An Improved JADE Hybridizing with Tuna Swarm Optimization for Numerical Optimization Problems

Abstract

In this paper, we propose IJADE-TSO, a novel hybrid algorithm in which an improved adaptive differential evolution with optional external archive (IJADE) has been combined with the tuna swarm optimization (TSO). The proposed algorithm incorporates the spiral foraging search and parabolic foraging search of TSO into the mutation strategy in IJADE to improve the exploration ability and population diversity. Additionally, to enhance the convergence efficiency, crossover factor (CR) ranking, CR repairing, top α r₁ selection, and population linear reduction strategies have been included in the algorithm. To evaluate the superiority of the proposed algorithm, IJADE-TSO has been benchmarked with its state-of-the-art counterparts using the CEC 2014 test set. Finally, to check the validity of IJADE-TSO, we apply it to photovoltaic (PV) parameter identification and compare its performance with those of other recently developed well-known algorithms. The statistical results reveal that IJADE-TSO outperforms the other compared algorithms.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

There are an increasing number of optimization problems regarding engineering design and industrial production calling for more effective optimization methods in the real world [1, 2]. One efficient approach is the stochastic heuristic algorithm, which can be divided into three categories [3]: evolution-based algorithms [4, 5], physical-phenomena-based algorithms [6, 7] and swarm-based algorithms [8, 9]. Differential evolution (DE) [10], one of the most excellent heuristic algorithms, has been widely used to solve complex optimization problems such as mechanical engineering [11], optimal power flow [12], parameter optimization [13], neural network training [14, 15], and complex control system [16]. The simplicity and efficiency of the DE algorithm has been attracting extensive exploration of it since its proposition by Storn and Price in 1995.

The performance of DE algorithm is sensitive to the mutation strategy and parameters and prone to fall into a local optimum. To fix these defects, a series of DE variants were proposed over the past two decades. These DE variants can be easily traced through the result of the competition organized by IEEE Congress on Evolutionary Computation (CEC), where DE-based variant algorithms have ranked the top 3 in successive years except for 2013. According to the ways of improvement, current improvement methods can be classified into three main categories: optimizing mutation scheme, adaptive setting of control parameters, and hybridization.

1.1. Optimizing Mutation Scheme

Zhang and Sanderson [17] proposed JADE which implements the current-to-best mutation strategy for the first time with adaptively set CR and mutant factors (F) achieved through evolving the mutation factors and crossover probabilities based on their historical record of success. Based on current-to-pbest strategy, V. Stanovov et al. [17] developed current-to-pbest/r, a new mutation strategy in LSHADE-RSP, which performs better in comparison with the alternative algorithms.

1.2. Adaptive Setting of Control Parameters

As an improvement upon the robustness of JADE, SHADE [18] enhances the adaptive performance of control parameters by adopting a diverse set of parameters to guide control parameter adaptation based on the historical memory of successful parameter settings of JADE. As an extension of the SHADE algorithm, LSHADE [19] incorporates the linear population size reduction strategy to balance the exploration and exploitation. iL-SHADE, an improved version of LSHADE, was proposed by J. Brest et al. [20], in which the memory update mechanism is modified from LSHADE and the initial value of CR is set to 0.8 to enhance population diversity.

1.3. Hybridization

Hybridization is another important strategy to modify DE as plentiful hybridized algorithms concerning DE have been developed. C. Zhang et al. [21] proposed DE-PSO algorithm, where DE is hybridized with PSO for enhancing the exploration ability with most new individuals generated by the modified DE operator. However, the algorithm was only tested on the basic benchmark functions. S. Y. Du and Z. G. Liu [22] proposed PSOJADE by introducing a multicrossover operation and JADE to enhance the global exploration and the local exploitation in PSO. W. Gong et al. [23] proposed DE/BBO, a hybrid DE with BBO, which combines the exploration of DE with the exploitation of BBO effectively, enabling it to generate the promising candidate solutions. A. W. Mohamed et al. [24] proposed a hybridization framework in which CMA-ES is hybridized with LSHADE through adaptive approach, and semiparameter adaptation is adopted to effectively adapt the values of F.

Nevertheless, conventional current-to-pbest mutation strategy in the above hybrid algorithms can only capture the interindividual differential information to form the trial vector, which is single and prone to stagnation. Therefore, it is judicious to integrate it with other algorithms with strong global exploitation capabilities to improve its performance. However, roughly hybridizing swarm-based algorithms with DE rarely performs well. Hence, it is crucial to find an effective way to hybridize the DE algorithm with other algorithms.

Tuna swarm optimization [25] is one of the excellent swarm-based algorithms proposed by L. Xie et al. It shows a strong global search ability due to spiral foraging and parabolic foraging search behavior. This paper introduces TSO into the mutation strategy instead of simply hybridizing TSO with IJADE. In the iteration of each generation, individuals in the population except the optimal individual are selected with a certain probability to generate offspring individuals by IJADE or TSO, respectively. Similarly, the mutation vectors generated by TSO are subject to crossover operations. In addition, a series of improvements are made to improve the convergence efficiency.

The main contributions of this paper are as follows: (1) A novel hybrid mutation strategy is proposed. In the mutation operator, spiral foraging and parabolic foraging are incorporated into classical “current-to-pbest” mutation strategy. (2) To enhance the convergence efficiency of JADE, CR sorting mechanism and CR repairing are introduced. (3) A top α r₁ selection strategy is proposed to accelerate convergence efficiency. (4) The linear population size reduction strategy is introduced to approximate the optimal individual in the final phase of the search.

2. Brief Review of DE and JADE

Optimization problems are common in the real world. The common form is shown as follows [10]:$$\begin{array}{c}\text{(1)}& \text{Objective\hspace{0.17em}function}:f\mathbf{x},\hspace{1em}\mathbf{x}=x_1,x_2,\dots ,x_D,\text{Constraint}:L_j\le x_j\le U_j.\end{array}$$

We have that $\mathbf{x}=x_1,x_2,\dots ,x_D$ is the optimization vector, D is the dimension of the optimization problem, and $L_j$ and $U_j$ are the boundaries.

2.1. Differential Evolution

At the beginning of the optimization problem, DE stochastically generates populations in the search space. The individuals in the population create the next generation in an evolutionary manner. When the individual explores a new location with a better fitness value, the individual moves to that location. There are four main operators in DE, that is, initialization, mutation, crossover, and selection operations. These operations will be discussed in detail as follows.

2.1.1. Initialization

Like other swarm intelligence optimization algorithms, the population is initialized first.$$\begin{array}{}\text{(2)}& {\mathbf{x}}_i|L_j\le x_{i,j}\le U_j,\hspace{1em}i=\mathrm{1,2},\dots ,\text{\hspace{0.17em}NP};j=\mathrm{1,2},\dots ,D,\end{array}$$where ${\mathbf{x}}_i$ is the ith individual, j represents the jth dimension, and NP is the initial population number.$$\begin{array}{}\text{(3)}& x_{i,j}=L_j+rand\mathrm{0,1}{U}_j-L_j.\end{array}$$

2.1.2. Mutation

For each individual ${\mathbf{x}}_i$ in the population, the mutant vector of DE is generated in the following way:$$\begin{array}{}\text{(4)}& \frac{\text{DE}/\text{rand}}{1}:{\mathbf{v}}_{i,g+1}={\mathbf{x}}_{r1,g}+F{\mathbf{x}}_{r2,g}-{\mathbf{x}}_{r3,g}.\end{array}$$

In the above formula, $r1$, $r2$, and $r3$ are randomly selected in the population and $r1\ne r2\ne r3$. Parameter F is used to control the amplification of the difference vector and $0\le F\le 2$.

