Cauchy distribution optimizer for photovoltaic model parameter extraction and numerical optimization

Abstract

The Cauchy distribution optimizer (CDO), a revolutionary nature-inspired metaheuristic method for engineering design, is developed in this study. To prevent early convergence, find a balance between exploration and exploitation in the answer space, CDO method primarily mimics the position parameter and scale parameter behavior of the Cauchy distribution. 29 common CEC2017 unconstrained benchmark functions were used to confirm the CDO’s efficacy and competitiveness. The experimental results show that CDO ranked first overall, with an average ranking of 1.862. Additionally, the approach is applied to five real-world restricted optimization problems including the extraction of solar system key parameters in order to further validate its efficacy. According to the simulation results, the created CDO is a very promising technology that can outperform other cutting-edge competing technologies. The source code for the CDO algorithm is publicly accessible at https://ww2.mathworks.cn/matlabcentral/fileexchange/182811-cauchy-distribution-optimizer.

Introduction

A group of effective problem-solving techniques known as metaheuristic algorithms seeks to approach optimum solutions to challenging optimization problems by mimicking biological activity, physical processes, or natural events^1–3. Compared with traditional accurate algorithms, meta-heuristic algorithm has stronger global search ability and adaptability, especially for NP-hard problems, multi-modal optimization or high-dimensional nonlinear scenarios^4–6. The core idea is to avoid falling into local optimal by balancing exploration and development, while taking into account computational efficiency. Meta-heuristic algorithms are widely used in engineering optimization (such as path planning, scheduling problems), artificial intelligence (neural network training, feature selection), economics (portfolio optimization), bioinformatics (gene sequence analysis), and industrial production (parameter tuning, resource allocation)^7,8. These algorithms start the search in the solution space according to the preset principles ^9,10. As shown in Fig. 1, MAs are primarily divided into two categories based on the principles that the algorithm abides by: swarm intelligence algorithms and evolutionary algorithms.

Fig. 1.

Classification of MAs.

One kind of optimization algorithm that draws inspiration from biological evolution is called an evolutionary algorithm. They look for the best answer in the solution space by mimicking processes like natural selection, inheritance, and variation¹¹. The core idea is "survival of the fittest", which gradually approaches the optimal solution of the problem by iteratively optimizing the individuals (candidate solutions) in the population. The most well-known kind of evolutionary algorithms is the Genetic Algorithm (GA) ¹², which was developed by John Holland in 1975 and is based on Darwin’s idea of biological evolution (natural selection and genetic variation). It achieves the optimization search of "survival of the fittest" in the computer by simulating the biological evolution process. The swarm intelligence algorithm is another category. It should be mentioned that most swarm intelligence optimizers can be classified as either physics-based, biologically-inspired, based on humans, or based on mathematics. Among these, biologically inspired algorithms mimic the biological behavior of fish, insects, and other species. The slime mould algorithm (SMA)¹³, grey wolf optimizer (GWO)¹⁴, and particle swarm optimization (PSO)^15,16 are important and new algorithms in the first class. Additionally, human-based techniques, such as the teaching–learning-based optimizer (TLBO)^17,18 imitate the rules governing human behavior in groups, mathematics-based techniques represented by bayesian optimization (BO)^19–21 and conformal mapping-based optimization (CMBO)²² real-world applications of mathematical laws, and physics-based methods such as gravitational search algorithm (GSA)^23,24, harmony search (HS)^25,26, and simulated annealing (SA)^27–29 simulate the laws of physics.

Even though MAs are widely utilized, not all optimization problems can always benefit from their superior performance, as the no-free-lunch theorem (NFL) indicates^30,31. As a result, researchers are encouraged to either propose new techniques or use some useful enhancements to improve the convergence performance of current algorithms³². The ultimate goal is to recognize the compromise between exploration as well as exploitation. In the context of metaheuristic algorithms, the exploration capability actually refers to the algorithm’s capacity grow and discover interesting areas inside the solution space^33,34. In order to encourage the search agents to jump out of several local optima, the exploration capacity is intended to be preserved when employing MAs to handle complicated multimodal circumstances³⁵. Conversely, the exploitation capability refers to the ability to quickly converge into the ideal global solution³⁶. However, the primary shortcoming of the majority of MAs remains the disparity between exploration and exploitation, which severely restricts algorithm performance. In order to balance exploration with exploitation, this article creates a new physics-based metaheuristic method called the Cauchy distribution optimizer (CDO).

Motivation

MAs have garnered a lot of attention since they are among the most effective instruments for handling optimization problems in the real world³⁷. According to the description given above, many academics have sought to improve their performances using various strategies, leading to the development of a sizable number of intriguing algorithms. However, as optimization issues become more complex, there is a constant demand for improved optimization methods³⁸. Furthermore, the practicality and efficacy of MAs motivate us to create a brand-new metaheuristic approach. Compared to the more well-known normal distribution, there are some differences between them. Both the Cauchy distribution and the normal distribution have bell-shaped curves, but there are significant differences between them. The tail of the Cauchy distribution is much thicker than that of the normal distribution, which indicates that the Cauchy distribution is more prone to extreme values. This way, when it finally reaches the minimum value, it will be more conducive to escaping from the extreme. Additionally, the CDO algorithm was developed to get beyond the drawbacks of the majority of MAs and ultimately strike a balance between the swarm’s variety and convergence.

Contribution

We present the Cauchy Distribution optimizer (CDO), a novel meta-heuristic method for solving numerical optimization and engineering design issues, which is inspired by the Cauchy distribution. The following is a summary of the primary contributions:

• A generalized Cauchy distribution optimization algorithm (CDO) based on mathematics for simulating the Cauchy probability distribution is proposed.

• 29 sample CEC2017 unconstrained benchmarks are used to validate the strength of the proposed technique, and simulation results are compared with other established methodologies.

• Statistical analysis such as Friedman’s and Wilcoxon’s tests are used to assess the strength of the increased CDO algorithm.

• The analysis of exploration and exploitation in CDO explains why the presented technique performs so well.

• The advantage of CDO over competing algorithms is further confirmed by its use to extract key characteristics from solar systems.

Remainder of the essay is structured in this manner. The Cauchy distribution optimizer is thoroughly explained in the section "Cauchy distribution optimizer". The results of the simulation are shown and examined in the section "Results and analysis of experiments". The application of the algorithm is divided into basic application in the section "Experimental study on finite optimization problems in engineering" and parameter extraction of photovoltaic in the section "Utilization for photovoltaic system parameter extraction". The section "Conclusion" displays conclusions and some upcoming projects.

Cauchy distribution optimizer

The CDO algorithm is proposed in this section and is broken down into the subsequent subsections. The CDO’s motivation is introduced in the first subsection. The CDO framework is then provided in the second subsection. Lastly, a description of how to use CDO for optimization is given.

Inspiration

CDO is inspired by Cauchy’s distribution theory. A continuous probability distribution with no mathematical prediction is the Cauchy distribution. When a random variable X satisfies its probability density function, it is said that X follows a Cauchy distribution. The Cauchy distribution, sometimes called the Cauchy–Lorentz distribution, is a continuous probability distribution that bears the names of Hendrik Lorentz and Augustin–Louis Cauchy. Its probability density function can be expressed as:

1

In the formula: x₀ is the positional parameter that defines the peak position of the distribution; is the scale parameter of half the width at half of the maximum value. Its significance in physics is largely attributed to the fact that it is the solution of the differential equation that describes forced resonance. In spectroscopy, it describes the shape of spectral lines that are broadened by resonance or other mechanisms. Its corresponding cumulative distribution function can be expressed as:

2

In the formula: x₀ is the positional parameter that defines the peak position of the distribution; is the scale parameter of half the width at half of the maximum value.

Figure 2 shows the Cauchy distribution under different x₀ and . It can be see that when keeps unchanged, the probability density curve will move towards the direction of increasing x₀ or decreasing x₀. In addition, according to Fig. 2b, when x₀ keeps unchanged, the probability density curve will expand towards the direction of increasing .

Fig. 2.

Cauchy distribution under different x₀ and γ.

In general, there are three steps in the search process for population-based optimization techniques^39,40. First, the scattered distribution contains all initialized individuals⁴¹. Then, under the supervision of the planned exploration and exploitation techniques, everyone begins to move in the direction of the global optimum solution⁴². At last, everyone congregates around the optimal solution that has been reached. In actuality, several Cauchy distributions can be used to characterize this search process. To be more precise, each person’s position can be thought of as a random variable with a Cauchy distribution. The ideal location and the positional parameter are farther apart in the first stage. All people’s positions have a significantly higher scale parameter. The gap between the mean position and the ideal position is progressively reduced in the second stage. Over time, the scale parameter of each individual’s position gradually decreases. The final stage allows the scale parameter of each person’s position to attain the minimal gap between the mean position and the optimal position.

The CDO algorithm’s initialization step, exploration stage, exploitation stage, and population mechanism will all be covered in the sections that follow.

Initialization stage

The iteration process in CDO begins with a swarm that is created at random. According to Eq. (3), the complete swarm is usually represented as an array with N rows and Dim columns, where N denotes the swarm’s size and Dim denotes the solution space’s dimensions.

3

The symbols L and U represent the lower and upper bounds of the solution space. respectively. Rand is a randomly generated number in the interval [0,1].

The framework of the proposed method

Figure 3 shows the framework of the proposed CDO. It can be seen that CDO has a very simple structure and the designed information sharing strategies in CDO consist of local exploitation and global exploration. Local exploitation is based on the built generalized Cauchy distribution model, which is guided by the current mean position, the direction established by the average position and optimal value, as well as the current best position. Global exploration involves randomly selecting one from four individuals (the top three in fitness ranking and the average of the top half in fitness ranking), the direction determined by the average location and the ideal value, as well as the Cauchy random number that changes with the number of repeats. The detailed explanation for the two learning strategies is as follows:

Fig. 3.

The framework of the proposed CDO.

Local exploitation

The process of locating better solutions inside the search space that includes each person’s current place is known as "local exploitation⁴³." Considering the connection between the population’s individual distribution and the Cauchy distribution, a generalized Cauchy distribution model for optimization can be built by

4

5

6

7

where M represents the attenuation rate, and its value is 0.3. represents the ith position parameters of the distribution peak position, shows a vector with a number that was selected at arbitrary from [0,1]. indicates the ith the scale parameter, indicates the position of the ith optimal solution, indicates the position of the ith average solution, is a random number between 0 and 1, N is the number of people, and Zi(t) represents the ith individual who participated in tth iteration.

The functions of the three parameters that were used—, , and M—in the planned local exploitation approach are then examined.

• Position parameters of the distribution peak position . There is useful information about the global optimal solution in the current best individual . As a result, the ith individual is drawn in the direction of the best individual , who has a higher probability of coming up with a superior solution. Keep in mind that even if is caught in local optima, all individuals continue to travel in that direction, which will cause the population as a whole to converge too soon. Equation (7) introduces the mean position of the current population in order to overcome this problem. This search algorithms are shown in Fig. 4 to demonstrate the impact of the mean location . The direction between the mean position and the best individual can be followed by common individuals in Fig. 4. Furthermore, the mean position varies throughout generations, which aids in the discovery of better solutions, even though the location of the present best individual may remain constant in some generations. In order to increase the likelihood of avoiding the local optima to some degree, the mean position is incorporated into the planned local exploitation strategy.

• Scale parameter . Scale parameter is employed to enhance the local search ability of the proposed CDO. Furthermore, according to Eqs. (6–7), scale parameter can be regarded as a random sequence between the optimal solution and the mean to perform a local search around position . Furthermore, according to Eq. (6), the generated random sequence fluctuates more when the distance between the location of the ith individual , the mean position , and the position of the best individual is greater. In other words, there is little chance that the individual will discover a better solution when their has an extremely low fitness value. Therefore, a random sequence with significant fluctuations can aid the person in finding a better answer. There is a high likelihood that the individual will discover a better solution when their fitness score is high. A random sequence with little fluctuation can therefore assist the individual in coming up with a better answer.

• Attenuation rate M. In the CDO algorithm, the attenuation rate M is introduced to reduce the influence of the current optimal solution on the search results.

Fig. 4.

The schematic view of local search strategies.

Global exploration

Search for a promising solution on a global scale is known as global exploration⁴⁴. From Fig. 5, the global exploration in CDO is based on fifth selected individuals, Cauchy random number and scale parameter, which might be stated as:

8

9

10

11

12

Fig. 5.

The schematic view of global search strategies.

where the distribution peak position’s ith position parameters are represented by . A vector containing random numbers is indicated by , which is a number selected at random from the Cauchy random number. indicates the ith the scale parameter, indicates the position of the ith optimal solution, indicates the position of the ith average solution, A number chosen at random that ranges from 0 to 1 is called ., N is the number of individuals, and Zi(t) stands for the ith individual who participated in tth iteration. Zc (t) represents the ith average value of the top half of the fitness ranking. represents the ith Cauchy random number, t represents the number of iterations in this round, The maximum number of iterations is denoted by Max_iter.

• Position parameters of the distribution peak position . This value is randomly selected from the average of the current optimal solution, the second-best solution, the third-best solution, and the first half of the solutions. This ensures that the group converges without prematurely clustering around a single point, thus avoiding getting stuck in a local optimal solution. As can be seen from Fig. 5, the updated position moves forward in the direction determined by , starting from the fixed

  • Scale parameter . Scale parameter is employed to enhance the local search ability of the proposed CDO. Furthermore, according to Eq. (10), scale parameter depends on the current solution and the distance between it and the optimal solution as well as the average solution. If the current value is too far from the optimal solution, the scale parameter will increase, ensuring that the current value can be searched within a larger scale towards the determined range. This can improve the search efficiency. When the current value is close to the optimal solution, the scale parameter is reduced, which ensures that the solution does not exceed the boundary.

