1. Introduction
With the progress of human society, the depleted stocks of fossil energy cannot meet the increasing energy requirement. Moreover, the applications of fossil energy have resulted in a series of environmental deterioration issues, including air pollution and the greenhouse effect [
1,
2,
3]. Therefore, the development and utilization of clean energy, such as solar, biomass, hydrogen, wind, water, and nuclear energy, will help alleviate the present energy crisis. Recently, the attention paid to solar energy, among numerous clean energy sources, has sharply increased because of its wide distribution and easy availability [
4]. In the electric power industry, photovoltaic (PV) generation systems enable the conversion of solar energy into electricity, where this transformation is implemented by solar cells. However, solar panels are susceptible to adverse weather conditions and environmental factors due to their year-round outdoor operation, including the shortening of the service life of solar cells and reduction in the output of power and energy conversion efficiency [
5,
6]. To achieve maximized and steady conversion efficiency for PV systems in diverse environments and complex scenarios, it becomes imperative to find a feasible method for accurately simulating, optimizing, and controlling corresponding PV models.
By constructing the mathematical representation of the PV generation system, significant progress has been achieved in recent years in understanding the operation function of PV systems [
7]. The bulk of models aim to realize the optimal fit to the actual measurement of current–voltage data obtained from PV cells [
8]. Most research focuses on constructing equivalent circuits with diodes to simulate the real behavior of PV cells due to the similarity between the output characteristics of the p-n junction of a diode and a PV cell [
9]. Among various PV models, the single diode model (SDM), double diode model (DDM), triple diode model (TDM), and PV module models have been applied extensively in practice [
10]. It is noteworthy that the dynamic behavior of these models depends on several unknown parameters, including photocurrent, diode saturation current and ideal factor, and shunt and series resistances. Additionally, these parameters are easily influenced by complex factors, such as device aging, malfunctions, and volatile operations. Thus, the accurate identification of these unknown parameters associated with PV models is an arduous but meaningful task for augmenting the performance of solar generation systems.
In recent years, numerous mature techniques have been presented and used in the parameter identification of PV models [
11,
12]. There are mainstream techniques, including analytical, iterative-based methods and meta-heuristic algorithms (MHAs) [
13]. The analytical method employs a series of mathematical formulas to identify model parameters. Although this method is implemented readily, it is highly dependent on the initial conditions and normally has a high computational cost. Accordingly, it is not efficient in solving the parameter identification of PV models with multi-modal and non-linear features [
14]. The second method, i.e., the iterative-based method, mainly comprises the Lambert W functions [
15] and Newton–Raphson [
16] methods, which are easily trapped in local optima of multi-modal functions due to excessive reliance on initial values and the gradient information of the problem [
17]. Fortunately, MHAs have surmounted these limitations since this algorithm remains unaffected by initial conditions and does not depend on the problem features [
18,
19]. MHAs also achieve high solution accuracy and competitive computational efficiency compared to traditional methods when tackling complex problems [
20]. Representative approaches are differential evolution (DE) [
21,
22], particle swarm optimization (PSO) [
23,
24], gaining-sharing knowledge algorithm (GSK) [
20,
25], whale optimization algorithm (WOA) [
26,
27], genetic algorithm (GA) [
28], artificial bee colony optimization (ABC) [
29,
30], Grey-wolf optimization [
31], and JAYA optimization [
32,
33], among many advanced algorithms [
34,
35,
36,
37,
38]. Many MHAs have obtained excellent performances for the parameter estimation of PV models. Actually, researchers are inclined to use modified versions of algorithms due to the finite performance of the originals when estimating parameters of PV models. A differential evolution using a novel penalty method (P-DE) was developed for the parameter identification of some multi-crystalline, mono-crystalline, and thin-film modules [
22]. Gao et al. [
21] presented a directional permutation DE (DPDE) to determine parameters in the SDM, DDM, TDM, and three other PV module models. The compared experiment results demonstrated its higher solving accuracy than that of other optimization algorithms. Recently, Gu et al. [
39] introduced an elite learning adaptive DE variant (ELADE) with a parameter adaptive strategy and an elite mutation strategy applied to obtain the characteristics of several PV models. Jordehi et al. [
40] developed a modified time-varying PSO algorithm by controlling the individual acceleration coefficients (TVACPSO) to achieve a trade-off for exploration and exploitation to accurately estimate the photovoltaic model parameters. Given the complex features of PV representations, a PSO variant incorporating a mutation idea from DE was employed to mitigate premature convergence for parameter estimation of the SDM, DDM, poly-crystalline Photo Watt-PWP 201, and multi-crystalline IFRI250-60 modules [
41]. A dual-population GSK algorithm (DPGSK) was developed for accurate parameter identification in PV system modeling [
42]. DPGSK employs a dual-population evolution strategy to balance exploration and exploitation to improve the convergence rate and population diversity. The results confirm that the parameter extraction accuracy of DPGSK outperforms that of the other methods. An improvement of GA, incorporating a novel convex crossover method, was presented to balance the population diversification and solution accuracy of GA [
