A Level-Based Learning Swarm Optimizer with Stochastic Fractal Search for Parameters Identification of Solar Photovoltaic Models

Abstract

As the most popular renewable energy, solar energy could be converted into electricity by photovoltaic (PV) systems directly. To maximize the effectiveness of the conversion, it is critical to find the precise and accurate parameters of the PV model. In this paper, we propose a level-based learning swarm optimizer with stochastic fractal search (LLSOF) to tackle the parameter estimation of several kinds of solar PV models. The population is separated into multiple levels according to their fitness at first. The individuals at the lower levels evolve through learning from the individuals at the higher levels. Benefiting from the interactive learning among levels, the population could approach the multiple optimal regions rapidly. To enhance the local search ability, stochastic fractal search is introduced to locate the optima accurately. Combination of both, the proposed LLSOF could achieve a good balance on both exploration and exploitation. To evaluate the performance of LLSOF, it is used to obtain the parameters of three PV models and compared with nine well-established algorithms. Comparative results validate the excellent performance of LLSOF. Moreover, the application manufactory’s data sheets report the superior efficiency and effectiveness of LLSOF for the parameter estimation of PV systems.

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

To cope with the predictable energy crisis and serious environmental problems, countries around the world endeavor to find renewable energy. Among the various renewable energies, solar energy draws much attention due to its advantages such as cleanliness, wide availability, acquisition, sustainability, and so on [1]. Solar energy could be obtained and utilized through the sunshine to electricity conversion. Photoelectric conversion, which is based on photovoltaic (PV) systems, is one of the well-known utilization of solar energy. Accurate PV modeling and parameter extraction could represent the nonlinear current-voltage (I-V) characteristics of solar cells [2]. To analyze the cell characteristics during various operating conditions, equivalent circuit models are applied. The most frequently used models are single diode model (SDM), double diode model (DDM), and three-diode model (TDM) [3]. The PV model is usually described by a set of nonlinear equations and the performance of the model depends mainly on the involving uncertain parameters. Therefore, it is critical to accurately identify these involving unknown parameters of the model. However, the circuit equation of the PV model is nonlinear, implicit, and multimodal; it is quite a challenge to identify the unknown parameters [4]. Moreover, PV systems are usually placed in the outdoors environment, and the temperature and radiation produce an effect on the PV systems. To improve the effectiveness and efficiency, it requires obtaining the best performance under different conditions, which depends on accurate and reliable parameter values. Therefore, a feasible optimization method to identify the parameters of the PV model is needed.

In recent years, many works have been dedicated to estimating the parameters of cells and modules, including metaheuristic and analytical methods. Analytical methods reduce the number of parameters to be solved by using a specific set of data to analyze. Although these methods could consume fewer computational resources, the accuracy is hard to guarantee. Metaheuristic techniques are a category of models in evolutionary computation inspired by biological patterns or real-world problems to mimic physical laws. Compared with the traditional analytical methods, metaheuristic methods show significant potential in dealing with complex optimization problems due to the obvious advantages such as (1) obtaining multiple optima during one run and (2) could handle nonlinear problems without gradient information or extra constraints. Through the literature review, it is seen that the studies using metaheuristics become more and more popular and show promising performance for parameter estimation of the PV models.

1.1. Related Works

Numerous approaches and their alternatives have already been used to estimate the parameters of the PV model. A comprehensive survey [3, 4] has been made on the related works of identification of unknown parameters of solar cell models, including the electrical circuit models of PV cells, parameters extraction of PV models under different conditions, performance criterion, the proposed approaches, and future research directions. The existing research can be roughly classified into the following four categories.

(1) Traditional evolutionary algorithms (EAs)-based methods. As one of the typical EAs, the genetic algorithm (GA) has been applied to identify the parameters of PV cells and/or modules [5, 6]. GA was also combined with the simulated annealing to extract the unknown variables of SDM of the PV cell or module [7]. Moreover, differential evolution (DE) and its variants [8–13] garnered a lot of interest in identifying the PV models’ parameters. In reference [14], DE is recommended for the evaluation of PV cell parameters under various operating conditions. Lambert W function and a preliminary metaheuristic step were introduced to select the optimal mutation and crossover rates of DE for each specific I-V characteristic of PV cells and modules. In order to fast and accurately estimate the parameters of PV models, Li et al. [15] proposed a memetic adaptive DE (MADE), in which the success-history-based adaptive DE is used for the global search while the Nelder–Mead simplex method is employed for the local search. The experiments showed MADE could obtain competitive results with fewer computational resources. Liang et al. [16], proposed a self-adaptive ensemble-based differential DE (SEDE), in which three different mutation strategies with different properties are combined. Gao et al. [17] proposed directional permutation DE (DPDE). Through fully utilizing the information arisen from the search population and the direction of differential vectors, DPDE could achieve robust and competitive results for parameter estimation on SDM, DDM, TDM, and PV modules.

