An adaptive differential evolution with decomposition for photovoltaic parameter extraction

1.   Introduction

With the development of science and technology in human society, traditional technology can no longer not be used to solve complex practical problems. Therefore, the meta-heuristic algorithm came into being. Meta-heuristic algorithms are widely used in these complex practical problems because of their simple concept and flexible parameters. At present, the proposed algorithms are mainly divided into four categories ^[1], The first category is swarm-based algorithms, such as Artificial Bee Colony (ABC) ^[2]. The second category is human-based algorithms, such as Teaching based learning algorithm (TBLA) ^[3]. The third category is physics-based algorithms, such as Fireworks Algorithm (FWA) ^[4]. The last category is evolutionary algorithms, such as Tree Growth Algorithm (TGA) ^[5]. The classification of the meta-heuristic algorithm is shown in Table 1.

However, not all problems can be solved ^[43] by these algorithms, especially the multidisciplinary problems that have become increasingly complex in recent years. Therefore, we need to find better algorithms for solving multidisciplinary and complex problems.

The improvement for AO is less because the time to propose the algorithm is not long, so far there is not enough research on AO algorithms. Wang et al. ^[44] proposed a hybrid improvement of the AO algorithm and the HHO algorithm and added hunting strategies and adversarial-based learning rules to them. It has been verified by experiments that the improved algorithm has a very good performance. Elaziz et al ^[45] proposed a Mobile Net V3 AO hybrid model. He uses the AO algorithm to reduce the dimensionality of the image, thereby improving the accuracy of the data. Zhang et al. ^[46] proposed the hybridization of the AO algorithm and the AOA algorithm. Zhao et al. ^[47] applied the JAYA algorithm to the parameter extraction of photovoltaic models and added a chaotic adaptive strategy. Zhang et al. ^[48] used the enhanced adaptive comprehensive learning hybrid algorithm of Rao-1 and JAYA algorithm for photovoltaic model parameter extraction. Zhou et al. proposed ^[49] metaphor-free dynamic spherical evolution for parameter estimation of photovoltaic modules, it helps the research on the application of swarm intelligence algorithm in photovoltaic modules. We believe that the enhanced AO algorithm can be used in photovoltaic model parameter extraction. In the future, we will also conduct relevant research. Simrandeep Singh et al. ^[50] proposed Arithmetic optimization based image segmentation. Hussien et al. ^[51] proposed comparative research and application of Harris Eagle optimization research progress. Wang et al. ^[52] proposed an enhanced remora optimization algorithm. Hashim et al. ^[53] proposed a snake optimizer. Zheng et al. ^[54] proposed an improved wild horse optimizer. Hussien et al. ^[55] proposed an enhanced COOT optimization algorithm for dimensionality reduction. Yu et al. ^[56] proposed an enhanced aquila optimizer for global optimization and constrained engineering problems. Yang et al. ^[57] proposed an efficient DBSCAN optimized by the arithmetic optimization algorithm. Cui et al. ^[58] proposed a modified slime mold algorithm via levy flight. Yang et al. proposed ^[59] an opposition learning and spiral modelling based arithmetic optimization algorithm. Hussien et al. ^[60] proposed boosting whale optimization with an evolution strategy and Gaussian random walks, and used it to image segmentation. Hussien et al. ^[61] proposed a comprehensive review of moth-flame optimization: variants, hybrids, and applications. Also, swarm intelligence algorithms are used to solve real-world engineering problems. Yu et al. ^[62] proposed laplace crossover and random replacement strategy boosted Harris hawks optimization. Qi et al. ^[63] proposed directional mutation and crossover for the immature performance of whale algorithm with application to engineering optimization. Zhao et al. ^[64] proposed opposition-based ant colony optimization with an all-dimension neighborhood search for engineering design. Zhou et al. ^[65] proposed advanced orthogonal learning and Gaussian barebone hunger games for engineering design.