43]. Afterward, a GA adopting non-uniform mutation and blend crossover operators (GAMNU) was constructed for the parameter extraction of two simple PV models and three other commercial solar cells [
44]. Chen et al. [
29] proposed an ABC algorithm combined with teaching–learning-based optimization (TLABC) to be employed for the parametric extraction of three common PV models. In [
45], an improved artificial bee colony optimization algorithm based on the chaotic map theory (CIABC) was developed to fortify the search capability of ABC at PV parameter extraction. In [
46], a hunter–prey optimization algorithm with reciprocity and sharing and learning interaction was presented to identify the unknown parameters of several PV models. Furthermore, Sharma et al. [
47] proposed an improved moth flame optimization technique with the opposite learning method and Lévy flight mechanism (OBLVMFO) to identify parameters of three PV panels, i.e., the STE 4/100 and SS2018P poly-crystalline, and LSM20 mono-crystalline modules. In addition to improving parameter identification accuracy, from a non-parametric statistical perspective, OBLVMFO exhibits significantly better optimization performance than the classical MFO. In [
48], eight optimization techniques were applied in the parametric extraction of the R.T.C. France PV cells, and the LSM20 and SS2018 PV modules. The relevant experimental results assessed the capabilities of each improved algorithm in constructing PV models, thereby enhancing energy conversion efficiency. Due to space limitations, research on applying more MHAs to resolve the parameter identification problem in PV models can be found in [
11,
12,
49,
50].
Among numerous MHAs, the DE algorithm, judged one of the most effective optimization methods, has been employed widely for the parameter identification of various PV models and performs considerably well [
21,
51]. Relative research has mainly focused on the equilibrium between the exploration and exploitation capabilities of DE. For example, a differential evolution using self-adaptive multiple mutation strategies of the random assignment method (SEDE) was proposed for individuals based on the iterative process [
52]. Afterward, an improved DE with one elite and obsolete dynamic learning, called DOLDE, used a dynamic oppositional learning mechanism to balance the global and local search abilities of individuals [
53]. However, one easily overlooked fact is that the algorithmic exploration and exploitation capabilities should be determined comprehensively based on both the iteration process and the search inclinations of individuals. In the early stages, it is reasonable that the entire population conducts a global search, while the population should shift towards a local search in the later iterations. Each individual should be assigned different search tasks based on their fitness values at different stages. Considering the complex characteristics of the nonlinear, multivariate, and non-convex nature of parameter identification for photovoltaic models, the single mutation strategy often causes the DE algorithm to become trapped in a local optima, thereby losing its optimization capabilities. Furthermore, individuals with better fitness should focus on a local search due to the finite computational source, while those with relatively poor fitness should execute a global search. These elites should be allocated more computational resources, thereby guiding the population evolution.
Based on the above discussion, this paper designs a modified differential evolution with a collaboration mechanism of dual mutation strategies and an orientation guidance mechanism (DODE) for the accurate parameter identification of photovoltaic models. Specifically, a collaboration mechanism of dual mutation strategies is developed to coordinate the search tendencies of each individual in the population and make a trade-off between the exploration and exploitation capabilities of DE. Moreover, considering the complex feature of the parameter identification problem of the PV model, it is also crucial to rationally allocate computational resources. Some elites with better fitness values should be allocated more computational resources. However, for DE, elites have single evolutionary directions and are easily stuck at local optima. To address this issue, an orientation guidance mechanism based on the population evolution trend is also developed to facilitate the evolution of elites, thereby effectively alleviating the state of elites being trapped in local optima.
The main contributions of this paper are as follows:
An improved differential evolution is proposed by incorporating a collaboration mechanism of dual mutation strategies and an orientation guidance mechanism into it. The proposed DODE algorithm can accurately estimate many PV models’ parameters due to its improved optimization ability.
Extensive comparisons with ten advanced algorithms, including five improved DE algorithms and five other representative meta-heuristic algorithms, are conducted in six different PV models, i.e., the single diode, double diode, triple diode models, and Photowatt-PWP201, mono-crystalline STM6-40/36, and poly-crystalline STP6-120/36 module models.
Experimental results show that the proposed DODE possesses the higher accuracy of parameter estimation by obtaining the minimum root mean square errors on these PV models.
The convergence curves on the six PV models indicate that the proposed DODE provides the faster convergence speed. Statistical analyses also verify the significant competitive superiorities of DODE compared to other optimization algorithms.
The proposed DODE algorithm yields a high exactitude in identifying parameters of PV models, with high similarity between the simulated data obtained by DODE and the experimentally measured data.
The structure of the remainder of this paper is arranged as follows:
Section 2 elaborates on the problem of parameter estimation of the common PV models and formulates mathematical expressions. The classical DE algorithm and its principles are introduced in
Section 3.
