# Optimal Identification of Unknown Parameters of Photovoltaic Models Using Dual-Population Gaining-Sharing Knowledge-Based Algorithm

Authors: Guojiang Xiong; Lei Li; Ali Wagdy Mohamed; Jing Zhang; Yao Zhang; Hao Chen
Journal: International Journal of Intelligent Systems
 (2023)

---

## Abstract

Establishing an accurate equivalent model is a critical foundation to describe the energy conversion characteristics of a photovoltaic system, which can support the research of fault analysis, output power prediction, and performance analysis of the photovoltaic system. However, the widely used equivalent models are highly nonlinear and have many unknown parameters, making it difficult to identify these parameters accurately. Our previous work found that the gaining-sharing knowledge-based algorithm (GSK) shows promising performance in solving this problem. But its efficacy is not enough to achieve accurate parameters within a relatively limited computing resource. In this context, a dual-population GSK algorithm (DPGSK), which introduces a dual-population evolution strategy for more excellent searchability, is proposed to address this issue. In each iteration, the population splits equally and randomly into two subpopulations, one of which performs the junior gaining-sharing phase while the other performs the senior gaining-sharing phase. Then two updated subpopulations merge to form a new population. This allows for a grand reconciliation of convergence speed and population diversity, giving DPGSK powerful optimization performance. Afterward, DPGSK is applied to five photovoltaic models and validated for performance against other advanced metaheuristics. Besides, the impact of different components on DPGSK is also investigated. Results and comparisons show that either component is indispensable to DPGSK, and DPGSK strengthens the convergence and achieves accurate and reliable results, demonstrating its superiority over other algorithms in solving this studied problem.


---

## Body

## 1. Introduction

As the most dominant contributor to current energy sources, fossil fuels, although cheap, versatile, and easy to store and transport, have resulted in climate change, environmental pollution, and even global warming as a result of their overuse [1–3]. Therefore, clean, green, and efficient renewable energy sources are urgently needed to curb the use of fossil energy. The renewable nature of solar, wind, geothermal, and biomass energy is self-explanatory. Among them, solar energy deserves to be used extensively as a universal, vast, and long-lasting clean energy due to its exclusive merits [4–6]. Photovoltaic (PV) is currently the primary form of utilization of solar power. To boost the efficiency of electrical energy conversion, it is highly essential to optimize the PV system accurately. As the electrical characteristics of a PV system are nonlinear, varying environments can affect the PV efficiency, leading to different performance parameters, and the parameters will also change in varying environments [7]. Thereby, accurate and effective modeling has become a significant and challenging problem in PV system optimization. To better design, predict, and estimate the performance of a PV system, an equivalent model to accurately characterize its energy transformation relationship is indispensable. The widely used equivalent models include the single/double diode models (SDM/DDM) [8–10]. SDM and DDM can help to understand the energy conversion behavior of PV systems and explain the dynamic voltage-current electrical characteristics. However, due to the existence of exponential and implicit functions, these two models are highly nonlinear and highly nonconvex. They contain five and seven unknown parameters, respectively, leading to difficulties in time consumption and low accuracy in achieving their optimal models. How to efficiently obtain accurate values for these models' unknown parameters is, therefore, a critical foundation for establishing the PV models.At present, researchers have proposed various solutions to address this problem. Deterministic methods and metaheuristics can basically cover these solutions [8, 11]. The deterministic methods are further classified into analytical and numerical methods. The former deploys some particular measured points such as the maximum power and short-circuit and open-circuit moments to solve nonlinear equations to get unknown parameters [12–14]. The increase in the number of parameters impacts the modeling complexity and is time-consuming in the calculation. The analytical methods include the Lambert-W-based method [15], OSMP-based method [16], and reduced space search [17]. The numerical methods try to reduce the dimensionality of system equations by iterating all the experimental current and voltage data successively to solve the problem quickly. However, it applies only to continuous, differentiable, and convex functions [18]. In addition, initial values are crucial and unsuitable ones can lead to large errors in the identification results because these methods easily fall into local optimization. Therefore, the use of numerical methods has some limitations. The common numerical methods contain the Gauss–Seidel method [19], Newton-Raphson method [20], and the least-square method [21].Deterministic methods rely heavily on functional models, which have severe defects such as sensitivity to initial solutions and easy to fall into local optimization. Thus, many metaheuristic algorithms derived from natural phenomena have been proposed to tackle various complex optimization problems including the studied problem [4, 22]. Each algorithm, however, has its own particular strengths and weaknesses. Particle swarm optimization (PSO) is a typical algorithm that is easy to realize and requires fewer parameters to coordinate, but it easily falls into local optimum and has insufficient search accuracy [23–25]. The cuckoo search algorithm (CS) is robust and not easily trapped in a local optimum, but it converges slowly [26]. Genetic algorithm (GA) converges fast but easily suffers from the problem of premature [27, 28]. Differential evolution (DE) is concise and valid but is strongly influenced by the algorithm parameters [29, 30]. Teaching-learning-based optimization (TLBO) is an easy-to-implement stochastic metaheuristic. However, its search capability is poor and the search accuracy is low [31]. The whale optimization algorithm (WOA) exploits the randomness of the best search agent to model the predation mechanisms. Nevertheless, its adaptive parameters depend on random distributions, and thereby it is prone to premature convergence [32]. Inspired by the intelligent behavior of bees, artificial bee colony optimization (ABC) avoids falling into local optima by employing operators to construct solutions randomly [33]. But it converges slowly and is hard to achieve satisfactory solutions within a limited resource. Motivated by the supply and demand mechanism, supply-demand-based optimization (SDO) combines different dynamic patterns of the spider web model organically to balance exploration and exploitation well [34]. However, its structure is slightly complex. In addition to the above-mentioned metaheuristic algorithms, many modified algorithms have also been proposed, such as the self-learning discrete jaya (SD-Jaya) [35], either-or TLBO (EOTLBO) [36], improved WOA (IWOA) [37], bee pollinator flower pollination algorithm (BPFPA) [38], classified perturbation mutation-based PSO (CPMPSO) [23], teaching-learning-based artificial bee colony (TLABC) [39], and Mixed-Variable Differential Evolution(MVDE) [40].Undeniably, both the basic metaheuristic algorithms and their improved versions have shown excellent performance in tackling the PV models’ parameter identification problem. Nevertheless, due to the problem’s complexity and importance and the fact that it is not easy to obtain accurate values within a given limited computing resource, it is still a hard nut to crack. Besides, the no free lunch (NFL) theorem [41] points out emphatically that there is still a need to try and propose more methods with better performance to solve this concerned, tough optimization problem. Meanwhile, unlike the heuristic methods that rely on the characteristics of the specific problem to be solved, the metaheuristic algorithms achieve heuristic guidance through the exchange of information between population individuals and require less computational effort for large neighborhood search, and thus, can improve the search efficiency with the population-based iterated greedy mechanism [42]. Therefore, they are more popular to be applied to different complex engineering problems including the studied PV parameter identification.Gaining-sharing knowledge-based algorithm (GSK) is another effective population-based metaheuristic algorithm [43]. Inspired by the two processes of knowledge gaining-sharing, GSK equips with two significant phases including junior/senior gaining-sharing, to prompt population individuals to evolve. The experimental results have shown the superior performance of GSK in solving benchmark optimization problems. Motivated by this, in one of our previous works, we applied the basic GSK successfully in tackling the PV models’ parameter identification problem for the first time [44]. The achieved results have demonstrated its good robustness and accuracy over other peer algorithms in this problem. However, we also found some shortcomings of GSK. On the one hand, the algorithm converges relatively slowly compared to other algorithms, especially in the early evolutionary stage. On the one hand, it is hard to achieve sufficiently accurate enough parameters within a relatively limited computing resource. The main reason is as follows: The original GSK algorithm relies primarily on the junior/senior gaining-sharing phases to weigh the exploration and exploitation. The first phase is mainly responsible for exploration, while the latter primarily supervises exploitation. In the early evolutionary stage, GSK tends to perform the junior phase, which leads to a slower convergence rate. In the later stage, the overwhelming adoption of the senior phase leads to a rapid decline in population diversity, which is not conducive to refining solutions with sufficient accuracy. Therefore, these two phases are not coordinated well enough to shape a powerful GSK in solving this problem.In this paper, inspired by the distinct functional characteristics of these two phases, we propose an improved variant of GSK, namely, dual-population GSK (DPGSK), to tackle the studied problem. At the beginning of each iteration, the whole population is chopped up into two equal subpopulations randomly. One subpopulation employs the junior phase to update the corresponding individuals, while the other subpopulation adopts the senior phase to arm them. Finally, these two subpopulations merge to obtain a new population. This iterative approach can improve the convergence speed in the early stage, as only the senior phase is used in the first subpopulation. In the later stage, the population diversity is maintained as only the junior phase is used in the second subpopulation. The DPGSK algorithm’s effectiveness is confirmed by comparing it with other algorithms in five models.The motivations behind DPGSK are as follows:(1)
The original GSK contains two phases, and two subpopulations can select mutually exclusive phases to perform. However, in the original version, they are not well coordinated in different stages, leading to slow convergence in the early stage and a lack of adequate solution-refining capability in the later stage.(2)
The dual-population evolution strategy can achieve higher efficiency by updating two subpopulations in two distinct phases simultaneously.(3)
The dual-population can balance the population diversity and the convergence rate to avoid. This can compensate for poor convergence speed in the early stage and poor solution accuracy in the later stage. Therefore, exploration and exploitation can be equilibrated well to achieve accurate results.The main contributions of this paper are listed as follows:(1)
An enhanced approach, namely, DPGSK is put forward to achieve accurate PV models’ parameters.(2)
A dual-population evolution strategy is designed for DPGSK, by which each subpopulation selects one of the two phases of GSK to generate heterogeneous individuals to elevate the searchability of DPGSK.(3)
The suggested DPGSK algorithm is implemented in five PV models. The results highly confirm the superiority of DPGSK over other algorithms in solving the studied problem.The remaining parts are outlined as follows: Section2 presents the mathematical model of the studied problem. Sections 3 and 4 introduce the basic GSK algorithm and the proposed DPGSK, respectively. In Section 5, the analysis and discussions are summarized. Finally, Section 6 gives the conclusions.


## 2. Description of the PV Models

This section describes the SDM, DDM, and PV module models in detail. To establish the PV models more accurately, the model parameter identification problem is also formulated in this section.


