	2.3451E−11	7.8698E−10
F27	3.0199E−11	5.4620E−06	4.1997E−10	3.0199E−11	1.9963E−05	1.9568E−10	9.1171E−01	3.0199E−11	1.6955E−02	1.6132E−10	2.3715E−10
F28	2.2668E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12	2.3638E−12
F29	3.0199E−11	2.1540E−06	3.0199E−11	4.9752E−11	8.7663E−01	3.0199E−11	8.5000E−02	3.0199E−11	1.3017E−03	3.0199E−11	3.0199E−11
F30	1.7479E−05	1.2860E−06	3.0199E−11	2.9215E−09	3.0199E−11	7.1186E−09	1.0937E−10	3.0199E−11	3.0199E−11	1.1077E−06	3.0199E−11

and

Table 7. The p-values for various algorithms on the CEC 2017 (dim = 100).

Item	RTH	SAO	GRO	SO	ESC	INFO	SBOA	GKSO	IGWO	HHWOA	ED
F1	6.2828E−06	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	1.0702E−09	3.0199E−11
F2	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11
F3	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	2.0023E−06	3.0199E−11	2.4581E−01	1.5292E−05	3.0199E−11
F4	8.9934E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11
F5	3.0199E−11	5.0114E−01	3.0199E−11	3.0199E−11	3.2651E−02	3.0199E−11	1.6238E−01	3.0199E−11	1.0277E−06	4.9426E−05	3.0199E−11
F6	3.0199E−11	3.8053E−07	3.0199E−11	3.0199E−11	2.5805E−01	3.0199E−11	3.0199E−11	3.0199E−11	4.5726E−09	3.0199E−11	3.0199E−11
F7	3.0199E−11	3.0199E−11	3.0199E−11	2.7548E−03	3.8053E−07	3.0199E−11	3.0199E−11	3.0199E−11	2.3985E−01	3.0199E−11	3.0199E−11
F8	3.0199E−11	7.8446E−01	3.0199E−11	3.0199E−11	3.4029E−01	4.1997E−10	4.6756E−02	3.0199E−11	5.5611E−04	3.8349E−06	3.0199E−11
F9	3.0199E−11	4.4205E−06	3.0199E−11	2.9215E−09	4.0772E−11	3.0199E−11	1.0105E−08	3.0199E−11	1.7290E−06	3.6439E−02	3.0199E−11
F10	4.5726E−09	6.9522E−01	1.4067E−04	3.0199E−11	3.0199E−11	3.1821E−04	8.9934E−11	6.6955E−11	2.3399E−01	1.6947E−09	3.0199E−11
F11	6.6955E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	8.1014E−10	3.0199E−11	4.5726E−09	3.0199E−11	3.0199E−11	3.0199E−11
F12	1.8575E−03	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11
F13	3.0199E−11	2.3715E−10	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11
F14	3.0199E−11	2.3168E−06	3.0199E−11	3.0199E−11	3.0199E−11	5.1877E−02	3.0199E−11	5.5999E−07	1.2057E−10	6.0658E−11	3.0199E−11
F15	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11
F16	1.0576E−03	1.5638E−02	3.3874E−02	4.2259E−03	5.0922E−08	2.5805E−01	3.1830E−03	1.8682E−05	4.4205E−06	1.4532E−01	3.4742E−10
F17	3.8249E−09	1.8090E−01	9.2344E−01	9.7052E−01	1.5964E−07	2.0283E−07	8.6499E−01	1.2870E−09	1.5014E−02	7.6588E−05	4.9980E−09
F18	4.0772E−11	7.3803E−10	2.2273E−09	8.4848E−09	3.0199E−11	1.0666E−07	3.0199E−11	1.7769E−10	1.1674E−05	4.9752E−11	3.0199E−11
F19	3.0199E−11	4.9752E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11
F20	1.4298E−05	1.0869E−01	1.8916E−04	4.5043E−11	6.0104E−08	5.8282E−03	2.4994E−03	6.0971E−03	2.8378E−01	8.4180E−01	3.0199E−11
F21	3.0199E−11	5.5923E−01	1.0702E−09	1.8500E−08	1.2212E−02	3.0199E−11	1.4067E−04	3.0199E−11	3.8307E−05	5.5727E−10	3.0199E−11
F22	2.0023E−06	7.1988E−05	5.3221E−03	3.0199E−11	3.0199E−11	5.8587E−06	1.1023E−08	3.8249E−09	1.1077E−06	2.4913E−06	3.0199E−11
F23	3.0199E−11	4.5043E−11	3.0199E−11	2.3243E−02	3.0199E−11	3.0199E−11	1.4643E−10	3.0199E−11	1.0702E−09	3.0199E−11	3.0199E−11
F24	3.0199E−11	2.1959E−07	3.0199E−11	8.1527E−11	2.2780E−05	3.0199E−11	8.7710E−02	3.0199E−11	5.3221E−03	3.0199E−11	3.0199E−11
F25	8.1465E−05	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0103E−07	3.0199E−11
F26	3.0199E−11	4.5726E−09	3.0199E−11	9.5207E−04	5.4617E−09	3.0199E−11	1.3732E−01	3.0199E−11	4.1178E−06	3.0199E−11	3.8202E−10
F27	4.9752E−11	8.8411E−07	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	8.3520E−08	3.0199E−11	2.2658E−03	3.0199E−11	3.0199E−11
F28	3.5547E−01	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.3520E−08	3.0199E−11
F29	7.6950E−08	1.5178E−03	1.6947E−09	1.7290E−06	5.3685E−02	4.5043E−11	4.6756E−02	3.0199E−11	4.6371E−03	1.8500E−08	1.0937E−10
F30	9.3519E−01	5.5727E−10	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	3.0199E−11	5.5727E−10	3.0199E−11

present experimental results across three dimensions for 11 comparison algorithms on the CEC2017 test set. The data reveal that MEED exhibits significant differences from the comparison algorithms on most test functions.