Section 4 describes the proposed DODE algorithm for the parameter identification of PV models. The experiment results and statistical analyses are presented in
Section 5. Finally,
Section 6 summarizes the sections above while outlining future research directions.
3. Differential Evolution
Differential evolution (DE) is a simple and effective intelligence optimization algorithm suitable for solving optimization problems in consecutive space. In the population of DE,
NP solutions form a population
Pop, which is indicated by [
x1, g,
x2, g, …,
xNP, g] at the
gth generation. Among them, each
xi,g (
i = 1, 2, …,
NP) is encoded as [
xi, 1, g,
xi, 2, g, …,
xi, D, g], where
D represents the decision variable dimension of the unsolved problem. After the random initialization, three operators, i.e., mutation, crossover, and selection, are repeatedly executed to produce the offspring of the whole population for subsequent iterations until satisfying a termination criterion. Specifically, the random initialization is conducted for each individual using Equation (10) at the first iteration.
where
rand expresses a random number sampled from the scope [0, 1];
g equals 1 at the first iteration.
j indicates an integer from 1 to
D;
xj,max and
xj,min respectively represent the top and bottom boundaries of the
jth decision variable.
Then, the individual
xi,g generates the own mutation vector
vi,g by implementing a mutation operator. Three prevalent mutation operators are given as Equations (11)–(13) [
64].
Among these equations, vi,g, produced by a mutation strategy, means the mutator of the individual xi,g. xbest,g indicates the individual with the best fitness function value at the gth iteration. r1, r2, and r3 are three mutually unequal integers sampled stochastically from 1 to NP. The parameter F is the scaling factor for magnifying differential vectors.
After the mutant operation, each
xi,g will generate its trial vector
ui,g, considered a combination of the mutant vector
vi,g and target vector
xi,g, through conducting the crossover operation. In general, a widely used binomial crossover operation can be described as follows [
64]:
In Equation (14), rand indicates a random number evenly sampled from (0, 1); the parameter CR represents the crossover rate, representing the number of components of the trial vector ui,g from the mutant vector vi,g; and jrand is an integer randomly chosen from 1 to D for guaranteeing the existing difference between ui,g and xi,g.
Eventually, the selection operation is employed to select an individual with a better fitness value from
ui,g and
xi,g. Considering the minimization nature of the objective function in the PV models, a widely used selection operator is introduced as defined in Equation (15) [
64].
where
f (
ui,g) and
f (
xi,g) respectively represent the fitness values of
ui,g and
xi,g.
5. Experiments and Discussions
In this section, six parameter extraction experiments on PV models, encompassing SDM, DDM, TDM, and three PV module models, are conducted to verify the effectiveness of the proposed DODE. The experimental data for SDM, DDM, and TDM, including 26 pairs of current–voltage (
I-
V) values [
73], are acquired by a 57 mm diameter commercial R.T.C. France silicon solar cell under the irradiance of 1000 W/m
2 at the ambient temperature of 33 °C. The remnant three modules, including Photowatt-PWP201, mono-crystalline STM6-40/36, and poly-crystalline STP6-120/36, assemble 36 poly-crystalline silicon cells in series (i.e.,
Ns = 36,
Np = 1) and have been tested at 45 °C, 51 °C, and 55 °C, respectively. The measured values of current–voltage of six PV models stem from [
74,
75]. The bottom and top limitations for each unidentified variable of six PV models are specified in
Table 1 [
52,
56].
Through a comprehensive comparison with some advanced algorithms, the achieved experiment results were extensively validated in terms of various aspects, encompassing accuracy of parameter identification, errors of extraction results, algorithmic convergence curves, and other relevant indexes. These compared algorithms comprise five DE algorithms (i.e., JADE [
64], CoDE [
72], MPEDE [
68], SEDE [
52], and DOLADE [
53]) and five other meta-heuristic algorithms (i.e., IWOA [
27], PGJAYA [
76], IGSK [
25], LaPSO [
62], and RTLBO [
77]). The necessary parameters, configured according to their original papers, and brief introductions of these algorithms, are enumerated in
Table 2. For the proposed DODE, the population size
NP and parameter
p are set to 30 and 0.2, respectively. It is notable that parameter tuning is a difficult task [
78]. Thus, a systematical analysis of parameter configurations will be a topic of our future research.
The proposed DODE and the other ten algorithms were compiled and independently executed 30 times using the MATLAB R2018b platform to ensure an even-handed comparison. The minimum RMSE value among the 30 runs is recorded in the table, along with the corresponding obtained PV model parameters. Additionally, this paper’s maximum fitness evaluation number of ten methods is 50,000, as used in many existing studies [
29,
52,
61].