### 2.1. SDM

Figure1(a) is the circuital diagram of SDM. The total output voltage is VL, and the output current IL is expressed as follows [45]:(1)IL=IPV−ID−Ish=IPV−IsdexpqVL+ILRsnkT−1−VL+ILRsRsh,where IPV means the photo-generated current, ID means the diode current, Ish means shunt resistance current, Isd means the saturation current, n means the diode ideal factors, k denotes the Boltzmann constant (1.3806503×10−23J/K), q denotes the electron charge (1.60217646×10−19C), T represents the PV cell temperature in Kelvin, Rs is the series resistance, and Rsh is the parallel resistance.Figure 1
Circuits of PV models: (a) SDM; (b) DDM; (c) PV module.
(a)
(b)
(c)As can be seen from the aforementioned parameters, the parameters identified in the SDM areIPV,Isd,Rs,Rsh,andn.


### 2.2. DDM

Figure1(b) is the circuital diagram of the DDM. The SDM ignores the recombination losses of current, while the DDM can solve this problem and guarantee a balance between simplicity and accuracy. The IL is expressed as follows [46, 47]:(2)IL=IPV−ID1−ID2−Ish=IPV−Isd1expqVL+ILRsn1kT−1−Isd2expqVL+ILRsn2kT−1−VL+ILRsRsh,where ID1 and ID2 mean the first and second diode currents and n1 and n2 mean their corresponding ideal factors, respectively.From (2), the parameters identified in the DDM are IPV,Isd1,Isd2,Rs,Rsh,n1,andn2.


### 2.3. PV Module

The circuital diagram of the PV module is given in Figure1(c). It has Na×Nb PV cells. IL is calculated as follows [48, 49]:(3)IL=IPVNb−IsdNbexpVL+ILRsNa/NbnNaVt−1−VL+ILRsNa/NbRshNa/Nb.From (3), the parameters identified in the PV module are IPV,Isd,Rs,Rsh,andn, which are the same as the SDM.


### 2.4. Objective Function

In this work, we convert the studied problem into a numerical optimization problem, and use the root mean square error (RMSE) between the measured data and the calculated data as the objective function [50, 51].(4)RMSEx=1N∑k=1NfVL,IL,x2,where N means the number of measured data. The error functions fVL,IL,x and the solution vector x are expressed as follows:for the SDM,(5)fVL,IL,x=IPV−IsdexpqVL+ILRsnkT−1−VL+ILRsRsh−IL,x=IPV,Isd,Rs,Ish,n,,for the DDM,(6)fVL,IL,x=IPV−Isd1expqVL+ILRsn1kT−1−Isd2expqVL+ILRsn2kT−1−VL+ILRsRsh−ILx=IPV,Isd1,Isd2,Rs,Ish,n1,n2,for the PV module,(7)fVL,IL,x=IPVNb−IsdNbexpVL+ILRsNa/NbnNaVt−1−VL+ILRsNa/NbRshNa/Nb−IL,x=IPV,Isd,Rs,Ish,n.


## 2.1. SDM

Figure1(a) is the circuital diagram of SDM. The total output voltage is VL, and the output current IL is expressed as follows [45]:(1)IL=IPV−ID−Ish=IPV−IsdexpqVL+ILRsnkT−1−VL+ILRsRsh,where IPV means the photo-generated current, ID means the diode current, Ish means shunt resistance current, Isd means the saturation current, n means the diode ideal factors, k denotes the Boltzmann constant (1.3806503×10−23J/K), q denotes the electron charge (1.60217646×10−19C), T represents the PV cell temperature in Kelvin, Rs is the series resistance, and Rsh is the parallel resistance.Figure 1
Circuits of PV models: (a) SDM; (b) DDM; (c) PV module.
(a)
(b)
(c)As can be seen from the aforementioned parameters, the parameters identified in the SDM areIPV,Isd,Rs,Rsh,andn.


## 2.2. DDM

Figure1(b) is the circuital diagram of the DDM. The SDM ignores the recombination losses of current, while the DDM can solve this problem and guarantee a balance between simplicity and accuracy. The IL is expressed as follows [46, 47]:(2)IL=IPV−ID1−ID2−Ish=IPV−Isd1expqVL+ILRsn1kT−1−Isd2expqVL+ILRsn2kT−1−VL+ILRsRsh,where ID1 and ID2 mean the first and second diode currents and n1 and n2 mean their corresponding ideal factors, respectively.From (2), the parameters identified in the DDM are IPV,Isd1,Isd2,Rs,Rsh,n1,andn2.


## 2.3. PV Module

The circuital diagram of the PV module is given in Figure1(c). It has Na×Nb PV cells. IL is calculated as follows [48, 49]:(3)IL=IPVNb−IsdNbexpVL+ILRsNa/NbnNaVt−1−VL+ILRsNa/NbRshNa/Nb.From (3), the parameters identified in the PV module are IPV,Isd,Rs,Rsh,andn, which are the same as the SDM.


## 2.4. Objective Function

In this work, we convert the studied problem into a numerical optimization problem, and use the root mean square error (RMSE) between the measured data and the calculated data as the objective function [50, 51].(4)RMSEx=1N∑k=1NfVL,IL,x2,where N means the number of measured data. The error functions fVL,IL,x and the solution vector x are expressed as follows:for the SDM,(5)fVL,IL,x=IPV−IsdexpqVL+ILRsnkT−1−VL+ILRsRsh−IL,x=IPV,Isd,Rs,Ish,n,,for the DDM,(6)fVL,IL,x=IPV−Isd1expqVL+ILRsn1kT−1−Isd2expqVL+ILRsn2kT−1−VL+ILRsRsh−ILx=IPV,Isd1,Isd2,Rs,Ish,n1,n2,for the PV module,(7)fVL,IL,x=IPVNb−IsdNbexpVL+ILRsNa/NbnNaVt−1−VL+ILRsNa/NbRshNa/Nb−IL,x=IPV,Isd,Rs,Ish,n.


## 3. The Basic GSK Algorithm

GSK is a novel approach proposed for tackling optimization problems. According to the two processes of knowledge gaining-sharing, this algorithm comprises junior and senior gaining-sharing phases accordingly [43].


### 3.1. Initialization

In GSK, the population is made up ofNp individuals. The i-th individual xi is denoted as xi=xi1,xi2,⋯,xiD, where D is the number of individual dimensions.


### 3.2. Dimension Partitioning

During the individual update process,DJ denotes the number of dimensions an individual uses in the junior phase. DS denotes the rest that the individual uses in the senior phase. For an individual, an empirical equation is used to determine DJ and DS as follows:(8)DJ=D×1−GGENK,(9)DS=D−DJ,where K means the knowledge rate, G means the generation number, and GEN means the required generation number.


### 3.3. Junior Gaining-Sharing Phase

For each individual in this phase, if the knowledge can be gained and shared, the updating formula is expressed as follows:(10)xi,new=xi+kf⋅xi−1−xi+1+xr−xi,iffxi>fxr,xi+kf⋅xi−1−xi+1+xi−xr,otherwise,,where kf means the knowledge factor. xr is a random individual different from xi−1, xi and xi+1. xi−1 and xi+1 are selected by the following method:Step 1: sort individuals in ascending order by their fitness valuesStep 2: each individualxi selects two adjacent individuals including xi−1 and xi+1 to gain and share knowledge. The best (xi−1) and worst (xi+1) individuals are considered to be its adjacent individuals. For the best individual, the latter adjacent individuals are selected as follows: x1,x2,andx3. For the worst individual, the former adjacent individuals are selected as follows: xNp−2,xNp−1,andxNp.After generatingxi,new, a parameter kr named knowledge ratio is adopted to regulate the probability of each update of an individual.


### 3.4. Senior Gaining-Sharing Phase

The update process in this phase is as follows:(11)xi,new=xi+kf⋅xp−best−xp−worst+xm−xi,iffxi>fxm,xi+kf⋅xp−best−xp−worst+xi−xm,otherwise,,where xp−best, xp−worst, and xm are gaining and sharing sources different from xi−1 and selected by the following method:Step 1: sort individuals in ascending order by their fitness values and divide them into three groups, which are the best group (the top100p% individuals), the middle group (the medial 1−2×100p% individuals), and the worst group (the bottom 100p% individuals), respectivelyStep 2: for each individualxi, the xp−best, xp−worst, and xm are randomly generated from the above three groups, respectivelyThe pseudocode of the GSK is given in Algorithm1. FEs denotes the amount of fitness function, and Max_FEs denotes the upper limit of FEs.Algorithm 1: Basic GSK.
Input: algorithm parameters: NP, kf, kr, K, and p
Output: optimal solution
(1)
Set FEs= 0 and G  = 1
(2)
Initialize a random population with NP individuals
(3)
Evaluate the objection function value for each individual
(4)
FEs = FEs + NP
(5)
WhileFEs < Max_FEs do
(6)
Fori to NPdo
(7)
//Junior gaining-sharing phase
(8)
CalculateDJ with Equation (8)
(9)
Forj = 1 to DJdo
(10)
Generatexi.new with Equation (10)
(11)
End
(12)
//Senior gaining-sharing phase
(13)
Calculate theDS with Equation (9)
(14)
Forj = 1 to DSdo
(15)
Generatexi.new with Equation (11)
(16)
End
(17)
Evaluate the objection function value forxi.new
(18)
FEs = FEs + 1
(19)
Iff (xi.new) ≤ f (xi) then
(20)
xi = xi.new
(21)
End
(22)
End
(23)
G = G + 1
(24)
End while


## 3.1. Initialization

In GSK, the population is made up ofNp individuals. The i-th individual xi is denoted as xi=xi1,xi2,⋯,xiD, where D is the number of individual dimensions.


## 3.2. Dimension Partitioning

During the individual update process,DJ denotes the number of dimensions an individual uses in the junior phase. DS denotes the rest that the individual uses in the senior phase. For an individual, an empirical equation is used to determine DJ and DS as follows:(8)DJ=D×1−GGENK,(9)DS=D−DJ,where K means the knowledge rate, G means the generation number, and GEN means the required generation number.


## 3.3. Junior Gaining-Sharing Phase

For each individual in this phase, if the knowledge can be gained and shared, the updating formula is expressed as follows:(10)xi,new=xi+kf⋅xi−1−xi+1+xr−xi,iffxi>fxr,xi+kf⋅xi−1−xi+1+xi−xr,otherwise,,where kf means the knowledge factor. xr is a random individual different from xi−1, xi and xi+1. xi−1 and xi+1 are selected by the following method:Step 1: sort individuals in ascending order by their fitness valuesStep 2: each individualxi selects two adjacent individuals including xi−1 and xi+1 to gain and share knowledge. The best (xi−1) and worst (xi+1) individuals are considered to be its adjacent individuals. For the best individual, the latter adjacent individuals are selected as follows: x1,x2,andx3. For the worst individual, the former adjacent individuals are selected as follows: xNp−2,xNp−1,andxNp.After generatingxi,new, a parameter kr named knowledge ratio is adopted to regulate the probability of each update of an individual.