Considering the powerful global exploration capability of the AO and early convergence ^[44] caused by insufficient exploration and exploitation in the later phases, an enhanced AO with velocity-aided global search mechanism and adaptive opposition-based learning, called VAIAO. The proposed VAIAO has better search ability than the original algorithm, and it is got better performance easily. This article was inspired by the velocity-aided global search mechanism ^[66], and this paper applies the velocity-aided global search Mechanism to the AO. On the other land, the adaptive opposition-based learning rule is introduced to avoid being trapped in the local optimum.

The performance of the VAIAO is tested by using 27 classical benchmark functions, the Wilcoxon statistical test, and five engineering optimization problems. In order to compare the various experimental results, IAO ^[67], AO ^[30] and some well-known algorithms including Arithmetic AOA ^[6], HHO ^[33], STOA ^[29], ChOA ^[68], SOA ^[28], PSO ^[24], DE ^[8] are introduced. The results of the experiment show that the proposed VAIAO is better than another algorithm.

The basic AO and improved strategies will be introduced later. And carry out simulation experiments and engineering optimization problems using VAIAO and comparison algorithms to solve. Finally, the conclusion is given at the end of the article.

2.   Aquila optimizer

When the Aquila is preying, it can switch four strategies in different situations when facing the prey in different situations to escape the eagle's predation strategy, which can be represented by the following four strategies.

2.1.   Strategy 1: Flying high for searching prey

In this strategy, Aquila flies through high-altitude hunting areas to find the best spot for prey, and once it finds the best spot, Aquila swoops vertically at prey. This behavior can be expressed by the formal.

where, ${X}_{best}\left(t\right)$ represent the global optimal position, ${X}_{M}\left(t\right)$ represents the current mean position, and $\mathrm{t}$ and $\mathrm{T}$ represent the current number of iterations and maximum number of iterations.

2.2.   Short glide attack and fight

In this strategy, Aquila will switch from flying at high altitude to hovering on the head of the prey, preparing for Aquila.

where, the Aquila's random position is ${X}_{R}\left(t\right)$, and $D$ is the size of dimension. $LF$ is Levy flight function. The $\mathrm{y}$ and $\mathrm{x}$ represents the shape of the search, which can be expressed by the formula:

where, ${\mathrm{r}}_{1}$ is Aquila's search cycles which number between $\left[\mathrm{1, 20}\right]$, ${\mathrm{D}}_{1}$ is a random integer from one to D dimensions, and ${\rm{ \mathsf{ ω} }}$ is a constant of 0.005.

2.3.   Approaching prey and attack

In this strategy, Aquila first finds and determines the approximate location of the prey through strategy 2. After finding the prey, Aquila descends vertically for initial hunting. This behavior can be expressed by the formula:

where ${\rm{ \mathsf{ α} }}$ and ${\rm{ \mathsf{ δ} }}$ are adjustment parameters during the development process, fixed at 0.1, $\mathrm{U}\mathrm{B}$ and $\mathrm{L}\mathrm{B}$ are the upper and lower bounds of the search space, respectively.

2.4.   Catching prey on land

In this strategy, Aquila will follow the prey's escape track to land and chase it, then attack it. This behavior be expressed by the formula:

where, ${G}_{1}$ is the random number which between [1, −1], and ${G}_{2}$ is the flight slope when Aquila tracking the prey between [2,0].

3.   The simplified IAO

The IAO ^[67] is a simplification of the AO ^[69]. Although the AO algorithm has better convergence speed and exploration ability, the AO algorithm has poor convergence performance in the later stage. For solving this problem, the IAO algorithm selects only strategies 1 and 2 based on the Aquila algorithm. In this way, IAO algorithm can avoid the situation that the convergence speed decreased in the later period. The simplified improved Aquila will be easier than ever.

4.   The proposed VAIAO algorithm

During exploration, individuals perform fast flights and hunts of Aquila in the search space. Even though the simplified algorithm speeds up the convergence and improves the search ability. Individuals can still easily fall into local optima. And the speed of convergence and exploration ability of AO can be strengthened That's why we should improve and overcome these shortcomings.