4.4.2. Friedman Mean Rank Test

The Friedman test is a non-parametric statistical test commonly used to compare the performance of multiple algorithms over a set of benchmark problems. It serves as the non-parametric alternative to the repeated-measures ANOVA and is particularly suitable for algorithmic performance comparison when the assumptions of normality and homoscedasticity are not satisfied.

In this subsection, the Friedman test was applied to evaluate the overall performance differences among all compared algorithms across multiple benchmark functions. For each problem, the algorithms were ranked according to their average fitness values, where a lower rank indicates better performance.

Table 8. Friedman mean rank test result.

Suites	CEC2017
Dimension	30		50		100
Algorithms	$\mathit{M}.\mathit{R}$	$\mathit{T}.\mathit{R}$	$\mathit{M}.\mathit{R}$	$\mathit{T}.\mathit{R}$	$\mathit{M}.\mathit{R}$	$\mathit{T}.\mathit{R}$
RTH	7.87	10	7.10	7	5.67	5
SAO	5.40	3	4.50	2	4.57	2
GRO	7.23	7	7.80	11	8.80	11
SO	7.83	9	7.70	9	7.30	8
ESC	6.10	5	6.63	6	7.60	10
INFO	7.40	8	7.77	10	7.30	7
SBOA	5.17	2	5.03	3	4.57	3
GKSO	8.23	11	7.63	8	7.53	9
IGWO	6.33	6	6.50	5	6.47	6
HHWOA	6.03	4	5.97	4	5.50	4
ED	8.83	12	9.47	12	9.50	12
MEED	1.57	1	1.90	1	3.20	1

presents the experimental results of MEED and its comparison algorithms across three dimensions on the CEC2017 test set. “$M.R$” represents the average ranking achieved by the algorithm across 30 test functions, while “$T.R$” denotes the final ranking obtained by the algorithm on one dimension of the test set.

As can be seen from the experiment results, in the 30-dimensional scenario, the MEED algorithm achieved an average ranking of 1.57 and ranked first overall, while SBOA—the relatively better-performing algorithm among others—had an average ranking of 5.17, significantly higher than MEED. In the 50-dimensional scenario, MEED’s average ranking was 1.90, and the second-best SAO had an average ranking of 4.5, still showing a noticeable gap compared to MEED. Even when dimensions increased to 100, MEED maintained optimal performance with an average rank of 3.2. Notably, compared to the unmodified ED, it achieved the worst performance across all three dimensions. In summary, MEED demonstrates superior optimization performance and robustness compared to algorithms like SAO, SBOA, and IGWO in complex multi-dimensional optimization problems, maintaining stable and excellent optimization results across different dimensional scenarios.

5. MEED for Photovoltaic Model Parameter Estimation

5.1. Photovoltaic Model

In the field of photovoltaic (PV) system modeling, the single-diode model (SDM) and the double-diode model (DDM) are the two most widely used photovoltaic models. The SDM is more commonly adopted due to its structural simplicity and computational efficiency, whereas the DDM is preferred when higher modeling accuracy is required, as it can better capture complex recombination and leakage effects. The SDM provides a compact representation of PV cell behavior using a single exponential term, effectively describing the current–voltage (I-V) characteristics under various environmental conditions with minimal complexity. However, to capture more detailed physical phenomena such as recombination losses within the depletion region, the DDM introduces an additional diode component, thereby enhancing modeling accuracy and parameter interpretability. Both models have been extensively employed in performance analysis, fault diagnosis, maximum power point tracking (MPPT), and parameter estimation of PV systems. Their widespread use and strong physical interpretability make them essential foundations for evaluating and validating optimization algorithms designed for PV parameter extraction. Therefore, this paper conducts an experimental analysis of these two models, with detailed specifics as follows:

5.1.1. Single-Diode Model (SDM)

The single-diode model (SDM) has become the most widely used representation of photovoltaic (PV) cell characteristics due to its excellent trade-off between model simplicity and physical accuracy. By employing only one exponential diode term along with a current source, series resistance, and shunt resistance, the SDM is capable of accurately describing the nonlinear current–voltage (I-V) relationship of PV cells under various temperature and irradiance conditions. Its low computational complexity enables efficient simulation and parameter extraction, making it particularly suitable for real-time control and optimization applications such as maximum power point tracking (MPPT) and system diagnostics. Furthermore, the parameters of the SDM possess clear physical interpretability, allowing for direct correlation with internal physical processes of the PV cell. The equivalent circuit of the single-diode model is shown in Figure 6 [35].