## 3.4. Senior Gaining-Sharing Phase

The update process in this phase is as follows:(11)xi,new=xi+kf⋅xp−best−xp−worst+xm−xi,iffxi>fxm,xi+kf⋅xp−best−xp−worst+xi−xm,otherwise,,where xp−best, xp−worst, and xm are gaining and sharing sources different from xi−1 and selected by the following method:Step 1: sort individuals in ascending order by their fitness values and divide them into three groups, which are the best group (the top100p% individuals), the middle group (the medial 1−2×100p% individuals), and the worst group (the bottom 100p% individuals), respectivelyStep 2: for each individualxi, the xp−best, xp−worst, and xm are randomly generated from the above three groups, respectivelyThe pseudocode of the GSK is given in Algorithm1. FEs denotes the amount of fitness function, and Max_FEs denotes the upper limit of FEs.Algorithm 1: Basic GSK.
Input: algorithm parameters: NP, kf, kr, K, and p
Output: optimal solution
(1)
Set FEs= 0 and G  = 1
(2)
Initialize a random population with NP individuals
(3)
Evaluate the objection function value for each individual
(4)
FEs = FEs + NP
(5)
WhileFEs < Max_FEs do
(6)
Fori to NPdo
(7)
//Junior gaining-sharing phase
(8)
CalculateDJ with Equation (8)
(9)
Forj = 1 to DJdo
(10)
Generatexi.new with Equation (10)
(11)
End
(12)
//Senior gaining-sharing phase
(13)
Calculate theDS with Equation (9)
(14)
Forj = 1 to DSdo
(15)
Generatexi.new with Equation (11)
(16)
End
(17)
Evaluate the objection function value forxi.new
(18)
FEs = FEs + 1
(19)
Iff (xi.new) ≤ f (xi) then
(20)
xi = xi.new
(21)
End
(22)
End
(23)
G = G + 1
(24)
End while


## 4. The Proposed DPGSK

Although the original GSK works well on many optimization problems as a new type of metaheuristic algorithm, there is still room for improvement on a specific problem [44, 52, 53]. For example, in the studied problem in this paper, GSK has proven to have comprehensive performance in our previous study but still suffers from insufficient convergence and insufficient accuracy of solutions. Therefore, we purposefully propose an improved scheme to further raise its performance in the PV parameter identification problem.It can be seen from the original GSK that an individual needs to refer to a lot of information to equip himself. Namely, theDJ and DS dimensions of an individual choose to perform the junior and senior phases, respectively, which will perplex the movement of the individual toward the optimal solution. In this context, exploration and exploitation are hard to be coordinated well enough to shape GSK with a powerful search ability to tackle the PV models’ parameter identification problem. This is mainly due to the relatively poor exploitation capacity in the early evolutionary stage when GSK mostly uses the junior phase. Moreover, GSK is gradually leaning towards the senior phase in the later stage, which leads to inadequate solution refining capability. Therefore, achieving highly-accurate parameters within a relatively limited computing resource is hard for GSK to solve this problem.To conquer the shortcomings of GSK and balance exploration and exploitation, this work introduces a dual-population evolution strategy to obtain a dual-population GSK (DPGSK). In DPGSK, we randomly divide the population into two equal subpopulationspop1 and pop2 in each iteration. The first subpopulation, pop1, performs the junior gaining-sharing phase, while the other subpopulation, pop2, performs the senior gaining-sharing phase. After updating, these two subpopulations merge to obtain a new population. In this way, all dimensions of an individual refer to the same gaining and sharing sources to unify the movement to harmonize convergence and population diversity. DPGSK’s flowchart is presented in Figure 2 and Algorithm 2.Figure 2
The flow chart of DPGSK.Algorithm 2: Proposed DPGSK.
Input: algorithm parameters: NP, kf, kr, and p
Output: optimal solution
(1)
Set FEs= 0 and G  = 1
(2)
Initialize a random population with NP individuals
(3)
Evaluate the objection function value for each individual
(4)
FEs = FEs + NP
(5)
WhileFEs < Max_FEs do
(6)
Divide the population intopop1 and pop2 randomly
(7)
//Junior gaining-sharing phase (forpop1)
(8)
Fori = 1 to NP/2do
(9)
Generatexpop1, i, new with Equation (10)
(10)
Evaluate the objection function value forxpop1, i, new
(11)
FEs = FEs + 1
(12)
Iff (xpop1, i, new) ≤ f (xpop1, i), then
(13)
xpop1, i = xpop1, i, new
(14)
End
(15)
End
(16)
//Senior gaining-sharing phase (forpop2)
(17)
Fori = 1 to NP/2do
(18)
Generatexpop2, i, new with Equation (11)
(19)
Evaluate the objection function value forxpop2, i, new
(20)
FEs = FEs + 1
(21)
Iff (xpop2,i,new) ≤ f (xpop2,i) then
(22)
xpop2,i = xpop2,i,new
(23)
End
(24)
End
(25)
Mergepop1 and pop2 to obtain a new population
(26)
G = G + 1
(27)
End while


## 5. Experimental Results

The proposed DPGSK algorithm is applied to five PV models, as shown in Table1. Table 2 shows the parameters’ range for these PV models [16, 54–56].Table 1
Five models’ information.
PV model
Solar cell
Irradiance (W/m2)
Temperature (°C)
Max_FEs
Type
Number
SDM/DDM
RT.C. France silicon solar cell
1
1000
33
10000
Photowatt-PWP201
Polysilicon cell
36
1000
45
10000
STM6-40/36
Monocrystalline silicon cell
36
1000
51
15000
STP6-120/36
Monocrystalline silicon cell
36
1000
55
15000Table 2
PV models’ parameters range.
Parameter
Single/double diode
Photowatt-PW201
STM6-40/36
STP6-120/36
LB
UB
LB
UB
LB
UB
LB
UB
I
ph (A)
0
1
0
2
0
2
0
8
I
sd, Isd1, Isd2 (μA)
0
1
0
50
0
50
0
50
R
s (Ω)
0
0.5
0
2
0
0.36
0
0.36
R
sh (Ω)
0
100
0
2000
0
1000
0
1500
n, n1, n2
1
2
1
50
1
60
1
50To better demonstrate DPGSK, we compare the algorithm with the basic GSK and other eight improved metaheuristic algorithms, which are the improved TLBO algorithm (ITLBO) [57], teaching-learning-based ABC algorithm (TLABC) [39], self-adaptive TLBO algorithm (SATLBO) [58], phasor PSO algorithm (PPSO) [59], comprehensive learning PSO algorithm (CLPSO) [60], improved JAYA optimization algorithm (IJAYA) [61], adaptive guided DE (AGDE) [62], and hybrid DE with WOA (DE_WOA) [63]. Table 3 shows the parameters setting for each algorithm. For the sake of fairness, all algorithms run individually 30 times in MATLAB R2018a.Table 3
Parameters setting.
Algorithm
Parameter setting
DPGSK
NP = 20,kr = 0.9, kf = 0.5, p=0.1
GSK
NP = 20,kr = 0.9, kf = 0.5, K = 10, p=0.1
ITLBO
NP = 20
SATLBO
NP = 20
TLABC
NP = 20,F = rand (0,1)
IJAYA
NP = 20
PPSO
NP = 20
CLPSO
NP= 20, gap = 7, c = 1.49445, w=0.9 to 0.4
AGDE
NP = 20, part_size = 0.1
DE_WOA
NP = 20,F = rand (0.1,1), CR = rand (0,1)