It is important to note that the group teaching optimization algorithm (GTOA), CDO, and TLBO have certain characteristics. The population of the next generation is produced by CDO, TLBO, and GTOA using the present ideal scenario and the mean result for the existing population. Furthermore, there is no work required to fine-tune the starting parameters for CDO, TLBO, and GTOA. CDO, TLBO, and GTOA differ significantly in the following ways, though:

• The mathematical models and inspirations of CDO, TLBO, and GTOA differ. The association between the Cauchy distribution rule and the search process of metaheuristics, traditional teaching phenomena, and group teaching phenomena, respectively, served as the inspiration for CDO, TLBO, and GTOA.

• In CDO, TLBO, and GTOA, the present optimal solution and the population mean have distinct functions. As seen in Eq. (4), the generalized mean position of each individual in CDO is generated using the present optimal solution and the mean solution of the current population. By raising the class mean, a teacher can raise the entire class to their level. In TLBO, the mean solution of the present population and the current optimal solution are introduced to the teacher phase. The entire population is split into two groups in GTOA: the average group and the remarkable group. During the teacher phase, the mean solution of the current population and the current optimal solution are used. By raising the mean of the entire outstanding group, a teacher can raise students from the exceptional group to their level.

• These three algorithms also differ in how their answers update their positions.

The implementation of the proposed method for optimization

This section presents the CDO implementation. The planned local exploitation and global exploration tactics serve as the foundation for the suggested CDO. These two strategies are of the same importance to CDO, and the probability of being selected needs to be adjusted dynamically. Moreover, like other population-based optimization algorithms, the population in CDO is initialized by Eq. (3). In this case, the random number is compared to the cumulative probability function P of the Cauchy distribution. If the random number is smaller than P, Eq. (4) is executed; otherwise, Eq. (8) is executed. The specific probability function P is shown as Eq. (13).

13

where equal to 0, equal to 1, t represents the number of iterations in this round. As seen in Fig. 6 in particular, the value of the Cauchy random number varies with the quantity of repetitions.

Fig. 6.

Trends of the P.

Algorithm 1 provides a summary of the CDO algorithm’s whole procedure.

Algorithm 1.

Algorithm 1 Cauchy Distribution Optimizer.

Time complexity of CDO

This subsection examines the Cauchy distribution optimizer’s (CDO) time complexity. Keep in mind that initialization, individual position updates, fitness evaluation, and fitness sorting make up the majority of the CDO algorithm. The temporal complexity of each component is shown as follows: initialization O(N ∗ Dim), individual position update O(N ∗ Dim), fitness evaluation O(N), and fitness sorting O(N ∗ logN). Among these, the dimensionality of the solution space is denoted by Dim. The following can then be used to estimate the temporal complexity: O(N ∗ Dim + N ∗ tmax ∗ (logN + Dim + 1)).

Results and analysis of experiments

This section validates the Cauchy distribution optimizer’s (CDO) strength using the CEC2017 test suite, real-world restricted optimization problems, and photovoltaic system parameter extraction. The following five components make up the entire experiment: (1) benchmark functions and experimental configurations; (2) CEC2017 test suite trials; (3) real-world restricted optimization problems; and (4) application to solar system parameter extraction.

Experimental configurations and benchmark functions

29 benchmark problems with dimensions of 10, 30, 50, and 100 were selected from the CEC2017 test set for experimental research. The 29 CEC 2017 unconstrained benchmark tests are classified in Table 1 based on their characteristics.

Table 1.

CEC2017 benchmark functions.

	No	Function		Minimum	Range
Unimodal functions	C20171	Shifted and Rotated Bent Cigar Function		100	[– 100,100]
	C20173	Shifted and Rotated Zakharov Function		300
Simple multimodal functions	C20174	Shifted and Rotated Rosenbrock’s Function		400
	C20175	Shifted and Rotated Rastrigin’s Function		500
	C20176	Shifted and Rotated Expanded Scaffer’s Function		600
	C20177	Shifted and Rotated Lunacek Bi_Rastrigin Function		700
	C20178	Shifted and Rotated Non-Continuous Rastrigin’s Function		800
	C20179	Shifted and Rotated Levy Function		900
	C201710	Shifted and Rotated Schwefel’s Function		1000
Hybrid functions	C201711	Hybrid-Function-1 (N = 3)		1100
	C201712	Hybrid-Function-2 (N = 3)		1200
	C201713	Hybrid-Function-3 (N = 3)		1300
	C201714	Hybrid-Function-4 (N = 4)		1400
	C201715	Hybrid-Function-5 (N = 4)		1500
	C201716	Hybrid-Function-6 (N = 4)		1600
	C201717	Hybrid-Function-6 (N = 5)		1700
	C201718	Hybrid-Function-6 (N = 5)		1800
	C201719	Hybrid-Function-6 (N = 5)		1900
Composition functions	C201720	Hybrid-Function-6 (N = 6)		2000
	C201721	Composition-Function-1 (N = 3)		2100
	C201722	Composition-Function-2 (N = 3)		2200
	C201723	Composition-Function-3 (N = 4)		2300
	C201724	Composition-Function-4 (N = 4)		2400
	C201725	Composition-Function-5 (N = 5)		2500
	C201726	Composition-Function-6 (N = 5)		2600
	C201727	Composition-Function-7 (N = 6)		2700
	C201728	Composition-Function-8 (N = 6)		2800
	C201729	Composition-Function-9 (N = 3)		2900
	C201730	Composition-Function-10 (N = 3)		3000

Because unimodal problems only have one global optimum, as shown in Table 1, they are frequently used to validate an algorithm’s exploitation capacity. However, fundamental multimodal issues are more challenging to solve than unimodal ones since there are several local optima. Furthermore, because hybrid and composition issues have distinct features at various locations, they are probably going to more closely resemble the actual search space. Consequently, the potential performance of methods to solve real-world optimization issues may be reflected by employing these standard benchmarks.

We use 50 search agents in our tests, and the termination threshold is 500 total iterations. Additionally, seven cutting-edge rivals are selected to compare with the Cauchy distribution optimizer (CDO). These include the Equilibrium Optimizer (EO), Grey Wolf Optimizer (GWO), Mayfly Algorithm (MA), Particle Swarm Optimization (PSO), Loin Swarm Optimization (LSO), Harris Hawk optimization (HHO), Whale Optimization Algorithm (WOA), and Marine Predators Algorithm (MPA). As shown in Table 2, the parameter configuration is maintained in accordance with the related references. Each algorithm is run separately 20 times on each test problem in order to reduce the impact of randomness on the experiment. All algorithms are implemented using MATLAB (Matrix Laboratory) R2018b on a 64-bit Windows 11 computer. The software can be obtained from the official website. The best algorithm’s result is indicated in bold.

Table 2.

Settings for comparison algorithms’ parameters.

Algorithms	Parameter	Value
MA	The male mayfly’s attractive factor
	The female mayfly’s attract factor
	The male mayfly’s eyesight factor
	Factor of motion
LSO	The likelihood of a lion baby’s actions
	The disturbance factor that influences lioness motion variety
	The disturbance factor that impacts the movement distance of the young lions
GWO	Convergence factor	Linearly decreasing from 2 to 0
WOA	Convergence parameter	Linearly decreasing from 2 to 0
	A randomly matrix
	Random digit
EO	The factor of exploration amount
	The factor of exploitation amount
MPA	Fish collection apparatus
HHO	Flight constant
CDO	Attenuation rate	M = 0.3

Test conducted using the CEC2017 test suite

Convergence behavior of CDO

By using multiple CEC2017 benchmark tests, this section investigates the convergence behavior of CDO in a two-dimensional parameter space. In particular, the searching history, trajectory graph, and convergence graph in the test’s first and second dimensions show the convergence actions of CDO. As shown in Fig. 7, The flat structure of unimodal issues like C2017₁ as well as C2017₃ is shown in the first column, which defines the parameter space. In contrast, complicated mixed problems accurately reflect the real answer area, whereas simple multimodal problems have several local optimal solutions.

Fig.7.

Convergence behavior of CDO.

The most popular indicator for confirming the effectiveness of metaheuristic methods is the convergence curve in the subsequent column. The method exhibits a quick rate of convergence on all eight benchmarks, according to the convergence graphs produced by CDO, as shown in Fig. 7. Because of individual interaction and learning, CDO provides a valuable exploitative feature to get closer to the worldwide optimal for unimodal situations. When managing straightforward hybrid and multimodal situations, with the help of the swarm’s elites, CDO occasionally momentarily enters the local optimal state, but the algorithm reaches an exceptionally high degree of accuracy. The third and fourth columns, respectively, display the trajectory diagram in first and second dimensions. Each person’s position history during the iteration process is then visually shown in the searching history graph’s fifth column. CDO recognizes a significant trade-off among the two processes, as evidenced by people’s tendency to choose attractive and promising locations in early rounds before ultimately clustering around the greatest potential worldwide. The CDO has long concentrated on using the left portion of the solution space, particularly for the hybrid issue C2017₂₄, even though the right portion of the solution space yields the greatest results. This really illustrates the improved exploration capability of the developed technique, which aids in avoiding local optima and maintains the swarm’s variety.

Statistical results

In this section, 10 dimensional, 30 dimensional, 50 dimensional, and 100 dimensional CEC2017 test sets were used to compare the performance of CDO with MPA, GWO, MA, PSO, LSO-HHO, WOA, and EO. Tables 3, 4, 5, and 6 describe the simulation results obtained through eight methods. Based on the mean and standard deviation, the following content has been explained:

• When the dimensions of the unimodal problems C2017₁ and C2017₃ are 10, the CDO algorithm performs better than all competing techniques. Specifically, CDO can directly find the global optimal solution of function C2017₃, which other algorithms cannot achieve. It should be pointed out that although WOA also achieved the best results on C2017₃, CDO performed better than them in terms of standard deviation. When the dimensions increased to 30, 50, and 100, the performance of CDO did not significantly decrease.

• For basic multimodal problems, in the case of 10 dimensions, CDO performs better than MPA, GWO, MA, PSO, LSO, HHO, WOA, and EO in 7, 6, 7, 7, 7, and 7 tasks (C2017₄-C2017₁₀). In the case of 30, 50, and 100 dimensions, CDO performs better than MPA, GWO, MA, PSO, LSO, HHO, WOA, and EO in 7, 6, 7, 7, 7, and 7 problems, respectively. CDO achieves the same advantages as dim = 10. It can be seen that in simple multimodal problems, CDO performs equally well in different dimensions.

• For hybrid problems C2017₁₁–C2017₂₀, in the case of 10Dim, CDO outperforms MPA, GWO, MA, PSO, LSO, HHO, WOA, and EO on 7, 10, 10, 9, 10, 9, and 9 problems, respectively. In the cases of 30, 50 and 100 dim, the performance of CDO also gains the same advantages with the increase of the dimension.

• For composition problems C2017₂₁–C2017₃₀, in the case of 10Dim, CDO outperforms MPA, GWO, MA, PSO, LSO, HHO,WOA, and EO on 9, 10, 9, 10, 10, 10,and 8 problems, respectively. In the cases of 30, 50 and 100 dim, the performance of CDO also gains the same advantages with the increase of the dimension.

Table 3.

Experimental results of eight algorithms on the CEC2017 test set (dimension 10).