The single-diode model comprises a current source that represents the photo-generated current induced by solar irradiation, a diode that characterizes the PN junction behavior of the semiconductor, series resistance $R_s$ reflecting the ohmic losses of electrodes, interconnections, and materials, and shunt resistance $R_{sh}$ accounting for leakage paths through the semiconductor structure. The output current can be represented by Equation (18):

$$I_{out}=I_{ph}-I_d-I_{sh},$$

(18)

where $I_{ph}$, $I_d$, and $I_{sh}$ denote the photocurrent, the diode current, and the current through the shunt resistor, respectively. The diode and shunt currents can be expressed as follows:

$$I_d=I_o·\left\{\exp \left[{\displaystyle \frac{q·\left(V_{\mathit{out} }+R_s·I_{\mathit{out} }\right)}{a·k·T}}\right]-1\right\},$$

(19)

$$I_{\mathit{sh}}={\displaystyle \frac{V_{\mathit{out} }+R_s·I_{\mathit{out} }}{R_{sh}}},$$

(20)

where $I_o$ is the diode reverse saturation current, $a$ is the diode ideality factor, $k$ is the Boltzmann constant $\left(1.3806503\times {10}^{-23} \mathrm{J}·{\mathrm{K}}^{-1}\right)$, $q$ is the electron charge $\left(1.60217646\times {10}^{-19}\mathrm{C}\right)$, and $T$ ambient temperature in Kelvin.

Substituting Equations (19) and (20) into Equation (18) yields the output current–voltage relationship of the single-diode model:

$$I_{out}=I_{ph}-I_o·\left\{\exp \left[{\displaystyle \frac{q·\left(V_{\mathit{out} }+R_s·I_{\mathit{out} }\right)}{a·k·T}}\right]-1\right\}-{\displaystyle \frac{V_{\mathit{out} }+R_s·I_{\mathit{out} }}{R_{sh}}},$$

(21)

Thus, the single-diode model is fully characterized by five parameters: $\left(I_{ph}, I_o, a, R_s, \mathrm{and} R_{sh}\right)$.

5.1.2. Double-Diode Model (DDM)

Although the single-diode model (SDM) provides a simple and effective representation of photovoltaic (PV) cell behavior, it exhibits certain limitations in accurately describing the recombination losses and non-ideal effects that occur within the depletion region and at the semiconductor interfaces. To overcome these deficiencies, the double-diode model (DDM) introduces an additional diode to account for recombination mechanisms in the junction region, thereby providing a more physically accurate and comprehensive description of the PV cell. The inclusion of the second diode significantly enhances the model’s capability to simulate PV performance under low-irradiance or non-ideal temperature conditions, where carrier recombination effects become more prominent. Moreover, the DDM improves the fitting accuracy of the current–voltage (I-V) curve, enabling more precise parameter estimation and performance prediction. Despite its slightly higher computational complexity, the DDM is widely adopted in high-precision PV modeling, fault diagnosis, and device characterization, where modeling fidelity is critical. The equivalent circuit of the double-diode model is illustrated in Figure 7 [35].

The current output of the double-diode model can be represented as:

$$I_{out}=I_{ph}-I_{d1}-I_{d2}-I_{sh},$$

(22)

By combining the diode current and shunt resistance expressions, the $\left(I_{out}-V_{out}\right)$ characteristic equation of the double-diode model can be written as:

$$\begin{array}{c}{I}_{out}=I_{ph}-I_{o1}·\left\{\exp \left[{\displaystyle \frac{q·\left(V_{\mathit{out} }+R_s·I_{\mathit{out} }\right)}{a_1·k·T}}\right]-1\right\}-I_{o1}·\left\{\exp \left[{\displaystyle \frac{q·\left(V_{\mathit{out} }+R_s·I_{\mathit{out} }\right)}{a_2·k·T}}\right]-1\right\}\\ -{\displaystyle \frac{V_{\mathit{out} }+R_s·I_{\mathit{out} }}{R_{sh}}},\end{array}$$

(23)

where $I_{d1}$ and $I_{d2}$ are the currents through diodes $D_1$ and $D_2$, respectively; $I_{o1}$ and $I_{o2}$ denote the saturation current and the diffusion current; and $a_1$ and $a_2$ are the ideality factor and diffusion factor.

As a result, the double-diode model necessitates determining seven unknown parameters: $\left(I_{ph}, I_{o1}, I_{o2}, a_1, a_2, R_s, \mathrm{and} R_{sh}\right)$.

5.2. Problem Formulation

The unknown parameters can be determined by formulating an optimization problem, where an objective function $g$ measures the difference between experimental data and the results predicted by the model. The aim of the optimization is to minimize this difference within a specified search domain, thus identifying the optimal model parameters. Typical error functions used in such cases are given as follows:

(1)

Objective function for the single-diode model (SDM):

$$\left\{\begin{array}{c}g\left(V_{\mathit{out} },I_{\mathit{out} },y\right)=N_p·I_{ph}-N_p·I_o·\left\{exp\left[{\displaystyle \frac{q·\left(\frac{V_{\mathit{out} }}{N_s}+R_s·\frac{I_{\mathit{out} }}{N_p}\right)}{a·k·T}}\right]-1\right\}-\\ N_p·{\displaystyle \frac{\frac{V_{\mathit{out} }}{N_s}+R_s·\frac{I_{\mathit{out} }}{N_p}}{R_{\mathit{sh} }}}-I_{\mathit{out} }\\ y=\left(I_{ph},I_o,a,R_s,R_{\mathit{sh} }\right)\end{array}\right.,$$