### 5.1. Results of the SDM

The identification results of the five parameters of the SDM are shown in Table4. After getting the solutions, they can be used to calculate the current data and individual absolute error values (IAE = |IL measured − IL calculated|), presented in Table 5. We can see that the DPGSK’s IAE (0.01769439) is consistently smaller than PPSO (0.01770360), AGDE (0.01770370), GSK (0.01770416), TLABC (0.01771991), ITLBO (0.01802719), and IJAYA (0.01806562). The current-voltage (I-V) and power-voltage (P-V) characteristic curves of DPGSK are plotted in Figure 3(a) and Figure 3(b), respectively, showing that a good agreement is found between the calculated data and the measured data. It demonstrates that DPGSK has superior accuracy in this model.Table 4
Parameter identification results of the SDM.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
0.76077553
0.32302079
0.03637709
53.71852013
1.48118358
GSK
0.76077553
0.32302085
0.03637709
53.71852481
1.48118360
ITLBO
0.76088942
0.30531731
0.03658155
51.11235228
1.47555381
SATLBO
0.76098523
0.33649797
0.03621845
53.24192945
1.48531569
TLABC
0.76074567
0.32115398
0.03639442
53.76527575
1.48060283
IJAYA
0.76083168
0.29010861
0.03678511
52.18389372
1.47039884
PPSO
0.76076441
0.33383560
0.03624545
54.58985322
1.48450689
CLPSO
0.76098396
0.370652361
0.03492495
57.25728241
1.49542141
AGDE
0.76077548
0.32303137
0.03637697
53.72031579
1.48118683
DE_WOA
0.76077552
0.32302447
0.03637705
53.71869823
1.48118474Table 5
IAE results in the SDM.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
−0.2057
0.7640
0.76402556
0.76408765
0.76436713
0.76405468
0.76423508
0.76408764
0.76408749
2
−0.1291
0.7620
0.76262328
0.76266264
0.76286952
0.76263091
0.76276822
0.76266264
0.76266253
3
−0.0588
0.7605
0.76133622
0.76135473
0.76149499
0.76132414
0.76142189
0.76135472
0.76135466
4
0.0057
0.7605
0.76015484
0.76015423
0.76023338
0.76012468
0.76018618
0.76015427
0.76015420
5
0.0646
0.7600
0.75907387
0.75905585
0.75907929
0.75902727
0.75905582
0.75905635
0.75905586
6
0.1185
0.7590
0.75807680
0.75804300
0.75801572
0.75801534
0.75801419
0.75804300
0.75804305
7
0.1678
0.7570
0.75713938
0.75709159
0.75701864
0.75706483
0.75703780
0.75709159
0.75709166
8
0.2132
0.7570
0.75620170
0.75614207
0.75602889
0.75611634
0.75606865
0.75614207
0.75614217
9
0.2545
0.7555
0.75515544
0.75508732
0.75494149
0.75506296
0.75500327
0.75508732
0.75508744
10
0.2924
0.7540
0.75373601
0.75366447
0.75349632
0.75364218
0.75358460
0.75366447
0.75366460
11
0.3269
0.7505
0.75145540
0.75138806
0.75121264
0.75136907
0.75133542
0.75138806
0.75138820
12
0.3585
0.7465
0.74740107
0.74734834
0.74718587
0.74733452
0.74735513
0.74734834
0.74734848
13
0.3873
0.7385
0.74012294
0.74009688
0.73997134
0.74009054
0.74020040
0.74009688
0.74009699
14
0.4137
0.7280
0.72738539
0.72739678
0.72733136
0.72740028
0.72762993
0.72739678
0.72739686
15
0.4373
0.7065
0.70690114
0.70695327
0.70695979
0.70696778
0.70732358
0.70695327
0.70695331
16
0.4590
0.6755
0.67520863
0.67529489
0.67536910
0.67531996
0.67577635
0.67529490
0.67529488
17
0.4784
0.6320
0.63078292
0.63088431
0.63100001
0.63091697
0.63140894
0.63088431
0.63088426
18
0.4960
0.5730
0.57199009
0.57208207
0.57220153
0.57211788
0.57256254
0.57208207
0.57208200
19
0.5119
0.4990
0.49943067
0.49949164
0.49957875
0.49952593
0.49984987
0.49949164
0.49949157
20
0.5265
0.4130
0.41347542
0.41349356
0.41352576
0.41352273
0.41368331
0.41349356
0.41349350
21
0.5398
0.3165
0.31724244
0.31721950
0.31719456
0.31724187
0.31724001
0.31721950
0.31721946
22
0.5521
0.2120
0.21215403
0.21210317
0.21203682
0.21211888
0.21199141
0.21210317
0.21210313
23
0.5633
0.1035
0.10277871
0.10272135
0.10264290
0.10273211
0.10254314
0.10272134
0.10272129
24
0.5736
−0.0100
−0.00920948
−0.00924886
−0.00930440
−0.00924053
−0.00941687
−0.00924887
−0.00924895
25
0.5833
−0.1230
−0.12438567
−0.12438137
−0.12437645
−0.12437260
−0.12445883
−0.12438137
−0.12438151
26
0.5900
−0.2100
−0.20924496
−0.20919304
−0.20912068
−0.20918205
−0.20915470
−0.20919304
−0.20919323
∑
I
A
E
0.01769439
0.01770416
0.01802719
0.01771991
0.01806562
0.01770360
0.01770370Figure 3
Characteristic curves of DPGSK in the SDM. (a) I-V. (b) P-V.
(a)
(b)Table6 shows the objective function values (RMSE) obtained by different algorithms in the SDM. To accurately verify the DPGSK’s effectiveness, the best, worst, mean, and standard deviation (Std) values are used as reference indexes. The optimum values are highlighted in bold. DPGSK, GSK, AGDE, and DE_WOA obtain the smallest RMSE value (9.86021878E-04), and DPGSK and GSK get the best mean value (9.86021878E-04). It is worth noting that DPGSK can get the optimal value (9.86021878E-04) in the worst index and the smallest value (5.01330110E-17) in the standard deviation. To analyze DPGSK’s performance more fully, we used Wilcoxon’s rank-sum test to validate it at a 0.05 confidence level. The R+, R−, and p value result in Table 7 present that R+ is notably higher than R−, which shows the superior status of the presented DPGSK compared with other algorithms. Furthermore, the p values are prominently below 0.05. In fact, the maximum p value is only 2.43367962E-09, which is much smaller than the threshold value of 0.05. Therefore, the results in all indexes prove that DPGSK has the best search performance in the SDM.Table 6
RMSE values in the SDM.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
5.01330110E − 17
GSK
9.86021878E − 04
9.86021881E − 04
9.86021878E − 04
5.77198278E − 13
ITLBO
9.86942857E − 04
2.75231746E − 03
1.33505454E − 03
3.77679086E − 04
SATLBO
9.97038389E − 04
4.00809225E − 03
1.73821140E − 03
7.96717306E − 04
TLABC
9.86040588E − 04
2.09497050E − 03
1.15942131E − 03
2.34420712E − 04
IJAYA
9.95090806E − 04
1.52426755E − 03
1.15901344E − 03
1.52354222E − 04
PPSO
9.86716093E − 04
3.96453377E − 03
1.65111091E − 03
7.01783183E − 04
CLPSO
1.31226121E − 03
4.67174715E − 03
2.48297876E − 03
7.74142094E − 04
AGDE
9.86021878E − 04
1.40212898E − 03
1.00260688E − 03
7.67404144E − 05
DE_WOA
9.86021878E − 04
9.86024119E − 04
9.86021990E − 04
4.37393539E − 10
The optimum values are highlighted in bold.Table 7
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms in the SDM.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1355.50
474.50
7.74308961E − 11
↑
ITLBO
1365.00
465.00
3.01418492E − 11
↑
SATLBO
1365.00
465.00
3.01418492E − 11
↑
TLABC
1365.00
465.00
3.01418492E − 11
↑
IJAYA
1365.00
465.00
3.01418492E − 11
↑
PPSO
1365.00
465.00
3.01418492E − 11
↑
CLPSO
1365.00
465.00
3.01418492E − 11
↑
AGDE
1364.00
466.00
3.33006361E − 11
↑
DE_WOA
1319.00
511.00
2.43367962E − 09
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.In addition, Figure4 gives all the above algorithms’ convergence performance. It shows that DE_WOA converges the fastest in the early stage (before about 800 evaluations). However, after that, DPGSK exceeds DE_WOA, and it can find a better global optimal value instead of falling into a local optimal value. Furthermore, compared with GSK, the convergence speed of DPGSK is significantly improved, which demonstrates that the proposed dual-population evolution strategy indeed boosts the convergence of GSK considerably. In short, the DPGSK algorithm has excellent convergence in the SDM.Figure 4
Convergence curves in the SDM.


### 5.2. Results of the DDM

The identification results of the DDM are provided in Table8. Table 9 and Figures 5(a) and 5(b) show the IAE results and characteristic curves achieved by DPGSK, respectively. We can see that the IAE value of DPGSK (0.01759995) is the smallest among all algorithms, indicating DPGSK is more competitive in the parameter identification accuracy of the DDM. Table 10 shows the RMSE values in the DDM, and DPGSK gets the best values in all the indexes (9.82484859E-04, 9.87885272E-04, 9.84815569E-04, and 1.68677548E-06, respectively). Table 11 shows the R+, R−, and p values of Wilcoxon’s rank-sum. Obviously, the values of R+ are higher than those of R− consistently for all algorithms, and the p values are all lower than 0.05 except for GSK. It indicates that DPGSK is significantly better than other algorithms and comparable to GSK. Therefore, it can conclude that DPGSK has a stronger search ability and robustness in the DDM. Moreover, the convergence curves in the DDM are presented in Figure 6. DE-WOA has the fastest convergence in the early stage but falls into local optimum quickly. DPGSK surpasses DE-WOA after about 2000 evaluations. The original GSK converges relatively slowly in the first half of the evolution, resulting in inaccurate values for the unknown parameters. In contrast, DPGSK has better overall convergence performance than other algorithms and searches for the global optimum more effectively.Table 8
Parameter identification results of the DDM.
Algorithm
Parameter
I
ph (A)
I
sd1 (μA)
I
sd2 (μA)
R
s (Ω)
R
sh (Ω)
n
1
n
2
DPGSK
0.76077168
0.25583590
0.11052916
0.03644723
54.01092626
1.46639593
1.64126723
GSK
0.76077847
0.17092238
0.26394496
0.03648583
54.28926304
1.76856035
1.46597779
ITLBO
0.76077553
0.14239269
0.18110179
0.03637850
53.72272176
1.48849728
1.47607649
SATLBO
0.76096271
0.40876719
0.16662876
0.03648058
54.25291092
1.70980158
1.43554851
TLABC
0.76084150
0.14106070
0.39227701
0.03679306
54.15110002
1.42305668
1.68539145
IJAYA
0.76079609
0.23469918
0.21099112
0.03651187
54.73346624
1.68421377
1.45209458
PPSO
0.76083767
0.10308949
0.60945205
0.03760498
52.22118743
1.39259550
1.75664548
CLPSO
0.76077382
0.04056545
0.16670401
0.04103186
51.35048578
1.31412732
1.58710667
AGDE
0.76084759
0.11467499
0.21581341
0.03643901
52.88962572
1.53128949
1.46569019
DE_WOA
0.76078731
0.38618205
0.10301104
0.03677098
54.69665473
1.62937670
1.40773189Table 9
IAE results in the DDM.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
−0.2057
0.7640
0.76404493
0.76405437
0.76408737
0.76412147
0.76404506
0.76422705
0.76421064
2
−0.1291
0.7620
0.76264663
0.76264432
0.76266247
0.76270783
0.76264646
0.76276121
0.76276331
3
−0.0588
0.7605
0.76134678
0.76135011
0.76135466
0.76141028
0.76136273
0.76141570
0.76143492
4
0.0057
0.7605
0.76015271
0.76016206
0.76015425
0.76021906
0.76018428
0.76018028
0.76021561
5
0.0646
0.7600
0.75906006
0.75907473
0.75905596
0.75912848
0.75910566
0.75904889
0.75910005
6
0.1185
0.7590
0.75805209
0.75807121
0.75804319
0.75812109
0.75810989
0.75800317
0.75807141
7
0.1678
0.7570
0.75710434
0.75712665
0.75709182
0.75717100
0.75717191
0.75701636
0.75710543
8
0.2132
0.7570
0.75615666
0.75618033
0.75614233
0.75621582
0.75623053
0.75602526
0.75614234
9
0.2545
0.7555
0.75510107
0.75512352
0.75508757
0.75514495
0.75517617
0.75492077
0.75507511
10
0.2924
0.7540
0.75366391
0.75369196
0.75366466
0.75369265
0.75374364
0.75344150
0.75364091
11
0.3269
0.7505
0.75138959
0.75140000
0.75138812
0.75137426
0.75144661
0.75111331
0.75135496
12
0.3585
0.7465
0.74733934
0.74733975
0.74734825
0.74728651
0.74737732
0.74704911
0.74730858
13
0.3873
0.7385
0.74007737
0.74006754
0.74009662
0.73999500
0.74009380
0.73983240
0.74005480
14
0.4137
0.7280
0.72737048
0.72735306
0.72739641
0.72728047
0.72736897
0.72725349
0.72735795
15
0.4373
0.7065
0.70688719
0.70690744
0.70695291
0.70686026
0.70691820
0.70701042
0.70692291
16
0.4590
0.6755
0.67527661
0.67526034
0.67529466
0.67526010
0.67527383
0.67558755
0.67527618
17
0.4784
0.6320
0.63087826
0.63086933
0.63088427
0.63091974
0.63089314
0.63135733
0.63087640
18
0.4960
0.5730
0.57208764
0.57208659
0.57208221
0.57217132
0.57212484
0.57260736
0.57208087
19
0.5119
0.4990
0.49950384
0.49950813
0.49949187
0.49959773
0.49955993
0.49991695
0.49949193
20
0.5265
0.4130
0.41350590
0.41351180
0.41349377
0.41357743
0.41357227
0.41370317
0.41349169
21
0.5398
0.3165
0.31722685
0.31723122
0.31721961
0.31725638
0.31729325
0.31717851
0.31721518
22
0.5521
0.2120
0.21210344
0.21210483
0.21210315
0.21208974
0.21216130
0.21186020
0.21209971
23
0.5633
0.1035
0.10271591
0.10271488
0.10272123
0.10267609
0.10276008
0.10239586
0.10272483
24
0.5736
−0.0100
−0.00925611
−0.00925748
−0.00924897
−0.00923317
−0.00922742
−0.00950125
−0.00923084
25
0.5833
−0.1230
−0.12438481
−0.12438340
−0.12438136
−0.12438285
−0.12437102
−0.12438554
−0.12434028
26
0.5900
−0.2100
−0.20918955
−0.20918379
−0.20919287
−0.20913412
−0.20918506
−0.20890333
−0.20912974
∑
I
A
E
0.01759995
0.01760029
0.01770340
0.01763368
0.01763637
0.01769859
0.01787209
The optimum values are highlighted in bold.Figure 5
Characteristic curves of DPGSK in the DDM: (a) I-V and (b) P-V.
(a)
(b)Table 10
RMSE values in the DDM.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
9.82484859E − 04
9.87885272E − 04
9.84815569E − 04
1.68677548E − 06
GSK
9.82681198E − 04
9.91166708E − 04
9.85246463E − 04
1.78610712E − 06
ITLBO
9.84345253E − 04
4.31097698E − 03
1.75430730E − 03
8.04653832E − 04
SATLBO
9.90006061E − 04
2.98794798E − 03
1.71833393E − 03
5.73888171E − 04
TLABC
9.85002657E − 04
2.31622995E − 03
1.32240610E − 03
3.65025331E − 04
IJAYA
9.86676609E − 04
2.12261011E − 03
1.31023130E − 03
2.60436562E − 04
PPSO
9.87968584E − 04
2.78834805E − 03
1.62052853E − 03
5.06325501E − 04
CLPSO
1.53872824E − 03
4.52312248E − 03
2.99124493E − 03
7.37512494E − 04
AGDE
9.82862593E − 04
1.84667496E − 03
1.07310473E − 03
2.00132037E − 04
DE_WOA
9.83101057E − 04
1.36161947E − 03
1.01475307E − 03
7.93377243E − 05
The optimum values are highlighted in bold.Table 11
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms in the DDM.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
974.00
856.00
3.87099778E − 01
≈
ITLBO
1333.00
497.00
6.72195436E − 10
↑
SATLBO
1365.00
465.00
3.01985936E − 11
↑
TLABC
1344.00
486.00
2.37146943E − 10
↑
IJAYA
1362.00
468.00
4.07716485E − 11
↑
PPSO
1365.00
465.00
3.01985936E − 11
↑
CLPSO
1365.00
465.00
3.01985936E − 11
↑
AGDE
1175.00
655.00
1.24770538E − 04
↑
DE_WOA
1177.00
653.00
1.10577260E − 04
↑
The sign “↑” and “≈” means that DPGSK is significantly superior and similar to the compared competitor.Figure 6
Convergence curves different algorithms in the DDM.