No	Metrics	CDO	MA	LSO	GWO	HHO	WOA	EO	MPA
Fun1	Mean	2.50E+03	1.18E+07	3.36E+08	6.38E+07	1.86E+10	1.02E+07	8.78E+09	3.07E+05
	Std	2.51E+03	4.17E+06	2.32E+08	1.47E+08	3.84E+10	8.48E+06	1.70E+09	1.24E+06
Fun3	Mean	3.00E+02	1.46E+04	5.18E+03	1.33E+03	2.05E+04	3.00E+03	1.32E+04	7.11E+03
	Std	7.53E-01	4.87E+03	1.98E+03	1.42E+03	1.61E+03	1.46E+02	1.92E+03	3.29E+03
Fun4	Mean	4.09E+02	4.11E+02	4.37E+02	4.15E+02	2.34E+03	4.52E+02	9.50E+02	4.45E+02
	Std	1.94E+01	1.40E+01	2.59E+01	1.33E+01	1.11E+03	4.41E+01	1.18E+02	4.34E+01
Fun5	Mean	5.25E+02	5.33E+02	5.34E+02	5.20E+02	6.52E+02	5.54E+02	6.07E+02	5.39E+02
	Std	8.22E+00	5.93E+00	8.59E+00	8.48E+00	2.28E+01	1.85E+01	1.29E+01	1.17E+01
Fun6	Mean	6.00E+02	6.02E+02	6.19E+02	6.01E+02	6.80E+02	6.37E+02	6.53E+02	6.19E+02
	Std	2.54E-01	5.36E-01	3.43E+00	1.63E+00	1.15E+01	1.15E+01	6.68E+00	1.12E+01
Fun7	Mean	7.33E+02	7.47E+02	7.70E+02	7.31E+02	8.69E+02	7.93E+02	8.22E+02	7.51E+02
	Std	1.04E+01	5.33E+00	1.62E+01	1.07E+01	1.12E+01	2.20E+01	1.11E+01	2.03E+01
Fun8	Mean	8.16E+02	8.31E+02	8.33E+02	8.09E+02	8.88E+02	8.45E+02	8.57E+02	8.49E+02
	Std	4.95E+00	5.88E+00	7.91E+00	9.49E+00	6.67E+00	1.17E+01	5.09E+00	1.43E+01
Fun9	Mean	9.22E+02	9.03E+02	1.05E+03	9.13E+02	2.43E+03	1.45E+03	1.66E+03	1.39E+03
	Std	4.95E+01	1.27E+00	9.79E+01	1.87E+01	1.35E+02	2.98E+02	1.39E+02	2.50E+02
Fun10	Mean	1.73E+03	2.27E+03	2.31E+03	1.68E+03	2.94E+03	2.22E+03	2.44E+03	1.82E+03
	Std	2.56E+02	1.91E+02	1.42E+02	3.66E+02	9.84E+01	2.81E+02	1.36E+02	2.41E+02
Fun11	Mean	1.12E+03	8.28E+03	2.10E+03	8.03E+05	4.34E+04	1.49E+03	2.24E+03	6.69E+03
	Std	9.67E+00	1.40E+04	1.08E+03	7.51E+05	1.86E+04	6.32E+02	6.52E+02	5.42E+03
Fun12	Mean	5.49E+04	3.25E+06	6.22E+06	8.94E+05	1.43E+09	2.99E+06	1.11E+08	4.39E+06
	Std	5.21E+04	2.52E+06	6.86E+06	1.29E+06	6.67E+08	3.18E+06	4.79E+07	4.10E+06
Fun13	Mean	8.33E+03	1.04E+04	1.70E+04	1.96E+04	1.21E+08	1.61E+04	1.77E+06	1.04E+04
	Std	5.76E+03	5.95E+03	2.14E+04	2.08E+04	1.29E+08	1.06E+04	1.66E+06	1.11E+04
Fun14	Mean	4.65E+03	1.82E+03	1.96E+03	3.03E+03	1.78E+03	2.29E+03	1.60E+03	2.33E+03
	Std	5.13E+03	4.56E+02	5.65E+02	1.86E+03	2.60E-01	1.13E+03	4.59E+01	1.15E+03
Fun15	Mean	4.89E+03	3.73E+03	7.67E+03	4.29E+03	1.21E+04	1.00E+04	8.13E+03	5.94E+03
	Std	6.49E+03	2.24E+03	5.20E+03	1.87E+03	2.60E+01	6.91E+03	1.72E+03	7.13E+03
Fun16	Mean	1.75E+03	1.68E+03	1.79E+03	1.68E+03	2.50E+03	1.94E+03	2.12E+03	1.76E+03
	Std	1.27E+02	3.68E+01	6.90E+01	6.91E+02	2.54E+02	1.48E+02	6.66E+01	1.38E+02
Fun17	Mean	1.75E+03	1.77E+03	1.78E+03	1.76E+03	2.11E+03	1.81E+03	1.85E+03	1.80E+03
	Std	4.56E+01	8.42E+00	1.46E+01	2.41E+01	8.17E+01	5.61E+01	2.75E+01	4.74E+01
Fun18	Mean	6.13E+03	3.02E+04	9.69E+04	3.39E+04	4.19E+08	1.59E+04	4.56E+07	3.60E+04
	Std	5.11E+03	1.46E+04	1.87E+05	2.63E+04	4.17E+08	1.32E+04	4.45E+07	1.53E+04
Fun19	Mean	1.18E+04	4.06E+03	1.48E+04	7.42E+03	2.73E+07	4.74E+04	4.12E+05	1.36E+04
	Std	1.07E+04	1.98E+03	5.51E+03	6.11E+03	6.13E+07	9.82E+04	4.06E+05	1.27E+04
Fun20	Mean	2.11E+03	2.32E+03	2.17E+03	2.06E+03	2.43E+03	2.18E+03	2.25E+03	2.13E+03
	Std	7.39E+01	3.09E+01	4.04E+01	2.97E+01	9.62E+01	8.01E+01	3.71E+01	5.81E+01
Fun21	Mean	2.31E+03	2.32E+03	2.28E+03	2.32E+03	2.43E+03	2.36E+03	2.30E+03	2.21E+03
	Std	3.78E+01	3.09E+01	4.68E+01	2.73E+01	2.59E+01	3.58E+01	3.58E+01	1.17E+00
Fun22	Mean	2.23E+03	2.31E+03	2.36E+03	2.30E+03	3.77E+03	2.36E+03	2.88E+03	2.31E+03
	Std	1.51E+01	1.29E+01	2.84E+01	2.75E+01	5.70E+02	2.19E+02	1.47E+02	1.06E+01
Fun23	Mean	2.63E+03	2.63E+03	2.65E+03	2.65E+03	2.80E+03	2.66E+03	2.72E+03	2.64E+03
	Std	1.05E+01	2.40E+01	8.84E+00	7.51E+00	1.36E+01	2.07E+01	1.12E+01	1.40E+01
Fun24	Mean	2.74E+03	2.77E+03	2.77E+03	2.75E+03	2.95E+03	2.78E+03	2.91E+03	2.73E+03
	Std	5.80E+01	5.50E+00	1.87E+01	1.17E+01	9.90E+01	4.66E+01	8.11E+01	8.31E+01
Fun25	Mean	2.94E+03	2.95E+03	2.97E+03	2.94E+03	4.07E+03	2.96E+03	3.40E+03	2.98E+03
	Std	1.07E+01	1.52E+01	1.18E+01	1.29E+01	4.63E+02	3.44E+02	2.03E+02	3.25E+01
Fun26	Mean	3.12E+03	2.96E+03	3.16E+03	3.00E+03	4.60E+03	3.36E+03	3.95E+03	3.01E+03
	Std	1.51E+02	3.65E+01	5.27E+01	1.10E+02	4.19E+02	4.22E+02	1.70E+02	6.73E+01
Fun27	Mean	3.11E+03	3.10E+03	3.11E+03	3.08E+03	3.36E+03	3.15E+03	3.20E+03	3.10E+03
	Std	2.09E+01	2.27E+00	3.13E+00	1.56E+01	8.80E+01	4.45E+01	2.92E+01	4.24E+00
Fun28	Mean	3.28E+03	3.36E+03	3.38E+03	3.28E+03	3.86E+03	3.42E+03	3.77E+03	3.34E+03
	Std	1.09E+01	7.19E+01	6.12E+01	1.10E+01	9.26E+01	1.47E+02	4.82E+01	8.13E+01
Fun29	Mean	3.15E+03	3.33E+03	3.25E+03	3.19E+03	3.83E+03	3.32E+03	3.46E+03	3.24E+03
	Std	7.59E+01	3.75E+01	2.89E+01	4.59E+01	1.30E+02	9.65E+01	4.61E+01	6.49E+01
Fun30	Mean	5.46E+05	8.16E+05	9.72E+05	4.93E+04	8.28E+08	8.10E+05	5.90E+06	6.35E+05
	Std	6.42E+05	5.62E+05	4.49E+05	3.59E+03	4.29E+07	8.41E+05	2.28E+06	7.56E+04

Table 4.

Experimental results of eight algorithms on the CEC2017 test set (dimension 30).

No	Metrics	CDO	MA	LSO	GWO	HHO	WOA	EO	MPA
Fun1	Mean	2.66E+03	1.86E+09	1.48E+10	3.74E+09	7.69E+10	2.15E+09	5.24E+10	8.63E+09
	Std	2.87E+03	4.05E+09	2.65E+09	2.35E+09	3.48E+09	9.01E+08	5.216E+09	4.92E+09
Fun3	Mean	7.87E+04	1.77E+05	7.11E+04	5.98E+04	9.42E+04	2.55E+05	1.33E+04	7.11E+03
	Std	3.36E+04	2.79E+04	8.43E+03	1.14E+04	1.25E+02	8.34E+04	8.68E+04	3.30E+03
Fun4	Mean	4.94E+02	4.12E+02	1.72E+03	6.76E+02	2.33E+03	4.52E+02	1.31E+04	9.54E+02
	Std	2.31E+01	1.41E+01	3.95E+02	1.81E+02	2.20E+04	1.00E+03	9.63E+02	3.80E+02
Fun5	Mean	6.00E+02	7.27E+02	8.01E+02	6.09E+02	1.01E+03	8.03E+02	9.39E+02	7.81E+02
	Std	2.98E+01	1.35E+01	2.28E+01	2.95E+01	1.87E+01	4.01E+01	1.50E+01	4.67E+01
Fun6	Mean	6.02E+02	6.20E+02	6.65E+02	6.09E+02	7.12E+02	6.78E+02	7.00E+02	6.63E+02
	Std	1.82E+00	2.54E+00	6.50E+00	5.29E+00	2.27E+00	2.13E+01	2.99E+00	8.64E+00
Fun7	Mean	8.60E+02	1.06E+03	1.20E+03	8.85E+02	1.57E+03	1.27E+03	1.50E+03	1.59E+03
	Std	3.642E+01	2.61E+01	5.29E+01	5.40E+01	3.42E+01	3.51E+02	2.70E+01	1.54E+02
Fun8	Mean	8.61E+02	1.04E+03	1.06E+03	9.06E+02	1.23E+03	1.15E+03	1.16E+03	1.10E+03
	Std	2.11E+01	1.73E+01	2.59E+01	3.19E+01	1.21E+01	6.84E+01	1.24E+01	6.48E+01
Fun9	Mean	1.78E+03	2.28E+03	7.26E+03	2.54E+03	1.62E+04	9.88E+03	1.17E+04	8.27E+03
	Std	6.46E+02	5.04E+02	1.17E+03	1.20E+03	1.92E+03	4.05E+03	8.00E+02	1.23E+03
Fun10	Mean	4.40E+03	8.70E+03	8.89E+03	5.56E+03	1.31E+04	7.03E+03	9.08E+03	5.60E+03
	Std	5.51E+02	3.13E+02	4.14E+02	1.27E+03	3.28E+02	8.52E+02	3.06E+02	7.45E+02
Fun11	Mean	2.16E+03	9.69E+05	9.01E+03	6.06E+05	1.62E+04	2.40E+05	7.73E+03	4.40E+05
	Std	2.69E+03	2.80E+05	3.29E+03	4.51E+05	2.83E+02	2.30E+05	1.13E+03	1.83E+05
Fun12	Mean	1.21E+06	9.84E+07	1.59E+09	1.02E+08	2.23E+10	3.27E+08	1.13E+10	1.45E+08
	Std	7.87E+05	4.48E+07	5.22E+08	1.14E+08	3.99E+09	2.71E+08	2.45E+09	1.32E+08
Fun13	Mean	2.12E+04	1.73E+07	2.81E+08	1.48E+06	2.03E+10	4.47E+06	5.85E+09	1.56E+07
	Std	2.23E+04	9.89E+06	1.32E+08	2.14E+06	6.46E+09	5.38E+06	1.42E+09	3.50E+07
Fun14	Mean	4.26E+05	4.67E+05	9.76E+05	4.54E+05	4.12E+08	1.76E+06	3.95E+06	6.04E+05
	Std	4.02E+05	5.34E+05	4.95E+05	6.38E+05	2.89E+07	2.50E+06	1.86E+06	6.19E+05
Fun15	Mean	8.45E+03	3.68E+06	1.14E+07	8.27E+05	3.38E+09	1.39E+06	6.25E+08	7.96E+04
	Std	1.07E+04	3.80E+06	1.20E+07	1.68E+06	9.79E+08	1.71E+06	2.29E+08	2.61E+04
Fun16	Mean	2.72E+03	3.28E+03	3.70E+03	2.83E+03	8.18E+03	4.12E+03	5.99E+03	3.13E+03
	Std	3.38E+02	3.09E+02	3.05E+02	4.66E+02	1.62E+03	4.90E+02	4.69E+02	3.68E+02
Fun17	Mean	2.22E+03	2.50E+03	2.58E+03	2.09E+03	2.16E+04	2.69E+03	4.30E+03	2.62E+03
	Std	2.09E+02	1.72E+02	2.24E+02	1.51E+02	2.68E+03	3.69E+02	4.91E+02	2.87E+02
Fun18	Mean	1.68E+06	7.59E+06	4.71E+06	2.95E+06	4.80E+08	1.09E+07	4.17E+07	3.16E+06
	Std	1.31E+06	4.66E+06	4.39E+06	6.58E+04	4.44E+08	1.06E+07	1.98E+07	3.82E+06
Fun19	Mean	8.96E+03	9.39E+06	1.40E+07	7.29E+05	3.11E+09	1.69E+07	4.15E+08	9.34E+05
	Std	7.64E+03	8.31E+06	1.64E+07	8.11E+05	3.87E+08	1.18E+07	1.15E+08	7.72E+04
Fun20	Mean	2.46E+03	2.77E+03	2. 60E+03	2.45E+03	3.59E+03	2.88E+03	3.08E+03	2.83E+03
	Std	1.94E+02	1.59E+02	1.05E+02	1.36E+02	1.08E+03	2.24E+02	1.20E+02	3.05E+02
Fun21	Mean	2.39E+03	2.53E+03	2.56E+03	2.42E+03	2.89E+03	2.62E+03	2.75E+03	2.56E+03
	Std	1.87E+01	1.21E+01	2.53E+01	3.92E+01	5.79E+01	5.49E+01	3.28E+01	3.09E+01
Fun22	Mean	3.42E+03	2.69E+03	7.47E+03	6.97E+03	1.15E+04	7.39E+03	9.39E+03	6.39E+03
	Std	1.82E+03	1.27E+01	2.02E+03	2.51E+03	3.87E+02	3.01E+03	4.09E+02	1.09E+03
Fun23	Mean	2.78E+03	2.88E+03	3.00E+03	2.79E+03	3.82E+03	3.13E+03	3.59E+03	2.93E+03
	Std	3.63E+01	1.39E+01	3.29E+01	4.69E+01	1.69E+02	1.17E+02	1.07E+02	5.49E+01
Fun24	Mean	2.94E+03	3.04E+03	3.19E+03	3.01E+03	4.19E+03	3.25E+03	3.76E+03	3.07E+03
	Std	3.47E+01	1.95E+01	5.21E+01	6.99E+01	1.94E+02	9.07E+01	1.30E+02	5.28E+02
Fun25	Mean	2.90E+03	3.03E+03	3.37E+03	3.01E+03	6.67E+03	3.137E+03	4.666E+03	3.603E+03
	Std	1.84E+01	4.97E+01	7.60E+01	6.01E+01	3.53E+02	6.207E+01	1.455E+02	3.294E+02
Fun26	Mean	4.74E+03	5.99E+03	7.39E+03	4.75E+03	1.44E+04	8.38E+03	1.13E+04	6.87E+03
	Std	8.15E+02	1.19E+02	4.02E+02	4.15E+02	7.08E+01	7.17E+02	4.06E+02	9.26E+02
Fun27	Mean	3.25E+03	3.25E+03	3.53E+03	3.20E+03	5.46E+03	3.48E+03	4.33E+03	3.32E+03
	Std	2.20E+01	1.07E+01	6.71E+01	4.39E-04	4.19E+02	1.94E+02	2.31E+02	5.56E+01
Fun28	Mean	3.22E+03	3.53E+03	4.71E+03	3.32E+03	9.28E+03	3.61E+03	7.29E+03	5.88E+03
	Std	1.94E+01	8.36E+01	2.84E+02	4.77E+01	1.82E+02	1.06E+02	4.53E+02	1.30E+01
Fun29	Mean	3.93E+03	4.46E+03	5.50E+03	3.71E+03	3.62E+04	5.40E+03	7.09E+03	4.58E+03
	Std	2.07E+02	2.11E+02	3.84E+02	2.195E+02	4.19E+04	4.82E+02	6.08E+02	6.34E+02
Fun30	Mean	1.30E+04	3.97E+06	1.32E+08	3.20E+06	3.37E+09	5.76E+07	1.52E+09	6.31E+06
	Std	6.18E+03	5.59E+05	7.70E+07	8.06E+06	1.38E+09	4.59E+07	4.85E+08	7.89E+06