(24)

(2)

Objective function for the double-diode model (DDM):

$$\{{\displaystyle \genfrac{}{}{0pt}{}{\begin{array}{c}g\left(V_{\mathit{out} },I_{\mathit{out} },y\right)=N_p·I_{ph}-N_p\cdot I_{o1}\cdot \left\{\mathit{exp}\left[\frac{q·\left(\frac{V_{\mathit{out} }}{N_s}+R_s·\frac{I_{\mathit{out} }}{N_p}\right)}{a_1·k·T}\right]-1\right\}\\ -N_p\cdot I_{o2}\cdot \left\{\mathit{exp}\left[\frac{q·\left(\frac{V_{\mathit{out} }}{N_s}+R_s·\frac{I_{\mathit{out} }}{N_p}\right)}{a_2·k·T}\right]-1\right\}\end{array}}{\begin{array}{c}-N_p·\frac{\frac{V_{\mathit{out} }}{N_s}+R_s·\frac{I_{\mathit{out} }}{N_p}}{R_{\mathit{sh} }}-I_{\mathit{out} }\\ y=\left(I_{ph},I_{o1},I_{o2},a_1,a_2,R_s,R_{sh}\right)\end{array}}},$$

(25)

For a single cell, set $N_s=1$ and $N_p=1$.

Finally, the total discrepancy between the experimental I-V curve and the model prediction is assessed using the root mean square error (RMSE):

$$RMSE\left(y\right)=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{n=1}^N{\left(V_{\mathit{out},n},I_{\mathit{out},n},y\right)}^2}{N}}},$$

(26)

where $N$ is the total number of measured data points $\left(V_{\mathit{out}}, n, I_{\mathit{out}}, n\right)$.

5.3. Experimental Parameter Setting and Simulation Analysis

In this section, we validate the effectiveness of the proposed MEED algorithm for extracting photovoltaic model parameters. First, we briefly introduce the experimental parameters. Then, we apply MEED to estimate unknown parameters in two photovoltaic models: the single-diode model (SDM) and the double-diode model (DDM). Detailed specifications are as follows:

5.3.1. Experimental Parameter Setting

Experimental data were collected from a Photowatt-PWP 201 photovoltaic module, consisting of 36 series-connected polycrystalline silicon cells. Under a temperature of 33 °C and an irradiance of 1000 W/m², 26 sets of current–voltage (I-V) measurements were recorded. These experiments aimed to determine the parameters of the SDM and DDM for RTC France photovoltaic cells, and the results were subsequently compared with those obtained using other advanced optimization algorithms.

All algorithms were implemented in MATLAB 2024b and executed on a personal computer with a 2.5 GHz CPU, 16 GB RAM, and Windows 11. Each algorithm was independently run 30 times per problem, with a maximum of 1000 iterations and a population size of 50. To highlight the differences between the proposed algorithm and other algorithms addressing similar problems, the Wilcoxon rank-sum test was employed to assess statistical significance. The ranges of unknown parameters for each model are summarized in

Table 9. The ranges of unknown parameters.

Parameters	Single-Diode PV Models		Double-Diode Models
	$\mathit{L}\mathit{b}$	$\mathit{U}\mathit{b}$	$\mathit{L}\mathit{b}$	$\mathit{U}\mathit{b}$
$I_{ph}\left(A\right)$	0	1	0	2
$I_d\left(\mu A\right)$	0	1	0	50
$R_s\left(\Omega \right)$	0	0.5	0	2
$R_{sh}\left(\Omega \right)$	0	100	0	2000
$n$	1	2	1	50
$I_{d1}\left(\mu A\right)$	0	1	0	50
$I_{d2}\left(\mu A\right)$	0	1	0	50
$n_1$	1	2	1	50
$n_1$	1	2	1	50

[35]. $Lb$ denotes the lower bound of the parameter, while $Ub$ denotes the upper bound of the parameter.

As mentioned earlier, the root mean square error (RMSE) offers a straightforward metric for quantifying the difference between experimental data and simulated results. A lower RMSE signifies a closer agreement between the simulated and measured data, highlighting the algorithm’s superior capability in estimating the unknown parameters of the photovoltaic system. In other words, the diode model obtained through this algorithm more accurately represents the actual behavior of solar cells and PV modules. Consequently, minimizing this error is of critical importance.

In addition, the absolute error (IAE) and relative error (RE) are used to assess the deviation at each measured voltage, defined as follows:

$$IAE=\left|I_{measure}-I_{simulate}\right|,$$

(27)

$$RE={\displaystyle \frac{I_{measure}-I_{simulate}}{I_{measure}}},$$

(28)

5.3.2. Experimental Analysis of SDM

In this subsection, we conducted experimental analysis using MEED and 11 other comparison algorithms. The experimental results are shown in

Table 10. Comparison among different algorithms on SDM.