### 5.3. Results of the PV Modules

The parameters identified for the Photowatt-PW201, STM6-40/36, and STP6-120/36 are listed in Tables12–14, respectively. The IAE values are shown in Tables 15–17, and the characteristic curves are presented in Figures 7(a)–7(c), respectively. Moreover, Tables 18–20 show different algorithms’ RMSE values, and Tables 21–23 show the R+, R−, and p values of Wilcoxon’s rank-sum test. Figures 8(a)–8(c) provide the convergence curves for these three PV modules, respectively.Table 12
Parameter identification results of the Photowatt-PW201 module.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
1.03051430
3.48226306
1.20127100
981.98224632
48.64283503
GSK
1.03051431
3.48226099
1.20127104
981.98120353
48.64283278
ITLBO
1.03044768
3.43239211
1.20299680
985.81188317
48.58697983
SATLBO
1.03066993
2.61812262
1.23945153
952.61293357
47.56009632
TLABC
1.03059985
3.41535302
1.20283782
957.61637732
48.56901287
IJAYA
1.03067358
3.32748556
1.20534996
923.06854887
48.47141869
PPSO
1.03066227
3.40455344
1.20359921
956.17388108
48.55670692
CLPSO
1.03194503
4.80166061
1.14178462
753.68578258
49.92457911
AGDE
1.03051430
3.48226269
1.20127102
981.98235599
48.64283461
DE_WOA
1.03051401
3.48247551
1.20126216
982.02716996
48.64306864Table 13
Parameter identification results of the STM6-40/36 module.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
1.66390478
1.73865689
0.00427377
15.92829400
1.52030292
GSK
1.66390491
1.73865480
0.00427377
15.92830421
1.52030275
ITLBO
1.66291802
2.32922408
0.00329262
17.67901268
1.55312223
SATLBO
1.66242121
3.24072656
0.00219364
19.81988076
1.59198324
TLABC
1.66496850
1.49591379
0.00474385
14.81882861
1.50398507
IJAYA
1.66566937
1.31619537
0.00500220
13.97331288
1.49035945
PPSO
1.66400834
1.66355701
0.00440936
15.70905258
1.51546361
CLPSO
1.68515659
1.33316783
0.00238545
14.63260110
1.78305059
AGDE
1.66390479
1.73866072
0.00427376
15.92829744
1.52030316
DE_WOA
1.66390477
1.73865738
0.00427377
15.92829743
1.52030295Table 14
Parameter identification results of the STP6-120/36 module.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
7.47252993
2.33499503
0.00459463
22.21989009
1.26010347
GSK
7.47252810
2.33498605
0.00459464
22.22282108
1.26010320
ITLBO
7.47867495
2.48328081
0.00456420
16.93389869
1.26533610
SATLBO
7.55010206
33.20410083
0.00293965
17.73907128
1.52961944
TLABC
7.47266894
2.33243162
0.00459544
22.08141044
1.26001308
IJAYA
7.51783186
5.12186124
0.00417303
12.29917328
1.32886699
PPSO
7.50376792
2.31232586
0.00317927
20.02371401
1.48672583
CLPSO
7.53505228
4.16487988
0.00278201
15.23020319
1.55760973
AGDE
7.47254343
2.33103110
0.00459542
22.14767462
1.25996110
DE_WOA
7.47253093
2.33497159
0.00459464
22.21826350
1.26010264Table 15
IAE results in the Photowatt-PW201 module.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
0.1248
1.0315
1.02919209
1.02919208
1.02905920
1.02917072
1.02918849
1.02923011
1.02923011
2
1.8093
1.0300
1.02738435
1.02738436
1.02732835
1.02738979
1.02734233
1.02744658
1.02744658
3
3.3511
1.0260
1.02574214
1.02574214
1.02569285
1.02570863
1.02560225
1.02576310
1.02576310
4
4.7622
1.0220
1.02405399
1.02406101
1.02406176
1.02403615
1.02387760
1.02408867
1.02408867
5
6.0538
1.0180
1.02228341
1.02228341
1.02224939
1.02218680
1.02198398
1.02223779
1.02223779
6
7.2364
1.0155
1.01988740
1.01990740
1.01989398
1.01979923
1.01956222
1.01984925
1.01984925
7
8.3189
1.0140
1.01635081
1.01635081
1.01634184
1.01622066
1.01596271
1.01627036
1.01627036
8
9.3097
1.0100
1.01049143
1.01049143
1.01050210
1.01036154
1.01009951
1.01041151
1.01041151
9
10.2163
1.0035
1.00067876
1.00067876
1.00071461
1.00056341
1.00031699
1.00061378
1.00061378
10
11.0449
0.9880
0.98465335
0.98465335
0.98471834
0.98456668
0.98435574
0.98461667
0.98461667
11
11.8018
0.9630
0.95969741
0.95969741
0.95979128
0.95964995
0.95949003
0.95969740
0.95969740
12
12.4929
0.9255
0.92304876
0.92304876
0.92316502
0.92304371
0.92294136
0.92308508
0.92308508
13
13.1231
0.8725
0.87258816
0.87258817
0.87271422
0.87261959
0.87257006
0.87265070
0.87265070
14
13.6983
0.8075
0.80739213
0.80739013
0.80743031
0.80736470
0.80735418
0.80738205
0.80738205
15
14.2221
0.7265
0.72795782
0.72795783
0.72805795
0.72801921
0.72803047
0.72802144
0.72802144
16
14.6995
0.6345
0.63646618
0.63646619
0.63653731
0.63652005
0.63653864
0.63650849
0.63650849
17
15.1346
0.5345
0.53569607
0.53569608
0.53573580
0.53573308
0.53575082
0.53571142
0.53571142
18
15.5311
0.4275
0.42881615
0.42881616
0.42882779
0.42883256
0.42884801
0.42880600
0.42880600
19
15.8929
0.3185
0.31866266
0.31866267
0.31865925
0.31866529
0.31868219
0.31863952
0.31863952
20
16.2229
0.2085
0.20785712
0.20785712
0.20783547
0.20783778
0.20786289
0.20781823
0.20781823
21
16.5241
0.1010
0.09835421
0.09835421
0.09832938
0.09832420
0.09836545
0.09831560
0.09831560
22
16.7987
−0.0080
−0.00816934
−0.00816935
−0.00818905
−0.00820433
−0.00819923
−0.00819817
−0.00819817
23
17.0499
−0.1110
−0.11096845
−0.11096845
−0.11097591
−0.11103032
−0.11090721
−0.11097918
−0.11097918
24
17.2793
−0.2090
−0.20910752
−0.20910754
−0.20910699
−0.20914695
−0.20910487
−0.20910337
−0.20910337
25
17.4885
−0.3030
−0.30202238
−0.30202236
−0.30198950
−0.30204274
−0.30187032
−0.30197816
−0.30197816
∑
I
A
E
—
0.04153981
0.04156890
0.04184719
0.04171346
0.04161306
0.04157073
0.04157073
The optimum values are highlighted in bold.Table 16
IAE results in the STM6-40/36 module.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
0.0000
1.6630
1.66345813
1.66345826
1.66260805
1.66443537
1.66507301
1.66354109
1.66345815
2
0.1180
1.6630
1.66325224
1.66325237
1.66242248
1.66421410
1.66483839
1.66333233
1.66325226
3
2.2370
1.6610
1.65955120
1.65955133
1.65908545
1.66023736
1.66062210
1.65957999
1.65955121
4
5.4340
1.6530
1.65391446
1.65391460
1.65398860
1.65418895
1.65421589
1.65386715
1.65391448
5
7.2600
1.6500
1.65056580
1.65056595
1.65093009
1.65061376
1.65044382
1.65047762
1.65056582
6
9.6800
1.6450
1.64543044
1.64543058
1.64610666
1.64521365
1.64481498
1.64529874
1.64543045
7
11.5900
1.6400
1.63923405
1.63923420
1.63999761
1.63888800
1.63838498
1.63909183
1.63923407
8
12.6000
1.6360
1.63371510
1.63371525
1.63440865
1.63336092
1.63286405
1.63358524
1.63371511
9
13.3700
1.6290
1.62728848
1.62728863
1.62784771
1.62697130
1.62652647
1.62718088
1.62728850
10
14.0900
1.6190
1.61831518
1.61831533
1.61867332
1.61807425
1.61772795
1.61824074
1.61831519
11
14.8800
1.5970
1.60306738
1.60306753
1.60312097
1.60295914
1.60278898
1.60304385
1.60306739
12
15.5900
1.5810
1.58158500
1.58158515
1.58131434
1.58163074
1.58168525
1.58161748
1.58158501
13
16.4000
1.5420
1.54232746
1.54232760
1.54171614
1.54254961
1.54292624
1.54242399
1.54232746
14
16.7100
1.5240
1.52122498
1.52122512
1.52052881
1.52149721
1.52200675
1.52134082
1.52122499
15
16.9800
1.5000
1.49920573
1.49920587
1.49847526
1.49950443
1.50012831
1.49933327
1.49920574
16
17.1300
1.4850
1.48527115
1.48527129
1.48454267
1.48557516
1.48626024
1.48540239
1.48527116
17
17.3200
1.4650
1.46564322
1.46564336
1.46494367
1.46594192
1.46670030
1.46577552
1.46564323
18
17.9100
1.3880
1.38759935
1.38759947
1.38723486
1.38776734
1.38869950
1.38770062
1.38759937
19
19.0800
1.1180
1.11837212
1.11837216
1.12017631
1.11756378
1.11833822
1.11819130
1.11837216
20
21.0200
0.0000
−0.00002133
−0.00002166
−0.00039888
0.00020655
0.00005339
−0.00002328
−0.00002136
∑
I
A
E
—
0.06429210
0.06432185
0.06740885
0.06733509
0.07094699
0.06453392
0.06436733