(2) Swarm intelligence (SI)-based methods. The SI technique is inspired by the intelligence and collective behavior of biological communities or the self-organized social processes. As one of the perspectives, particle swarm optimization (PSO) and its variants [18–23] have been proposed for estimation of PV models’ parameters. An improved mutated PSO (MPSO) [24] with an adaptive mutation strategy is proposed to identify the optimal parameters of different photovoltaic models. Liang et al. [25] proposed a classified perturbation mutation-based PSO algorithm which is evaluated by extracting parameters of five different photovoltaic models under different conditions. In reference [26], fractional chaotic ensemble PSO (FC-EPSO) was proposed to model solar PV modules under different environmental conditions. In reference [27], PSO and grey wolf optimization (GWO) were hybrid to extract unknown parameters of photovoltaic SDM, in which a new cost function based on the values of the datasheet was proposed. In addition, the salp swarm inspired algorithm (SSA) [28]and the opposition-based learning modified salp swarm algorithm (OLMSSA) [29] were proposed for extracting the parameters of the electrical equivalent circuit of PV cells. Moreover, the whale optimization algorithm (WOA) [30, 31], fireworks algorithm [32], artificial bee colony (ABC) algorithm [33], the bacterial foraging optimization approach [34], cuckoo search (CS) [35], tunicate swarm algorithm (TSA) [36], bird collective-behavioral based optimization (COOT) [37], and other swarm intelligence-based approaches are also applied to accurately model the solar photovoltaic cell properties.

(3) Physics and objective law inspired technique. In reference [38], a physics-based meta-heuristic algorithm called resistance-capacitance optimization algorithm (RCOA) based on the concept of resistance-capacitance circuit response was proposed. Combined with an improved version of the Newton–Raphson method, RCOA could find the optimal PV parameters in fewer iterations. In reference [39], Gaussian and Cauchy mutation-based hunger games search optimizer and enhanced Newton–Raphson method were proposed to solve the behavior of the current-voltage relation of the three-diode equivalent model. Moreover, as a newly developed optimization algorithm, the gradient-based optimization algorithm (GBO) has been successfully applied to real-world engineering problems. To quickly distinguish between the exploration and exploitation phases and converges on more accurate results, Premkumar with his/her team have been dedicated a lot of work. In reference [40], an improved gradient-based optimization algorithm with chaotic drifts (CGBO) algorithm was proposed to derive the parameters of PV modules while offering precise I-V and P-V curves. In reference [41], an enhanced version of GBO named CCNMGBO, in which the crisscross (CC) algorithm and Nelder–Mead simplex (NMs) strategy are hybridized with GBO to estimate the uncertain parameters of various PV models. Another enhanced variant of GBO, called opposition decided GBO with balance analysis and diversity maintenance (OBGOA) [42] was proposed for parameter identification of solar photovoltaic models. Furthermore, the arithmetic optimization algorithm was improved [43, 44] to extract the solar cell/panel parameters as well. In reference [45], a novel chaotic northern goshawk optimization algorithm based on the adoption of modified Levenberg–Marquardt is proposed to extract the parameter of the single-diode PV model of the cell/module globally.

(4) The others. Due to the advantage of free parameters, low computational time, and low complexity, JAYA draw a lot of attention since it was developed by Venkata Rao [46]. Later, many variants of JAYA have been proposed, including improved JAVA (IJAVA) [47], performance-guided JAYA (PGJAYA) [48], enhanced chaotic JAYA algorithm [49], and so on. Meantime, improved Rao-1 (IRao-1) algorithm [50], comprehensive learning Rao-1 optimization (CLRao-1) [51], and Rao-2 (R-II) and Rao-3 (R-III) algorithm [52] were proposed to estimate the PV cell parameters. Moreover, the supply-demand-based optimization algorithm (SDOA) [53], reinforcement learning technique [54], teaching-learning-based optimization (STLBO) [55], and honey badger algorithm (HBA) [56] were also employed to extract the unknown parameters of PV models.

1.2. Motivation and Contribution

Through the literature review, it is seen that most meta-heuristic algorithms have the inherent disadvantages, such as parameter dependency, premature convergence, sensitive to the initial population, and complicated processes. Moreover, due to the parameter extraction problem of PV models is a multimodal optimization problem, these meta-heuristic algorithms need to be strengthened. Consequently, developing for accurate, reliable, and efficient metaheuristic algorithms to extract the parameters of solar PV cell models is still ongoing.

A level-based learning swarm optimizer (LLSO) [57] was proposed by Yang in 2018. It was inspired from the mixed-level learning methodology in which teachers treat students differently in accordance with their aptitude. Instead of using the historically best positions (such as pbest, $g\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}$, or nbest) to update particles, particles in different levels are utilized to direct the learning of particles. Since particles in different levels have diverse potentials in exploration and exploitation, they possess diverse evolutionary information to evolve the swarm. Therefore, the diversity of the algorithm could be promoted. However, one of the main drawbacks of many meta-heuristic algorithms is that they are easy to fall into the local optima and could be avoided to some extent. On the other hand, through the attraction of high levels, the individuals could approach the optimal region rapidly. Moreover, since the individual in the lower level learns from individuals who are randomly selected from all levels better than its level instead of the first level, the population could cover multiple optimal regions at the same time. However, the first level is not updated in the original LLSO; it will slow down the convergence speed of the algorithm and influence the accuracy of the results.

Stochastic fractal search (SFS) [58] is motivated by diffusion property turned up in fractals. It has been successfully applied in solving engineering problems [59]. Especially, it has a wonderful local search ability that could achieve a solution that has the least error compared with the globally optimum solution within a minimal number of iterations. Therefore, in this paper, the stochastic fractal search (SFS) is applied to the individuals of the first level in LLSO to enhance the local search ability. In this way, the exploration and exploitation could be well balanced through the combination of LLSO and SFS. This paper is a seminal attempt to apply an LLSO-based algorithm to the field of renewable energy. The major contributions of this paper are as follows.