4.1.   Velocity-Aided global search mechanism

When Aquila individuals move in the search space without considering the two features of velocity and acceleration, these movements do not always occur continuously or smoothly. The movement of search agents in the search space may break down, then leading to potential drift. Due to this reason, search agents may drift when executing search strategies. Considering the velocity term can avoid premature convergence by helping the search agent maintain its unique trajectory, enhancing the search agent's exploration capabilities, and balancing the exploration and development phases of the optimization process. In VAIAO, an initial random velocity is defined for each Aquila search agent as it performs a search policy move, and an initial random position is defined for each search agent in the search space. Therefore, each search agent has a velocity and position in each dimension of the optimization problem. This idea be expressed by this formula:

where, ${V}_{j}^{t}$ represent the velocity of the Aquila search agent in the $j$th dimension. $sgn$ is the sign function. ${A}_{j}^{t}$ represents the acceleration terms of the search agents, and ${A}_{j}^{t}$ be expressed by this formula:

where, ${r}_{1}$ represents a random number between 1 and 0, Furthermore, a is a linearly decreasing parameter determined as follows:

${D}_{j}^{t}$ a represents the modified distance between the focused search agent and the leading Aquila. These distances in each dimension can be calculated as follows:

where, ${C}_{j}^{t}$ can take these uncertainties into account and help the algorithm better perform the exploration phase. Especially in the early iterations of the optimization process. $c$ is an adaptively determined parameter as follows. ${r}_{2}$ represents a random number between 0 and 1.

$k$ acts as a tuning parameter, acting as an inertia weight, while facilitating a proper and reliable transition from exploration to exploitation, and is iterated through iterative computation as follows:

Finally, the next position of the search agent can be expressed by this formula:

The new position of search agent is calculated by this formula:

where, is the position of the ith search agent (Aquila) in the jth dimension in the (t+1)th iteration.

4.2.   Adaptive opposition-based learning

The starting point of introducing reverse learning into IAO is to get rid of the precocious dilemma when it falls into local convergence through reverse learning. Introducing the reverse learning probability of reverse learning requires a large amount of experimental data to determine its value in a hybrid algorithm designed by the original reverse learning and other algorithms. Hussien et al. ^[70] proposed opposition-based learning and chaotic local search strategy are used in Harris Hawks optimization algorithm for global optimization and feature selection. Hussien et al. ^[71] An enhanced opposition-based Salp Swarm Algorithm. Therefore, we find that the adaptive opposition-based learning rules can help the algorithm jump out of the local optimum. That's why we introduce adaptive opposition-based learning rules to improve the IAO algorithm. It can be expressed by this formula:

where, ${\mathrm{L}\mathrm{B}}_{\mathrm{j}}$ is the lower bound of the current problem in the jth dimension, and ${\mathrm{U}\mathrm{B}}_{\mathrm{j}}$ is the upper bound of the current problem in the jth dimension.

4.3.   Overview of the proposed VAIAO algorithm

The flowchart of the VAIAO algorithm is given in Figure 1. The pseudo-code for the VAIAO algorithm is shown in Table 2. The computational time complexity is O(T $\times$ D $\times$ N).

5.   Experiments and results

In this section, the 27 classic functions, the Wilcoxon test and Friedman is used to test the performance of VAIAO algorithm. The results of this experimental show that the VAIAO algorithm has better performance than the original AO algorithm.

5.1.   Experimental setup

In this section, some well-known algorithms are selected to verify the performance of the proposed VAIAO algorithm, such as IAO ^[67], AO ^[30], AOA ^[6], HHO ^[33], STOA ^[29], ChOA ^[68], SOA ^[28], PSO ^[24], DE ^[8]. To ensure the fairness of the experimental, set the population data to 30, and set the number of iterations is 500. The worst value, average value, optimal value, median value, and standard deviation of the output are compared to set the parameters of the algorithm according to the original version. The parameter settings for the comparative algorithms are shown in Table 3.

5.2.   Test function

In this section, there are 27 typical benchmark selection functions, including 9 single-peak functions (shown in Table 4), 9 two-dimensional multi-channel functions (shown in Table 5), and 9 multidimensional multi-peak functions (shown in Table 6). These benchmark functions are used to test the basic performance of the proposed OLAO algorithm. All simulation experiments will be conducted on Intel(R) Core (TM) i5-10300H CPU @2.50GHz 3106×2 core and MATLAB 2017b software will be used.