5.1. Simulation Results on Solar Cells
For three solar cells (i.e., SDM, DDM, and TDM), the minimum RMSE values and corresponding extracted parameters of all 11 algorithms are recorded in
Table 3,
Table 4 and
Table 5 after 30 independent experiments, and the optimal RMSE is marked in bold. In
Table 3, five distinct parameters, i.e.,
Iph,
Isd,
Rs,
Rsh, and
n, are identified using the different algorithms. It is clear that DODE yields the optimal RMSE value compared to others. In addition, other algorithms (e.g., SEDE, DOLDE, and IGSK) showed good competitiveness but are not up to the level of the proposed DODE. The same observations can be found in
Table 4 and
Table 5, and DODE is still superior to its competitors regarding the RMSE values. It is worth mentioning that although there is little difference in the RMSE among these results, further statistical analysis will reveal the magnitude of the differences between the experimental results of DODE and other algorithms, as shown in
Section 5.4.
To further show the solution accuracy of DODE, absolute errors (AEs) of current and power between measured data and identified data are reported in
Table 6 and
Table 7, where AE = |
IL −
Iide| or |
P −
Pide|.
VL and
IL represent the measured current–voltage values.
P indicates the power according to the measured data.
Iide and
Pide are the current and power calculated after substituting
VL and the extracted parameters into the corresponding model. The last row of these tables records the sum of AE for this set of identified data.
In
Table 6, the cumulative AEs of current of the DODE algorithm are 0.017704 A, 0.017318 A, and 0.017319 A for SDM, DDM, and TDM, respectively.
Table 7 indicates that the cumulative AEs of power of three solar cells obtained by DODE are 0.006584 W, 0.006554 W, and 0.006554 W, respectively. All these experimental data demonstrate that DODE achieves higher accuracy in parameter extraction for SDM, DDM, and TDM.
As a further analysis,
Figure 7a–c depict the fitting results using the parameters identified by DODE for each model. These curves represent the fitted
I-
V and
P-
V characteristics, while the solid dots represent the measured data. The current and power results identified through DODE are highly consistent with the measured results at different voltage levels, which explicitly demonstrates that the unidentified parameter values obtained by DODE are also highly accurate.
5.2. Simulation Results on PV Models
Based on the previous description, five common parameters must be accurately identified for three PV module models (i.e., Photowatt-PWP201, STM6-40/36, and STP6-120/36). The minimum RMSE and the associated parameter identification results are recorded in
Table 8,
Table 9 and
Table 10. For the first module, DODE yields the optimal RMSE value of 2.42507486809489 × 10
−3, while IGSK and DOLADE rank second and third with RMSE values of 2.42507486809494 × 10
−3 and 2.42507486809496 × 10
−3, respectively. DODE still performs the best among all 11 algorithms for the last two module models. Furthermore, SEDE, DOLADE, and IGSK obtain RMSE values closest to the best result of DODE, while MPEDE, IWOA, RTLBO, and JADE perform poorly. Subsequent statistical tests also support this observation.
Table 11 and
Table 12 report the AEs of current and power between the identified and measured values for these three PV modules. Intuitively, the cumulative AE increases accordingly as the model becomes more complex. The maximum AE of current and power is 0.277976 A and 3.911436 W, respectively, obtained from the simulation results of the third model.
Additionally,
Figure 7d–f respectively depict the curves of
I-
V and
P-
V according to the extracted parameters through DODE for the Photowatt-PWP201, STM6-40/36, and STP6-120/36 module models, illustrating that, even with complex models, DODE can closely approximate the actual experimental data. Therefore, this provides a reference and research foundation for accurately obtaining PV model parameters and ensuring the construction of PV models and the secure and steady work of PV systems.
5.3. Convergence Characteristic Analysis
According to the above discussion, it can be found that the proposed DODE performs better on the six PV models above than other methods. To gain a deeper insight into the dynamic search process of all tested algorithms, the convergence characteristic graphs of all tested algorithms are illustrated in
Figure 8, with a varying number of fitness evaluations, from which the proposed DODE converges faster than all competitors in six cases. It is noteworthy that the convergence characteristic of DODE shows a slow convergence rate in the early iterations and, after that, it quickly converges to the global optimum. The main reason for this phenomenon is that DODE fully exerts the superiorities of the collaboration and orientation guidance mechanisms, increasing the chances of discovering the global optimum via the guidance of elites when realizing an equilibrium between exploration and exploitation, thus enhancing its search performance.
5.4. Statistical Results
In this subsection, the statistical results based on the above experiment data are also elaborated to further verify the performance of DODE. The worst, best, and average values and the standard deviation of the RMSE among thirty independent tests are respectively represented by Max, Min, Mean, and SD for SDM, DDM, TDM, Photowatt-PWP201, STP6-120/36, and STM6-40/36 module models, as reported in
Table 13, and minimum values are marked in bold. Furthermore, the Wilcoxon rank-sum test [
79], a non-parametric test method, is conducted to judge whether there is statistical significance between DODE and its opponents under a significance level
α = 0.05. To facilitate observation, the signs, i.e., “+”, “−”, and “=”, respectively signify that the performance of DODE is significantly better than, inferior to, and not significantly different from the compared algorithm. The
p-value is used for assessing the difference between the proposed DODE and its competitor. If the
p-value is greater than or equal to
α, there is a non-significant difference between DODE and ten competitors, and vice versa. Observations can be made from
Table 13, as follows:
(1) For SDM, DODE yields the minimum Min and Mean values, indicating that it performs the best among 30 independent runs. DOLADE obtains the best Max and SD values, suggesting that it possesses the best robustness compared to others. Moreover, Wilcoxon rank-sum test results indicate that DODE is significantly superior to other competitors.