(1) A new LLSOF algorithm for identifying the unknown PV cell/module parameters is proposed.

(2) Instead of using the historically best positions to update particles, particles in different levels are utilized to direct the learning of particles. In this way, the diversity of the population would be maintained and multiple optima regions could be considered at the same time.

(3) The SFS is introduced to improve the local search ability. The solutions in the first level approach the global/local optima accurately through Gaussian random walk.

(4) Extensive experimentation and comparisons on parameter estimation problems for the PV models including SDM, DDM, TDM, and PV module model validate the effectiveness and efficiency of the proposed LLSOF algorithm.

The rest of this paper is arranged accordingly. Section 2 reviews the electrical equivalent of various PV models and the problem formulation of PV models. Section 3 presents the traditional LLSO algorithm, the SFS algorithm, and the proposed LLSOF. Section 4 analyzes and shows the results on various PV models of the cell and module. In addition, the experimental results of the manufacturer’s data sheet are provided to validate the performance of the proposed LLSOF algorithm. Lastly, Section 5 includes conclusions.

2. PV Models and Problem Formulation

PV model has the characteristics of a nonlinear output, so it is necessary to use an accurate mathematical model to describe it. Existing research has developed a variety of photovoltaic models, including the SDM, DDM, TDM, and PV module model. All variables and relevant parameters of the photovoltaic cell model are given and detailed in a list, as shown in Table 1.

Table 1

Relevant parameters of the PV cell model.

$I_d,I_{d1},I_{d2},I_{d3}$	Diode current (μA)	$n,n_1,n_2,n_3$	Diode ideal factor
IL	Output current of PV cell (A)	$k$	Boltzmann constant, 1.3806503 × 10−23 J/K
Iph	Photo-generated current (A)	$q$	Electronic charge, 1.60217646 × 10−19 C
Isd	Reverse saturation current of diode (A)	Np	The number of solar cells in parallel
Isd1	Diffusion current (A)	Ns	The number of solar cells in series
Isd2, Isd3	Saturation current (A)	T	Temperature of junction (K)
Ish	Shunt resistor current (A)	VL	Cell output voltage (V)
Rsh	Shunt resistor (Ω)	Vt	Junction thermal voltage (V)
Rs	Series resistance (Ω)	$D,D_1,\cdots ,D_s$	Different ideal diodes

2.1. Single Diode Model

As shown in Figure 1(a), the SDM is the most widely used PV model, which includes a current source, a parallel diode, a series resistor to represent the loss of load current, and a shunt resistor to denote the leakage current. PV cell output current in the single diode model can be derived as follows [4]:$$\begin{array}{}\text{(1)}& I_L=I_{ph}-I_{sd}\exp \frac{q{V}_t+I_L{R}_s}{nkT}-1-\frac{V_L+I_L{R}_s}{R_{sh}}\text{.}\end{array}$$

[figure(s) omitted; refer to PDF]

There are five unknown parameters (I_ph, I_sd, $n$, R_s, and R_sh) that need to be determined in the single diode based model of the PV cell.

2.2. Double Diode Model

The equivalent circuit of the DDM is shown in Figure 1(b), which includes a photo-generated current source, two parallel diodes, a series resistor, and a parallel resistor [11]. Similarly, the PV cell output current in the double diode-based model can be derived as follows [29]:$$\begin{array}{}\text{(2)}& I_L=I_{ph}-I_{sd1}\exp \frac{q{V}_L+I_L{R}_s}{n_{1}kT}-1-I_{sd2}\exp \frac{q{V}_L+I_L{R}_s}{n_{2}kT}-1-\frac{V_L+I_L{R}_s}{R_{sh}}\text{.}\end{array}$$

There are seven unknown parameters (I_ph, I_sd1, I_sd2, n₁, n₂, R_s, and R_sh) that need to be determined in the DDM of the PV cell.

2.3. Triple Diode Model

As shown in Figure 1(c), the TDM uses a third diode to simulate the leakage current in the grain boundaries of commercial solar cells. The output current I_L can be calculated as follows:$$\begin{array}{c}\text{(3)}& I_L=I_{ph}-I_{sd1}\exp \frac{q{V}_L+I_L{R}_s}{n_{1}kT}-1-I_{sd2}\exp \frac{q{V}_L+I_L{R}_s}{n_{2}kT}-1-I_{sd3}\exp \frac{q{V}_L+I_L{R}_s}{n_{3}kT}-1-\frac{V_L+I_L{R}_s}{R_{sh}}\text{.}\end{array}$$

2.4. PV Module Model

As shown in Figure 1(d), the PV module model usually consists of several PV cells connected in series and/or in parallel. The output current can be expressed as follows:$$\begin{array}{}\text{(4)}& I_L=I_{ph}{N}_p-I_{sd}{N}_p\exp \frac{q{V}_L/N_s+I_L{R}_s/N_p}{n_{1}kT}-1-\frac{V_L{N}_p/N_s+I_L{R}_s}{R_{sh}}\text{.}\end{array}$$

Same as in the SDM, there are five unknown parameters (I_ph, I_sd, n, R_s, and R_sh) that need to be estimated.