In addition, the mutant vector can be generated in other ways [26].$$\begin{array}{c}\text{(5)}& \frac{\text{DE}/\text{best}}{1}\text{:}{\mathbf{v}}_{i,g+1}={\mathbf{x}}_{best,g}+F{\mathbf{x}}_{r1,g}-{\mathbf{x}}_{r2,g},\frac{\text{DE}/\text{current}-\text{to}-\text{best}}{1}\text{:}{\mathbf{v}}_{i,g+1}={\mathbf{x}}_{i,g}+\lambda {\mathbf{x}}_{\text{best},g}-x_{i,g}+F{\mathbf{x}}_{r1,g}-{\mathbf{x}}_{r2,g},\\ \frac{\text{DE}/\text{best}}{2}\text{:}{\mathbf{v}}_{i,g+1}={\mathbf{x}}_{best,g}+F{\mathbf{x}}_{r1,g}-{\mathbf{x}}_{r2,g}+{\mathbf{x}}_{r3,g}-{\mathbf{x}}_{r4,g},\\ \frac{\text{DE}/\text{rand}}{2}\text{:}{\mathbf{v}}_{i,g+1}={\mathbf{x}}_{r5,g}+F{\mathbf{x}}_{r1,g}-{\mathbf{x}}_{r2,g}+{\mathbf{x}}_{r3,g}-{\mathbf{x}}_{r4,g}.\end{array}$$

2.1.3. Crossover

The trial vector ${\mathbf{u}}_{i,g}$ is obtained through replacing some components of target vector ${\mathbf{x}}_{i,g}$ with corresponding mutant vector ${\mathbf{v}}_{i,g}$.$$\begin{array}{}\text{(6)}& u_{i,j,g}=\begin{array}{c}{v}_{i,j,g}\text{if\hspace{0.17em}rand}<\text{CR\hspace{0.17em}or\hspace{0.17em}randi}1,D=j,\\ x_{i,j,g}\text{\hspace{0.17em}else.}\end{array}\end{array}$$

randi(1, D) generate a random integer between 0 and D. $\text{CR}\in \mathrm{0,1}$ is the crossover factor which decides the proportion of replaced components in ${\mathbf{x}}_{i,g}$.

2.1.4. Selection

In the selection operation, according to the greedy strategy, the individual of next generation is selected by comparing the trail vector ${\mathbf{u}}_{i,g}$ and the target vector ${\mathbf{x}}_{i,g}$ in DE. The selection method is as follows:$$\begin{array}{}\text{(7)}& {\mathbf{x}}_{i,g+1}=\begin{array}{cc}{\mathbf{u}}_{i,g},& iff{u}_i<f{x}_i,\\ {\mathbf{x}}_{i,g},& \text{else.}\end{array}\end{array}$$

2.2. JADE

The JADE algorithm is a variant of DE that was proposed by Zhang J. et al. [27] in 2009. In recent years, many excellent algorithms have been established based on JADE, and their advantages are as follows.

2.2.1. DE/Current-to-pbest Strategy

The most important improvement is that the new mutation strategy DE/current-to-pbest is implemented with the optional external archive, which uses historical information to provide evolution direction information.

Mutation strategy without external archive is$$\begin{array}{}\text{(8)}& {\mathbf{v}}_{i,g}={\mathbf{x}}_{i,g}+F_i{\mathbf{x}}_{\text{best},g}^p-{\mathbf{x}}_{i,g}+F_i{\mathbf{x}}_{r1,g}-{\mathbf{x}}_{r2,g}.\end{array}$$

Mutation strategy with external archive is$$\begin{array}{}\text{(9)}& {\mathbf{v}}_{i,g}={\mathbf{x}}_{i,g}+F_i{\mathbf{x}}_{\text{best},g}^p-{\mathbf{x}}_{i,g}+F_i{\mathbf{x}}_{r1,g}-{\tilde{x}}_{r2,g}.\end{array}$$

The difference between the above two mutation strategies is the element ${\tilde{x}}_{r2,g}$. In the strategy with external archive, there is a certain possibility for ${\tilde{x}}_{r2,g}$ to be selected from either the current group or the external archive, which enhances the diversity of the population.

2.2.2. Adaptive Control Parameters of F and CR

The control parameters F and CR are updated adaptively to improve the optimization performance.

CR is generated according to the normal distribution with mean ${\mu }_{CR}$ and standard deviation equal to 0.1, and F is generated according to the Cauchy distribution with mean ${\mu }_F$ and standard deviation equal to 0.1. ${\mu }_{CR}$ and ${\mu }_F$ are initially set to 0.5.

S_CR is a collection for storing the CR value of each generation of successful individuals. Similarly, S_F is a collection for storing the F value of each generation of successful individuals.

After each iteration, the values of ${\mu }_{CR}$ and ${\mu }_F$ are updated as follows:$$\begin{array}{c}\text{(10)}& {\mu }_{\text{CR}}=1-c\cdot {\mu }_{\mathrm{CR}}+c\cdot {\text{mean}}_A{S}_{\text{CR}},{\mu }_F=1-c\cdot {\mu }_F+c\cdot {\text{mean}}_L{S}_F.\end{array}$$

We have that c is the learning rate, which is set to 0.1, mean_A(.) refers to the arithmetic mean, and mean_L(.) represents the Lehmer mean, which is defined as follows:$$\begin{array}{}\text{(11)}& {\text{mean}}_L{\mathbf{S}}_F=\frac{{\displaystyle {\sum }_{F\in {\mathbf{S}}_F}{F}^2}}{{\displaystyle {\sum }_{F\in {\mathbf{S}}_F}F}}.\end{array}$$

In general, the control parameters of the DE algorithm are not adaptive [28] and the convergence performance is undesired. Although the convergence speed of JADE is high [29], the single mutation strategy results in poor population diversity and liability of falling into local optimum.

3. IJADE-TSO

3.1. Tuna Swarm Optimization

The JADE algorithm uses a single mutation strategy, giving rise to a greater possibility of the algorithm falling into a local optimum. To compensate for this, two foraging search strategies in tuna swarm optimization (TSO) [25] have been introduced into the mutation operation of IJADE. The two mutation strategies account for a respective percentage of the population to improve population diversity and avoid local optimum.

Tuna swarm optimization is a novel swarm-based metaheuristic algorithm for global optimization. The inspiration for TSO comes from the cooperative foraging behavior of tuna swarm. It mainly consists of two foraging behaviors of tuna swarm: spiral foraging and parabolic foraging. Its global exploration capacity excels its exploitation capacity.

3.1.1. Spiral Foraging

When foraging, the tuna group will form a spiral formation to round up the target in the face of the rapid position change of the target fish group. If a small part of tuna group moves firmly towards a certain direction, the surrounding tuna group will gradually adjust their directions and follow to round up. Each tuna will follow the previous tuna while chasing the target, forming a chain of information transmission. The position of the most important lead tuna is updated by the mutation strategy of IJADE, which augments the foraging efficiency. The mathematical model of the spiral foraging strategy is as follows:$$\begin{array}{c}\text{(12)}& {\mathbf{v}}_{i,g}={\alpha }_1\cdot {\mathbf{x}}_{\text{best},g}+\beta \cdot {\mathbf{x}}_{\text{best},g}-{\mathbf{x}}_{i,g}+{\alpha }_2\cdot {\mathbf{x}}_{i-1,g},\hspace{1em}i=\mathrm{2,3},\dots ,NP\text{,}{\alpha }_1=a+1-a\cdot \frac{g}{g_{\max }},\\ {\alpha }_2=1-a-1-a\cdot \frac{g}{g_{\max }},\\ \beta =e^{bl}\cdot \cos 2\pi b,\\ l=e^{3\cos g_{\max }+1/g-1\pi }.\end{array}$$

We have that ${\mathbf{x}}_{i,g}$ is the ith individual of the $g$+1 generation, ${\mathbf{x}}_{\text{best},g}$ is the current optimal individual (lead tuna), ${\alpha }_1$ and ${\alpha }_2$ are weight coefficients that control the tendency of individuals to move towards either the optimal individual or the previous individual, a is a constant used to determine the extent to which the tuna follows the optimal individual or the previous individual in the initial phase, $g$ denotes the number of current iterations, $g_{\max }$ is the maximum number of iterations, and b is a random number uniformly distributed between 0 and 1.