The optimum values are highlighted in bold.Table 17
IAE results in the STP6-120/36 module.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
0.0000
7.4800
7.47698179
7.47698019
7.47665638
7.47111081
7.51527601
7.50310413
7.47098999
2
9.0600
7.4500
7.45253756
7.45253743
7.45451827
7.45259805
7.48530660
7.49450105
7.45251333
3
9.4700
7.4200
7.44934550
7.44934543
7.45111948
7.44940349
7.48101491
7.48368056
7.44932094
4
10.3200
7.4400
7.43909225
7.43909232
7.44039457
7.43914567
7.46785784
7.47142399
7.43906828
5
11.1700
7.4100
7.42026504
7.42026524
7.42101522
7.42031503
7.44495300
7.44796198
7.42024411
6
11.8100
7.3800
7.39587321
7.39587350
7.39613741
7.39592154
7.41644421
7.40403264
7.39585681
7
12.3600
7.3700
7.36326491
7.36326527
7.36305632
7.36331247
7.37943806
7.36924502
7.36325428
8
12.7400
7.3400
7.33148326
7.33148365
7.33091921
7.33153057
7.34413104
7.33486735
7.33147771
9
13.1600
7.2900
7.28413020
7.28413063
7.28315199
7.28417735
7.29245408
7.28686643
7.28413125
10
13.5900
7.2300
7.21776129
7.21776173
7.21634963
7.21780812
7.22130851
7.21603695
7.21776995
11
14.1700
7.1000
7.08813761
7.08813803
7.08617271
7.08818289
7.08511764
7.08444912
7.08815694
12
14.5800
6.9700
6.95844943
6.95844981
6.95616503
6.95849205
6.95118750
6.95847554
6.95847554
13
14.9300
6.8300
6.81486058
6.81486089
6.81239172
6.81489929
6.80465661
6.74814273
6.81489101
14
15.3900
6.5800
6.56792998
6.56793018
6.56540330
6.56796057
6.55542580
6.50237572
6.56796260
15
15.7100
6.3600
6.34872814
6.34872825
6.34632033
6.34875084
6.33602884
6.28894274
6.34875907
16
16.0800
6.0000
6.03749323
6.03749322
6.03541018
6.03750464
6.02623996
5.99024242
6.03751827
17
16.3400
5.7500
5.77681472
5.77681465
5.77508163
5.77681719
5.76764593
5.74212427
5.77683304
18
16.7600
5.2700
5.27376618
5.27376604
5.27278276
5.27375386
5.26946723
5.26494555
5.27376968
19
16.9000
5.0700
5.08193494
5.08193479
5.08123945
5.08191802
5.07951765
5.08277576
5.08193271
20
17.1000
4.7900
4.78583409
4.78583395
4.78556827
4.78581135
4.78615088
4.80074242
4.78582332
21
17.2500
4.5600
4.54629057
4.54629048
4.54635205
4.54626429
4.54858123
4.57184242
4.54627337
22
17.4100
4.2900
4.27393013
4.27393010
4.27433549
4.27390109
4.27813192
4.31024324
4.27390624
23
17.6500
3.8300
3.83228334
3.83228345
3.83317162
3.83225270
3.83869284
3.88291542
3.83225035
24
19.2100
0.0000
0.00116402
0.00116712
0.00184104
0.00124967
-0.02787477
0.00762152
0.00115712
∑
I
A
E
—
0.27460349
0.27460538
0.28986916
0.28038683
0.48079071
0.64413128
0.28045679
The optimum values are highlighted in bold.Figure 7
Characteristic curves of DPGSK: (a) Photowatt-PW201 module, (b) STM6-40/36 module, and (c) STP6-120/36 module.
(a)
(b)
(c)Table 18
RMSE values in the Photowatt-PW201 module.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
2.42507487E − 03
2.42507487E − 03
2.42507487E − 03
4.75324029E − 17
GSK
2.42507487E − 03
2.43124074E − 03
2.42529952E − 03
1.12527868E − 06
ITLBO
2.42613506E − 03
6.78078310E − 03
2.83290424E − 03
8.11128363E − 04
SATLBO
2.43587695E − 03
6.03229203E − 03
3.04534170E − 03
8.69364652E − 04
TLABC
2.42618247E − 03
4.05234846E − 03
2.67703585E − 03
3.83080237E − 04
IJAYA
2.42566534E − 03
2.95825827E − 03
2.53272931E − 03
1.44953051E − 04
PPSO
2.42508316E − 03
8.83755053E − 03
2.94600003E − 03
1.24746322E − 03
CLPSO
2.83408533E − 03
6.75827193E − 03
4.01222813E − 03
9.56232818E − 04
AGDE
2.42507487E − 03
2.42507628E − 03
2.42507492E − 03
2.56868962E − 10
DE_WOA
2.42507489E − 03
3.54660632E − 03
2.56291129E − 03
2.64120863E − 04
The optimum values are highlighted in bold.Table 19
RMSE values in the STM6-40/36 module.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
6.69268329E − 18
GSK
1.72981371E − 03
1.72981372E − 03
1.72981371E − 03
1.50670485E − 12
ITLBO
1.80302370E − 03
5.04532930E − 02
5.08181795E − 03
8.64964315E − 03
SATLBO
2.00426673E − 03
2.98051206E − 02
8.68645469E − 03
7.72801077E − 03
TLABC
1.73011828E − 03
6.71432512E − 02
5.15375645E − 03
1.18096851E − 02
IJAYA
1.80656350E − 03
3.38695808E − 03
2.69248270E − 03
4.76741812E − 04
PPSO
1.73296413E − 03
3.08607469E − 02
4.78248606E − 03
5.06632794E − 03
CLPSO
5.51270979E − 03
2.82327010E − 02
1.59673399E − 02
7.07623601E − 03
AGDE
1.72981371E − 03
1.90393613E − 03
1.73951550E − 03
3.26588994E − 05
DE_WOA
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
7.83732302E − 13
The optimum values are highlighted in bold.Table 20
RMSE values in the STP6-120/36 module.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
1.66006031E − 02
1.66006031E − 02
1.66006031E − 02
3.64215013E − 16
GSK
1.66006031E − 02
1.66006379E − 02
1.66006043E − 02
6.34321846E − 09
ITLBO
1.66026542E − 02
1.82932081E − 01
4.09561836E − 02
3.36881282E − 02
SATLBO
1.67283597E − 02
5.94301196E − 02
4.09933264E − 02
1.18675783E − 02
TLABC
1.66007077E − 02
2.27284953E − 01
3.51585815E − 02
4.39960971E − 02
IJAYA
1.73920763E − 02
2.74919656E − 02
2.09806168E − 02
3.00496502E − 03
PPSO
1.66391981E − 02
1.14874892E − 01
4.06962594E − 02
1.85185406E − 02
CLPSO
3.52931825E − 02
8.67923901E − 01
1.77656627E − 01
1.93537380E − 01
AGDE
1.66006031E − 02
1.34812942E − 01
2.06460413E − 02
2.15653571E − 02
DE_WOA
1.66006031E − 02
1.66007829E − 02
1.66006139E − 02
4.10680224E − 08Table 21
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms for the Photowatt-PW201 module.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1365.00
465.00
2.96913841E − 11
↑
ITLBO
1365.00
465.00
2.96913841E − 11
↑
SATLBO
1365.00
465.00
2.96913841E − 11
↑
TLABC
1365.00
465.00
2.96913841E − 11
↑
IJAYA
1365.00
465.00
2.96913841E − 11
↑
PPSO
1365.00
465.00
2.96913841E − 11
↑
CLPSO
1365.00
465.00
2.96913841E − 11
↑
AGDE
1327.50
502.50
1.09035598E − 09
↑
DE_WOA
1365.00
465.00
2.96913841E − 11
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.Table 22
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms for the STM6-40/36 module.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1356.50
473.50
4.42956883E − 11
↑
ITLBO
1365.00
465.00
1.98787859E − 11
↑
SATLBO
1365.00
465.00
1.98787859E − 11
↑
TLABC
1365.00
465.00
1.98787859E − 11
↑
IJAYA
1365.00
465.00
1.98787859E − 11
↑
PPSO
1365.00
465.00
1.98787859E − 11
↑
CLPSO
1365.00
465.00
1.98787859E − 11
↑
AGDE
1365.00
465.00
1.98787859E − 11
↑
DE_WOA
1341.00
489.00
2.03566531E − 10
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.Table 23
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms for the STP6-120/36 module.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1363.00
467.00
3.29327551E − 11
↑
ITLBO
1365.00
465.00
2.69269726E − 11
↑
SATLBO
1365.00
465.00
2.69269726E − 11
↑
TLABC
1365.00
465.00
2.69269726E − 11
↑
IJAYA
1365.00
465.00
2.69269726E − 11
↑
PPSO
1365.00
465.00
2.69269726E − 11
↑
CLPSO
1365.00
465.00
2.69269726E − 11
↑
AGDE
1365.00
465.00
2.69269726E − 11
↑
DE_WOA
1354.50
475.50
7.59190233E − 11
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.Figure 8
Convergence curves: (a) Photowatt-PW201 module, (b) STM6-40/36 module, and (c) STP6-120/36 module.
(a)
(b)
(c)


#### 5.3.1. Analysis of the IAE Values

Figures7(a)–7(c) illustrate that the currents calculated by DPGSK fit the measured currents to a high degree. For the IAE results, DPGSK reaches the smallest IAE values in these modules. To be specific, the IAE values are 0.04153981, 0.06429210, and 0.27460349, respectively, demonstrating that DPGSK has higher accuracy in the parameter identification of these three PV modules than other algorithms.