(2) For DDM, DODE computes the best Max and Min values, while SEDE obtains the best Mean and SD. Wilcoxon rank-sum test results indicate that DODE significantly outdoes other algorithms.
(3) For TDM, we can observe that DODE attains the minimum on all four indexes. Moreover, all p-values indicate that DODE is significantly superior to others.
(4) For the Photowatt-PWP201 module model, DODE obtains the optimal values on Max, Min, and Mean, while SEDE possesses the same Mean as DODE and finds the best SD. In addition, the statistical results indicate that DODE is tied with SEDE, DOLADE, and IGSK, while is better than the remaining models.
(5) For the STM6-40/36 module model, DODE still possesses the best Max, Min, and Mean on four indexes, but SEDE, DOLADE, and IGSK achieve the Max and Mean values, which is the same as DODE. DOLADE has the best SD value. All
p-values indicate that DODE significantly outperforms the others.
Table 13.
Statistical results on six types of PV models obtained by all 11 tested algorithms.
Table 13.
Statistical results on six types of PV models obtained by all 11 tested algorithms.
Model | Algorithm | Max | Min | Mean | SD | Rank-Sum | p-Value |
---|
SDM | DODE | 9.86021877891504 × 10−4 | 9.86021877891317 × 10−4 | 9.86021877891411 × 10−4 | 4.76436810281140 × 10−17 | NA | NA |
JADE | 1.15321521879939 × 10−3 | 9.86021877891517 × 10−4 | 1.00716648820523 × 10−3 | 3.77565829307664 × 10−5 | (+) | 1.4295 × 10−11 |
CoDE | 9.86021877891586 × 10−4 | 9.86021877891456 × 10−4 | 9.86021877891509 × 10−4 | 4.79538261673457 × 10−17 | (+) | 7.4114 × 10−10 |
MPEDE | 9.86021909426073 × 10−4 | 9.86021877891538 × 10−4 | 9.86021879042075 × 10−4 | 5.64988986709267 × 10−12 | (+) | 1.3764 × 10−11 |
SEDE | 9.86021877891605 × 10−4 | 9.86021877891340 × 10−4 | 9.86021877891462 × 10−4 | 7.47323394301810 × 10−17 | (+) | 8.3835 × 10−3 |
DOLADE | 9.86021877891487 × 10−4 | 9.86021877891336 × 10−4 | 9.86021877891437 × 10−4 | 3.86051697701479 × 10−17 | (+) | 1.3967 × 10−2 |
IWOA | 9.87725092480884 × 10−4 | 9.86023447010448 × 10−4 | 9.86542108579345 × 10−4 | 5.30057786071925 × 10−7 | (+) | 1.4295 × 10−11 |
PGJAYA | 9.87580920047622 × 10−4 | 9.86021877975800 × 10−4 | 9.86076703117466 × 10−4 | 2.79467318577159 × 10−7 | (+) | 1.4215 × 10−11 |
IGSK | 9.86021877891559 × 10−4 | 9.86021877891351 × 10−4 | 9.86021877891483 × 10−4 | 4.56882160846755 × 10−17 | (+) | 3.6312 × 10−7 |
LaPSO | 9.86021877893221 × 10−4 | 9.86021877891438 × 10−4 | 9.86021877891614 × 10−4 | 4.30215181759160 × 10−16 | (+) | 4.6566 × 10−09 |
RTLBO | 9.95102691642195 × 10−4 | 9.86021887149475 × 10−4 | 9.87840663850941 × 10−4 | 2.66267998863458 × 10−6 | (+) | 1.4295 × 10−11 |
DDM | DODE | 9.86021877891492 × 10−4 | 9.82484851784979 × 10−4 | 9.83192257013672 × 10−4 | 1.41481043888143 × 10−6 | NA | NA |
JADE | 1.46814244601502 × 10−3 | 1.01024811601081 × 10−3 | 1.13980772164520 × 10−3 | 1.19243382122062 × 10−4 | (+) | 1.5080 × 10−11 |
CoDE | 1.01499195548530 × 10−3 | 9.82485401810740 × 10−4 | 9.85111165917372 × 10−4 | 5.77241006989757 × 10−6 | (+) | 7.0983 × 10−6 |
MPEDE | 1.34000615699996 × 10−3 | 9.83127480606161 × 10−4 | 1.02442669329253 × 10−3 | 9.08861171340222 × 10−5 | (+) | 1.9593 × 10−08 |
SEDE | 9.86021877891738 × 10−4 | 9.82484851785165 × 10−4 | 9.82818237549593 × 10−4 | 8.76073669368551 × 10−7 | (+) | 7.9047 × 10−5 |
DOLADE | 9.86021877891541 × 10−4 | 9.82484851785086 × 10−4 | 9.83310157876682 × 10−4 | 1.49599433374805 × 10−6 | (+) | 8.4759 × 10−3 |
IWOA | 1.13335738158299 × 10−3 | 9.83565133991052 × 10−4 | 1.02006932208289 × 10−3 | 3.83500358428075 × 10−5 | (+) | 4.9563 × 10−11 |