2.5. Objective Function Definition

The main purpose of identifying the PV model parameters is to improve the parameter accuracy and reduce the difference between experimental data and simulated data. The error functions are defined as follows:

For SDM:$$\begin{array}{}\text{(5)}& \begin{array}{c}{f}_k{V}_L,I_L,X=I_{ph}-I_{sd}\exp \frac{q{V}_L+I_L{R}_s}{nkT}-\frac{V_L+I_L{R}_s}{R_{sh}}-I_L\text{,}\\ X=I_{ph},I_{sd},R_s,R_{sh},z\text{.}\end{array}\end{array}$$

For DDM:$$\begin{array}{}\text{(6)}& \begin{array}{c}\begin{array}{c}{f}_k{V}_L,I_L,X=I_{ph}-I_{sd1}\exp \frac{q{V}_L+I_L{R}_s}{n_{1}kT}-1-I_{sd2}\exp \frac{q{V}_L+I_L{R}_s}{n_{2}kT}-1\\ -\frac{V_L+I_L{R}_s}{R_{sh}}-I_L,\end{array}\\ X=I_{ph},I_{sd1},I_{sd2},R_s,R_{sh},n_1,n_2\text{.}\end{array}\end{array}$$

For TDM:$$\begin{array}{}\text{(7)}& \begin{array}{c}f{V}_L,I_L,x=I_{ph}-I_{sd1}\exp \frac{V_L+I_L{R}_s}{n_1{V}_t}-1-I_{sd2}\exp \frac{V_L+I_L{R}_s}{n_2{V}_t}-1\\ -I_{sd3}\exp \frac{V_L+I_L{R}_s}{n_3{V}_t}-1-\frac{V_L+I_L{R}_s}{R_{sh}}-I_L\text{,}\\ x=I_{ph},I_{sd1},I_{sd2},I_{sd3},R_s,R_{sh},n_1,n_2,n_3\text{.}\end{array}\end{array}$$

For PV module model:$$\begin{array}{}\text{(8)}& \begin{array}{c}f{V}_L,I_L,x=I_{ph}-I_{ph}\exp \frac{V_L+I_L{R}_s{N}_s}{n_{1}kT}-1-\frac{V_L+I_L{N}_s}{R_{sh}{N}_s}-I_L\text{,}\\ x=I_{ph},I_{sd},R_s,R_{sh},n\text{.}\end{array}\end{array}$$

There are many objective functions used to describe the error, such as absolute error (AE), mean absolute error (MAE), sum squared error (SSE), and root mean square error (RMSE) [3]. Among them, RMSE is the most adopted function which is described as follows:$$\begin{array}{}\text{(9)}& \mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}X=\sqrt{\frac{1}{N}{{\displaystyle \sum }}_{k=1}^N{f}_k{V_L,I_L,X}^2,}\end{array}$$where $f_k{V}_L,I_L,X$ is the error function of the measured and simulated values of the PV model, N stands for the size of the experimental data, and X represents the parameters required to be estimated.

3. The Proposed Method

3.1. Basic Level-Based Learning Swarm Optimizer (LLSO)

The LLSO is proposed by Yang [57] in 2018 for solving large-scale optimization problems. The basic idea is individuals are usually in different evolution states during evolution, and have different potentials in global and local search space. Algorithm 1 gives the pseudocode of the LLSO. The population P is separated into a number of levels $L_i$$1\le i\le NL$, where $NL$ represents several levels, according to their fitness value, and then, the individuals in the lower level update by learning from the higher level as follows:$$\begin{array}{c}\text{(10)}& v_{i,j}^d\leftarrow r_1{v}_{i,j}^d+r_2{x}_{r{l}_1,k_1}^d-x_{i,j}^d+\varphi r_3{x}_{r{l}_2,k_2}^d-x_{i,j}^d\text{,}{x}_{i,j}^d\leftarrow x_{i,j}^d+v_{i,j}^d\text{,}\end{array}$$where $X_{i,j}=x_{i,j}^1,\cdots ,x_{i,j}^d,\cdots x_{i,j}^D$ is the location of the jth individual from the ith level and $V_{i,j}=v_{i,j}^1,\cdots ,v_{i,j}^d,\cdots v_{i,j}^D$ is the velocity. $r{l}_1$ and $r{l}_2$ represent the selected two different higher level indexes in the range of $1,i-1$, $k_1$ and $k_2$ denote as the randomly selected individual index within $1,LS$, where $LS$ is the number of individuals in each level. Through the cooperation of levels, diversity could be enhanced. Figure 2 presents the illustration of the level-based learning strategy. The individuals in the second level learn from the first level, the individuals of the third level learn from the first and second levels, and the individuals in the ith level learn from the first to (i − 1)th levels.

Algorithm 1: LLSO.