In the initial stage of foraging, the prey tracked by the lead fish is not necessarily the fish group that provides the most abundant food. Therefore, the whole tuna group needs to scatter to find the location of the fish group with the most abundant food. It is more efficient to stochastically select a point in the space and spirally search around it. The specific mathematical model is described as follows:$$\begin{array}{}\text{(13)}& {\mathbf{v}}_{i,g}={\alpha }_1\cdot {\mathbf{x}}_{rand,g}+\beta \cdot {\mathbf{x}}_{rand,g}-{\mathbf{x}}_{i,g}+{\alpha }_2\cdot {\mathbf{x}}_{i-1,g},\hspace{1em}i=\mathrm{2,3},\dots ,NP.\end{array}$$

We have that ${\mathbf{x}}_{rand,g}$ is a randomly generated reference point in the search space.

Metaheuristic algorithms usually perform extensive global exploration in the early stage and then gradually transition to precise local exploitation. Therefore, TSO changes the reference points of spiral foraging from random individuals to optimal individuals with the increase in iteration. In summary, the final mathematical model of the spiral foraging strategy is as follows:$$\begin{array}{}\text{(14)}& {\mathbf{v}}_{i,g}=\begin{array}{cc}{\alpha }_1\cdot {\mathbf{x}}_{\text{best},g}+\beta \cdot {\mathbf{x}}_{\text{best},g}-{\mathbf{x}}_{i,g}+{\alpha }_2\cdot {\mathbf{x}}_{i-1,g},\hspace{1em}i=\mathrm{2,3},\dots ,NP,& \text{if\hspace{0.17em}rand}<\frac{t}{t_{\max }},\\ {\alpha }_1\cdot {\mathbf{x}}_{rand,g}+\beta \cdot {\mathbf{x}}_{rand,g}-{\mathbf{x}}_{i,g}+{\alpha }_2\cdot {\mathbf{x}}_{i-1,g},\hspace{1em}i=\mathrm{2,3},\dots ,NP,& \text{if\hspace{0.17em}rand}\ge \frac{t}{t_{\max }}.\end{array}\end{array}$$

3.1.2. Parabolic Foraging

To prevent preys from escaping, in addition to spiral foraging, tunas also adopt parabolic foraging. Tuna forms a parabolic formation with prey as a reference point. Meanwhile, tuna forage preys by searching areas around themselves. The two approaches are performed simultaneously, with the assumption that the selection probabilities are 50% for both. The specific mathematical model is described as follows:$$\begin{array}{}\text{(15)}& {\mathbf{v}}_{i,g}=\begin{array}{cc}{\mathbf{x}}_{\text{best},g}+\text{rand}\cdot {\mathbf{x}}_{\text{best},g}-{\mathbf{x}}_{i,g}+TF\cdot p^2\cdot {\mathbf{x}}_{\text{best},g}-{\mathbf{x}}_{i,g},& \text{if\hspace{0.17em}rand}<\text{0.}5\text{,}\\ TF\cdot p^2\cdot {\mathbf{x}}_{i,g},& \text{if\hspace{0.17em}rand}\ge \text{0.}5\text{.}\end{array}\text{(16)}& p={1-\frac{t}{t_{\max }}}^{t/t_{\max }},\end{array}$$where TF is a random number with a value of 1 or −1.

Tuna work cooperatively with the two foraging strategies to hunt for the prey. In each iteration, each individual randomly executes one of the two foraging strategies.

3.2. Crossover Rate Sorting Mechanism and CR Repairing

In order to establish the relationship between CR and the individual fitness values, the CR sorting mechanism [30] is introduced. Firstly, the CR values are generated by Gaussian distribution and then are sorted in an ascending order. This is shown as follows:$$\begin{array}{c}\text{(17)}& {\text{CR}}^{\prime }=\text{sort}\text{CR},{}^{\prime }ascen{d}^{\prime },\text{index}=\text{sort}fX,{}^{\prime }ascen{d}^{\prime },\\ \text{CR}index={\text{CR}}^{\prime }.\end{array}$$

A variant of JADE is introduced in [31], which modifies the crossover factor according to the real crossover rate of each generation. In this way, the adaption of the crossover factor is improved. Its crossover operation becomes$$\begin{array}{c}\text{(18)}& b_{i,j}=\begin{array}{cc}1,& ifrand<\text{CR\hspace{0.17em}or\hspace{0.17em}}randi1,D=j,\\ 0,& \text{else,}\end{array}{\text{CR}}_i=\frac{{\displaystyle {\sum }_{j=1}^D{b}_{i,j}}}{D}.\end{array}$$

By sorting the CR values, the individuals with better fitness are given smaller CR, so their next generation can retain more characteristics of the parent individuals. Meanwhile the poor individuals will be given larger CR and a larger proportion of components will be replaced by the mutated individuals. This helps to improve the exploration efficiency.

With the CR sorting mechanism, the issue of neglecting the relationship between CR values and the individual fitness values in JADE can be effectively addressed. Through CR correction, the actual crossover rate is introduced into the calculation of the crossover factor in the next generation.

3.3. Top α r₁ Selection

In LSHADE-RSP [17], a ranking-based approach was proposed for the selection of r₁ and r₂. In the JADE algorithm, the selection of the r₁ individual is random. To improve the convergence efficiency of the algorithm, we use the top α r₁ selection strategy. The selection of r₁ is shown as follows:$$\begin{array}{}\text{(19)}& r_1=\text{floor}1+\alpha \cdot NP\cdot rand.\end{array}$$

We have that $\alpha \cdot NP$ is the number of candidates for the selection of r₁, and rand is a random value selected in [0, 1]. The individuals with better fitness values will be more likely to be selected. In this way, it is easier to form a difference vector that evolves towards the current optimal individual and expedite convergence.

3.4. Linear Population Size Reduction

The linear population size reduction strategy is introduced to accomplish depth exploration around the optimal individual and facilitate algorithm convergence in the final stage of the search. The mathematical formula is as follows:$$\begin{array}{}\text{(20)}& NP=\text{round}\frac{N{P}_{\min }-N{P}_{\text{init}}}{FE{s}_{\max }}\times FEs+N{P}_{\text{init}}.\end{array}$$

NP_min and NP_init represent the minimum population and the number of initial populations, respectively. However, the increasing number of populations will result in a decrease in population diversity, making it liable to be trapped in local optimum. Therefore, rather than setting NP_min to 4 in [19], NP_min is set to 50 in this paper.

The algorithm pseudocode is shown as Table 1 and the corresponding flowchart of IJADE-TSO algorithm is shown in Figure 1.

Table 1

Pseudocode of IJADE-TSO algorithm.