Algorithm	${\mathit{I}}_{\mathit{p}\mathit{h}}\mathbf{\left(}\mathit{A}\mathbf{\right)}$	${\mathit{I}}_{\mathit{d}}\mathbf{\left(}\mathit{\mu }\mathit{A}\mathbf{\right)}$	${\mathit{R}}_{\mathit{s}}\mathbf{\left(}\mathbf{\Omega }\mathbf{\right)}$	${\mathit{R}}_{\mathit{s}\mathit{h}}\mathbf{\left(}\mathbf{\Omega }\mathbf{\right)}$	$\mathit{n}$	$\mathit{R}\mathit{S}\mathit{M}\mathit{E}$	$\mathit{s}\mathit{i}\mathit{g}$
RTH	7.6078E−01	3.2302E−07	3.6377E−02	5.3719E+01	1.4812E+00	9.8615E−04	/
SAO	7.6076E−01	3.3180E−07	3.6270E−02	5.4453E+01	1.4839E+00	9.8735E−04	+
GRO	7.6076E−01	3.3812E−07	3.6194E−02	5.4964E+01	1.4858E+00	9.8987E−04	+
SO	7.6084E−01	1.0000E−06	3.1385E−02	1.0000E+02	1.6045E+00	2.4000E−03	+
ESC	7.6081E−01	5.2676E−07	3.4357E−02	9.1803E+01	1.5318E+00	1.7000E−03	+
INFO	7.6078E−01	3.2302E−07	3.6377E−02	5.3719E+01	1.4812E+00	9.8654E−04	+
SBOA	7.6075E−01	3.2100E−07	3.6407E−02	5.3949E+01	1.4805E+00	9.8633E−04	+
GKSO	7.6078E−01	3.2302E−07	3.6377E−02	5.3719E+01	1.4812E+00	9.8678E−04	+
IGWO	7.6102E−01	3.0477E−07	3.6644E−02	4.9105E+01	1.4753E+00	1.1000E−03	+
HHWOA	7.6078E−01	3.2299E−07	3.6377E−02	5.3712E+01	1.4812E+00	9.8674E−04	+
ED	7.6088E−01	3.2042E−07	3.6415E−02	5.3109E+01	1.4804E+00	9.8870E−04	+
MEED	7.6073E−01	3.2236E−07	3.6345E−02	5.3716E+01	1.4803E+00	9.8602E−04	+

. As shown in Table 10, the proposed MEED algorithm achieves the lowest RMSE (9.8602E−04) among all comparative algorithms, indicating its superior capability in accurately identifying the parameters of the single-diode model (SDM). Moreover, the extracted parameters are physically consistent and exhibit minimal deviations from reference values, demonstrating both precision and stability. The statistical significance test further confirms that MEED performs comparably or better than the best existing algorithms, highlighting its robustness and high convergence efficiency in photovoltaic parameter estimation.

It should be noted that the measured I-V data may contain noise caused by sensor inaccuracies, temperature variation, and irradiance fluctuations. To evaluate the robustness of the proposed algorithm, noisy measurement conditions were considered. The results indicate that MEED maintains stable convergence behavior and accurate parameter estimation performance even in the presence of measurement noise.

Figure 8 illustrates the convergence behavior of MEED and eleven comparative algorithms on the SDM parameter estimation problem. As observed, the proposed MEED (red curve) demonstrates the fastest convergence rate and achieves the lowest final fitness value. At the early stage (within the first 50 iterations), MEED rapidly decreases the objective function value, indicating its strong exploration capability. In the middle iterations (100–300), the curve continues to decline smoothly, reflecting an effective transition from exploration to exploitation. In contrast, algorithms such as ED, SO, and GRO exhibit slower convergence and tend to stagnate at higher fitness values, suggesting premature convergence or weaker local search abilities. Although some algorithms (e.g., IGWO and SBOA) perform relatively well, their convergence speed and final precision are still inferior to MEED. Overall, the convergence curve confirms that MEED not only achieves the lowest RMSE (as shown in Table 10) but also maintains high convergence efficiency and stability, validating its superior balance between global exploration and local exploitation.

Figure 9 illustrates the current–voltage (I-V) and power–voltage (P-V) characteristics of the single-diode model (SDM) estimated via the MEED method. The blue solid lines correspond to measured data, while the orange dots represent estimated results. It is apparent that the estimated I-V and P-V curves match the measured counterparts remarkably well over the entire voltage range. Specifically, in the I-V characteristic (Figure 9a), the estimated current aligns closely with the measured current, including the flat region at low voltages and the sharp decline near the open-circuit voltage. For the P-V characteristic (Figure 9b), the estimated power curve accurately tracks the measured curve, capturing the linear increase at low voltages, as well as the peak power point (MPP) and the subsequent decline. This high degree of agreement validates the effectiveness of the MEED method in identifying SDM parameters, which is essential for photovoltaic system analysis and optimization.

Table 11. IAE of MEED on SDM.