Table 5.

Experimental results of eight algorithms on the CEC2017 test set (dimension 50).

No	Metrics	CDO	MA	LSO	GWO	HHO	WOA	EO	MPA
Fun1	Mean	5.78E+03	1.79E+10	5.19E+10	1.29E+10	1.29E+11	1.17E+10	1.12E+11	6.00E+10
	Std	6.48E+03	3.09E+09	7.10E+09	6.82E+09	1.87E+09	2.57E+09	3.81E+09	2.04E+10
Fun3	Mean	2.18E+05	3.67E+05	1.58E+05	2.00E+05	2.79E+05	3.09E+05	2.06E+05	3.54E+05
	Std	6.62E+04	3.90E+04	3.92E+04	8.26E+04	9.58E+02	1.30E+05	1.40E+04	6.01E+04
Fun4	Mean	5.52E+02	2.03E+03	8.69E+03	1.93E+03	5.39E+04	3.06E+03	3.59E+04	7.05E+03
	Std	5.64E+01	5.08E+02	2.33E+02	9.01E+02	1.76E+03	9.01E+02	2.35E+03	6.00E+03
Fun5	Mean	6.90E+02	9.99E+02	1.01E+03	7.78E+02	1.30E+03	1.11E+03	1.20E+03	1.06E+03
	Std	3.59E+01	3.05E+01	4.79E+01	5.29E+01	1.10E+01	9.14E+01	3.05E+01	7.01E+01
Fun6	Mean	6.08E+02	6.37E+02	6.85E+02	6.30E+02	7.07E+02	6.92E+02	7.01E+02	6.70E+02
	Std	4.63E+00	7.16E+00	6.29E+00	8.19E+00	1.08E+00	9.04E+00	4.38E+00	9.12E+00
Fun7	Mean	9.97E+02	1.72E+03	1.73E+03	1.13E+03	2.16E+03	1.79E+03	2.09E+03	2.80E+03
	Std	5.45E+01	1.19E+02	6.08E+01	9.03E+01	1.60E+01	1.02E+02	1.54E+01	2.34E+02
Fun8	Mean	1.00E+03	1.29E+03	1.40E+03	1.08E+03	1.59E+03	1.37E+03	1.51E+03	1.37E+03
	Std	3.95E+01	3.03E+01	3.02E+01	8.55E+01	1.48E+01	6.84E+01	1.40E+01	8.02E+01
Fun9	Mean	3.12E+03	1.02E+04	2.87E+04	1.08E+04	5.09E+04	3.62E+04	3.94E+04	2.16E+04
	Std	1.45E+03	2.53E+03	3.12E+03	5.04E+03	1.83E+03	1.03E+04	1.70E+03	5.53E+03
Fun10	Mean	6.49E+03	1.52E+04	1.49E+04	9.89E+03	1.70E+04	1.27E+04	1.55E+04	9.57E+03
	Std	9.13E+02	5.30E+02	6.11E+02	3.10E+03	4.244E+02	1.23E+03	3.63E+02	8.86E+02
Fun11	Mean	1.78E+04	3.96E+06	2.31E+04	8.18E+06	3.67E+04	8.49E+03	2.33E+05	1.43E+06
	Std	2.98E+04	1.17E+06	4.08E+03	8.04E+05	6.13E+03	3.54E+03	2.09E+03	7.52E+05
Fun12	Mean	3.43E+06	1.59E+09	1.00E+10	3.34E+09	1.23E+11	2.10E+09	8.19E+10	1.88E+09
	Std	2.24E+06	5.06E+08	3.02E+09	2.58E+09	1.66E+10	8.07E+08	7.03E+09	9.91E+08
Fun13	Mean	6.64E+03	1.67E+08	2.07E+09	1.72E+08	6.86E+10	2.37E+08	4.68E+10	1.74E+08
	Std	5.51E+03	6.29E+06	9.38+08	1.34E+08	1.94E+10	1.71E+08	9.08E+09	2.09E+08
Fun14	Mean	1.02E+06	3.22E+06	7.80E+06	2.27E+06	3.00E+08	4.55E+06	6.99E+07	4.40E+06
	Std	9.73E+05	2.09E+06	4.48E+06	3.04E+06	1.49E+08	3.83E+06	3.14E+07	4.30E+06
Fun15	Mean	1.12E+04	3.07E+07	4.28E+08	9.60E+06	1.89E+10	1.67E+07	7.09E+09	2.20E+06
	Std	7.20E+03	2.00E+07	2.02E+08	1.24E+07	5.34E+09	1.71E+07	1.30E+09	1.19E+06
Fun16	Mean	3.49E+03	5.28E+03	5.48E+03	3.43E+03	1.39E+04	6.18E+03	9.36E+03	4.51E+03
	Std	4.38E+02	3.71E+02	4.89E+02	3.69E+02	2.63E+03	7.70E+02	7.47E+02	5.99E+02
Fun17	Mean	2.94E+03	4.23E+03	4.50E+03	2.91E+03	1.03E+05	4.51E+03	9.66E+03	4.48E+03
	Std	3.74E+02	3.05E+02	4.18E+02	2.83E+02	2.95E+04	4.08E+02	2.17E+02	5.14E+02
Fun18	Mean	3.50E+06	2.71E+07	3.59E+07	1.58E+07	7.80E+08	4.11E+07	1.64E+08	1.28E+07
	Std	2.53E+06	1.13E+07	1.29E+07	1.69E+07	2.47E+08	3.13E+07	3.37E+07	7.29E+06
Fun19	Mean	2.04E+04	1.89E+07	1.69E+08	1.00E+07	8.86E+09	6.32E+07	3.32E+09	3.43E+06
	Std	7.13E+03	1.09E+07	9.41E+07	2.06E+07	1.50E+09	6.65E+07	8.50E+08	4.67E+06
Fun20	Mean	3.08E+03	4.17E+03	3.77E+03	3.26E+03	4.73E+03	4.00E+03	4.30E+03	3.51E+03
	Std	3.63E+02	1.99E+02	2.17E+02	5.20E+02	4.96E+01	4.02E+02	1.68E+02	3.47E+02
Fun21	Mean	2.48E+03	2.80E+03	2.90E+03	2.57E+03	3.47E+03	3.00E+03	3.24E+03	2.87E+03
	Std	3.58E+01	2.47E+01	4.69E+01	5.70E+01	1.06E+02	9.81E+01	3.62E+01	9.32E+01
Fun22	Mean	8.11E+03	1.68E+04	1.69E+04	1.22E+04	1.88E+04	1.46E+04	1.68E+04	1.16E+04
	Std	1.71E+03	3.39E+02	4.58E+02	4.05E+03	4.09E+02	9.23E+02	3.91E+02	8.79E+02
Fun23	Mean	3.00E+03	3.24E+03	3.50E+03	3.05E+03	4.95E+03	3.86E+03	4.43E+03	3.40E+03
	Std	6.92E+01	1.97E+01	6.00E+01	8.92E+01	1.99E+02	2.12E+02	1.10E+02	9.75E+01
Fun24	Mean	3.17E+03	3.38E+03	3.73E+03	3.36E+03	5.59E+03	3.80E+03	4.93E+03	3.58E+03
	Std	6.97E+01	4.11E+01	7.93E+01	1.11E+02	4.55E+02	1.62E+02	3.08E+02	1.19E+02
Fun25	Mean	3.10E+03	4.43E+03	7.76E+03	3.99E+03	1.89E+04	4.45E+03	1.48E+04	9.88E+03
	Std	2.61E+01	3.63E+02	6.81E+02	4.50E+02	4.54E+02	3.79E+02	8.01E+02	2.60E+02
Fun26	Mean	6.64E+03	8.88E+03	1.30E+04	7.15E+03	1.93E+04	1.48E+04	1.68E+04	1.06E+04
	Std	1.56E+03	2.85E+02	9.59E+02	6.89E+02	1.14E+02	1.60E+03	4.43E+02	8.74E+02
Fun27	Mean	3.52E+03	3.60E+03	4.97E+03	3.26E+03	8.70E+03	4.43E+03	7.02E+03	3.87E+03
	Std	1.09E+02	6.51E+01	4.03E+02	2.74E+01	1.03E+03	4.28E+02	4.29E+02	2.26E+02
Fun28	Mean	3.36E+03	6.56E+03	8.25E+03	3.62E+03	1.91E+04	5.53E+03	1.32E+04	9.44E+03
	Std	3.23E+01	1.22E+03	7.18E+02	7.86E+02	7.36E+02	4.27E+02	4.48E+02	1.735E+03
Fun29	Mean	4.32E+03	5.98E+03	9.85E+03	4.40E+03	1.07E+06	8.88E+03	4.88E+04	6.86E+03
	Std	3.83E+02	3.92E+02	1.05E+03	4.58E+02	9.59E+05	2.06E+03	2.06E+04	1.01E+03
Fun30	Mean	9.27E+05	1.20E+08	5.01E+08	1.50E+07	1.46E+10	3.22E+08	6.99E+09	5.72E+07
	Std	1.24E+05	4.31E+07	1.61E+08	1.50E+07	4.51E+09	1.35E+08	1.37E+09	2.91E+07

Table 6.

Experimental results of eight algorithms on the CEC2017 test set (dimension 100).