Algorithm: IJADE-TSO

Initialise population

${\mu }_{CR}=\text{0.}5,{\mu }_F=0.8,A=\varnothing ,p=0.11,Ar=2.6,N{P}_{\min }=50,\alpha =0.6$

for $g$ = 1 to $g_{\max }$ do

for i = 1:NP

${\text{CR}}_i=rand{n}_i{\mu }_{CR},0.1$, $F_i=rand{c}_i{\mu }_F,0.1$

End

$\text{CR}=\text{sort}\text{CR}$

for i = 1:NP

if rand < 0.9 or i = 1

Generate $r_1,r_2,{\mathbf{x}}_{\text{best}}^p$

${\mathbf{v}}_{i,g}={\mathbf{x}}_{i,g}+F_i{\mathbf{x}}_{\text{best},g}^p-{\mathbf{x}}_{i,g}+F_i{\mathbf{x}}_{r1,g}-{\tilde{x}}_{r2,g}$

Else

if rand < 0.5

generate ${\mathbf{v}}_{i,g}$ According to equation (19)

else

generate ${\mathbf{v}}_{i,g}$ According to equation (20)

end

end

if $rand<\text{CR\hspace{0.17em}or\hspace{0.17em}}randi1,D=j$

$u_{i,j,g+1}=v_{i,j,g},b_{i,j}=1$

else

$u_{i,j,g+1}=x_{i,j,g},b_{i,j}=0$

end

${\text{CR}}_i\prime ={\displaystyle {\sum }_{j=1}^D{b}_{i,j}}/D$, $\text{CR}=\text{CR}\prime$

if $f{u}_i\le f{x}_i$

$x_{i,G+1}=u_{i,G},x_{i,G}⟶A,\hspace{1em}{\text{CR}}_i⟶S_{\text{CR}},F_i⟶S_F$

else

$x_{i,G+1}=x_{i,G}$

end

end

Update ${\mu }_{CR}$ , ${\mu }_F$ and NP

Update archive size by removing worst solutions

Update population size by removing worst solutions

end

[figure(s) omitted; refer to PDF]

4. Numerical Experiment and Discussion

4.1. CEC 2014

In this section, we use the 2014 IEEE CEC test suite to verify the performance of the IJADE-TSO by comparing it with state-of-the-art algorithms, including LSHADE [19], iLSHADE [20], SPS-LSHADE-EIG [32], CPI-JADE [33], GEDGWO [34], and DOLTLBO [35]. Among them, LSHADE won a prize at the 2014 IEEE CEC. The first four are all improved algorithms based on JADE. GEDGWO is a variant of GWO [8] applying the Gauss probability model to estimate the distribution. DOLTLBO is an enhanced teaching-learning-based optimization algorithm proposed in 2019. The parameters are the same as the recommended settings in the original works, as reported in Table 2.

Table 2

Parameter settings for the seven algorithms in the CEC 2014 test.

Algorithms	Parameter settings
IJADE-TSO	$N{P}_{\min }=50,{\mu }_{CR}=0.5,{\mu }_F=0.8,p=0.11,r^{arc}=2.6,\alpha =0.6$
SPS-LSHADE-EIG	$N{P}_{\min }=4,E{R}_{init}=1,p=0.11,H=6,M_F=0.5,M_{CR}=0.5,Q=64$
LSHADE	$N{P}_{init}=18\cdot D,N{P}_{\min }=4,r^{arc}=2.6,p=0.11,H=6,M_F=0.5,M_{CR}=0.5$
CPI-JADE	$N{P}_{init}=500,{\mu }_{CR}=0.5,{\mu }_F=0.5,c=0.1,p=0.05$
iLSHADE	$N{P}_{init}=12\cdot D,N{P}_{\min }=4,Ar=2.6,p_{init}=0.2,p_{\min }=0.1,H=6,M_F=0.5,M_{CR}=0.8$
GEDGWO	${\text{NP}}_{\text{init}}=500,\hspace{1em}r3\in \mathrm{0,1},r4\in \mathrm{0,1},r5\in \mathrm{0,1}$
DOLTLBO	${\text{NP}}_{init}=500,jr=0.3,w=10$

The definitions and optimal values of CEC 2014 test suite are provided in [36]. The number of maximum function evaluations (MaxFEs) is set to D × 10000, and D is set to 30 in this paper. Each function is evaluated 51 times independently for statistics. The error value between the best obtained solution and the optimum solution is recorded as the result.

All experiments are performed on a computer with AMD R7 4800U (1.80 GHz) processor and 16 GB of RAM. The programs are implemented by MATLAB 2016B platform.

From Table 3, It is apparent that, for unimodal functions, IJADE-TSO outperforms all other comparison algorithms, and the optimal value can be stably obtained, which confirms the excellent performance of IJADE-TSO in solving unimodal functions. As for the multimodal and hybrid functions, IJADE-TSO ranked in the top four except F5 and F12. IJADE-TSO ranks in the top two except F26 in solving composition functions. It is worth noting that the latter two groups of test functions are more complex, so the superiority of IJADE-TSO in solving complex optimization problems is better demonstrated. SPS-LSHADE-EIG performs well in multimodal functions except for F14, unimodal and multimodal functions, and hybrid functions but performs poorly in composition functions except F26. LSHADE and iLSHADE are similar to SPS-LSHADE-EIG, but LSHADE shows poor performance in F19 and iLSHADE performs poorer than SPS-LSHADE-EIG for composition functions. The performances of CPI-JADE and GEDGWO are poor. DOLTLBO performs well in F23, F25, F26, and F28 of composition functions but poor in other functions.

Table 3

Comparison of statistical results derived from seven algorithms for CEC 2014 test suite.

No.