#### 5.3.2. Analysis of the RMSE Values

For the best RMSE index, in the Photowatt-PW201 module, DPGSK, GSK and AGDE yield the optimum result (2.42507487E-03). In the other two modules, four algorithms including DPGSK, GSK, AGDE, and DE_WOA, obtain the optimum values, which are 1.72981371E-03 and 1.66006031E-02, respectively.For the worst RMSE index, DPGSK gets the optimum values in the Photowatt-PW201 and STP6-120/36, and the values are 2.42507487E-03 and 1.66006031E-02, respectively. For the STM6-40/36, both DPGSK and DE_WOA achieve the same optimum value (1.72981371E-03).For the mean RMSE index, DPGSK yields the optimum values, i.e., 2.42507487E-03 and 1.66006031E-02, respectively, in the Photowatt-PW201 and STP6-120/36 modules. For the STM6-40/36 module, DPGSK, GSK, and DE_WOA reach the same optimum value (1.72981371E-03).For the standard deviation, only DPGSK provides the best performance in all the three PV modules. The values are 4.75324029E-17, 6.69268329E-18, and 3.64215013E-16, respectively. Particularly, the optimal standard deviation values of DPGSK are significantly better than that of others.The above comparisons fully indicate that DPGSK obtains the most accurate parameters’ values for the three PV modules and shows the strongest robustness in searching for the global optimal solutions.


#### 5.3.3. Analysis of the Convergence Performance

In the Photowatt-PW201 module, DPGSK and DE-WOA converge faster than other algorithms during the initial stage (before about 1000 evaluations). Moreover, compared with GSK, DPGSK improves the convergence performance effectively. In the last two modules, DE-WOA converges fastest, and DPGSK converges similarly to GSK. In the STM6-40/36 module, DPGSK surpasses DE-WOA at approximately 4000 evaluations. In the STP6-120/36 module, DPGSK completely surpasses GSK at approximately 3000 evaluations and surpasses DE-WOA at approximately 6000 evaluations. DE-WOA falls into local convergence in all the PV modules, while DPGSK can avoid the local optimum effectively and surpass GSK to search for the global optimum. Therefore, it proves again that DPGSK has stronger competitiveness in convergence performance.


#### 5.3.4. Analysis of the Test Results

The test outcomes provided in Tables21–23 reveal that all the R+ values are clearly higher than the R− values. For most methods, the R+ is 1365, and only the R+ of DE-WOA is 1327.5 in the Photowatt-PW201 module. The R+ of GSK and DE-WOA in the STM6-40/36 module are 1356.5 and 1341, respectively. In the STP6-120/36 module, GSK and DE-WOA have an R+ of 1356.5 and 1341, respectively. Obviously, in these three modules, the p values are considerably below 0.05, indicating that DPGSK significantly outperforms all the other methods. It concludes that the suggested DPGSK is statistically more reliable than other competitors.


### 5.4. Whole Performance

The DPGSK algorithm is compared with other algorithms in different models separately above. However, its performance could not be verified comprehensively by the experimental results in a single model. Therefore, the Friedman test is employed here for the statistical analysis of multiple models simultaneously at a confidence level of 0.05. The test results, as displayed in Figure9, demonstrate that DPGSK ranks first with an average ranking of 1.40, followed by GSK, DE_WOA, AGDE, IJAYA, TLABC, PPSO, ITLBO, SATLBO, and CLPSO, which further indicates the superior status of the presented algorithm over others.Figure 9
Friedman test result.


### 5.5. Analysis of the Components

It is known that GSK contains two phases, i.e., the junior and the senior. In the proposed DPGSK, it adopts the dual-population evolution strategy to divide the senior and junior phases into two random subpopulations of equal size. In this subsection, the effect of each phase on DPGSK is evaluated. Two variants are considered. One variant that only uses the junior phase is IGSK1, and the other variant that only performs the senior phase is IGSK2.


#### 5.5.1. Analysis of the RMSE Values

For the best RMSE indicator, all four algorithms get the optimum values 9.86021878E-04 and 2.42507487E-03 in the SDM and the Photowatt-PW201 module, respectively, as shown in Table24. In the DDM, IGSK2 gets the best value (9.82484852E-04), and DPGSK’s value (9.82484859E-04) stays close behind IGSK2. In the STM6-40/36 and STP6-120/36 modules, DPGSK, GSK, and IGSK2 obtain the same optimum values, which are 1.72981371E-03 and 1.66006031E-02, respectively.Table 24
RMSE values for different variants of DPGSK.
Model
Algorithm
RMSE
Best
Worst
Mean
Std
SDM
DPGSK
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
5.01330110E − 17
GSK
9.86021878E − 04
9.86021881E − 04
9.86021878E − 04
5.77198278E − 13
IGSK1
9.86021878E − 04
1.14970749E − 03
1.00334452E − 03
3.09917403E − 05
IGSK2
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
4.95508657E − 17
DDM
DPGSK
9.82484859E − 04
9.87885272E − 04
9.84815569E − 04
1.68677548E − 06
GSK
9.82681198E − 04
9.91166708E − 04
9.85246463E − 04
1.78610712E − 06
IGSK1
9.83129226E − 04
3.35318293E − 03
1.69916657E − 03
5.92923212E − 04
IGSK2
9.82484852E − 04
9.90885053E − 04
9.85386108E − 04
1.82893145E − 06
Photowatt-PW201
DPGSK
2.42507487E − 03
2.42507487E − 03
2.42507487E − 03
4.75324029E − 17
GSK
2.42507487E − 03
2.43124074E − 03
2.42529952E − 03
1.12527868E − 06
IGSK1
2.42507487E − 03
2.43852468E − 03
2.42617462E − 03
2.72385502E − 06
IGSK2
2.42507487E − 03
2.42774735E − 03
2.42516515E − 03
4.87738798E − 07
STM6-40/36
DPGSK
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
6.69268329E − 18
GSK
1.72981371E − 03
1.72981372E − 03
1.72981371E − 03
1.50670485E − 12
IGSK1
1.73191719E − 03
7.07659606E − 03
3.38453446E − 03
1.41010667E − 03
IGSK2
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
2.49962266E − 15
STP6-120/36
DPGSK
1.66006031E − 02
1.66006031E − 02
1.66006031E − 02
3.64215013E − 16
GSK
1.66006031E − 02
1.66006379E − 02
1.66006043E − 02
6.34321846E − 09
IGSK1
1.66333141E − 02
3.40593689E − 02
2.35744620E − 02
6.13100930E − 03
IGSK2
1.66006031E − 02
1.66026785E − 02
1.66006723E − 02
3.78914430E − 07
The optimum values are highlighted in bold.For the worst RMSE indicator, DPGSK and IGSK2 get the optimum values in the SDM and STM6-40/36 module, which are 9.86021878E-04 and 1.72981371E-03, respectively. In the DDM, Photowatt-PW201 module, and STP6-120/36 module, only DPGSK achieves the optimum values, i.e., 9.87885272E-04, 2.42507487E-03, and 1.66006031E-02, respectively.For the mean RMSE indicator, DPGSK, GSK, and IGSK2 get the optimum values in the SDM and STM6-40/36 module and the values are 9.86021878E-04 and 1.72981371E-03, respectively. In the DDM, Photowatt-PW201 module, and STP6-120/36 module, only DPGSK achieves the optimum values, i.e., 9.84815569E-04, 2.42507487E-03, and 1.66006031E-02, respectively.For the standard deviation, DPGSK achieves the optimum values in the DDM, Photowatt-PW201, STM6-40/36, and STP6-120/36 modules, which are 1.68677548E-06, 4.75324029E-17, 6.69268329E-18, and 3.64215013E-16, respectively. Although IGSK2 gets the optimum value of 4.95508657E-17 in the SDM, DPGSK’s value (5.01330110E-17) is only a little short of IGSK2.Summarizing the above comparisons, we can see that DPGSK obtains the optimal values the most often. When it does not, the difference between its value and the best value is very slight. IGSK2 obtains the optimal value the second most often, GSK follows, and IGSK1 is at the end. Overall, DPGSK proves to be more effective in identifying accurate and reliable parameters for these PV models.


#### 5.5.2. Analysis of the Test Results

The test results in Table25 show that the R+ exceeds the R− for all models. In the SDM, the significance indicator exceeds 0.05 only for IGSK1, and thus, the other two algorithms are significantly worse than DPGSK. In the DDM, the p-values of GSK and IGSK2 exceed 0.05, indicating they are statistically comparable to DPGSK. In the other three PV modules, the significance indicators are considerably less than 0.05, meaning that DPGSK is significantly superior to the rest algorithms. Considering the above, DPGSK is statistically better than GSK, IGSK1, and IGSK2.Table 25
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. its variants for the five PV models.
Model
DPGSK vs.
R
+
R
−
p value
Sig.
SDM
GSK
1355.50
474.50
7.74308961E − 11
↑
IGSK1
1365
465
3.01418492E − 11
↑
IGSK2
954
875
5.64171398E − 01
≈
DDM
GSK
974.00
856.00
3.87099778E − 01
≈
IGSK1
1326
504
1.28703830E − 09
↑
IGSK2
948
882
6.30876292E − 01
≈
Photowatt-PW201
GSK
1365.00
465.00
2.96913841E − 11
↑
IGSK1
1346.5
483.5
1.82527310E − 10
↑
IGSK2
1167
663
1.95503446E − 04
↑
STM6-40/36
GSK
1356.50
473.50
4.42956883E − 11
↑
IGSK1
1365
465
1.98787859E − 11
↑
IGSK2
1346
484
8.29091595E − 11
↑
STP6-120/36
GSK
1363.00
467.00
3.29327551E − 11
↑
IGSK1
1365
465
2.69269726E − 11
↑
IGSK2
1245
585
8.06050812E − 07
↑
The signs “↑” and “≈” mean that DPGSK is significantly superior and similar to the compared competitor, respectively.