PGJAYA | 1.00293802399594 × 10−3 | 9.82509638915954 × 10−4 | 9.87716183405791 × 10−4 | 6.01377634281004 × 10−6 | (+) | 5.2220 × 10−7 |
IGSK | 9.86021877891651 × 10−4 | 9.82484851785200 × 10−4 | 9.83604169353349 × 10−4 | 1.56896272503491 × 10−6 | (+) | 3.5835 × 10−5 |
LaPSO | 9.86021877891546 × 10−4 | 9.82484851785202 × 10−4 | 9.82947554729584 × 10−4 | 1.12627926284723 × 10−6 | (+) | 2.1580 × 10−4 |
RTLBO | 1.22611480866827 × 10−3 | 9.85473186054530 × 10−4 | 1.01370580449482 × 10−3 | 5.43795142813846 × 10−5 | (+) | 2.7848 × 10−10 |
TDM | DODE | 9.86012141842487 × 10−4 | 9.82484851784993 × 10−4 | 9.82779670496747 × 10−4 | 9.09172775217468 × 10−7 | NA | NA |
JADE | 2.16528593256465 × 10−3 | 1.00957044428532 × 10−3 | 1.31035319578759 × 10−3 | 2.80955042829955 × 10−4 | (+) | 1.4323 × 10−11 |
CoDE | 1.60121397248637 × 10−3 | 9.83297746428071 × 10−4 | 1.13233460449822 × 10−3 | 1.65683169893251 × 10−4 | (+) | 5.1287 × 10−11 |
MPEDE | 1.91307457676052 × 10−3 | 9.82812864303851 × 10−4 | 1.22768988322227 × 10−3 | 2.56066083663204 × 10−4 | (+) | 1.1000 × 10−10 |
SEDE | 9.87089713674037 × 10−4 | 9.82484851786456 × 10−4 | 9.83811328146404 × 10−4 | 1.35211644958480 × 10−6 | (+) | 6.0062 × 10−08 |
DOLADE | 9.86022468500695 × 10−4 | 9.82484851785125 × 10−4 | 9.82880739811767 × 10−4 | 9.74800440804199 × 10−7 | (+) | 6.7632 × 10−4 |
IWOA | 1.40409074266789 × 10−3 | 9.83899673192525 × 10−4 | 1.13647517747175 × 10−3 | 1.00393239077011 × 10−4 | (+) | 1.7933 × 10−11 |
PGJAYA | 1.02121272228188 × 10−3 | 9.82489030043196 × 10−4 | 9.89231006684404 × 10−4 | 7.25801161236792 × 10−6 | (+) | 5.4929 × 10−10 |
IGSK | 9.86260741972316 × 10−4 | 9.82484851785564 × 10−4 | 9.83175863594395 × 10−4 | 1.03902873723312 × 10−6 | (+) | 2.3125 × 10−7 |
LaPSO | 9.93795049283600 × 10−4 | 9.82794807995161 × 10−4 | 9.85009173705836 × 10−4 | 2.74816286005806 × 10−6 | (+) | 3.3486 × 10−09 |
RTLBO | 1.37091043098428 × 10−3 | 9.86107279748943 × 10−4 | 1.06361165053575 × 10−3 | 9.15672794543958 × 10−5 | (+) | 1.4323 × 10−11 |
PWP201 | DODE | 2.42507486809506 × 10−3 | 2.42507486809489 × 10−3 | 2.42507486809502 × 10−3 | 4.10461626212945 × 10−17 | NA | NA |
JADE | 4.98633818165731 × 10−3 | 2.42507486810290 × 10−3 | 3.37970959170571 × 10−3 | 8.99516975676339 × 10−4 | (+) | 1.4780 × 10−11 |
CoDE | 2.42507486870952 × 10−3 | 2.42507486809513 × 10−3 | 2.42507486811983 × 10−3 | 1.09710461288146 × 10−13 | (+) | 1.4486 × 10−11 |
MPEDE | 3.10284947721046 × 10−3 | 2.42511061997362 × 10−3 | 2.62786165767249 × 10−3 | 2.16785092840004 × 10−4 | (+) | 1.4215 × 10−11 |
SEDE | 2.42507486809511 × 10−3 | 2.42507486809497 × 10−3 | 2.42507486809502 × 10−3 | 3.35024202198999 × 10−17 | (≈) | 1.8029 × 10−1 |
DOLADE | 2.42507486809510 × 10−3 | 2.42507486809496 × 10−3 | 2.42507486809503 × 10−3 | 3.40672816423479 × 10−17 | (≈) | 4.8226 × 10−1 |
IWOA | 2.91760303354987 × 10−3 | 2.42561743308710 × 10−3 | 2.58985585802751 × 10−3 | 1.07274004071403 × 10−4 | (+) | 1.4780 × 10−11 |
PGJAYA | 2.44840305719686 × 10−3 | 2.42507609902024 × 10−3 | 2.43149698978649 × 10−3 | 8.73626139992579 × 10−6 | (+) | 1.4550 × 10−11 |
IGSK | 2.42507486809511 × 10−3 | 2.42507486809494 × 10−3 | 2.42507486809503 × 10−3 | 3.87004392342166 × 10−17 | (≈) | 7.4480 × 10−1 |
LaPSO | 3.04703699752841 × 10−3 | 2.42507486809501 × 10−3 | 2.44580693907619 × 10−3 | 1.11645619026492 × 10−4 | (+) | 4.7641 × 10−08 |
RTLBO | 2.70490127586173 × 10−3 | 2.42889439046789 × 10−3 | 2.50904999738576 × 10−3 | 7.63295710664747 × 10−5 | (+) | 1.4780 × 10−11 |