Types

IJADE-TSO

SPS-LSHADE-EIG

LSHADE

CPI-JADE

GEDGWO

iLSHADE

DOLTLBO

Mean SD

Mean SD

Mean SD

Mean SD

Mean SD

Mean SD

Mean SD

F01

Unimodal functions

1.42E − 14

8.04E − 15

1.50E − 14

8.72E − 15

1.73E − 14

7.69E − 15

1.10E − 04

5.66E − 05

9.88E − 03

1.15E − 02

2.01E − 14

1.34E − 14

8.82E + 05

7.11E + 05

F02

0.00E + 00

0.00E + 00

0.00E + 00

0.00E + 00

1.11E − 15

5.57E − 15

1.57E − 09

1.23E − 09

3.18E − 04

1.12E − 03

1.11E − 15

5.57E − 15

4.96E + 02

7.91E + 02

F03

0.00E + 00

0.00E + 00

0.00E + 00

0.00E + 00

3.34E − 15

1.35E − 14

5.74E − 11

4.39E − 11

1.27E − 05

9.49E − 06

3.34E − 15

1.35E − 14

3.50E + 00

3.32E + 00

F04

Simple multimodal functions

4.90E − 14

3.41E − 14

1.05E − 13

9.33E − 14

8.92E − 14

4.44E − 14

6.21E − 01

6.24E − 01

5.62E − 04

3.95E − 03

5.24E − 14

2.50E − 14

5.78E + 01

3.75E + 01

F05

2.06E + 01

2.40E − 01

2.01E + 01

6.58E − 02

2.01E + 01

3.05E − 02

2.06E + 01

4.91E − 02

2.09E + 01

5.55E − 02

2.01E + 01

1.27E − 01

2.09E + 01

5.95E − 02

F06

0.00E + 00

0.00E + 00

1.93E − 02

1.38E − 01

0.00E + 00

0.00E + 00

1.24E + 01

2.64E + 00

1.09E − 01

3.08E − 01

1.93E − 02

1.38E − 01

1.73E + 01

3.26E + 00

F07

0.00E + 00

0.00E + 00

0.00E + 00

0.00E + 00

0.00E + 00

0.00E + 00

0.00E + 00

0.00E + 00

1.60E − 09

3.72E − 09

0.00E + 00

0.00E + 00

1.62E − 02

1.97E − 02

F08

2.49E − 01

1.78E + 00

1.56E − 13

6.81E − 14

1.92E − 13

7.71E − 14

2.83E + 01

2.10E + 00

4.11E + 01

1.06E + 01

1.14E − 13

0.00E + 00

7.35E + 01

1.71E + 01

F09

7.97E + 00

1.67E + 00

7.96E + 00

3.49E + 00

6.47E + 00

1.68E + 00

7.93E + 01

5.43E + 00

4.71E + 01

1.35E + 01

7.59E + 00

1.69E + 00

8.49E + 01

2.36E + 01

F10

6.16E + 00

3.41E + 00

4.90E − 03

1.15E − 02

3.27E − 03

7.65E − 03

1.17E + 03

1.18E + 02

1.75E + 03

5.03E + 02

7.35E − 03

1.37E − 02

2.93E + 03

6.12E + 02

F11

1.94E + 03

3.05E + 02

1.09E + 03

4.35E + 02

1.24E + 03

2.02E + 02

4.14E + 03

2.46E + 02

2.90E + 03

5.77E + 02

1.08E + 03

2.58E + 02

3.48E + 03

7.63E + 02

F12

1.78E + 00

5.81E − 01

1.02E − 01

3.49E − 02

1.63E − 01

2.25E − 02

9.10E − 01

1.06E − 01

1.42E + 00

5.52E − 01

1.87E − 01

1.51E − 01

1.82E + 00

2.33E − 01

F13

1.27E − 01

1.85E − 02

4.79E − 02

9.91E − 03

1.19E − 01

1.64E − 02

2.51E − 01

3.41E − 02

2.49E − 01

5.35E − 02

9.90E − 02

2.08E − 02

3.65E − 01

7.23E − 02

F14

2.28E − 01

3.98E − 02

2.80E − 01

3.75E − 02

2.34E − 01

2.92E − 02

2.39E − 01

2.56E − 02

2.52E − 01

4.05E − 02

1.66E − 01

3.39E − 02

2.35E − 01

3.75E − 02

F15

2.85E + 00

3.27E − 01

2.91E + 00

7.26E − 01

2.15E + 00

2.33E − 01

8.72E + 00

7.85E − 01

3.82E + 00

1.14E + 00

1.89E + 00

3.20E − 01

7.97E + 00

2.77E + 00

F16

9.96E + 00

5.94E − 01

7.21E + 00

9.30E − 01

8.57E + 00

4.37E − 01

1.13E + 01

2.64E − 01

1.01E + 01

7.26E − 01

7.85E + 00

7.79E − 01

1.16E + 01

2.99E − 01

F17

Hybrid functions

2.31E + 02

1.21E + 02

1.65E + 02

9.74E + 01

1.71E + 02

9.15E + 01

7.58E + 02

1.76E + 02

2.56E + 02

2.58E + 02

2.01E + 02

1.20E + 02

5.10E + 03

4.38E + 03

F18

5.84E + 00

2.88E + 00

5.78E + 00

2.65E + 00

6.21E + 00

2.48E + 00

4.25E + 01

5.98E + 00

3.95E + 01

2.37E + 01

4.84E + 00

2.03E + 00

2.44E + 02

5.01E + 01

F19

3.46E + 00

7.68E − 01

2.86E + 00

7.08E − 01

3.77E + 00

5.85E − 01

4.48E + 00

4.63E − 01

2.75E + 00

7.52E − 01

2.84E + 00

8.20E − 01

1.07E + 01

8.34E + 00

F20

3.48E + 00

2.58E + 00

2.86E + 00

1.26E + 00

2.97E + 00

1.25E + 00

1.67E + 01

1.91E + 00

1.80E + 01

1.49E + 01

2.59E + 00

1.05E + 00

1.78E + 02

5.44E + 01

F21

1.52E + 02

1.12E + 02

7.23E + 01

7.11E + 01

8.00E + 01

7.29E + 01

4.21E + 02

1.40E + 02

2.51E + 02

2.33E + 02

1.06E + 02

8.55E + 01

2.78E + 03

1.82E + 03

F22

3.58E + 01

1.91E + 01

2.78E + 01

2.87E + 01

2.91E + 01

2.34E + 01

1.07E + 02

4.66E + 01

1.99E + 02

1.03E + 02

3.77E + 01

4.28E + 01

2.13E + 02

9.12E + 01

F23

Composition functions

2.00E + 02

1.89E − 01

3.15E + 02

4.16E − 13

3.15E + 02

4.02E − 13

3.15E + 02

4.02E − 13

2.00E + 02

1.28E + 00

3.15E + 02

4.02E − 13

2.00E + 02

0.00E + 00

F24

2.00E + 02

0.00E + 00

2.24E + 02

1.32E + 00

2.24E + 02

1.00E + 00

2.22E + 02

6.45E − 01

2.00E + 02

2.85E − 03

2.14E + 02

1.07E + 01

2.00E + 02

5.03E − 11

F25

2.00E + 02

4.10E − 03

2.03E + 02

5.26E − 02

2.03E + 02

4.40E − 02

2.03E + 02

2.80E − 02

2.02E + 02

9.64E − 01

2.03E + 02

1.34E − 01

2.00E + 02

0.00E + 00

F26

1.00E + 02

1.88E − 02

1.00E + 02

1.40E − 02

1.00E + 02

1.62E − 02

1.00E + 02

2.95E − 02

1.00E + 02

5.30E − 02

1.00E + 02

2.77E − 02

1.00E + 02

6.91E − 02

F27

2.00E + 02

1.10E − 02

3.00E + 02

1.11E − 13

3.00E + 02

2.56E − 13

3.59E + 02

4.98E + 01

3.65E + 02

4.80E + 01

3.04E + 02

1.47E + 01

2.08E + 02

1.88E + 01

F28

2.02E + 02

1.23E + 01

8.40E + 02

1.35E + 01

8.45E + 02

1.55E + 01

8.44E + 02

1.57E + 01

5.51E + 02

3.03E + 02

8.46E + 02

1.96E + 01

2.00E + 02

2.51E − 01

F29

6.48E + 02

1.00E + 02

7.16E + 02

3.16E + 00

7.16E + 02

3.44E + 00

7.16E + 02

1.71E + 00

1.41E + 02

8.70E + 01

7.16E + 02

3.67E + 00

2.64E + 04

1.77E + 05

F30

8.38E + 02

8.18E + 02

1.27E + 03

5.75E + 02

1.23E + 03

5.65E + 02

9.30E + 02

1.52E + 02

4.34E + 02

6.02E + 01

2.22E + 03

1.01E + 03

1.95E + 03

5.92E + 02

Figure 2 shows the ranking of the algorithm in each test function. The closer to the center of the circle, the higher the ranking, and vice versa. The area enclosed by the curves shows the overall ranking of the algorithms.

[figure(s) omitted; refer to PDF]

4.1.1. Friedman Test

The Friedman test is used [37] to analyze the performance difference of algorithms on the 30 test functions. Table 4 shows the average ranking results of each algorithm.

Table 4

Friedman test.

Algorithm	IJADE-TSO	SPS-LSHADE-EIG	LSHADE	CPI-JADE	GEDGWO	iLSHADE	DOLTLBO
Rank	2.92	2.93	3.23	5.22	4.77	3.10	5.83

Friedman test: p value is 4.43e-11; Chi-square is 60.03.

The ranking value of IJDAE-TSO is 2.92, ranking the top among all algorithms. The rankings of the other six algorithms are as follows: SPS-LSHADE-EIG, iLSHADE, LSHADE, GEDGWO, CPI-JADE, and DOLTLBO. Note that, compared with the improved variant of JADE, CPI-JADE, which ranks sixth, IJDAE-TSO ranks the first, proving the effectiveness of the improvement strategies. The p value calculated by Iman-Davenport test is 4.43e-11, showing that there is a significant difference between algorithms.

4.1.2. Wilcoxon Signed-Rank Test

Reference [38] shows that it is insufficient to analyze and compare the algorithms only according to the average value. Because, in some tests, the results produced by the algorithms are slightly different, and the occurrence of these differences may be accidental, to verify whether the statistical difference between IJADE-TSO and the comparison algorithms is accidental or not, the Wilcoxon signed-rank test is used to analyze the performances of the IJADE-TSO and the other six algorithms. Table 5 lists the Wilcoxon signed-rank test [39] results of each competitor and IJADE-TSO with the significance level set to 0.05. In Table 5, “+” indicates that the optimization result of IJADE-TSO is better than the competitor, “−” denotes the opposite, and “=” represents that the optimization results are similar. “R+” denotes the magnitude with which IJADE-TSO surpasses the comparison algorithm, and “R–” indicates the opposite result. The statistical results of the test are also given in the last row of the table. According to the statistical results in Table 5, among the 30 test functions of CEC 2014 test set, IJADE-TSO shows a performance that is similar to those of SPS-LSHADE-EIG, LSHADE, and iLSHADE and precedes CPI-JADE in all test functions except F5, F7, F12, and F14.