No	Metrics	CDO	MA	LSO	GWO	HHO	WOA	EO	MPA
Fun1	Mean	5.44E+06	1.24E+11	1.77E+11	6.74E+10	2.90E+11	7.84E+10	2.68E+11	2.38E+11
	Std	4.73E+06	1.09E+10	1.02E+10	1.16E+10	4.34E+09	8.924E+09	4.185E+09	2.228E+10
Fun3	Mean	5.88E+05	8.50E+05	3.62E+05	1.02E+06	3.67E+05	9.83E+05	3.55E+05	9.17E+05
	Std	1.13E+05	6.09E+04	3.04E+04	3.11E+05	1.58E+02	1.19E+05	8.91E+03	1.98E+05
Fun4	Mean	8.48E+02	1.71E+04	3.08E+04	8.64E+03	1.52E+05	1.34E+04	1.04E+05	4.92E+04
	Std	5.33E+01	3.49E+03	3.88E+03	2.40E+03	5.60E+03	2.85E+03	7.15E+03	1.48E+04
Fun5	Mean	9.71E+02	1.82E+03	1.98E+03	1.30E+03	2.28E+03	1.83E+03	2.12E+03	2.00E+03
	Std	5.63E+01	5.29E+01	7.06E+01	6.17E+01	2.56E+01	1.16E+01	2.83E+01	1.48E+02
Fun6	Mean	6.19E+02	6.69E+02	7.03E+02	6.47E+02	7.19E+02	7.07E+02	7.15E+02	6.79E+02
	Std	4.83E+00	5.18E+00	5.27E+00	4.02E+00	7.00E-01	1.09E+01	2.09E+00	5.06E+00
Fun7	Mean	1.46E+03	5.52E+03	3.62E+03	2.23E+03	4.20E+03	3.76E+03	4.15E+03	5.73E+03
	Std	1.30E+02	3.31E+02	1.28E+02	1.58E+02	3.67E+01	1.47E+02	6.02E+02	6.29E+02
Fun8	Mean	1.26E+03	2.10E+03	2.34E+03	1.62E+03	2.72E+03	2.29E+03	2.62E+03	2.37E+03
	Std	6.17E+01	6.63E+01	4.13E+01	1.09E+02	1.50E+01	1.12E+02	2.15E+01	2.56E+02
Fun9	Mean	1.31E+04	5.15E+04	7.83E+04	4.55E+04	9.49E+04	7.19E+04	8.22E+04	4.89E+04
	Std	2.82E+03	7.28E+03	5.13E+03	1.61E+04	3.69E+03	1.72E+04	3.72E+03	7.96E+03
Fun10	Mean	1.40E+04	3.29E+04	3.27E+04	2.63E+04	3.38E+04	2.87E+04	3.18E+04	2.03E+04
	Std	1.14E+03	8.04E+02	8.03E+02	6.89E+03	2.96E+02	1.42E+03	7.22E+02	1.60E+03
Fun11	Mean	1.38E+06	8.30E+07	5.28E+05	1.37E+06	5.73E+05	3.69E+07	2.84E+05	1.03E+07
	Std	8.88E+05	2.08E+07	4.00E+05	1.84E+06	3.01E+03	3.47E+07	3.23E+04	3.71E+06
Fun12	Mean	4.26E+07	2.00E+10	7.08E+10	1.88E+10	2.51E+11	1.59E+10	2.01E+11	4.42E+10
	Std	1.35E+07	5.21E+09	2.06E+10	6.13E+09	1.27E+10	3.05E+09	4.26E+09	1.39E+10
Fun13	Mean	1.17E+04	1.40E+09	1.05E+10	2.11E+09	6.12E+10	9.45E+08	4.54E+10	2.35E+09
	Std	7.10E+03	3.81E+08	3.12E+09	1.83E+09	4.00E+09	4.29E+08	3.84E+09	1.50E+09
Fun14	Mean	2.61E+06	5.12E+07	2.01E+07	9.57E+06	2.41E+08	1.71E+07	5.28E+07	3.06E+07
	Std	1.47E+06	1.63E+07	7.63E+06	5.12E+06	6.12E+07	7.05E+06	1.24E+07	1.38E+07
Fun15	Mean	3.65E+03	2.12E+08	2.83E+09	3.46E+08	3.74E+10	1.96E+08	2.51E+10	4.19E+08
	Std	2.20E+03	8.84E+07	2.13E+08	3.79E+08	2.37E+09	1.58E+08	2.61E+09	3.91E+08
Fun16	Mean	5.21E+03	1.16E+04	1.30E+04	8.21E+03	3.28E+04	1.62E+04	2.44E+04	9.29E+03
	Std	6.24E+02	4.41E+02	7.36E+02	2.02E+03	3.76E+03	2.90E+03	9.91E+02	2.52E+03
Fun17	Mean	5.10E+03	8.74E+03	1.91E+04	6.62E+03	3.86E+07	1.45E+04	7.68E+06	1.28E+04
	Std	5.94E+02	2.22E+02	1.58E+04	1.42E+03	2.75E+07	1.23E+04	4.46E+06	1.01E+04
Fun18	Mean	3.41E+06	7.38E+07	2.97E+07	1.19E+07	9.64E+08	1.39E+07	2.00E+08	5.39E+07
	Std	1.60E+06	2.59E+07	1.04E+07	9.69E+06	2.96E+08	7.25E+06	6.12E+07	1.82E+07
Fun19	Mean	5.34E+03	2.80E+08	2.28E+09	3.12E+08	3.69E+10	1.93E+08	2.40E+10	4.82E+08
	Std	5.02E+03	1.62E+08	1.01E+09	2.31E+08	2.99E+09	1.18E+08	1.96E+09	4.22E+08
Fun20	Mean	4.85E+03	8.00E+03	7.23E+03	6.19E+03	9.12E+03	6.87E+03	7.93E+03	6.57E+03
	Std	4.96E+02	3.12E+02	4.84E+02	2.49E+03	3.52E+02	8.34E+02	2.40E+02	4.05E+02
Fun21	Mean	2.84E+03	3.68E+03	3.88E+03	3.23E+03	5.36E+03	4.37E+03	4.86E+03	3.96E+03
	Std	9.66E+01	5.58E+01	1.06E+02	8.06E+01	3.89E+02	2.95E+02	1.20E+02	2.06E+02
Fun22	Mean	1.66E+04	3.50E+04	3.50E+04	2.54E+04	3.77E+04	3.09E+04	3.50E+04	2.36E+04
	Std	1.22E+03	8.56E+02	8.39E+02	6.08E+03	1.74E+02	1.16E+03	4.18E+02	1.27E+03
Fun23	Mean	3.27E+03	4.08E+03	4.96E+03	3.85E+03	8.13E+03	5.16E+03	7.02E+03	4.16E+03
	Std	7.52E+01	5.70E+01	1.91E+02	1.25E+02	6.28E+02	2.29E+02	5.00E+02	1.34E+02
Fun24	Mean	3.91E+03	4.20E+03	6.92E+03	4.68E+03	1.33E+04	6.58E+03	1.08E+04	5.38E+03
	Std	9.08E+01	7.13E+02	2.87E+02	1.31E+02	1.12E+03	4.20E+02	6.40E+02	3.54E+02
Fun25	Mean	3.51E+03	2.14E+04	1.69E+04	7.66E+03	3.37E+04	8.62E+03	2.77E+04	2.95E+04
	Std	5.97E+01	3.12E+03	1.60E+03	9.90E+02	1.24E+03	8.41E+02	8.96E+02	4.76E+03
Fun26	Mean	1.31E+04	2.07E+04	3.47E+04	1.99E+04	5.81E+04	3.64E+04	5.15E+04	2.66E+04
	Std	1.86E+03	1.04E+02	1.41E+03	1.57E+03	6.29E+02	3.02E+03	1.14E+03	2.12E+03
Fun27	Mean	3.59E+03	4.40E+03	8.09E+03	3.21E+03	1.63E+04	6.12E+03	1.38E+04	4.40E+03
	Std	9.16E+01	2.53E+02	7.74E+02	3.18E+01	1.58E+03	1.09E+03	1.06E+03	2.91E+02
Fun28	Mean	3.67E+03	2.03E+04	2.21E+04	4.41E+03	3.31E+04	1.24E+04	2.85E+04	2.53E+04
	Std	4.47E+02	1.55E+03	1.44E+03	2.79E+03	4.52E+02	1.11E+03	6.79E+02	2.41E+03
Fun29	Mean	6.40E+03	1.21E+04	2.55E+04	7.65E+03	5.06E+06	1.90E+04	4.09E+05	1.51E+04
	Std	4.97E+02	7.56E+02	6.28E+03	9.03E+02	1.98E+06	2.96E+03	1.75E+05	4.52E+03
Fun30	Mean	6.27E+04	6.01E+08	9.74E+09	1.18E+09	5.63E+10	1.52E+09	4.04E+10	6.34E+08
	Std	4.44E+04	1.56E+08	2.23E+09	1.52E+09	4.11E+09	6.19E+08	1.89E+09	3.92E+08

These eight methods are subjected to the Wilcoxon test using the 10-dimensional CEC2017 test suites. The obtained p values are often less than 5%, as shown by the test results in Tables 7.The notation " = " indicates that the performance difference between the competitor and CDO algorithms is not statistically significant for each competitive function in this table, " + " indicates that the performance of the suggested algorithm is better, and "-" indicates that the competitor algorithm performs worse than CDO. This suggests that the developed CDO performs statistically better than its rivals.

Table 7.

The Wilcoxon test results of the 10 dimensional CEC2017 test set were compared.

Fun	MA	LSO	GWO	HHO	WOA	EO	MPA
Fun1	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	2.536E-04/-
Fun3	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-
Fun4	1.500E-03/-	8.918E-04/-	1.700E-03/-	8.858E-05/-	7.796E-04/-	8.858E-05/-	5.934E-04/-
Fun5	5.934E-04/-	5.167E-04/-	7.089E-01/=	8.858E-05/-	1.034E-04/-	8.858E-05/-	1.034E-04/-
Fun6	8.858E-05/-	8.858E-05/-	2.932E-04/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-
Fun7	6.806E-04/-	8.858E-05/-	4.330E-01/=	8.858E-05/-	8.858E-05/-	8.858E-05/-	4.500E-03/-
Fun8	1.034E-04/-	8.858E-05/-	5.016E-01/=	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-
Fun9	2.322E-01/=	5.934E-04/-	6.813E-01/=	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-
Fun10	1.034E-04/-	8.858E-05/-	4.781E-01/=	8.858E-05/-	1.890E-04/-	8.858E-05/-	0.550E-01/=
Fun11	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-	8.858E-05/-
Fun12	8.858E-05/-	8.858E-05/-	7.796E-04/-	8.858E-05/-	1.401E-04/-	8.858E-05/-	8.858E-05/-
Fun13	4.330E-01/=	1.913E-01/=	2.510E-02/-	8.858E-05/-	1.000E-02/-	8.858E-05/-	8.228E-01/=
Fun14	4.500E-03/+	6.400E-03/+	5.755E-01/=	2.200E-03/+	7.310E-02/=	3.902E-04/+	1.870E-02/+
Fun15	7.652E-01/=	2.060E-02/-	4.330E-01/=	1.500E-03/-	1.000E-02/-	2.200E-03/-	1.560E-01/=
Fun16	8.000E-03/+	2.627E-01/=	4.000E-02/+	8.858E-05/-	4.493E-04/-	8.858E-05/-	9.405E-01/=
Fun17	1.259E-01/=	7.310E-02/=	1.790E-01/=	8.858E-05/-	7.796E-04/-	1.890E-04/-	1.700E-03/-
Fun18	7.200E-03/-	1.370E-02/-	6.400E-03/-	8.858E-05/-	7.652E-01/=	8.858E-05/-	1.890E-04/-
Fun19	3.600E-03/+	2.959E-01/=	1.454E-01/=	8.858E-05/-	4.380E-02/-	8.858E-05/-	8.813E-01/=
Fun20	8.858E-05/-	3.600E-05/-	5.690E-02/=	1.304E-04/-	1.370E-02/-	1.629E-04/-	2.322E-01/=
Fun21	1.084E-01/=	6.740E-02/=	3.135E-01/=	8.858E-05/-	1.000E-03/-	1.169E-01/=	1.401E-04/+
Fun22	1.110E-02/-	8.858E-05/-	8.590E-02/=	8.858E-05/-	1.200E-03/-	8.858E-05/-	1.700E-03/-
Fun23	6.274E-01/=	1.204E-04/-	1.354E-01/=	8.858E-05/-	3.902E-04/-	8.858E-05/-	3.330E-02/-
Fun24	2.600E-05/-	5.700E-03/-	1.084E-01/=	8.858E-05/-	1.300E-03/-	1.034E-04/-	2.627E-01/=
Fun25	2.180E-01/=	8.858E-05/-	1.454E-01/=	8.858E-05/-	1.084E-02/=	8.858E-05/-	8.858E-05/-
Fun26	2.060E-02/-	7.200E-03/-	2.959E-01/=	8.858E-05/-	9.000E-03/-	1.401E-04/-	3.317E-01/=
Fun27	8.858E-05/+	2.471E-01/=	5.934E-04/+	8.858E-05/-	6.806E-04/-	1.034E-04/-	8.858E-05/+
Fun28	7.310E-02/=	2.760E-02/-	7.369E-01/=	8.858E-05/-	2.500E-03/-	8.858E-05/-	2.043E-01/=
Fun29	2.471E-01/=	8.813E-01/=	2.932E-04/-	8.858E-05/-	4.790E-02/-	8.858E-05/-	9.108E-01/=
Fun30	2.322E-01/=	3.330E-02/-	8.858E-05/-	8.858E-05/-	2.790E-01/=	8.858E-05/-	4.330E-01/=
+/=/−	4/10/15	1/7/21	2/17/10	1/0/28	0/4/25	1/1/27	3/11/15

Friedman’s test was also used to confirm the reliability of credit default swaps (CDOs). After evaluating through these eight algorithms, it was found that CDO scored the highest, as shown in Table 8.

Table 8.

Eight techniques are ranked on 29 unconstrained CEC2017 benchmarks using Friedman’s test.

No	CAO	MA	LSO	GWO	HHO	WOA	EO	MPA
Fun1	1	4	6	5	8	3	7	2
Fun3	1	7	4	2	8	3	6	5
Fun4	1	2	4	3	8	6	7	5
Fun5	1	3	4	2	8	6	7	5
Fun6	1	3	5	2	8	6	7	4
Fun7	1	3	5	2	8	6	7	4
Fun8	1	3	4	2	8	5	7	6
Fun9	1	2	4	3	8	6	7	5
Fun10	2	5	6	1	8	4	7	3
Fun11	1	6	3	8	7	2	4	5
Fun12	1	4	6	2	8	3	7	5
Fun13	1	3	5	6	8	4	7	2
Fun14	8	3	4	7	2	5	1	6
Fun15	2	1	5	3	8	7	6	4
Fun16	3	1	5	2	8	6	7	4
Fun17	1	3	4	2	8	6	7	5
Fun18	1	3	6	4	8	2	7	5
Fun19	3	1	5	2	8	6	7	4
Fun20	2	7	4	1	8	5	6	3
Fun21	3	6	2	5	8	7	4	1
Fun22	1	4	5	2	8	6	7	3
Fun23	1	2	5	4	8	6	7	3
Fun24	2	4	5	3	8	6	7	1
Fun25	1	3	5	2	8	4	7	6
Fun26	4	1	5	2	8	6	7	3
Fun27	5	2	4	1	8	6	7	3
Fun28	1	4	6	2	8	7	5	3
Fun29	1	6	4	2	8	5	7	3
Fun30	2	6	7	1	8	5	3	4
Mean rank	1.862	3.517	4.724	2.862	7.759	5.138	6.276	3.862
Result	1	3	5	2	8	6	7	4

Convergence analysis

The convergence graphs obtained by MPA, GWO, MA, PSO, LSO, HHO, WOA, EO, and CDO on 12 common benchmark problems with 10Dim are shown in Fig. 8. Keep in mind that the two best algorithms for handling unconstrained benchmarks are GWO and MA. Even if GWO and MA outperform it while handling some challenging benchmarks, The CDO method has achieved high convergence speed and accuracy, as seen in Fig. 8. Additionally, CDO offers faster speeds than its rivals, including MPA, PSO, LSO, and HHO.