MEED	$\mathit{V}\left(\mathit{V}\right)$	$\mathit{I}\left(\mathit{A}\right)$	${\mathit{I}}_{\mathit{s}\mathit{i}\mathit{m}}\left(\mathbf{A}\right)$	$\mathit{I}\mathbf{A}{\mathit{E}}_{\mathit{I}}\left(\mathbf{A}\right)$	${\mathit{P}}_{\mathit{s}\mathit{i}\mathit{m}}\left(\mathit{W}\right)$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{p}}\left(\mathbf{A}\right)$
1	−2.0570E−01	7.6409E−01	7.6400E−01	8.7704E−05	−1.1480E−04	1.8041E−05
2	−1.2910E−01	7.6266E−01	7.6200E−01	6.6309E−04	−8.7019E−04	8.5604E−05
3	−5.8800E−02	7.6136E−01	7.6050E−01	8.5531E−04	−1.1247E−03	5.0292E−05
4	5.7000E−03	7.6015E−01	7.6050E−01	3.4601E−04	4.5498E−04	1.9723E−06
5	6.4600E−02	7.5906E−01	7.6000E−01	9.4479E−04	1.2431E−03	6.1034E−05
6	1.1850E−01	7.5804E−01	7.5900E−01	9.5765E−04	1.2617E−03	1.1348E−04
7	1.6780E−01	7.5709E−01	7.5700E−01	9.1654E−05	−1.2108E−04	1.5380E−05
8	2.1320E−01	7.5614E−01	7.5700E−01	8.5864E−04	1.1343E−03	1.8306E−04
9	2.5450E−01	7.5509E−01	7.5550E−01	4.1313E−04	5.4683E−04	1.0514E−04
10	2.9240E−01	7.5366E−01	7.5400E−01	3.3612E−04	4.4579E−04	9.8282E−05
11	3.2690E−01	7.5139E−01	7.5050E−01	8.9097E−04	−1.1872E−03	2.9126E−04
12	3.5850E−01	7.4735E−01	7.4650E−01	8.5385E−04	−1.1438E−03	3.0611E−04
13	3.8730E−01	7.4012E−01	7.3850E−01	1.6172E−03	−2.1899E−03	6.2635E−04
14	4.1370E−01	7.2738E−01	7.2800E−01	6.1777E−04	8.4859E−04	2.5557E−04
15	4.3730E−01	7.0697E−01	7.0650E−01	4.7265E−04	−6.6900E−04	2.0669E−04
16	4.5900E−01	6.7528E−01	6.7550E−01	2.1985E−04	3.2546E−04	1.0091E−04
17	4.7840E−01	6.3076E−01	6.3200E−01	1.2417E−03	1.9648E−03	5.9404E−04
18	4.9600E−01	5.7193E−01	5.7300E−01	1.0716E−03	1.8702E−03	5.3153E−04
19	5.1190E−01	4.9961E−01	4.9900E−01	6.0702E−04	−1.2165E−03	3.1073E−04
20	5.2650E−01	4.1365E−01	4.1300E−01	6.4879E−04	−1.5709E−03	3.4159E−04
21	5.3980E−01	3.1751E−01	3.1650E−01	1.0101E−03	−3.1915E−03	5.4526E−04
22	5.5210E−01	2.1215E−01	2.1200E−01	1.5494E−04	−7.3085E−04	8.5542E−05
23	5.6330E−01	1.0225E−01	1.0350E−01	1.2487E−03	1.2065E−02	7.0339E−04
24	5.7360E−01	−8.7175E−03	−1.0000E−02	1.2825E−03	1.2825E−01	7.3562E−04
25	5.8330E−01	−1.2551E−01	−1.2300E−01	2.5074E−03	−2.0385E−02	1.4626E−03
26	5.9000E−01	−2.0847E−01	−2.1000E−01	1.5277E−03	7.2746E−03	9.0133E−04

further demonstrates the robustness of MEED in the SDM by presenting the absolute error (IAE) results. For the 26 measurement points, the IAE generally varies between 1.8041E−5 and 1.4626E−3 A. Most data points show minimal deviations: for instance, at 0.2924 V, the IAE is merely 9.8282E−05 A. Slightly larger errors are observed in the high-voltage (0.5833 V) and reverse-voltage regions, with a maximum of 1.4626E−3 A, confirming MEED’s excellent error control capability.

5.3.3. Experimental Analysis of DDM

Table 12. Comparison among different algorithms on DDM.

Algorithm	${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$	${\mathit{R}}_{\mathit{s}}\left(\mathbf{\Omega }\right)$	${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathbf{\Omega }\right)$	${\mathit{I}}_{\mathit{d}1}\left(\mathit{\mu }\mathit{A}\right)$	${\mathit{n}}_1$	${\mathit{I}}_{\mathit{d}2}\left(\mathit{\mu }\mathit{A}\right)$	${\mathit{n}}_2$	$\mathit{R}\mathit{S}\mathit{M}\mathit{E}$	$\mathit{s}\mathit{i}\mathit{g}$
RTH	7.6078E−01	3.6740E−02	5.5486E+01	2.2597E−07	1.4510E+00	7.4937E−07	2.0000E+00	9.8230E−04	/
SAO	7.6078E−01	3.6856E−02	5.5615E+01	5.3628E−07	1.7551E+00	1.4557E−07	1.4217E+00	9.8651E−04	+
GRO	7.6076E−01	3.6046E−02	5.6964E+01	1.7258E−07	1.8766E+00	3.1710E−07	1.4818E+00	9.8665E−04	+
SO	7.6057E−01	3.4549E−02	6.2743E+01	4.4473E−07	1.5164E+00	4.3184E−08	1.6974E+00	9.8638E−04	+
ESC	7.6038E−01	3.5857E−02	9.6426E+01	1.1478E−07	1.6041E+00	3.2207E−07	1.4928E+00	9.8470E−04	+
INFO	7.6078E−01	3.6747E−02	5.5514E+01	2.2496E−07	1.4506E+00	7.5672E−07	2.0000E+00	9.8747E−04	+
SBOA	7.6096E−01	3.7396E−02	5.2086E+01	6.7906E−07	1.7579E+00	9.8279E−08	1.3906E+00	9.8320E−04	+
GKSO	7.6078E−01	3.6692E−02	5.5312E+01	2.3590E−07	1.4546E+00	6.6801E−07	2.0000E+00	9.9661E−04	+
IGWO	7.5964E−01	3.5817E−02	9.0315E+01	5.1663E−07	1.9376E+00	3.0406E−07	1.4798E+00	9.9032E−04	+
HHWOA	7.6071E−01	3.6344E−02	5.5017E+01	3.1739E−07	1.4800E+00	6.0228E−08	1.9452E+00	9.8509E−04	+
ED	7.6075E−01	3.6556E−02	5.7131E+01	4.3946E−07	1.8720E+00	2.3677E−07	1.4568E+00	9.8611E−04	+
MEED	7.6078E−01	3.6740E−02	5.5485E+01	7.4935E−07	2.0000E+00	2.2597E−07	1.4510E+00	9.8228E−04	+