Table 5

Wilcoxon signed-rank test.

No.	SPS-LSHADE-EIG				LSHADE				CPI-JADE				GEDGWO				iLSHADE				DOLTLBO
	p value	R+	R−	Win	p value	R+	R−	Win	p value	R+	R−	Win	p value	R+	R−	Win	p value	R+	R−	Win	p value	R+	R−	Win
F1	6.27E − 01	193	158	=	3.56E − 02	170.5	60.5	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	1.53E − 02	140	31	+	5.15E − 10	1326	0	+
F2	1.00E + 00	0	0	=	5.00E − 01	3	0	=	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.00E − 01	3	0	=	5.15E − 10	1326	0	+
F3	1.00E + 00	0	0	=	2.50E − 01	6	0	=	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	2.50E − 01	6	0	=	5.15E − 10	1326	0	+
F4	4.62E − 05	529.5	65.5	+	5.82E − 05	421.5	43.5	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	6.82E − 01	115	95	=	5.15E − 10	1326	0	+
F5	5.15E − 10	0	1326	−	5.15E − 10	0	1326	−	4.04E − 01	574	752	=	2.80E − 09	1297	29	+	1.67E − 09	20	1306	−	2.65E − 09	1298	28	+
F6	1.00E + 00	1	0	=	1.00E + 00	0	0	=	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	1.00E + 00	1	0	=	5.15E − 10	1326	0	+
F7	1.00E + 00	0	0	=	1.00E + 00	0	0	=	1.00E + 00	0	0	=	5.15E − 10	1326	0	+	1.00E + 00	0	0	=	5.15E − 10	1326	0	=
F8	1.69E − 09	7	1218	−	2.51E − 09	7	1169	−	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	1.62E − 09	0	1176	−	5.15E − 10	1326	0	+
F9	8.37E − 01	641	685	=	1.08E − 04	250	1076	−	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	3.99E − 01	573	753	=	5.15E − 10	1326	0	+
F10	5.15E − 10	0	1326	−	5.15E − 10	0	1326	−	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	0	1326	−	5.15E − 10	1326	0	+
F11	9.66E − 09	51	1275	−	5.46E − 10	1	1325	−	5.15E − 10	1326	0	+	9.87E − 10	1315	11	+	5.46E − 10	1	1325	−	6.15E − 10	1323	3	+
F12	5.15E − 10	0	1326	−	5.15E − 10	0	1326	−	4.42E − 09	37	1289	−	1.64E − 03	327	999	−	5.46E − 10	1	1325	−	9.55E − 01	657	669	=
F13	5.15E − 10	0	1326	−	1.27E − 02	397	929	−	5.15E − 10	1326	0	+	5.46E − 10	1325	1	+	3.77E − 07	121	1205	−	5.15E − 10	1326	0	+
F14	1.87E − 07	1219	107	+	3.30E − 01	767	559	=	1.13E − 01	832	494	=	3.45E − 03	975	351	+	1.97E − 08	64	1262	−	1.96E − 01	801	525	=
F15	5.93E − 01	720	606	=	1.32E − 09	16	1310	−	5.15E − 10	1326	0	+	2.21E − 06	1168	158	+	5.46E − 10	1	1325	−	5.15E − 10	1326	0	+
F16	5.15E − 10	0	1326	−	7.35E − 10	6	1320	−	5.46E − 10	1325	1	+	1.11E − 01	833	493	=	5.46E − 10	1	1325	−	5.15E − 10	1326	0	+
F17	1.86E − 03	331	995	−	2.01E − 02	415	911	−	5.15E − 10	1326	0	+	9.18E − 01	652	674	=	1.71E − 01	517	809	=	5.15E − 10	1326	0	+
F18	8.37E − 01	641	685	=	2.85E − 01	777	549	=	5.15E − 10	1326	0	+	1.11E − 09	1313	13	+	8.98E − 02	482	844	=	5.15E − 10	1326	0	+
F19	6.91E − 04	301	1025	−	2.63E − 02	900	426	+	3.03E − 08	1254	72	+	2.79E − 05	216	1110	−	5.06E − 04	292	1034	−	5.15E − 10	1326	0	+
F20	4.88E − 01	589	737	=	5.42E − 01	598	728	=	5.15E − 10	1326	0	+	1.18E − 09	1312	14	+	1.77E − 01	519	807	=	5.15E − 10	1326	0	+
F21	1.00E − 04	248	1078	−	3.19E − 04	279	1047	=	1.05E − 09	1314	12	+	4.69E − 02	875	451	+	2.63E − 02	426	900	−	5.15E − 10	1326	0	+
F22	5.85E − 07	130	1196	−	2.29E − 07	111	1215	−	2.10E − 09	1302	24	+	7.35E − 10	1320	6	+	2.04E − 03	334	992	−	5.15E − 10	1326	0	+
F23	5.13E − 10	1326	0	+	5.13E − 10	1326	0	+	5.13E − 10	1326	0	+	1.10E − 05	1132	194	+	5.13E − 10	1326	0	+	2.40E − 09	0	1128	−
F24	5.14E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.14E − 10	1326	0	+
F25	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	1.63E − 09	0	1176	−
F26	5.15E − 10	0	1326	−	2.51E − 02	424	902	=	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	4.74E − 05	229	1097	−	5.15E − 10	1326	0	+
F27	5.12E − 10	1326	0	+	5.12E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.14E − 10	1326	0	+	5.46E − 10	1325	1	+
F28	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	5.15E − 10	1326	0	+	6.93E − 10	1321	5	+	5.15E − 10	1326	0	+	1.49E − 06	139	1136	−
F29	2.68E − 05	1111	215	+	6.02E − 05	1091	235	+	3.87E − 05	1102	224	+	5.46E − 10	1	1325	−	4.55E − 04	1037	289	+	7.80E − 10	1319	7	+
F30	3.89E − 03	971	355	+	2.96E − 03	980	346	+	2.06E − 02	910	416	+	6.56E − 03	373	953	−	6.36E − 08	1240	86	+	2.18E − 07	1216	110	+
+/−/=	9/12/9				10/11/9				26/1/3				24/4/2				8/13/9				24/3/3

4.1.3. Convergence Curve of Algorithms

To further illustrate the convergence performance of IJADE-TSO, the average error convergence curves in solving the test function are given in Figure 3.

[figure(s) omitted; refer to PDF]

For most unimodal functions, simple multimodal functions, and hybrid functions, IJADE-TSO shows better convergence accuracy and higher convergence speed compared to GEDGWO, CPI-JADE, and DLOTLBO and slower convergence speed compared to LSHADE, iLSHADE, and SPS-LSHADE-EIG. For the composite function, the convergence curve of IJADE-TSO drops slightly at the initial stage of iteration and then falls into local optimization, but it breaks through local optimization at the later stage of iteration and shows a significant decline. For composition functions, LSHADE, SPS-LSHADE-EIG, and iLSHADE converge quickly, but their accuracy is significantly worse than that of IJADE-TSO, and they are apt to fall into the local optimum. As mentioned above, IJADE-TSO possesses both strong convergence ability and the capacity to break through local optimization.

4.1.4. Box Graph of the Results

The following box graphs are drawn to analyze the distribution characteristics of the results for CEC 2014 test functions.

As shown in Figure 4, the center mark of each box represents the median value of the results of 51 runs, and the bottom and upper edges of each box represent the third and fourth quartiles, respectively. Outliers are drawn separately with “+.” IJADE-TSO has no outliers in 13 test functions (F2-F3, F5–F7, F9, F13-F14, F17, F19, F24, F26, and F29). The distribution of the IJADE-TSO solutions is more centralized, with rarely produced outliers, except for F30. Thus, it is safe to conclude that IJADE-TSO is more robust than the competing algorithms.