Fig. 8.

Convergence graphs obtained by eight methods on twelve 10-dimensional CEC2017 benchmark problems.

Boxplot analysis

As shown in Fig. 9, the distribution of research results is presented in a box plot for comparing different methods on the functions. For the method executed 20 times, the " + " symbol outside the box plot represents outliers. When these symbols are below the lower limit, it indicates that the algorithm performs efficiently in search space exploration. As illustrated in Fig. 9, The intermediate result of CDO algorithm is superior to the results of the other eight competing algorithms. Nevertheless, for the entire function set, CDO exhibits minimal variation between its upper and lower limits. CDO features fewer ( +) symbols, with no significant disparity between its upper and lower limits. This demonstrates that CDO can still yield satisfactory optimization results even under challenging conditions. Furthermore, the box plot further confirms the above analysis results, once again proving the consistency and stability of CDO.

Fig. 9.

Boxplot on a 10 dimensional test function.

Scalability analysis

In many practical optimization problems, there are multiple choice variables, which may make these problems computationally intensive. Scale analysis can be used to quantify the impact of variable dimensions on optimization performance and identify the limitations of algorithms in solving problems of immense size⁴⁵. In this section, the performance of the CDO algorithm was evaluated by conducting scalability tests on multiple functions with different dimensions, including various testing scenarios. In this algorithm performance validation, we selected the CEC-2017 standard test set and conducted comparative experiments on four typical scales. In this experiment, we set a function evaluation limit of up to 50,0 times for each algorithm and ensured the stability of the results through 20 independent repeated runs. As shown in Fig. 10, the fitness of these algorithms was systematically compared and demonstrated in four dimensions of test functions.

Fig.10.

Scalability evaluation of various techniques.

The experimental results show that, except for the Fun27 function, the CDO algorithm can generate the optimal solution in most test functions. This performance confirms its unique optimization strategy, which enables it to maintain efficient convergence accuracy even when the complexity of the problem significantly increases due to the increase in function dimensions.

Sensitivity analysis

The values of specified algorithm parameters can significantly affect the method’s efficacy in optimization. Searching the optimal parameter values poses significant difficulties and requires extensive experimentation and adjustments. This may be a lengthy and complex process, as the algorithm needs to be run multiple times for different parameter combinations to make parameter adjustments. Additionally, different scenarios could call for different tuning runs since the optimal parameter choices might change based on the particular issue that has to be resolved. Sensitivity analysis, as a systematic evaluation method, focuses on quantifying the impact of algorithm parameter changes on solution quality and constructing an evaluation framework for algorithm effectiveness. Specifically, this method first establishes a parameter solution quality mapping relationship and identifies the contribution of parameter perturbations to the output results through sensitivity indicators. On this basis, combined with preset optimization objectives such as convergence speed, stability, or accuracy, the overall performance of the algorithm is qualitatively or quantitatively evaluated. Finally, through parameter sensitivity sorting, the “key parameters” that significantly affect algorithm performance can be identified, providing a theoretical basis for subsequent algorithm optimization or parameter selection. We used parameter sensitivity analysis in this work to assess how attenuation rate(M) affects algorithm performance. In the test function, Through experimental design, we evaluated the effects of four function test parameters. Up to 500 function evaluations were performed in each of our 20 separate runs. We can evaluate the role of parameters by altering value individually while maintaining the same values for the other parameters. For various values of M {0.2, 0.25, 0.3, 0.35, 0.4}, we executed the CDO algorithm. The algorithm’s overall fitness values at various M values are displayed in Table 9. We found that when M = 0.3, the best results were achieved in f3, f9, and f18, while the second-best result was obtained in f27. However, experts may decide to employ different M values based on the particular problem.

Table 9.

Results obtained for various test functions under different M parameter settings.

M	Index	Functions
		Fun3	Fun9	Fun18	Fun27
0.2	Mean	3.027E+02	9.282E+02	1.728E+04	3.112E+03
	SD	5.312E+00	2.252E+01	1.682E+04	1.393E+02
0.25	Mean	3.405E+02	9.297E+02	2.153E+04	3.115E+03
	SD	9.380E+01	1.351E+01	1.457E+04	1.087E+01
0.3	Mean	3.000E+02	9.223E+02	6.134E+03	3.114E+03
	SD	7.534E-01	4.950E+01	5.042E+03	2.090E+01
0.35	Mean	3.058E+02	9.224E+02	1.421E+04	3.118E+03
	SD	1.115E+01	1.758E+01	9.256E+04	1.647E+01
0.4	Mean	3.129E+02	9.265E+02	1.368E+04	3.112E+03
	SD	2.964E+01	5.012E+01	1.086E+04	2.292E+01

Performance comparison between CDO and the latest methods

We contrasted the CDO optimizer with the latest algorithms in the past three years including CAO⁴⁵, PO⁴⁶, PKO⁴⁷, MSO⁴⁸, and SAO⁴⁴ to test its superiority in practical applications. Table 10 contains the parameter configurations of new algorithms other than CDO. This study adopted a uniform initial configuration for all algorithms: the initial population size was set to 50, and the maximum number of iterations was set to 500. We ran each optimizer thirty times for every assessment function By controlling the random factors and reducing their interference to the experiment. We have followed the method described in the previous section and presented the experimental results in Table 11. The Friedman ranking shows that CDO is the third best algorithm after CAO and PO, followed by CAO and PKO. At the same time, the algorithm complexity of CDO is the lowest, which indicates that CDO should be regarded as a powerful optimizer.

Table 10.

The new algorithm’s parameter configurations for the last three years.

Name	Parameter	Value
CAO	position inertia
PO	PF1	0.5
	PF2	0.5
	PF3	0.3
	u	0.2
	l	The score of the F function is 0.9, and the score of the C function is 0.7
	a	2 for the Function, 1 for C function
PKO	Beating factor (BF)	8
	PEmax, PEmin	0.5, 0
SAO	Melting speed of snow

Table 11.

The CDO and new algorithms’ statistical outcomes.

No	Index	CDO	CAO	PO	PKO	SAO
Fun1	Mean	2.504E+03	3.343E+03	3.225E+03	3.588E+07	4.075E+03
	Std	2.513E+03	3.389E+03	2.318E+08	1.570E+06	3.004E+03
	Rank	1	3	2	4	5
Fun3	Mean	3.000E+02	3.000E+02	3.091E+03	1.330E+03	3.000E+02
	Std	7.534E-01	0	1.463E+01	1.422E+03	0
	Rank	3	1	4	5	1
Fun4	Mean	4.086E+02	4.029E+02	4.033E+02	4.360E+02	4.023E+02
	Std	1.940E+01	7.318E-01	2.058E+00	1.334E+01	1.549E+01
	Rank	5	2	3	4	1
Fun5	Mean	5.250E+02	5.138E+02	5.188E+02	5.197E+02	5.127E+02
	Std	8.219E+00	6.678E+00	9.612E+00	8.685E+00	5.812E+00
	Rank	5	2	3	4	1
Fun6	Mean	6.001E+02	6.002E+02	6.000E+02	6.012E+02	6.000E+02
	Std	2.543E-01	3.641E-03	6.380E-04	1.401E+00	0
	Rank	3	4	2	5	1
Fun7	Mean	7.334E+02	7.412E+02	7.288E+02	7.305E+02	7.182E+02
	Std	1.042E+01	4.361E+00	8.570E+00	1.176E+01	3.203E+00
	Rank	4	5	2	3	1
Fun8	Mean	8.162E+02	8.138E+02	8.163E+02	8.119E+02	8.133E+02
	Std	4.947E+00	6.451E+00	5.307E+00	9.254E+00	5.912E+00
	Rank	4	3	5	1	2
Fun9	Mean	9.223E+02	9.228E+02	9.002E+02	9.081E+02	9.004E+02
	Std	4.950E+01	3.170E+00	6.707E+01	1.002E+00	1.708E+00
	Rank	4	5	1	3	2
Fun10	Mean	1.725E+03	1.738E+03	1.747E+03	1.646E+03	1.515E+03
	Std	2.545E+02	2.679E+02	1.824E+02	2.783E+02	2.294E+02
	Rank	3	4	5	2	1
Fun11	Mean	1.118E+03	1.128E+03	4.113E+05	1.119E+03	1.105E+03
	Std	9.669E+00	8.567E+01	2.166E+05	7.119E+02	5.027E+00
	Rank	2	4	5	3	1
Fun12	Mean	5.490E+04	1.335E+04	1.223E+04	8.158E+04	1.075E+04
	Std	5.207E+04	1.006E+04	1.453E+04	1.063E+04	1.101E+04
	Rank	4	3	2	5	1
Fun13	Mean	8.326E+03	1.174E+04	1.321E+03	3.307E+04	1.216E+04
	Std	5.753E+03	8.674E+03	9.119E+00	9.804E+03	9.512E+03
	Rank	2	3	1	5	4
Fun14	Mean	4.651E+03	8.953E+03	1.405E+03	2.324E+03	5.343E+03
	Std	5.145E+03	9.106E+03	3.650E+00	1.856E+03	5.858E+03
	Rank	3	5	1	2	4
Fun15	Mean	4.883E+03	8.573E+03	1.503E+03	1.982E+03	2.752E+03
	Std	6.487E+03	9.212E+03	1.745E+00	1.057E+03	1.042E+03
	Rank	4	5	1	2	3
Fun16	Mean	1.762E+03	1.723E+03	1.723E+03	1.998E+03	1.657E+03
	Std	1.265E+02	1.106E+02	1.243E+02	9.627E+00	5.706E+01
	Rank	4	2	3	5	1
Fun17	Mean	1.751E+03	1.769E+03	1.717E+03	1.817E+03	1.727E+03
	Std	4.561E+01	4.819E+01	9.218E+00	2.928E+01	1.695E+01
	Rank	3	4	1	5	2
Fun18	Mean	6.134E+03	1.500E+04	1.822E+03	6.666E+03	1.821E+04
	Std	5.042E+03	1.043E+04	6.545E+00	2.834E+03	9.612E+03
	Rank	2	4	1	3	5
Fun19	Mean	1.183E+04	1.022E+04	1.905E+03	2.345E+03	1.084E+04
	Std	1.065E+04	1.047E+04	7.769E-01	6.250E+00	1.216E+04
	Rank	3	1	4	5	2
Fun20	Mean	2.102E+03	2.082E+03	2.008E+03	2.058E+03	2.064E+03
	Std	7.432E+01	6.553E+01	1.002E+01	1.052E+01	7.097E+01
	Rank	5	4	1	2	3
Fun21	Mean	2.307E+03	2.292E+03	2.229E+03	2.316E+03	2.308E+03
	Std	3.774E+01	4.728E+01	4.853E+01	4.725E+01	2.589E+01
	Rank	3	1	2	5	4
Fun22	Mean	2.229E+03	2.301E+03	2.299E+03	2.303E+03	2.297E+03
	Std	1.502E+01	1.881E+00	1.233E+01	2.718E+01	1.839E+01
	Rank	1	4	3	5	2
Fun23	Mean	2.626E+03	2.627E+03	2.619E+03	2.654E+03	2.616E+03
	Std	1.049E+01	6.868E+00	5.546E+00	7.024E+00	8.926E+00
	Rank	3	4	2	5	1
Fun24	Mean	2.744E+03	2.732E+03	2.704E+03	2.806E+03	2.736E+03
	Std	5.811E+01	5.507E+01	9.652E+01	1.163E+01	5.621E+01
	Rank	4	2	1	5	3
Fun25	Mean	2.944E+03	2.937E+03	2.929E+03	2.939E+03	2.927E+03
	Std	1.070E+01	3.052E+01	2.277E+01	1.720E+01	2.429E+01
	Rank	5	3	2	4	1
Fun26	Mean	3.053E+03	3.065E+03	2.937E+03	2.998E+03	2.916E+03
	Std	1.509E+02	3.207E+02	4.459E+01	1.142E+02	8.659E+01
	Rank	4	5	2	3	1
Fun27	Mean	3.114E+03	3.104E+03	3.092E+03	3.157E+03	3.093E+03
	Std	2.090E+01	2.853E+01	2.006E+00	1.952E+01	2.868E+00
	Rank	4	3	1	5	2
Fun28	Mean	3.280E+03	3.380E+03	3.252E+03	3.288E+03	3.282E+03
	Std	1.094E+01	9.584E+01	1.383E+02	1.102E+01	1.485E+02
	Rank	2	5	1	4	3
Fun29	Mean	3.151E+03	3.217E+03	3.176E+03	3.167E+03	3.172E+03
	Std	7.653E+01	5.361E+01	2.203E+01	4.001E+01	4.010E+01
	Rank	1	5	4	2	3
Fun30	Mean	5.453E+05	4.128E+05	3.342E+05	3.245E+05	4.143E+05
	Std	6.415E+05	6.887E+05	4.128E+05	7.012E+03	5.292E+05
	Rank	4	5	2	1	3
Mean rank		3.294	3.301	2.345	3.759	2.241
Result		3	4	2	5	1

Experimental study on finite optimization problems in engineering

Unlike the previously reported CEC 2017 unconstrained benchmarks, This section validates the robustness of CDO by selecting five practical constrained optimization problems from different engineering fields. These intricate nonconvex issues may be divided into two groups: challenges in process synthesis and design, as well as challenges In the field of mechanical engineering, constraints are usually defined in two forms: equality constraints (satisfying specific equality relationships) and inequality constraints (satisfying specific inequality relationships). Firstly, we selected five typical engineering optimization problems for verification. In addition, constraint management is achieved through the use of penalty functions. All algorithms will have 50 participants, and each algorithm will undergo 20 independent tests with a termination condition of 500 iterations.