(1) Initialize the population P randomly and calculate the function values of individuals

(2) FES = Np;

(3) X_best ← the best individual in the P;

(4) while FES < MAXFES do

(5)  Sort the population in ascending order and separate them into NL levels;

   //Update the individuals in L_NL, …, L₃

(6)   For j = {1, …, LS} do

(7)    Select two levels from the top (i − 1) levels: rl₁, rl₂:

(8)    If (rl₂ < rl₁) then

(9)     Swap (rl₁, rl₂);

(10)    End If

(11)    Randomly select two particles from rl₁, rl₂: X_rl1,k1, X_rl2,k2;

(12)    Update particle X_{i, j} according to equations (8) and (9);

(13)    Calculate the fitness value f (X_{i, j}) of this particle;

(14)    If (f (X_{i, j}) < f (X_best)) then

(15)     X_best = X_{i, j};

(16)    End If

(17)   End for

(18)   FES = FES + LS;

(19)  End for

   //Update the second level

(20)   For j = {1, 2026, LS} do

(21)    Select two particles from the first level: X_{1, k1}, X_{1, k2};

(22)    If (f (X_{1, k2}) < f (X_{1, k1})) then

(23)     Swap (X_{1, k1}, X_{1, k2});

(24)    End If

(25)    Update particle X_{2, j} according to equations (8) and (9);

(26)    Calculate the fitness value f (X_{2, j}) of this particle;

(27)    If (f (X_{2, j}) < f (X_best)) then

(28)     X_best = X_{2, j};

(29)    End If

(30)   End for

(31)   FES = FES + LS;

(32) End While

[figure(s) omitted; refer to PDF]

3.2. Stochastic Fractal Search

The stochastic fractal search [31] was inspired by the natural phenomenon of growth [32]. There are two main processes including diffusion and updating. The diffusion operation takes the responsibility of exploration while the updating process performs the neighborhood search for the exploitation task. In the diffusion process, two types of Gaussian random walk are applied to perform the diffusion procedure on each individual, and in the updating process, the new position for every current solution is generated according to the location of other solutions. The updating procedures are defined as follows:$$\begin{array}{}\text{(11)}& \begin{array}{c}\begin{array}{cc}{X}^{\prime }=X_i-\epsilon \times X_{r1}-X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}},& \epsilon \le 0.5,\end{array}\\ \begin{array}{cc}{X}^{\prime }=X_i+\epsilon \times X_{r1}-X_{r2},& \epsilon >0.5,\end{array}\end{array}\end{array}$$where $X_{r1}$ and $X_{r2}$ are randomly selected particles from population, $X_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ is the best particle, $\epsilon$ is a random number which is uniformly distributed between [0, 1].

3.3. The Proposed Algorithm: LLSOF

Through the learning among levels, the LLSO could maintain a good diversity and has promising exploration ability. However, the individuals in the first level do not go through any updating process at all in the original LLSO, which constrains the local search ability. Therefore, the updating procedure of stochastic fractal search is adopted to improve the local search ability of the individuals in the first level in the proposed algorithm. Figure 3 shows the flowchart of the proposed level-based learning swarm optimizer with stochastic fractal search (LLSOF). The population is separated into NL levels at first, and each of them has LS individuals. The individuals in L_k ∈ {L_NL, …, L₃} evolve through learning from {L_k−1, …, L₁}, and individuals in L₂ update through learning from L₁, and individuals in L₁ locally search through stochastic fractal search. The interaction through different levels maintains the diversity and the local search of the first level promotes convergence and accuracy.

[figure(s) omitted; refer to PDF]

It is worth noting that since the updating procedure of stochastic fractal search can only be implemented on the individuals in the first level, the $X_{r1}$ and $X_{r2}$ from equation (11) are the individuals randomly selected from the first level.

3.4. Complexity Analysis

The proposed LLSOF takes the maintenance of the algorithmic simplicity of LLSO, which takes O (NP × log (NP) + NP), to rank the swarm and divide the swarm into NL levels at each generation, and takes O (NP × Dim) to update of particles in all levels, where Dim is the number of variables to be optimized. Overall, the time complexity of the proposed algorithm LLSOF is O (NP × log (NP) + NP + NP × Dim). Therefore, the time complexity is approximately O (NP × log (NP) + NP) due to Dim is much smaller than NP in the parameter identification of PV models. As for the space complexity, LLSOF takes O (NP × Dim) space.

4. Experiments and Discussion

In this section, both SDM and DDM are adopted to validate the performance of the proposed algorithm to extract the PV optimal parameters. The data for the current voltage of SDM, DDM, and TDM were measured on 57 mm diameter commercial silicon R. T. C France solar sold under the irradiance of 1,000 W/m² at 33°C. For Photo Watt-PWP201, which has 36 connected cells in series under a temperature of 45°C with the irradiance of 100 W/m², and experimental data coming from several manufacturer’s data sheets are considered as well in this paper. Table 2 represents the lower and upper bounds of the involved parameters of the PV model. Seven state-of-the-art metaheuristic algorithms are chosen for comparison, including biogeography-based learning, particle swarm optimization (BLPSO) [34], PGJAYA [6], logistic chaotic JAYA algorithm (LCJAYA) [35], multiple learning backtracking spiral algorithm (MLBSA) [36], SEDE [7], HBA [23], COOT [21], and TSA [20]. Table 3 represents the controlling parameters of the related algorithms. It is worth noting that all involved parameters are in accordance with the original recommendations in the proposed paper. As to the proposed LLSOF, the population size is set to 50, and the number of levels NL is adaptively adjusted through a pool S = [4, 6, 8, 10, 20, 50] and $\varphi$ is set to 0.4 according to LLSO. Each experiment runs 30 independent times. All experiments are implemented on a PC with 1.80 GHz Intel (R) Core (TM) i7-8565U CPU and 16 GB of RAM with MATLAB R2020a. Furthermore, the maximum number of function evaluations (MaxFES) sets to $5\times {10}^4$ for all competitors.

Table 2

Parameters of different PV models.