5.3.   Experimental series 1: Intensification capability experiments

In this section, the test function contains an unimodal test function and a multimodal test function. The algorithm tests the single peak function to verify the algorithm's exploration ability is effective, because the unimodal test function only has one global optimal value. The results of the experiment are shown in Table 7. In most instances, the VAIAO algorithm has the best convergence performance in almost all case and has better performance on unimodal functions than other algorithm.

The multimodal test functions include many local optimal solutions and global optimal solutions. With the increase of the dimension, the algorithm with poor performance can easily fall into local optimum. Therefore, it is very effective to use these functions to test the diversity of algorithms. The results of the multimodal two-dimensional in this section are shown in Tables 8 and 9. In the low-dimensional diversification experiment, the VAIAO algorithm proposed in this paper can converge to 0 many times, and obtain the optimal value through the data in Tables 8 and 9.

5.4.   Experiment series 3: Accelerated convergence experiments

In order to understand the performance of the algorithm, this section will perform accelerated convergence analysis of these algorithms on 27 classic benchmark functions to compare the performance of the algorithm, the results are shown in Figures 2–5. The results in Figure 2 show the VAIAO algorithm has faster convergence s peed, higher convergence accuracy, and more stable convergence curve.

5.5.   Experiment series 4: Scalability experiments

One of the problems which the algorithm must solve is the high-dimensional problem. The dimensions are selected as 60,100,300,500 to solve this problem, and the results are shown in Tables 10–13.

5.6.   Wilcoxon statistical sign-rank test and Friedman test on the 27 benchmark Functions

The Wilcoxon statistical tests are often introduced to verify the performance of the proposed algorithm. If the proposed algorithm value is smaller than another algorithm, it's proved the proposed algorithm is superior. This experiment is carried out 30 times, and the results of the Wilcoxon statistical test on Table 14. Table 14 shows that the Wilcoxon statistical test results for 27 functions are less than 0.05 in most cases. The results of the Wilcoxon sign-rank test and Friedman test on 27 functions are shown in Tables 15–18. That also means the VAIAO algorithm is better than the others.

5.7.   Ablation experiments

Since the proposed VAIAO introduced two improvement strategies, velocity-aided global search mechanism and adaptive opposition-based learning. Therefore, it is necessary to improve these strategies by ablation experiments.

In VAIAO, there are two variants. The first variant is VAIAO-2 which is with only velocity-aided global search mechanism. The second variant is VAIAO-3 which with adaptive opposition-based learning. Three single-peak functions (F1, F2, F3), three two-dimensional multi-channel functions (F12, F13, F16), and 3 multidimensional multi-peak functions (F24, F25, F26) are tested. The results of the ablation experiments are shown in Table 19.

6.   Engineering design problem

Engineering design problems are used to test the performance of algorithm, the results of the engineering design problem in real world can reflect the advantages and disadvantages of an algorithms. It is important to use the traditional engineering design problem to test proposed VAIAO algorithm.

6.1.   Three-bar truss design problem

The three-bar truss is a classic optimization problem which is an engineering design problem. The schematic model of three-bar truss is shown in Figure 6. The pressure of each Truss member is $\sigma$. the cross-sectional area is ${A}_{1}( = {x}_{1})$ and ${A}_{2}( = {x}_{2})$. The two parameters are optimized to minimize the total weight in the case of satisfying three limiting conditions. The three constraints are as follows:

Objective:

Subject to:

Variable range:

Wheel $l = 100cm, P = 2KN/c{m}^{2}, \sigma = 2KN/c{m}^{2}$, Table 20 shows the results of three bar truss design problem. T It can be seen from the data results in the table 20 that the best value of VAIAO smaller than that of other algorithms. From this reason, it can be concluded that VAIAO can better solve the engineering problem.

6.2.   Compression spring design problem

The compression spring design problem is one of the classic engineering optimization problems in mechanical engineering. The schematic model of compression spring design problem is shown in Figure 7. The essence of the compression spring design problem is to minimize the weight of the tension and compression spring by optimizing three parameters (wire diameter (d), average coil diameter (D) and effective coil number (N)).