Design issues of pressure vessels

We have shown the external shape of the pressure vessel through Fig. 11. The project’s goal is to determine the total expense of the initial supplies needed to make pressure vessels. We need to arrange various parameters reasonably to minimize the price⁴⁵. This study aims to find the optimal solutions for the four decision variables⁴⁵ in order to achieve the set goals. This optimization problem can be expressed mathematically in the following way:

Fig. 11.

Diagrammatic representation of pressure vessel issues with design.

Independent variable

Minimize

Conditional on

Variable scope

To verify the viability and efficacy of the solution, we compare the CDO algorithm with other algorithms and identify the ideal value as the most effective approach to this problem. Table 12 illustrates how CDO outperformed other algorithms overall based on the outcomes of numerous testing. In addition, the mathematical statistical results of each method can be obtained from Table 13, which highlights the significant advantages of the CDO algorithm in several dimensions. A convergence curve constraint has been added.

Table 12.

The best answers to eight algorithms for pressure vessel problems with design.

Name	Ts	Th	R	L	Best value
CDO	0.7781686984	0.3846491734	40.31961873	200	5885.332789
MA	0.7781686952	0.3846497265	40.31961874	199	5885.335723
LSO	12.520945789	6.1862579034	40.52035128	198	5885.335475
GWO	0.7784057150	0.3854974375	40.32307346	197	5890.100385
HHO	0.8412702385	0.4185982962	43.56480023	160	6015.089537
WOA	0.8159039536	0.4491632018	41.34256028	187	6213.502630
EO	1.1925843287	0.6116802358	63.99028642	97	11,547.25489
MPA	0.9007213864	0.4448562830	46.62750376	126	6133.982157

Table 13.

Statistical outcomes of different metaheuristic algorithms on the pressure vessel minimal cost problem.

Name	Optimal solution	Worst solution	Average solution	Standard deviation
CDO	5885.332789	5885.404682	5885.343369	1.79E-01
MA	5885.335723	6197.254168	5956.214952	9.12E+01
LSO	5885.335475	6108.036127	6046.061285	5.26E+02
GWO	5890.100385	6799.287102	6279.258743	4.56E+02
HHO	6015.089537	8006.547213	7124.368201	4.19E+02
WOA	6213.502630	9685.176320	7998.034128	9.88E+02
EO	11,547.25489	20,036.48725	16,324.36857	6.23E+03
MPA	6133.982157	7785.387520	6958.365478	4.04E+02

Welded beam design

The design issue of welded beams is also a classic problem. As shown in Fig. 12, the welded beam mainly combines multiple basic components together through welding.

Fig. 12.

Design sketch chart for welded beams.

Selecting the four ideal design parameters—steel thickness (b), steel length (l), weld depth (t), and steel height (h)—is the best method to lower the total cost of beams⁴⁵. The optimization model can be expressed in the following way:

Already know: ,

Objective function:.

Domain of variables

Conditional on:

Where P = 6000 lb, L = 14 in, in, psi, psi, psi, psi.

The expressions for each function in the constraint conditions are as follows:

, ,

,

,,,

’.

lb, in, in, psi,

psi, psi, psi.

After 20 runs, all of the findings are compiled in Table 14. CDO has the best results among the four metrics. CDO performs more stably than other algorithms. Table 15 lists the parameter values that correspond to each algorithm’s optimal result.

Table 14.

Outcomes of the welded beam design issue.

Name	Best	Mean	Worst	Std	Rank
CDO	1.724528	1.8912584	2.1024865	0.1036289	1
MA	1.724540	2.0246609	2.2271258	0.2376005	2
LSO	1.724720	1.9528803	2.1445280	0.1802265	4
GWO	1.733479	2.6590058	3.1280562	0.7820321	7
HHO	1.726301	1.9964306	2.8731124	0.5209705	5
WOA	1.733479	2.6589732	3.1495631	0.7801787	8
EO	1.729839	1.9275905	2.4980264	0.3089879	6
MPA	1.724847	2.2615601	2.8092036	0.4308692	3

Table 15.

The welded beam design issue combines the best variables.

Name	Variables when taking the optimal value				Optimum Cost
	h	l	t	b
CDO	0.187156	3.470488	9.036623	0.205731	1.724528
MA	0.187156	3.744548	9.173438	0.205057	1.724540
LSO	0.203687	3.470489	9.036624	0.205729	1.724720
GWO	0.205701	3.479005	9.036874	0.205732	1.733479
HHO	0.203137	3.744548	9.173438	0.205057	1.726301
WOA	0.205700	3.657587	9.176331	0.205111	1.733479
EO	0.204368	3.856979	3.856979	0.212148	1.729839
MPA	0.205729	3.512662	8.997062	0.207548	1.724847

Three span truss design

This three span truss structure’s design aims to satisfy specific limiting requirements including deflection, buckling, and strain while minimizing the weight of the building as much as feasible. There are two parameters that need to be adjusted in this problem, as shown in Fig. 13. In terms of math, it is represented as:

Fig. 13.

Design sketch of three span truss.

Consider: ,

Minimize: ,

Conditions of restriction:

Variables range:

where cm, kN/cm², kN/cm².

The construction issue of a three-span truss was calculated using the CDO method along with other algorithms, and the outcomes were compared. Table 16 displays the variable values corresponding to the best outcomes of eight algorithms, while Table 17 offers an in-depth examination of the CDO algorithm’s performance with other approaches. The data in Table 17 shows that the CDO algorithm outperforms different algorithms on this topic.

Table 16.

The three span truss design’s optimal variables.

Name	Variables when taking the optimal value		Optimum Cost
	A1	A2
CDO	0.78867510	0.40824820	263.8958433
MA	0.78867509	0.40824840	263.8958434
LSO	0.78866541	0.40827578	263.8958434
GWO	0.78867513	0.40824828	263.8958434
HHO	0.78856500	0.40855970	263.8958522
WOA	0.78867514	0.40824829	263.8958436
EO	0.78867520	0.40824821	263.9035847
MPA	0.78867513	0.40824828	263.8958435

Table 17.

The three span truss design’s outcomes.

Name	Best	Mean	Worst	Std	Rank
CDO	263.8958433	263.8959042	263.8959632	0.00796	1
MA	263.8958434	263.8959107	263.8959782	0.00964	8
LSO	263.8958434	263.8968301	263.8974365	0.03027	3
GWO	263.8958434	263.8969258	263.8981364	0.03351	4
HHO	263.8958522	263.9023824	263.9128522	0.12096	6
WOA	263.8958436	264.0862710	264.5958436	0.54898	2
EO	263.9035847	264.0632804	264.6035847	0.49729	5
MPA	263.8958435	263.9624782	264.1958435	0.39421	7

Spring design

The spring’s production is depicted in Fig. 14 is also a classic engineering design problem, and we also use this problem to test the performance of CDO. Reducing spring pressure is a spring’s optimization objective. Shear stress, deformation, as well as pulsation frequency are the external scope of this engineering design problem. The ideal parameter selections in this design method are the wire size (d), average coil diameter (D), and the overall amount of effective coils (N). The next vector displays this design scheme’s optimization parameters:, in which d, D, and N are represented by the variables × 1, × 2, and × 3. The following is a description of the mathematical equation that resolves the optimal design problem:

Fig. 14.

A tension spring’s schematic structural diagram.

Objective function:  

Conditions of restriction:

Variables range:

Table 18 summarizes the statistical results of CDO algorithm and other mainstream metaheuristic algorithms in this design challenge. From the comprehensive evaluation of indicators such as best solution, average performance, worst case scenario, and standard deviation, the CDO algorithm shows significant advantages, and its optimization effect is superior to the comparative methods. This result indicates that under the same number of iterations and search agent size, the CDO algorithm has higher reliability and success rate in dealing with such design problems. Regarding this design issue, the suggested CDO algorithm’s effectiveness in terms of variable values and target costs is contrasted with that of competing algorithms in Table 19. The findings show that the optimal design of this kind of spring can be achieved by the CDO method at a cost of 0.0127, which is typically only marginally better than the comparative optimization strategies.

Table 18.

Outcomes of the spring design issue.

Name	Best	Mean	Worst	Std	Rank
CDO	0.0127	0.0128	0.0129	0.0001	1
MA	0.0132	0.0210	0.0278	0.0087	4
LSO	0.0127	0.0137	0.0152	0.0029	2
GWO	0.0127	0.0143	0.0186	0.0030	3
HHO	0.0145	0.0174	0.0213	0.0038	5
WOA	0.0184	0.6927	1.0258	0.5201	7
EO	0.0156	0.0708	0.1129	0.0435	6
MPA	0.0193	0.0681	0.0796	0.0153	8

Table 19.

The spring design problem’s finest variables.

Name	Variables when taking the optimal value			Optimum Cost
	d	D	N
CDO	0.0544	0.4320	7.9897	0.0127
MA	0.0739	0.8479	4.8849	0.0132
LSO	0.0534	0.3959	9.5725	0.0127
GWO	0.0530	0.3781	9.9821	0.0127
HHO	0.0540	0.4106	9.5623	0.0145
WOA	0.0623	0.6542	10.641	0.0184
EO	0.0601	0.5992	9.8241	0.0156
MPA	0.0609	0.7512	8.3478	0.0193

Speed reducer design problem

A well-known real-world example for assessing the effectiveness of optimization methods is the reducer’s design (see Fig. 15 for the structural schematic).This design contains seven decision parameters, constituting a challenging optimization problem. Among them, four key constraints need to be considered to achieve the weight reduction goal, includes axial bearing stress, lateral axial distortion, surface contact stress, and gear tooth bending stress. Seven key design parameters were selected for this optimization problem: tooth modulus (b), number of teeth (z), end face width (m), first shaft diameter (d1), second shaft diameter (d2), first shaft bearing spacing (l1), and second shaft bearing spacing (l2). These parameters cover the core geometric features and dimensional elements of the gear system. To solve this optimization problem, parameter vectors are used for mathematical modeling, with specific variable definitions as follows:

Fig. 15.

A speed reducer design’s schematic structural diagram.

Consider variable: z = [z1 z2 z3 z4 z5 z6 z7] = [b m z l1 l2 d1 d2].

This engineering design will be carried out under the following constraints:

where the variety of values for these variables’ design parameters b, m, z, l1, l2, d1, and d2 are , , , , , and , respectively.

Table 20 shows the performance statistical comparison between CDO algorithm and other optimization methods in reducer design problems. From the statistical results, it can be seen that CDO technology outperforms other metaheuristic algorithms in key indicators such as convergence accuracy and stability, and obtains the optimal response value among all compared methods. This result indicates that the CDO algorithm has higher effectiveness in solving such complex engineering optimization problems.

Table 20.

The results obtained through various algorithms.

Name	Best	Mean	Worst	Std
CDO	2994.507	2994.513	2994.516	2.13E-02
MA	3142.017	3565.048	4290.853	3.57E+02
LSO	3047.530	3112.902	3189.069	7.62E+01
GWO	2999.108	3010.263	3017.082	5.09E+00
HHO	3013.772	3174.114	3459.027	2.14E+02
WOA	3012.462	3194.095	3507.314	3.92E+02
EO	2995.148	3004.283	3009.256	3.56E+00
MPA	3033.419	3125.960	3189.410	6.85E+01

Table 21 provides a systematic comparison of the design and cost optimization schemes proposed by the CDO algorithm in the speed reduction design challenge with other optimization methods previously discussed. The data in this table shows that the CDO algorithm exhibits significant advantages in solving this problem, with its design cost reaching the lowest value (about 2994.507), which is superior to all compared algorithms. This result confirms that the CDO algorithm can effectively determine the optimal design solution for the engineering optimization problem.

Table 21.

Comparison of multiple meta-heuristic algorithms’ optimization outcomes.

Algorithms		Optimal Values for Variables						Optimum Cost
	b	m	z	l1	l2	d1	d2
CDO	3.500	0.7	17	7.303	7.715	3.351	5.288	2994.507
MA	3.598	0.7	17	7.925	8.102	3.379	5.342	3142.017
LSO	3.546	0.7	17	7.808	8.303	3.354	5.306	3047.530
GWO	3.501	0.7	17	7.404	7.755	3.359	5.287	2999.108
HHO	3.502	0.7	17	7.929	7.738	3.374	5.289	3013.772
WOA	3.500	0.7	17	7.948	7.967	3.351	5.293	3012.462
EO	3.500	0.7	17	7.314	7.716	3.350	5.288	2995.148
MPA	3.549	0.7	17	7.837	8.259	3.357	5.299	3033.419

Utilization for photovoltaic system parameter extraction

This part uses a more thorough and critical actual engineering approach to assess CDO’s benefits over other well-known competitors. Photovoltaic (PV) systems, which directly convert solar energy into electrical energy, are very effective instruments in the field of energy from natural sources^49–51. Therefore, creating a precise and effective model for PV systems using characteristics derived from measured current–voltage data is an essential task⁵². The key parameters for PV systems are extracted using CDO and the remaining eight techniques. Three traditional photovoltaic models were adopted: photovoltaic module model (PVMM), single diode model (SDM), and dual diode model (DDM)^53–55. The electronic designs for these three variants are displayed in Fig. 16.