Parameters	SDM/DDM/TDM	PV module
	[LB, UB]	[LB, UB]
$I_{ph}A$	[0, 1]	[0, 2]
$I_{sd}$, $I_{sd1}$, $I_{sd2}$, $I_{sd3}$$\mu A$	[0, 1]	[0, 50]
$R_s\mathrm{\Omega }$	[0, 0.5]	[0, 2]
$R_{sh}\mathrm{\Omega }$	[0, 100]	[0, 1000]
$n$, $n_1$, $n_2$, $n_3$	[1, 2]	[1, 50]

Table 3

Parameter setting of different algorithms.

Years	Algorithms	Descriptions	Control parameters
2018	MLBSA	Multiple learning backtracking search algorithm	NP = 50
2019	PGJAYA	Performance-guided JAYA algorithm	NP = 20
2019	LCJAYA	Logistic chaotic JAYA algorithm	NP = 20
2020	TSA	Tunicate swarm algorithm	NP = 80; Pmin = 1; Pmax = 4
2020	SEDE	Self-adaptive ensemble-based DE;	NP = 30; F = [1.0, 1.0, 0.8]; Cr = [0.1, 0.9, 0.2]
2021	COOT	Coot optimization algorithms	NP = 30; $p=0.5$; NL = 10; Ncoot = 20
2021	DPDE	Directional permutation DE	NP = 18 × Dim; $p=0.11$; H = 5
2021	CAJDE	Chaotic local search-based JADE	NP = 30; L = 50; r = 0.01; $p=0.05$
2022	HBA	Honey badger algorithm	NP = 30; C = 2; β = 6
	LLSOF	Level-based learning swarm optimizer with stochastic fractal search	NP = 50; $\varphi$ = 0.4

4.1. Experimental Results on SDM

The parameters and the best RMSE values obtained by the compared algorithms on SDM are presented in Table 4. It is seen that LLSOF and SEDE are tied for the first place among the ten algorithms. The results obtained by SEDE, MLBSA, DPDE, and CAJDE are pretty competitive as well. Moreover, the measured data and the simulated data are plotted in Figure 4. It is observed from Figure 4 that the difference between the measured data and the estimated data by LLSOF is almost negligible, which illustrates that LLSOF could extract the parameters for SDM.

Table 4

Comparison of estimation results of the RTC France solar cell using the SDM.

Algorithms	$I_{ph}\mathrm{A}$	$I_{sd}\mu \mathrm{A}$	$R_s\mathrm{\Omega }$	$R_{sh}\mathrm{\Omega }$	$n$	RMSE
PGJAYA	0.7607	0.3230	0.0363	53.7158	1.4811	9.86060444E − 04
LCJAYA	0.7607	0.3237	0.0363	53.7710	1.4814	9.91598597E − 04
MLBSA	0.7607	0.3235	0.0363	53.7532	1.4813	9.86021877E  −  04
SEDE	0.7607	0.3230	0.0363	53.7185	1.4811	9.86021877E  −  04
HBA	0.7607	0.3230	0.0363	53.7300	1.4812	3.07128673E − 03
COOT	0.7609	0.4330	0.0359	99.8371	1.5104	1.28535049E − 03
TSA	0.7836	0.2979	0.0197	56.6268	1.4829	1.70175001E − 03
DPDE	0.7608	0.3230	0.0364	53.7185	1.4811	9.86021877E  −  04
CAJDE	0.7608	0.3230	0.0364	53.7185	1.4811	9.86021877E  −  04
LLSOF	0.7607	0.3230	0.0363	53.7185	1.4811	9.86021877E  −  04

The best results are marked in bold.

[figure(s) omitted; refer to PDF]

4.2. Experimental Results on DDM

The fitness landscape of the DDM is much more complex than that of the SDM. Therefore, extracting the parameters of DDM throws a bigger challenge to the algorithm. Table 5 lists the best RMSE and the involved parameters obtained by the compared algorithms. From Table 5, LLSOF ranks the first compared with the other competitors. It is seen that MLBSA, SEDE, HBA, DPDE, and CAJDE get similar results, which are relatively small as well. Same with DDM, the estimated results and the measured data are plotted in Figure 5; similar results could be observed. The estimated results are entirely consistent with the measured data. These results fully validate LLSOF and are effective for extracting the parameters of the DDM.

Table 5

Comparison of estimation results of the RTC France solar cell using the DDM.

Algorithms	$I_{ph}\mathrm{A}$	$I_{sd}\mu \mathrm{A}$	$R_s\mathrm{\Omega }$	$R_{sh}\mathrm{\Omega }$	$n_1$	$I_{sd2}\mu \mathrm{A}$	$n_2$	RMSE
PGJAYA	0.7608	0.2116	0.0368	55.7920	1.4455	0.8740	2.0000	9.82605320E − 04
LCJAYA	0.7607	0.22596	0.0367	55.4815	1.4518	0.74640	2.0000	9.82503561E − 04
MLBSA	0.7608	0.2273	0.0670	55.4612	1.4515	0.73835	2.0000	9.82484851E  −  04
SEDE	0.7608	0.7493	0.0367	55.4854	2.0000	0.2260	1.4510	9.82484851E  −  04
HBA	0.7597	0.9998	0.0359	93.3887	1.9124	0.2265	1.4574	9.82970552E − 04
COOT	0.7652	0.8187	0.0297	45.1569	1.6081	0.7228	1.7719	1.67982102E − 03
TSA	0.7632	0.3106	0.0455	20.4536	1.2800	0.5291	1.8140	3.11825221E − 03
DPDE	0.7608	0.0000	0.0367	55.4854	2.0000	0.0000	1.4510	9.82484851E  −  04
CAJDE	0.7608	0.7493	0.0367	55.4854	2.0000	0.2260	1.4510	9.82484851E  −  04
LLSOF	0.7607	0.3200	0.0363	53.7185	1.4811	0.2937	1.4811	9.82484851E  −  04