[figure(s) omitted; refer to PDF]

4.2. Parameters Identification of Photovoltaic Models

Global warming has become an increasingly serious issue threatening the survival of mankind nowadays. In response to the environmental impact of global warming, solar energy has become widely exploited as a clean energy source [40]. Solar energy has many advantages, such as being nonpolluting, renewable, cheap, and easy to obtain. Photovoltaic systems are commonly used in daily life as an important device for collecting solar energy. PV models are crucial for PV systems, as they are used to imitate the response of real PV cells, fitting measured current-voltage (I-V) data under all operating conditions. The most widely used PV systems are single-diode model (SDM), double-diode model (DDM), and PV module model [41]. It is necessary to identify the parameters in the established photovoltaic models to make the model more consistent with the measured data. In this section, the proposed IJADE-TSO algorithm is used to identify parameters of photovoltaic models.

4.2.1. Problem Statement

In [42], there are three PV models, single-diode model, double-diode model, and PV module model, which are not repeated in this paper. The PV parameter identification models referenced from [41, 42] are as follows.

In general, parameters extraction problem is usually transformed into a class of optimization problem to leverage the optimization algorithms. Similar to [42], the root mean square error (RMSE) is also used as the objective function in this study, which is defined as follows:$$\begin{array}{}\text{(21)}& \text{RMSE}x=\sqrt{\frac{1}{N}{\displaystyle \sum _{k=1}^{N}f{V_k,I_k,\mathbf{x}}^2}},\end{array}$$where N is the number of measured I-V data samples and x is a vector that concludes the unknown parameters to be extracted. For different PV models, their objective functions are represented as follows.

For single-diode model,$$\begin{array}{c}\text{(22)}& fV,I,\mathbf{x}=I_{ph}-I_o\exp \frac{V+I{R}_s}{a{V}_t}-1-\frac{V+I{R}_s}{R_{sh}}-I,\mathbf{x}=I_{ph},I_o,R_s,R_{sh},a.\end{array}$$

For double-diode model,$$\begin{array}{c}\text{(23)}& fV,I,\mathbf{x}=I_{ph}-I_{o1}\exp \frac{V+I{R}_s}{a_1{V}_t}-1-I_{o2}\exp \frac{V+I{R}_s}{a_2{V}_t}-1-\frac{V+I{R}_s}{R_{sh}}-I\mathbf{x}=I_{ph},I_{o1},I_{o1},R_s,R_{sh},a_1,a_2.\end{array}$$

For PV module based on the SDM,$$\begin{array}{c}\text{(24)}& fV,I,\mathbf{x}=I_{ph}-I_o\exp \frac{V+I{R}_s}{a{V}_t}-1-\frac{V+I{R}_s}{R_{sh}}-I,\mathbf{x}=I_{ph},I_o,R_s,R_{sh},a.\end{array}$$

4.2.2. Experimental Results and Analysis

To evaluate the performance of IJADE-TSO, we employ it to extract parameters of SDM, DDM, and PV module model. The I-V data of SDM and DDM are obtained from [43], which are measured on a 57 mm diameter commercial (R.T.C France) silicon solar cell under 1000 W/m² at 33°C.

(1) Results for the Three Models. In order to further show the quality of the results, we have introduced individual absolute error (IAE).$$\begin{array}{}\text{(25)}& \text{IAE}=I_m-I_c.\end{array}$$

I_m is the measured current and I_c is the simulated current. IAE represents the absolute error between them.

The I-V characteristics obtained by IJADE-TSO and the individual absolute error (IAE) between the experimental data and simulated data are shown in Tables 6–8 and the I-V characteristics are plotted in Figure 5. In addition, Tables 6–8 present the individual absolute error (IAE) between the measured data and calculated data. From Table 6, all the IAE values of single-diode model are smaller than 2.5434E − 03. From Table 7, all the IAE values of double-diode model are smaller than 1.6284E − 03. From Table 8, all the IAE values of double-diode model are smaller than 4.8328E − 03. The results show that the calculated data obtained by IJADE-TSO are remarkably in accordance with the measured data over the whole voltage range, which indicates that IJADE-TSO identifies highly accurate parameters.

Table 6

IAE of IJADE-TSO for each measurement on single-diode model.

Item	Vm (V)	Im (A)	Ic (A)	IAE
1	−0.2057	0.764	0.7641	8.7704E − 05
2	−0.1291	0.762	0.7627	6.6309E − 04
3	−0.0588	0.7605	0.7614	8.5531E − 04
4	0.0057	0.7605	0.7602	3.4601E − 04
5	0.0646	0.76	0.7591	9.4479E − 04
6	0.1185	0.759	0.7580	9.5765E − 04
7	0.1678	0.757	0.7571	9.1654E − 05
8	0.2132	0.757	0.7561	8.5864E − 04
9	0.2545	0.7555	0.7551	4.1313E − 04
10	0.2924	0.754	0.7537	3.3612E − 04
11	0.3269	0.7505	0.7514	8.9097E − 04
12	0.3585	0.7465	0.7474	8.5385E − 04
13	0.3873	0.7385	0.7401	1.6172E − 03
14	0.4137	0.728	0.7274	6.1777E − 04
15	0.4373	0.7065	0.7070	4.7265E − 04
16	0.459	0.6755	0.6753	2.1985E − 04
17	0.4784	0.632	0.6308	1.2417E − 03
18	0.496	0.573	0.5719	1.0716E − 03
19	0.5119	0.499	0.4996	6.0702E − 04
20	0.5265	0.413	0.4136	6.4879E − 04
21	0.5398	0.3165	0.3175	1.0101E − 03
22	0.5521	0.212	0.2122	1.5494E − 04
23	0.5633	0.1035	0.1023	1.2487E − 03
24	0.5736	−0.01	−0.0087	1.2825E − 03
25	0.5833	−0.123	−0.1255	2.5074E − 03
26	0.59	−0.21	−0.2085	1.5277E − 03

Table 7

IAE of IJADE-TSO for each measurement on double diode model.

Item	Vm (V)	Im (A)	Ic (A)	IAE
1	−0.2057	0.764	0.7640	1.6588E − 05
2	−0.1291	0.762	0.7626	6.0410E − 04
3	−0.0588	0.7605	0.7613	8.3770E − 04
4	0.0057	0.7605	0.7602	3.2621E − 04
5	0.0646	0.76	0.7591	8.9232E − 04
6	0.1185	0.759	0.7581	8.7858E − 04
7	0.1678	0.757	0.7572	1.8861E − 04
8	0.2132	0.757	0.7562	7.5639E − 04
9	0.2545	0.7555	0.7552	3.2270E − 04
10	0.2924	0.754	0.7537	2.7765E − 04
11	0.3269	0.7505	0.7514	8.9913E − 04
12	0.3585	0.7465	0.7473	8.0144E − 04
13	0.3873	0.7385	0.7400	1.5107E − 03
14	0.4137	0.728	0.7272	7.5305E − 04
15	0.4373	0.7065	0.7069	3.5030E − 04
16	0.459	0.6755	0.6752	2.8946E − 04
17	0.4784	0.632	0.6308	1.2392E − 03
18	0.496	0.573	0.5720	1.0053E − 03
19	0.5119	0.499	0.4997	7.0614E − 04
20	0.5265	0.413	0.4137	7.3367E − 04
21	0.5398	0.3165	0.3175	1.0462E − 03
22	0.5521	0.212	0.2121	1.2300E − 04
23	0.5633	0.1035	0.1022	1.3367E − 03
24	0.5736	−0.01	−0.0088	1.2082E − 03
25	0.5833	−0.123	−0.1255	2.5434E − 03
26	0.59	−0.21	−0.2084	1.6284E − 03

Table 8

IAE of IJADE-TSO for each measurement on PV module model.