Fig. 16.

Photovoltaic cell equivalent circuit diagrams.

As illustrated in Fig. 16a, the source of photocurrent (Ipℎ), reverse saturation current (Isd), series resistance (Rs), parallel resistance (Rsℎ), and the ideal coefficient of the diode (n) are the five essential parameters that must be obtained in SDM⁴⁴. The following is a summary of the mathematical equation that these parameters satisfy:

14

15

16

where Id stands for the diode current determined by Eq. (15), I_L for the output current, VL for the output voltage, T for the battery temperature in Kelvin, q for the electronic charge (1.6022 × 10⁻¹⁹), and k for the Boltzmann’s constant (1.3807 × 10 − 23 J/k)⁴⁴. The I_L calculation formula is then shown in SDM in Eq. (17).

17

Unlike SDM, the DDM is increased by including the effect of recombination loss in depletion zones. The corresponding circuit diagram is shown in Fig. 16b. Next, Eq. (18) provides the formula for calculating IL in DDM.

18

where the diffusion current is represented by Is_d1 and the saturation current by I_sd2. The ideality factors of diodes are denoted by n₁ and n₂⁵⁶. DDM is intended to extract seven core parameters: I_pℎ, I_sd1, R_s, R_sℎ, n₁, Is_d2, n₂.

Multiple identical PV cells placed in series or parallel make up the single-diode-based PVMM, as shown in Fig. 16c. Equation (19) provides the mathematical expression for I_L in PVMM.

19

where the numbers of cells connected in series and parallel are denoted by N_p and N_s, respectively. Five key parameters (I_pℎ, I_sd, R_s, R_sℎ, n) are intended to be derived from this model⁵⁷. The bounds of numerous variables in three PV models are reported in Table 22.

Table 22.

Limits of the different parameters of three PV models.

Name	SDM/DDM		PVMM
	lb	ub	lb	ub
Ipℎ (A)	0	1	0	2
Isd1, Isd2, Isd (μA)	0	1	0	50
n1, n2, n	0	2	0	50
Rsℎ (Ω)	0	100	0	2000
Rs (Ω)	0	0.5	0	2

Reducing the error among measuring and test data is the primary objective of this problem. The root mean square error (RMSE)^58,59, which has a mathematical description in Eq. (20), is typically used as the fitness function in complex nonlinear optimization problems.

20

The number of measured data is indicated by N, whereas the solution vector, which includes the unknown core parameters, is denoted by X. Next, each model’s objective function is stated as follows:

For SDM:

21

For DDM:

22

For PVMM:

23

Easwarakhanthan et al.⁴⁴ recorded the reference current voltage data for this study using an official RTC French silicon solar panel via France that had a diameter of 57 mm and produced a maximum of 1000 watts per square meter at 33 °C. The efficacy of the suggested approach for parameter extraction is frequently assessed using this standard set. Additionally, 50 individuals are employed, and the termination criterion is 500 total iterations. Tables 23, 24, 25 show the best outcomes that eight algorithms produced after 20 separate runs on SDM, DDM, and PVMM, respectively. The following are some findings from observations:

Table 23.

Eight techniques achieved the best results during 20 separate runs on SDM.

Methods	Optimal variables					RMSE
	Ipℎ(A)	Isd1(μA)	Rs(Ω)	Rsℎ(Ω)	n
CDO	0.760776	0.323021	0.0363771	53.7185	1.48119	9.8602E−04
MA	0.762523	0.385614	0.0352405	42.7495	1.49953	1.5287E−03
LSO	0.760621	0.413816	0.0353887	62.3453	1.50653	1.0946E−03
GWO	0.760776	0.323021	0.0363771	53.7185	1.48119	9.8602E−04
HHO	0.760059	0.239914	0.0372831	54.3757	1.45171	1.4178E−03
WOA	0.760776	0.323021	0.0363771	53.7185	1.48119	9.8602E−04
EO	0.760639	0.438642	0.0351438	65.9151	1.51259	1.1551E−03
MPA	0.724216	0.607958	0.0516663	27.0936	1.56212	5.9041E−02

Table 24.

Over 20 separate runs on DDM, eight strategies produced the greatest results.

Methods	Optimal variables							RMSE
	Ipℎ(A)	Isd1(μA)	Rs(Ω)	Rsℎ(Ω)	n1	Isd2(μA)	n2
CDO	0.760781	0.749340	0.0367404	55.4854	2.00000	0.225974	1.45102	9.8248E−04
MA	0.760781	0.606518	0.0366365	55.6499	1.96837	0.236752	1.45534	6.0583E−02
LSO	0.760779	0.576481	0.0365678	55.6797	1.99998	0.255896	1.46176	9.8372E−04
GWO	0.760781	0.749340	0.0367404	55.4854	2.00000	0.225974	1.45102	9.8248E−04
HHO	0.760201	0.434402	0.0351989	76.6717	1.53249	0.039298	1.44290	1.2450E−03
WOA	0.760779	0.542563	0.0364397	55.6783	1.99994	0.256900	1.46169	9.8370E−04
EO	0.760142	0.703767	0.0374001	58.1968	1.90395	0.258735	1.45925	1.1942E−03
MPA	0.760224	0.432685	0.0373124	55.5148	1.45264	0.360097	1.45670	1.9274E−03

Table 25.

Eight techniques produced the greatest results during 20 separate PVMM tests.

Methods	Optimal variables					RMSE
	Iph (A)	Isd1 (μA)	Rs (Ω)	Rsh (Ω)	n
CDO	1.030514	3.482263	1.201271	981.9822	48.6429	2.4251E–03
MA	1.127198	1.266695	1.474165	1281.5935	44.9737	9.3316E–02
LSO	1.022033	0.138684	1.545601	996.4929	38.7129	1.2888E–02
GWO	1.030514	3.482263	1.201271	981.9822	48.6429	2.4251E–03
HHO	1.029579	3.996734	1.189785	1207.1769	49.3763	2.4943E–03
WOA	1.030514	3.482263	1.201271	981.9822	48.6429	2.4251E–03
EO	1.030594	3.433815	1.202718	967.1523	48.5894	2.4251E–03
MPA	1.030276	3.298946	1.208045	986.7730	48.4343	2.4358E–03

In the arena of SDM optimization problems, the CDO algorithm took the lead with a stunning score of 9.8602 × 10⁻⁴, followed closely by the equally impressive GWO and WOA algorithms (both reaching 9.8602 × 10⁻⁴), forming the first tier together. The MA algorithm ranks second with a score of 1.5287 × 10⁻³, while LSO (1.0946 × 10⁻³) and HHO (1.4178 × 10⁻³) form the third tier. Surprisingly, the EO algorithm failed to enter the top three with a score of 1.1551 × 10⁻³, while the MPA algorithm lagged behind other contestants with a score of 5.9041 × 10⁻², becoming the “pursuer” in this optimization competition. The CDO algorithm may get the best RMSE when compared to other rivals, with the exception of GWO and WOA.

In the 20 independent competitions of the nine optimization algorithms DDM, CDO, MA, LSO, GWO, HHO, WOA, EO, and MPA, their RMSE (root mean square error) scores are as follows: DDM leads with a slight advantage of 9.8248 × 10⁻⁴, followed closely by MA (9.8372 × 10⁻⁴) and LSO (9.8248 × 10⁻⁴) to form the first group; GWO (1.2445 × 10⁻³) and WOA (1.1942 × 10⁻³) form the second tier; EO (1.9274 × 10⁻³) and MPA (6.0583 × 10⁻²) form a clear watershed—especially the MPA algorithm, whose error value is two orders of magnitude higher than the optimal solution, becoming the “laggard” in this accuracy competition. The CDO method may get the best RMSE when compared to other rivals, with the exception of GWO.

In the PVMM optimization competition, eight algorithms including CDO, MA, LSO, GWO, HHO, WOA, EO, and MPA competed fiercely. The final transcript shows that CDO, GWO, and WOA are tied for the championship with an excellent score of 2.4251 × 10⁻³, while MA lags significantly behind with a score of 9.3316 × 10⁻². LSO (1.2888 × 10⁻²) and HHO (2.4943 × 10⁻³) form the second tier, while EO (2.4255 × 10⁻³) and MPA (2.4358 × 10⁻³) follow closely behind with a small gap. It is worth noting that the error value of the MA algorithm is nearly 40 times higher than the optimal solution, making it the “laggard” in this accuracy competition. The CDO algorithm may get the best RMSE when compared to other rivals, with the exception of GWO and WOA.

Table 26 presents a statistical comparison of RMSE (root mean square error) for 8 algorithms on three photovoltaic models. Data analysis shows that the CDO algorithm significantly outperforms other algorithms in overall performance, with outstanding error control capabilities. Of particular note is that CDO has demonstrated strong competitiveness in comparison to GWO and WOA, and even achieved better error suppression effects in some models. Additionally, Friedman’s test is applied to eight algorithms based on three PV models. CDO achieves a decent rating among these eight algorithms, as seen in Table 27. In particular, based on average performance, CDO is comparable to GWO and WOA and outperforms MA, LSO, HHO, EO, and MPA.

Table 26.

RMSE statistical results obtained using nine approaches on three PV models.

RMSE	Item	Methods
		GCDO	MA	LSO	GWO	HHO	WOA	EO	MPA
SDM	Best	9.8602E−04	1.5287E−03	1.0946E−03	9.8602E−04	1.4178E−03	9.8602E−04	1.1551E−03	5.9041E−02
	Mean	9.8602E−04	1.7649E−03	5.5717E−03	9.8602E−04	2.2895E−03	9.8602E−04	1.6459E−03	1.7921E−01
	SD	8.6382E−10	2.5927E−02	4.2614E-03	4.0259E−08	1.0653E−03	6.2481E−06	6.0257E−04	9.8973E−02
DDM	Best	9.8248E−04	6.0583E−02	9.8372E−04	9.8248E−04	1.2445E−03	9.8366E−04	1.1942E−03	1.9274E−03
	Mean	1.0301E−03	1.134E−01	3.8246E−03	1.1509E−03	1.9411E−03	2.4752E−03	4.1067E−03	3.3915E−03
	SD	1.4328E−04	4.2549E−02	3.9625E−03	6.5662E−04	6.8524E−04	2.6521E−03	3.278E−03	2.493E−03
PVMM	Best	2.4251E−03	9.3316E−02	1.2888E−02	2.4251E−03	2.4943E−03	2.4251E−03	2.4255E−03	2.4358E−03
	Mean	2.4252E−03	1.0579E−01	2.4823E−02	2.7425E−03	5.4386E−02	2.4287E−03	2.4789E−03	3.8827E−03
	SD	6.9584E−07	1.9247E−02	1.3049E−02	2.6895E−04	4.9862E−02	5.9624E−05	5.8069E−05	1.6480E−03

Table 27.

Friedman’s test is used to rank eight approaches on three PV models.

Methods	SDM	DDM	PVMM	Ranking
GCDO	1	1	1	1
MA	5	8	8	7
LSO	7	7	6	6.66
GWO	2	2	4	2.66
HHO	6	3	7	5.33
WOA	3	4	2	3
EO	4	5	3	4
MPA	8	6	5	6.33

Figures 17, 18, 19 respectively show the absolute error distribution between the predicted results of the CDO algorithm and the measured data of the SDM/DDM model. Through residual analysis, the accuracy difference of the algorithm in photovoltaic system modeling is visually presented. The CDO algorithm’s prediction accuracy is confirmed by Figures 17, 18, 19, which unequivocally demonstrate that the experimental results produced by the algorithm is very consistent with the data collected.

Fig. 17.

Validation of the prediction accuracy of the CDO method in SDM modeling.

Fig. 18.

Validation of the prediction accuracy of the CDO method in DDM modeling.

Fig. 19.

Validation of the prediction accuracy of the CDO method in PVMM modeling.

Conclusion

A novel Cauchy Distribution Optimization (CDO) approach has been put forth to extract photovoltaic model parameters. The Cauchy distribution theory is the source of inspiration for CDO. By creating a Cauchy state distribution model, each person’s position is updated. The most notable characteristic of CDO, in contrast to the majority of current meta-heuristic algorithms, is that it requires no effort to fine-tune the starting parameters. To assess the effectiveness of CDO, we selected 29 functions from CEC2017, five traditional engineering scenarios, and extracted parameters for three solar models. The testing results demonstrate that CDO performs more accurately and efficiently than the compared algorithms. Additionally, enhancing control and optimization of the real photovoltaic system is the aim of parameter extraction for solar models. For the optimization and management of the real solar system, the solution generated by the suggested CDO is therefore a superior option compared to any of the comparative algorithms, based on the experimental data.

CDO is a straightforward and effective method that doesn’t need the starting settings to be adjusted. As a result, CDO has a lot of potential for application in different kinds of optimization challenges. Future studies will concentrate on applying CDO to additional engineering optimization issues, like logistics transportation and unmanned aircraft group path planning. This algorithm will also be developed in binary and multi-objective variants.

Author contributions

Da Fang: Conceptualization, Software, Writing – original draft, Visualization. Quan Zhou: Validation, Writing – review & editing. Jun Yan: Software, Writing. Suping Guo: Draw picture & editing. Ping Guo: Software. (1) This material is the authors’ own original work, which has not been previously published elsewhere. (2) The paper is not currently being considered for publication elsewhere. (3) The paper reflects the authors’ own research and analysis in a truthful and complete manner.

Funding

There was no Funding.

Data availability

Data are available on request. The source code for the CDO algorithm is publicly accessible at https://ww2.mathworks.cn/matlabcentral/fileexchange/182811-cauchy-distribution-optimizer.

Declarations

Competing interests

The authors declare no competing interests.