[figure(s) omitted; refer to PDF]

4.3. Experimental Results on TDM

There are nine parameters that need to be estimated in TDM; therefore, the complexity of the model is noticeably higher. Table 6 shows the best RMSE and the involved parameters obtained by the compared algorithms. From Table 6, it can be concluded that the LLSOF achieves the best results with PGJAYA, MLBSA, SEDE, HBA, DPDE, and CAJDE. The power with voltage characteristics gained by LLSOF is illustrated in Figure 6. From Figure 6, the LLSOF’s test values of power and current fit the real model well.

Table 6

Comparison of estimation results of the RTC France solar cell using the TDM.

Algorithms	$I_{ph}\mathrm{A}$	$I_{sd1}\mu \mathrm{A}$	$R_s\mathrm{\Omega }$	$R_{sh}\mathrm{\Omega }$	$n_1$	$I_{sd2}\mu \mathrm{A}$	$n_2$	$I_{sd3}\mu \mathrm{A}$	$n_3$	RMSE
PGJAYA	0.7608	0.3801	0.0367	55.4854	1.4510	0.5580	1.5892	0.0000	2.0000	9.82484851E − 04
LCJAYA	0.7607	0.2536	0.0365	56.2576	1.4612	0.5843	2.0000	0.0012	1.8572	9.85756778E  − 04
MLBSA	0.7608	0.0971	0.0365	52.9063	1.0179	0.2112	1.9318	0.0000	1.4852	9.82484851E − 04
SEDE	0.7608	0.2260	0.0367	55.4854	2.0000	0.2260	1.4510	0.1914	2.0000	9.82484851E − 04
HBA	0.7608	0.2260	0.0367	55.4854	1.4510	0.7493	2.0000	0.0000	1.7683	9.82484851E − 04
COOT	0.7608	0.2133	0.0368	55.8349	1.4462	0.8609	2.0000	0.0017	1.9995	1.23160248E  − 03
TSA	0.7608	0.6264	0.0367	55.1815	1.9951	0.1942	1.4739	0.0511	1.7163	1.03033980E  − 03
DPDE	0.7608	0.7493	0.0367	55.4854	2.0000	0.2259	1.4510	0.0139	1.4510	9.82484851E − 04
CAJDE	0.7608	0.7493	0.0367	55.4854	1.4510	0.0000	1.3825	0.7493	2.0000	9.82484851E − 04
LLSOF	0.7608	0.7493	0.0367	55.4854	2.0000	0.2120	1.4510	0.0139	1.4510	9.82484851E − 04

[figure(s) omitted; refer to PDF]

4.4. Experimental Results on the PV Module Model

The best RMSE value and the related parameters obtained by the algorithms on the PV module are listed in Table 7. It is observed that LLSOF and SEDE achieved the first place among the other competitors. Except for COOT and TSA, the remaining algorithms could obtain very competitive RMSE results. In addition, Figure 7 shows the obvious I-V and P-V fitting curves of the measured data and the estimated results of LLSOF. It demonstrates that the estimated results achieved by LLSOF are almost equivalent to the measured data. In summary, LLSOF can obtain very accurate parameters.

Table 7

Comparison of estimation results of the photowatt-PWP201 solar module.

Algorithms	$I_{ph}\mathrm{A}$	$I_{sd}\mu \mathrm{A}$	$R_s\mathrm{\Omega }$	$R_{sh}\mathrm{\Omega }$	$n$	RMSE
BLPSO	1.0305	3.4854	1.2013	986.6686	48.6460	2.42512012E − 03
PGJAYA	1.0305	3.4632	1.2019	975.4490	48.6219	2.42507488E − 03
LCJAYA	1.0341	4.8709	1.1743	898.3252	49.9711	2.42877254E − 03
MLBSA	1.0305	3.6200	1.1968	999.8284	48.7928	2.42844709E − 03
STLBO	1.0305	3.4883	1.2011	984.1533	48.6495	2.42507526E − 03
SEDE	1.0305	3.4823	1.2013	981.9823	48.6428	2.42507486E  −  03
HBA	1.0514	6.2250	0.1970	889.0754	82.7648	2.42507536E − 03
COOT	1.0687	8.5084	0.0763	999.8402	86.1776	1.51913356E − 02
TSA	1.0815	4.7211	0.1187	283.8852	79.8895	1.39303804E − 02
DPDE	1.0305	3.4823	0.0333	27.2772	1.3511	2.42507486E  −  03
CAJDE	1.0305	3.4823	0.0333	27.2772	1.3511	2.42507486E  −  03
LLSOF	1.0305	3.4823	1.2013	981.9823	48.6428	2.42507486E  −  03

The best results are marked in bold.