STM6-40/36 | DODE | 1.72981370994070 × 10−3 | 1.72981370994064 × 10−3 | 1.72981370994068 × 10−3 | 1.27675873707146 × 10−17 | NA | NA |
JADE | 3.27263799793268 × 10−3 | 1.72981370994262 × 10−3 | 2.55458258766248 × 10−3 | 4.68082127878847 × 10−4 | (+) | 1.4295 × 10−11 |
CoDE | 1.72981370994073 × 10−3 | 1.72981370994069 × 10−3 | 1.72981370994071 × 10−3 | 9.79125207905825 × 10−18 | (+) | 5.5889 × 10−11 |
MPEDE | 2.19736357651583 × 10−3 | 1.72981371268042 × 10−3 | 1.85867139058643 × 10−3 | 1.17777494798869 × 10−4 | (+) | 1.3773 × 10−11 |
SEDE | 1.72981370994070 × 10−3 | 1.72981370994066 × 10−3 | 1.72981370994068 × 10−3 | 1.24745772198282 × 10−17 | (+) | 2.0390 × 10−2 |
DOLADE | 1.72981370994070 × 10−3 | 1.72981370994066 × 10−3 | 1.72981370994068 × 10−3 | 9.16570947960752 × 10−18 | (+) | 6.8532 × 10−3 |
IWOA | 1.89009612817313 × 10−3 | 1.73243299926313 × 10−3 | 1.80107707717986 × 10−3 | 3.34272034354920 × 10−5 | (+) | 1.4295 × 10−11 |
PGJAYA | 1.74876658470105 × 10−3 | 1.72981397145254 × 10−3 | 1.73077791266686 × 10−3 | 3.45844480461756 × 10−6 | (+) | 1.3531 × 10−11 |
IGSK | 1.72981370994070 × 10−3 | 1.72981370994066 × 10−3 | 1.72981370994068 × 10−3 | 1.43583193270945 × 10−17 | (+) | 1.8744 × 10−2 |
LaPSO | 1.46802616521865 × 10−1 | 1.72981370994068 × 10−3 | 6.83425884513958 × 10−3 | 2.59981175065994 × 10−2 | (+) | 3.3622 × 10−6 |
RTLBO | 2.03259457471086 × 10−3 | 1.72981674269726 × 10−3 | 1.75707325040703 × 10−3 | 5.52642113734253 × 10−5 | (+) | 1.4295 × 10−11 |
STP6-120/36 | DODE | 1.66006031250855 × 10−2 | 1.66006031250846 × 10−2 | 1.66006031250850 × 10−2 | 2.04948114449183 × 10−16 | NA | NA |
JADE | 3.23869693230732 × 10−2 | 1.66006031304504 × 10−2 | 2.35124969859339 × 10−2 | 4.76162310942712 × 10−3 | (+) | 1.5080 × 10−11 |
CoDE | 1.66006031250892 × 10−2 | 1.66006031250856 × 10−2 | 1.66006031250862 × 10−2 | 7.62895279964384 × 10−16 | (+) | 1.5070 × 10−11 |
MPEDE | 1.82620932807556 × 10−2 | 1.66022624048459 × 10−2 | 1.69172833234824 × 10−2 | 4.21939732308916 × 10−4 | (+) | 1.5080 × 10−11 |
SEDE | 1.66006031250856 × 10−2 | 1.66006031250850 × 10−2 | 1.66006031250853 × 10−2 | 1.56419096552865 × 10−16 | (+) | 4.3699 × 10−7 |
DOLADE | 1.66006031250854 × 10−2 | 1.66006031250848 × 10−2 | 1.66006031250852 × 10−2 | 1.53948604759203 × 10−16 | (+) | 7.1962 × 10−4 |
IWOA | 1.79420615507602 × 10−2 | 1.66007104108898 × 10−2 | 1.70674957253113 × 10−2 | 3.73519403638798 × 10−4 | (+) | 1.5080 × 10−11 |
PGJAYA | 1.67838656900354 × 10−2 | 1.66006033230863 × 10−2 | 1.66077600606031 × 10−2 | 3.27754428313028 × 10−5 | (+) | 1.4957 × 10−11 |
IGSK | 1.66006031250856 × 10−2 | 1.66006031250849 × 10−2 | 1.66006031250853 × 10−2 | 1.46366533003527 × 10−16 | (+) | 9.2828 × 10−7 |
LaPSO | 1.00180624317972 | 1.66006031250853 × 10−2 | 5.90955682568764 × 10−2 | 1.82562599818994 × 10−1 | (+) | 4.1546 × 10−11 |
RTLBO | 4.53742241915738 × 10−2 | 1.66006281589215 × 10−2 | 1.78173047683371 × 10−2 | 5.23759995710919 × 10−3 | (+) | 1.5080 × 10−11 |
(6) For the STP6-120/36 module model, DODE obtains the best Min and Mean compared to others. DOLADE obtains the best Max value, while IGSK possesses the best SD. From the statistical results, it is observed that DODE still maintains a competitive advantage compared to other algorithms.
As a further comparison, the Friedman test [
80] is adopted to obtain the overall rankings of all 11 tested algorithms on the six PV problems. In the Friedman test, the null hypothesis postulates that the average values of all tested methods are equal. Then, all algorithms are ranked based on their average values across each problem, and the average ranking of each algorithm among all problems is calculated to evaluate their respective performance. A lower average ranking indicates that the corresponding algorithm performs better. The overall ranking results of all compared algorithms are illustrated in