presents the comparison results of MEED and eleven other algorithms on the double-diode model (DDM). It can be observed that the proposed MEED algorithm achieves the lowest RMSE value of 9.8228E−04, demonstrating its superior accuracy in parameter estimation. The extracted parameters ($I_{ph}, R_s, R_{sh}, I_{d1}, I_{d2}, n_1, \mathrm{and} n_2$) obtained by MEED are highly consistent with reference values, confirming its ability to provide physically meaningful and stable solutions. In comparison, traditional and metaheuristic algorithms such as SO, GRO, and IGWO show relatively higher RMSE values, indicating weaker fitting accuracy and a tendency toward local optima. Although several algorithms (e.g., SBOA and ESC) also reach competitive results, their estimated parameters exhibit slightly larger deviations. This suggests that these methods may lack sufficient balance between global exploration and local exploitation. Overall, MEED demonstrates both high precision and excellent stability in identifying DDM parameters. The results confirm that the proposed multi-strategy enhancement effectively improves convergence reliability and the global search capability of the baseline algorithm, enabling MEED to achieve more accurate modeling of photovoltaic characteristics.

Figure 10 illustrates the convergence behaviors of twelve algorithms on the double-diode model (DDM). As shown, the proposed MEED algorithm exhibits the fastest and most stable convergence trend among all competitors. In the early stage (within the first 100 iterations), MEED rapidly reduces the objective function value, reaching a near-optimal region far earlier than other algorithms. This demonstrates its strong global exploration ability and efficient search guidance in the initial phase. As iterations proceed, MEED continues to refine the solution with minimal oscillation, ultimately achieving the lowest best score (≈10⁻³). In contrast, other algorithms such as SO, ESC, and IGWO show slower convergence and higher stagnation levels, indicating a higher tendency to be trapped in local optima. The relatively smooth convergence curve of MEED also highlights its excellent balance between exploration and exploitation, preventing premature convergence while maintaining steady progress toward the global optimum. Overall, the convergence analysis verifies that the multi-strategy enhancements embedded in MEED significantly accelerate convergence speed, improve search stability, and enhance robustness when addressing complex nonlinear parameter estimation problems in photovoltaic modeling.

Figure 11 illustrates the comparison between the measured and estimated I-V and P-V characteristics of the photovoltaic (PV) cell based on the double-diode model (DDM) using the MEED algorithm. As shown in Figure 9a, the estimated current–voltage curve (red circles) almost perfectly coincides with the experimental data (blue line), indicating that the identified parameters accurately reproduce the nonlinear behavior of the PV cell under various voltage conditions. Similarly, in Figure 9b, the power–voltage curve obtained by the proposed MEED method closely follows the measured values across the entire operating range. The near-perfect overlap between the two curves confirms that the estimated parameters provide a highly precise power prediction, especially around the maximum power point (MPP). These results demonstrate that the MEED algorithm exhibits strong parameter identification capability, effectively minimizing the root mean square error (RMSE) and maintaining excellent fitting accuracy. Consequently, MEED can achieve superior modeling performance and ensure the reliable representation of photovoltaic system characteristics.

Table 13. IAE of MEED on DDM.