Item	Vm (V)	Im (A)	Ic (A)	IAE
1	0.1248	1.0315	1.0291	2.3808E − 03
2	1.8093	1.03	1.0274	2.6189E − 03
3	3.3511	1.026	1.0257	2.5820E − 04
4	4.7622	1.022	1.0241	2.1072E − 03
5	6.0538	1.018	1.0223	4.2918E − 03
6	7.2364	1.0155	1.0199	4.4307E − 03
7	8.3189	1.014	1.0164	2.3631E − 03
8	9.3097	1.01	1.0105	4.9615E − 04
9	10.2163	1.0035	1.0006	2.8710E − 03
10	11.0449	0.988	0.9845	3.4516E − 03
11	11.8018	0.963	0.9595	3.4783E − 03
12	12.4929	0.9255	0.9228	2.6612E − 03
13	13.1231	0.8725	0.8726	9.9663E − 05
14	13.6983	0.8075	0.8073	2.2574E − 04
15	14.2221	0.7265	0.7283	1.8365E − 03
16	14.6995	0.6345	0.6371	2.6380E − 03
17	15.1346	0.5345	0.5362	1.7131E − 03
18	15.5311	0.4275	0.4295	2.0113E − 03
19	15.8929	0.3185	0.3188	2.7448E − 04
20	16.2229	0.2085	0.2074	1.1105E − 03
21	16.5241	0.101	0.0962	4.8328E − 03
22	16.7987	−0.008	−0.0083	3.2539E − 04
23	17.0499	−0.111	−0.1109	6.3517E − 05
24	17.2793	−0.209	−0.2092	2.4727E − 04
25	17.4885	−0.303	−0.3009	2.1364E − 03

[figure(s) omitted; refer to PDF]

(2) Statistical Results and Convergence Speed. To validate the superior performance of IJADE-TSO, the comparisons are carried out with other algorithms including MLHADE [44], improved JAYA [45], JADE [27], crow search algorithm (CSA) [46], and multiple learning backtracking search algorithm (MLBSA) [47]. Among the above algorithms, MLSHAPE, MLBSA, IJAYA, and CSA are excellent algorithms that have been applied in the field of photovoltaic parameter identification in recent years. JADE is the prototype of the improved algorithm in this paper. For fair comparison, the value of function evaluations of all algorithms is set to 50000 for all problems. Meanwhile, each algorithm is tested 30 times independently. The population sizes of all compared algorithms are set to 50. Other algorithm parameter settings are the same as the references.

The statistical results are reported in Table 9. For single-diode model, IJADE-TSO has the same accuracy as MLSHADE and MLBSA. The Min RMSE searched by all algorithms have no difference except IJAYA. As for standard deviation, IJADE-TSO is better than its competitors. For double-diode model, only IJADE-TSO and MLSHADE algorithms obtain the same Min RMSE, but the Mean and SD RMSE of IJADE-TSO ranks the first. For PV module model, the minimum RMSE of PV model obtained by all algorithms are no different. Only MLBSA, IJADE-TSO, and JADE ranked the first in the Mean RMSE. The SD value obtained by IJADE-TSO ranks the first. Overall, IJADE-TSO has better search ability and robustness than the comparison algorithms and can steadily obtain the minimum RMSE value in all three PV parameter identification models.

Table 9

Statistical results of RMSE of different algorithms for three models.

Model	Algorithm	IJADE-TSO	MLSHADE	MLBSA	IJAYA	JADE	CSA
Single-diode model	Min	9.8602E − 04	9.8602E − 04	9.8602E − 04	9.8613E − 04	9.8602E − 04	9.8602E − 04
	Mean	9.8602E − 04	9.8602E − 04	9.8602E − 04	1.0030E − 03	9.8624E − 04	1.0362E − 03
	SD	1.9126E − 17	3.6556E − 17	7.0565E − 11	6.1010E − 05	1.1128E − 06	1.0178E − 04
Double-diode model	Min	9.8248E − 04	9.8248E − 04	9.8249E − 04	9.8354E − 04	9.8863E − 04	9.8467E − 04
	Mean	9.8296E − 04	9.8336E − 04	9.8647E − 04	1.0128E − 03	1.1220E − 03	1.3258E − 03
	SD	1.2228E − 06	1.4528E − 06	5.1996E − 06	6.7236E − 05	1.6138E − 04	5.1016E − 04
PV module model	Min	2.4251E − 03	2.4251E − 03	2.4251E − 03	2.4251E − 03	2.4251E − 03	2.4251E − 03
	Mean	2.4251E − 03	2.4373E − 03	2.4251E − 03	2.4377E − 03	2.4251E − 03	2.6458E − 03
	SD	1.9641E − 17	4.6443E − 05	1.6392E − 08	2.9308E − 05	1.9823E − 17	6.2071E − 04

The convergence curves among all competitors for three PV models are shown in Figure 6. Obviously, for the single-diode model, the descent curve of the IJADE-TSO algorithm is significantly higher compared to the other algorithms in the initial parts, and then it declines fastest and obtains the best results after 10,000 function evaluations. Although MLSHADE and MLBSA also obtain the best results, they are significantly slower than IJADE-TSO. For the double-diode model, IJADE-TSO achieves the best results with the least number of function evaluations. For the PV model, IJADE-TSO, MLBSA, and JADE all achieve the best results, but IJADE-TSO uses the least number of function evaluations and its iteration curve declines the fastest. Among the three PV models, the descent curve of the IJADE-TSO declines the fastest compared with other algorithms and achieves the best results.

[figure(s) omitted; refer to PDF]

In order to comprehensively compare the performances of all algorithms in the three PV models, we conducted a Friedman test with a significance of 0.05, and the results are shown in Table 10.

Table 10

Friedman test results.

Algorithms	IJADE-TSO	MLSHADE	MLBSA	IJAYA	JADE	CSA
Min ranksa	1	2	3.67	5	4	5.33
Mean ranksb	1	2.67	3	4.67	3.67	6
SD ranksc	1	3	3	4.33	3.67	6

^aFriedman test: p value is 0.0305; Chi-square is 12.33. ^bFriedman test: p value is 0.0262; Chi-square is 12.71. ^cFriedman test: p value is 0.0382; Chi-square is 11.76.

In terms of Min, Mean, and SD of RMSE, IJADE-TSO excels all other algorithms, ranking the first, which shows that IJADE-TSO has an excellent performance in terms of accuracy, robustness, and computational efficiency.

5. Conclusions

This paper proposed a novel hybrid algorithm, named IJADE-TSO. The spiral foraging search and parabolic foraging search of TSO are introduced into the mutation strategy in IJADE to improve the exploration ability and population diversity. Meanwhile, this paper integrates JADE with three strategies: CR sorting mechanism and CR repairing, top α r₁ selection, and population linear reduction.

The proposed IJADE-TSO is evaluated using CEC 2014 benchmarks with dimensionality of 30 and photovoltaic parameter identification problem. Although the statistical results of the CEC 2014 benchmarks demonstrate that IJADE-TSO is slightly superior to SPS-LSHADE-EIG and iLSAHDE, IJADE-TSO is significantly better than other comparison algorithms such as MLHADE in solving photovoltaic parameter identification problem. This shows that the two mutant strategies combined contributed significantly to expanding the search scope and avoiding local optimization, and three other strategies are conducive to improve the convergence efficiency. IJADE-TSO achieves a good balance in exploration and exploitation and demonstrates great value in solving practical engineering problems.

The current spiral foraging and parabolic foraging strategies are selected with a stationary probability, which means the selected strategy may not be optimal and an adaptive selection strategy needs to be developed in the future.

Acknowledgments

The authors acknowledge funding received from the following science foundations: the National Natural Science Foundation of China (no. 62101590), the Science Foundation of the Shanxi Province, China (2021JM-223, 2022JQ-584, and 2021JM-224), and Military Scientific Research Project (KJ2020C009002).

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