Consider:

Objective:

Subject to:

Variable ranges:

The experimental results are shown in Table 21. The schematic model of the Compression Spring Design Problem is shown in Figure 7. Table 21 shows the final results of VAIAO and comparative algorithms after solving this engineering problem. As shown in the table 16, VAIAO achieves the minimum weight of the compression spring. IAO and AO are in all cases applied to this engineering problem the algorithm presents the worst objective value.

6.3.   Pressure vessel design problem

The pressure vessel design is also an engineering problem which needs to optimize cost, including material cost, molding cost and welding cost, to minimize the total cost. This problem has four issues that need to be optimized (the thickness of the shell (X1), the thickness of the head (X2), the inner radius (X3), the length of the vessel section (X4). There are 4 constraints in this problem.

Consider:

Objective:

Subject to:

Variable ranges:

The schematic model of pressure vessel is shown in Figure 8. Table 22 shows that the best value of VAIAO is the smallest, which means that the optimized solution is obtained by VAIAO.

6.4.   Speeds reducer design problem

The speeds reducer design problem needs to optimize 7 variables to get the minimize the weight of the reducer. Face width parameter is b ( = ${x}_{1}$), Tooth modulus is m ( = ${x}_{2}$), the number of teeth in the pinion is z ( = ${x}_{3}$). Between bearings, the length of the first shaft is l1 ( = ${x}_{4}$) and the length of the second shaft l2 ( = ${x}_{5}$). the diameter of first shafts is d1 ( = ${x}_{6}$), and the diameter of second shafts is d2 ( = ${x}_{7}$). At the same time, the speed reducer design problem has 11 constraints:

Consider:

Objective:

Subject to:

Variable ranges:

The schematic model of speeds reducer design problem is shown in Figure 9. The experimental results are shown in Table 23. Table 23 shows the final results of the VAIAO and comparison algorithms after solving the speed reducer design problem. As shown in Table 21, VAAIO has the smallest best value compared to another algorithm.

6.5.   Gear design problem

The gear design problem is also a classic engineering optimization problem. The problem is to achieve the minimum transmission ratio cost by optimizing the number of teeth. The four gears are A(${x}_{1}$), B(${x}_{2}$), C(${x}_{3}$), A(${x}_{4}$). This engineering optimization problems has no constraints.

Consider:

Objective:

Variable ranges:

The schematic model of gear design problem is shown in Figure 10. The results of gear design problem are shown in Table 24 The results of VAIAO algorithm are also the best compared with other algorithms. As shown in Table 24, Compared with IAO, AO and other objective algorithms, the most competitive results.

7.   Conclusions and future works

In this paper, an enhanced AO algorithm is proposed to improve the exploration ability and convergence speed of IAO. Inspired by the velocity-Aided Global Search Mechanism, the velocity parameters and acceleration parameters are introduced into the AO algorithm to help the search agent update the position and prevent a number of good positions from being missed during the optimization process. In addition, introduced the adaptive opposition-based learning rule is introduced to improve the local optimum. The performance proposed VAIAO algorithm and the comparison algorithms are tested by 27 classical benchmark functions, and five engineering optimization problems. The results of the experiment show that the VAIAO algorithm is easier to obtain better global exploration and exploitation capabilities, faster convergence speed and higher convergence accuracy than any other algorithm. Because the effect of the proposed VAIAO algorithm is better than the original algorithm in terms of optimization. In future work, the proposed VAIAO will be used to solve complex multidisciplinary problems and real-meaning engineering problems, such as Photovoltaic module model parameter extraction, image segmentation and engineering problems of multi-objective optimization.

Acknowledgments

This work was supported by the Innovation Fund for Industry-University-Research in Chinese Universities with grant number 2021ZYA11012, the open project of Hubei Engineering Research Center for Specialty Flowers Biological Breeding with grant number 2022ZD006, the educational teaching research project of Jingchu university of technology with grant number JX 2022-016, the science research project of Jingchu university of technology with grant number YY202203 and the general funded science and technology research project of Jingmen with grant number 2022YFYB152.

Conflict of interest

All authors declare no conflicts of interest in this paper.