[figure(s) omitted; refer to PDF]

4.5. Comparison with Original LLSO

Since LLSOF is modified from LLSO, their comparisons with SDM, DDM, TDM, and PV modules are carried out. Figure 8 shows the obtained RMSE results of LLSOF and LLSO with SDM, DDM, TDM, and PV modules in 30 independent runs. It is observed that the results obtained by LLSOF are much more stable than LLSO with all PV models. Meanwhile, the best RMSE values obtained by LLSOF are better than LLSO in almost all instances. The reason is that the introduced random walk could promote the local search and help the population jump out of the local optima. Compared with basic LLSO, the results achieved by LLSOF have smaller variations for different runs, which shows the robustness of LLSOF.

[figure(s) omitted; refer to PDF]

4.6. Comparation on CPU Time

The rank of average CPU time for all runs under uniform function evaluations (FEs) is listed in Figure 9. The LLSOF consumes the least amount of CPU calculation time due to its concise and efficient structure. Most of the other algorithms consume more execution time than the LLSOF.

[figure(s) omitted; refer to PDF]

4.7. Results of Experimental Data from the Manufacturer’s Datasheet

To test the practicability of the proposed algorithm, the manufacturer’s data sheet of three PV modules which includes thin-film (ST40), monocrystalline (SM55), and multicrystalline (KC200GT) [24] is adopted to further evaluate the performance of LLSOF. The used experimental data are directly extracted from the current-voltage curves given in the manufacturer’s data sheets at five different irradiation levels (1000 W/m², 800 W/m², 600 W/m², 400 W/m², and 200 W/m²) and temperature at 25°C.

The best estimated parameters for the three types of PV modules at different levels of irradiation are shown in Table 8. From Table 8, it can be observed that for the PV models, $I_{ph}$ and $I_{sd}$ will be increased while $R_s$, $R_{sh}$, and $n$ fluctuate as the irradiance increases in results of RMSE are quite small for all the situations. Moreover, to evaluate the estimated optimal parameters of the PV modules, the obtained current and the I-V characteristics are plotted in Figure 10. It is observed that the calculated data are perfectly matched with the data from the manufacturer. As a whole, the LLSOF could effectively identify the parameters of the PV systems.

Table 8

At various irradiance levels and T = 25°C, LLSOF extracted parameter values for PV modules.

Parameters	$I_{ph}\mathrm{A}$	$I_{sd}\mu \mathrm{A}$	$R_s\mathrm{\Omega }$	$R_{sh}\mathrm{\Omega }$	$n$	RMSE
Thin-film (ST40)
200 $w/m^2$	0.5331	1.4296E − 06	1.1857	344.9765	1.4975	4.7720089E − 04
400 $w/m^2$	1.0667	1.5442E − 06	1.0368	379.2209	1.5539	7.7929510E − 04
600 $w/m^2$	1.6044	2.0816E − 06	1.1041	353.9972	1.5031	6.9396278E − 04
800 $w/m^2$	2.1336	2.1112E − 06	1.0750	408.0886	1.5364	2.1003116E − 03
1000 $w/m^2$	2.6733	2.3800E − 06	1.0925	406.1483	1.5328	1.3936515E − 03
Monocrystalline (SM55)
200 $w/m^2$	0.6915	1.5118E − 07	0.2779	448.9890	1.3835	3.2138535E − 04
400 $w/m^2$	1.3822	1.6746E − 07	0.3281	444.8756	1.3943	9.4193343E − 04
600 $w/m^2$	2.0696	2.0808E − 07	0.3105	492.0842	1.4123	1.1437149E − 03
800 $w/m^2$	2.7563	5.3919E − 07	0.2598	473.2318	1.4970	3.4516198E − 03
1000 $w/m^2$	3.4436	6.2859E − 07	0.2804	1095.3159	1.5119	6.5007079E − 03
Multicrystalline (KC200GT)
200 $w/m^2$	1.6432	2.2863E − 08	0.1537	861.2929	1.1221	3.0553221E − 03
400 $w/m^2$	3.2812	3.3691E − 08	0.2248	1726.2371	1.2063	7.8213473E − 03
600 $w/m^2$	4.9241	4.8503E − 08	0.2797	4829.3845	1.2305	7.8202473E − 03
800 $w/m^2$	6.5686	5.4811E − 09	0.2563	4918.3611	1.2831	3.0239818E − 03
1000 $w/m^2$	8.2154	7.8511E − 08	0.2816	4169.3555	1.2479	2.7433792E − 03

[figure(s) omitted; refer to PDF]

5. Conclusion

In this paper, an improved and level-based learning with stochastic fractal search (LLSOF) has been proposed to identify the unknown parameters in complex nonlinear PV systems. The proposed algorithm is easy to implement and is applied to solve nonlinear optimization problems. The LLSOF is able to search for better solutions while maintaining a good diversity of the population. Experimental results on a bunch of benchmark problems and manufactures’ data illustrate that LLSOF can find better and more stable solutions than the basic LLSO and the other nine peer methods. However, there are parameters that need to be determined in the proposed algorithm. Therefore, to further improve our work, LLSOF with self-adaptive parameters would be a promising direction. Moreover, the other local search strategy such as the simulated annealing method is worth a try. In our future work, the proposed algorithm would be implemented in the parameter identification for PV cells in the application of illumination and heating for tobacco factory.

Acknowledgments

This research work was funded by the Henan Science and Technology Project (222102210037).

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