Figure 9. It is evident that the proposed DODE algorithm is the top-performing algorithm in comparison to all opponents by yielding the best average ranking of 1.67 for six PV models, followed by SEDE and DOLADE, with average rankings of 2.58 and 2.67, respectively.
Additionally, the computational cost of each tested algorithm on the above six PV models is illustrated in
Figure 10. This shows that, despite the consistency in the maximum number of fitness evaluations, the total computation time varies among the algorithms due to differences in their computational complexity. Specifically, IGSK possesses the shortest computation cost, while RTLBO possesses the longest. The computational costs of the remaining algorithms are nearly equivalent, and the slight increase in the computation cost of DODE is acceptable.
6. Conclusions
To determine optimal parameters in the operation of PV systems, this paper develops an improved differential evolution algorithm, termed DODE, to accurately and efficiently extract some unknown parameters in various PV models. DODE innovatively integrates a novel collaboration mechanism of dual mutation strategies and an orientation guidance mechanism, effectively leveraging individual information at different stages and population evolution orientation to enhance the optimization capability of DE. The equivalent circuits are provided based on the diode, and the mathematical formulations of problems are constructed. The RMSE is utilized as one of the standards for evaluating the algorithmic quality. Extensive comparisons are made with existing state-of-the-art algorithms on PV model parameter identification, and the simulated data and measured data are presented. I-V and P-V characteristic curves are plotted. The convergence curves depict that DODE yields the optimal values of identified parameters, and the corresponding RMSE values are 9.86021877891317 × 10−4, 9.82484851784979 × 10−4, 9.82484851784993 × 10−4, 2.42507486809489 × 10−3, 1.72981370994064 × 10−3, and 1.66006031250846 × 10−2, 6.0600 × 10−4 for the six different PV modules. Further, the statistical analysis based on two non-parametric test methods (i.e., the Wilcoxon rank-sum and the Friedman tests), shows that the DODE algorithm effectively addresses these issues and achieves highly competitive experimental results.
To summarize, DODE shows a powerful performance compared with these advanced algorithms regarding the accuracy and reliability of identified parameters, the convergence characteristics, and some statistical indexes. However, a few constraints still need to be further considered in future work. Firstly, DODE should be applied to the parameter identification problem of other PV scenarios to thoroughly verify its optimization capabilities. In particular, its performance can be further demonstrated under different experimental conditions, such as variations in ambient temperature, irradiance, wind speed, shading condition, and other ever-changing and uncontrollable factors. Then, the feasibility and adaptability of DODE can be further improved by combining the design principles of other optimization algorithms. The authors believe that DODE cannot be considered a ubiquitous method because, according to the description of the “No Free Lunch” theorem, no one method can solve all optimization issues.