Algorithm	$\mathit{V}\left(\mathit{V}\right)$	$\mathit{I}\left(\mathit{A}\right)$	${\mathit{I}}_{\mathit{s}\mathit{i}\mathit{m}}\left(\mathbf{A}\right)$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{I}}\left(\mathbf{A}\right)$	${\mathit{P}}_{\mathit{s}\mathit{i}\mathit{m}}\left(\mathit{W}\right)$	$\mathit{I}\mathit{A}{\mathit{E}}_{\mathit{p}}\left(\mathbf{A}\right)$
1	−2.0570E−01	7.6400E−01	7.6398E−01	1.6594E−05	−1.5715E−01	3.4135E−06
2	−1.2910E−01	7.6200E−01	7.6260E−01	6.0409E−04	−9.8452E−02	7.7988E−05
3	−5.8800E−02	7.6050E−01	7.6134E−01	8.3770E−04	−4.4767E−02	4.9257E−05
4	5.7000E−03	7.6050E−01	7.6017E−01	3.2621E−04	4.3330E−03	1.8594E−06
5	6.4600E−02	7.6000E−01	7.5911E−01	8.9232E−04	4.9038E−02	5.7644E−05
6	1.1850E−01	7.5900E−01	7.5812E−01	8.7858E−04	8.9837E−02	1.0411E−04
7	1.6780E−01	7.5700E−01	7.5719E−01	1.8862E−04	1.2706E−01	3.1650E−05
8	2.1320E−01	7.5700E−01	7.5624E−01	7.5639E−04	1.6123E−01	1.6126E−04
9	2.5450E−01	7.5550E−01	7.5518E−01	3.2269E−04	1.9219E−01	8.2125E−05
10	2.9240E−01	7.5400E−01	7.5372E−01	2.7764E−04	2.2039E−01	8.1182E−05
11	3.2690E−01	7.5050E−01	7.5140E−01	8.9914E−04	2.4563E−01	2.9393E−04
12	3.5850E−01	7.4650E−01	7.4730E−01	8.0145E−04	2.6791E−01	2.8732E−04
13	3.8730E−01	7.3850E−01	7.4001E−01	1.5107E−03	2.8661E−01	5.8508E−04
14	4.1370E−01	7.2800E−01	7.2725E−01	7.5305E−04	3.0086E−01	3.1154E−04
15	4.3730E−01	7.0650E−01	7.0685E−01	3.5029E−04	3.0911E−01	1.5318E−04
16	4.5900E−01	6.7550E−01	6.7521E−01	2.8946E−04	3.0992E−01	1.3286E−04
17	4.7840E−01	6.3200E−01	6.3076E−01	1.2392E−03	3.0176E−01	5.9286E−04
18	4.9600E−01	5.7300E−01	5.7199E−01	1.0053E−03	2.8371E−01	4.9861E−04
19	5.1190E−01	4.9900E−01	4.9971E−01	7.0613E−04	2.5580E−01	3.6147E−04
20	5.2650E−01	4.1300E−01	4.1373E−01	7.3367E−04	2.1783E−01	3.8628E−04
21	5.3980E−01	3.1650E−01	3.1755E−01	1.0462E−03	1.7141E−01	5.6474E−04
22	5.5210E−01	2.1200E−01	2.1212E−01	1.2300E−04	1.1711E−01	6.7907E−05
23	5.6330E−01	1.0350E−01	1.0216E−01	1.3367E−03	5.7549E−02	7.5298E−04
24	5.7360E−01	−1.0000E−02	−8.7917E−03	1.2083E−03	−5.0429E−03	6.9305E−04
25	5.8330E−01	−1.2300E−01	−1.2554E−01	2.5434E−03	−7.3229E−02	1.4836E−03
26	5.9000E−01	−2.1000E−01	−2.0837E−01	1.6284E−03	−1.2294E−01	9.6076E−04

further demonstrates the robustness of MEED in the DDM by presenting the absolute error (IAE) results. For the 26 measurement points, the IAE generally varies between 3.4135E−06 and 9.6076E−04 A. Most data points show minimal deviations: for instance, at 5.7000E−03 V, the IAE is merely 1.8594E−06 A. Slightly larger errors are observed in the high-voltage (5.9000E−01 V) and reverse-voltage regions, with a maximum of 9.6076E−04 A, confirming MEED’s excellent error control capability.

From the quantitative results, MEED consistently achieves lower RMSE values for both the single-diode model (SDM) and the double-diode model (DDM). Compared with the baseline ED and other competing metaheuristic algorithms, the improvement in RMSE obtained by MEED is nontrivial, demonstrating its stronger global search capability and more accurate parameter identification performance. This improvement is also reflected in the I-V and P-V characteristic curves, where the curves generated by MEED exhibit closer agreement with the measured data across the entire operating range. This robustness can be attributed to the combined effects of hybrid initialization and the historical trend-based update mechanism, which enhance search stability and prevent premature convergence.

Overall, these results demonstrate that MEED not only improves fitting accuracy but also provides statistically significant and robust performance advantages over existing algorithms in photovoltaic parameter estimation problems.

6. Conclusions

This paper proposes a multi-strategy enhanced enterprise development (MEED) optimization algorithm suitable for photovoltaic model parameter estimation. First, a hybrid initialization method based on chaotic mapping and adversarial learning mechanism is introduced to enhance the initial population quality, enabling the algorithm to explore the potential search space more effectively. Subsequently, a trend position update method based on historical information is introduced to accelerate convergence toward the global optimum. Furthermore, the boundary control method based on mirror reflection significantly enhances the algorithm’s exploration capability, effectively avoiding local optima traps. Comparative evaluations against 11 other algorithms using the IEEE CEC2017 test set, combined with statistical analysis, validate the superiority of MEED. Experimental results demonstrate that MEED exhibits significant advantages. Additionally, the MEED method was applied to parameter identification for single-diode (SDM) and dual-diode (DDM) photovoltaic models. Experiments based on real measurement data demonstrate the effectiveness and reliability of MEED in addressing complex engineering optimization problems, providing a robust and efficient solution for precise modeling and optimization of photovoltaic systems.

Future studies could further broaden the application scope of MEED, including its utilization in parameter optimization for wind energy systems and electricity load prediction within other complex renewable energy scenarios. Moreover, coupling adaptive parameter adjustment strategies with multi-objective optimization frameworks is expected to greatly improve the algorithm’s flexibility in handling dynamic and multi-constraint optimization tasks.