### 5.6. Discussions

In this paper, we boost the efficacy of the original GSK algorithm using a dual-population evolution strategy. The resultant DPGSK algorithm to identify the PV models’ parameters is compared with various excellent metaheuristics in five models. The effect of different components on DPGSK is also investigated. From the results, we have reached the following summations:(1)
The characteristic curves of I-V and P-V show that DPGSK obtains superior accuracy and reliability in parameter identification. The IAE values for DPGSK in the five PV models are 0.01769439, 0.01759995, 0.04153981, 0.06429210, and 0.27460349, respectively, which are all the best compared with other algorithms.(2)
The RMSE values show that DPGSK achieves optimal values in the five models, and in particular, it outperforms others evidently in terms of standard deviation. Besides, the testR+, R−, and p values present that DPGSK gets the most accurate values and shows the strongest robustness in searching for the global optimal solutions. It proves that the dual-population evolution strategy does raise the accuracy of GSK in solving the parameter identification of PV models.(3)
With regard to the convergence curves, DPGSK and DE_WOA have faster convergence at the beginning of the iteration. Although DE_WOA converges fastest in the DDM, STM6-40/36, and STP6-120/36, it tends to fall into local extrema. DPGSK can avoid local convergence and search for the global optimum effectively. Besides, it is worth mentioning that DPGSK can address the issue of slow convergence of the GSK algorithm in early iterations.(4)
DPGSK’s overall performance is verified more comprehensively through the Friedman test. DPGSK ranks first with an average ranking of 1.40, followed by GSK, DE_WOA, AGDE, IJAYA, TLABC, PPSO, ITLBO, SATLBO, and CLPSO. The ranking results demonstrate the marked superiority of DPGSK over other algorithms in tackling this studied problem.(5)
The effect of the junior phase and the senior phase on DPGSK is analyzed. The results show that these two phases do affect the performance of DPGSK. Either component is indispensable to DPGSK. Nevertheless, the proposed dual-population evolution strategy can exhibit the power to coordinate them well. Namely, both exploration and exploitation can be balanced well in DPGSK.


## 5.1. Results of the SDM

The identification results of the five parameters of the SDM are shown in Table4. After getting the solutions, they can be used to calculate the current data and individual absolute error values (IAE = |IL measured − IL calculated|), presented in Table 5. We can see that the DPGSK’s IAE (0.01769439) is consistently smaller than PPSO (0.01770360), AGDE (0.01770370), GSK (0.01770416), TLABC (0.01771991), ITLBO (0.01802719), and IJAYA (0.01806562). The current-voltage (I-V) and power-voltage (P-V) characteristic curves of DPGSK are plotted in Figure 3(a) and Figure 3(b), respectively, showing that a good agreement is found between the calculated data and the measured data. It demonstrates that DPGSK has superior accuracy in this model.Table 4
Parameter identification results of the SDM.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
0.76077553
0.32302079
0.03637709
53.71852013
1.48118358
GSK
0.76077553
0.32302085
0.03637709
53.71852481
1.48118360
ITLBO
0.76088942
0.30531731
0.03658155
51.11235228
1.47555381
SATLBO
0.76098523
0.33649797
0.03621845
53.24192945
1.48531569
TLABC
0.76074567
0.32115398
0.03639442
53.76527575
1.48060283
IJAYA
0.76083168
0.29010861
0.03678511
52.18389372
1.47039884
PPSO
0.76076441
0.33383560
0.03624545
54.58985322
1.48450689
CLPSO
0.76098396
0.370652361
0.03492495
57.25728241
1.49542141
AGDE
0.76077548
0.32303137
0.03637697
53.72031579
1.48118683
DE_WOA
0.76077552
0.32302447
0.03637705
53.71869823
1.48118474Table 5
IAE results in the SDM.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
−0.2057
0.7640
0.76402556
0.76408765
0.76436713
0.76405468
0.76423508
0.76408764
0.76408749
2
−0.1291
0.7620
0.76262328
0.76266264
0.76286952
0.76263091
0.76276822
0.76266264
0.76266253
3
−0.0588
0.7605
0.76133622
0.76135473
0.76149499
0.76132414
0.76142189
0.76135472
0.76135466
4
0.0057
0.7605
0.76015484
0.76015423
0.76023338
0.76012468
0.76018618
0.76015427
0.76015420
5
0.0646
0.7600
0.75907387
0.75905585
0.75907929
0.75902727
0.75905582
0.75905635
0.75905586
6
0.1185
0.7590
0.75807680
0.75804300
0.75801572
0.75801534
0.75801419
0.75804300
0.75804305
7
0.1678
0.7570
0.75713938
0.75709159
0.75701864
0.75706483
0.75703780
0.75709159
0.75709166
8
0.2132
0.7570
0.75620170
0.75614207
0.75602889
0.75611634
0.75606865
0.75614207
0.75614217
9
0.2545
0.7555
0.75515544
0.75508732
0.75494149
0.75506296
0.75500327
0.75508732
0.75508744
10
0.2924
0.7540
0.75373601
0.75366447
0.75349632
0.75364218
0.75358460
0.75366447
0.75366460
11
0.3269
0.7505
0.75145540
0.75138806
0.75121264
0.75136907
0.75133542
0.75138806
0.75138820
12
0.3585
0.7465
0.74740107
0.74734834
0.74718587
0.74733452
0.74735513
0.74734834
0.74734848
13
0.3873
0.7385
0.74012294
0.74009688
0.73997134
0.74009054
0.74020040
0.74009688
0.74009699
14
0.4137
0.7280
0.72738539
0.72739678
0.72733136
0.72740028
0.72762993
0.72739678
0.72739686
15
0.4373
0.7065
0.70690114
0.70695327
0.70695979
0.70696778
0.70732358
0.70695327
0.70695331
16
0.4590
0.6755
0.67520863
0.67529489
0.67536910
0.67531996
0.67577635
0.67529490
0.67529488
17
0.4784
0.6320
0.63078292
0.63088431
0.63100001
0.63091697
0.63140894
0.63088431
0.63088426
18
0.4960
0.5730
0.57199009
0.57208207
0.57220153
0.57211788
0.57256254
0.57208207
0.57208200
19
0.5119
0.4990
0.49943067
0.49949164
0.49957875
0.49952593
0.49984987
0.49949164
0.49949157
20
0.5265
0.4130
0.41347542
0.41349356
0.41352576
0.41352273
0.41368331
0.41349356
0.41349350
21
0.5398
0.3165
0.31724244
0.31721950
0.31719456
0.31724187
0.31724001
0.31721950
0.31721946
22
0.5521
0.2120
0.21215403
0.21210317
0.21203682
0.21211888
0.21199141
0.21210317
0.21210313
23
0.5633
0.1035
0.10277871
0.10272135
0.10264290
0.10273211
0.10254314
0.10272134
0.10272129
24
0.5736
−0.0100
−0.00920948
−0.00924886
−0.00930440
−0.00924053
−0.00941687
−0.00924887
−0.00924895
25
0.5833
−0.1230
−0.12438567
−0.12438137
−0.12437645
−0.12437260
−0.12445883
−0.12438137
−0.12438151
26
0.5900
−0.2100
−0.20924496
−0.20919304
−0.20912068
−0.20918205
−0.20915470
−0.20919304
−0.20919323
∑
I
A
E
0.01769439
0.01770416
0.01802719
0.01771991
0.01806562
0.01770360
0.01770370Figure 3
Characteristic curves of DPGSK in the SDM. (a) I-V. (b) P-V.
(a)
(b)Table6 shows the objective function values (RMSE) obtained by different algorithms in the SDM. To accurately verify the DPGSK’s effectiveness, the best, worst, mean, and standard deviation (Std) values are used as reference indexes. The optimum values are highlighted in bold. DPGSK, GSK, AGDE, and DE_WOA obtain the smallest RMSE value (9.86021878E-04), and DPGSK and GSK get the best mean value (9.86021878E-04). It is worth noting that DPGSK can get the optimal value (9.86021878E-04) in the worst index and the smallest value (5.01330110E-17) in the standard deviation. To analyze DPGSK’s performance more fully, we used Wilcoxon’s rank-sum test to validate it at a 0.05 confidence level. The R+, R−, and p value result in Table 7 present that R+ is notably higher than R−, which shows the superior status of the presented DPGSK compared with other algorithms. Furthermore, the p values are prominently below 0.05. In fact, the maximum p value is only 2.43367962E-09, which is much smaller than the threshold value of 0.05. Therefore, the results in all indexes prove that DPGSK has the best search performance in the SDM.Table 6
RMSE values in the SDM.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
5.01330110E − 17
GSK
9.86021878E − 04
9.86021881E − 04
9.86021878E − 04
5.77198278E − 13
ITLBO
9.86942857E − 04
2.75231746E − 03
1.33505454E − 03
3.77679086E − 04
SATLBO
9.97038389E − 04
4.00809225E − 03
1.73821140E − 03
7.96717306E − 04
TLABC
9.86040588E − 04
2.09497050E − 03
1.15942131E − 03
2.34420712E − 04
IJAYA
9.95090806E − 04
1.52426755E − 03
1.15901344E − 03
1.52354222E − 04
PPSO
9.86716093E − 04
3.96453377E − 03
1.65111091E − 03
7.01783183E − 04
CLPSO
1.31226121E − 03
4.67174715E − 03
2.48297876E − 03
7.74142094E − 04
AGDE
9.86021878E − 04
1.40212898E − 03
1.00260688E − 03
7.67404144E − 05
DE_WOA
9.86021878E − 04
9.86024119E − 04
9.86021990E − 04
4.37393539E − 10
The optimum values are highlighted in bold.Table 7
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms in the SDM.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1355.50
474.50
7.74308961E − 11
↑
ITLBO
1365.00
465.00
3.01418492E − 11
↑
SATLBO
1365.00
465.00
3.01418492E − 11
↑
TLABC
1365.00
465.00
3.01418492E − 11
↑
IJAYA
1365.00
465.00
3.01418492E − 11
↑
PPSO
1365.00
465.00
3.01418492E − 11
↑
CLPSO
1365.00
465.00
3.01418492E − 11
↑
AGDE
1364.00
466.00
3.33006361E − 11
↑
DE_WOA
1319.00
511.00
2.43367962E − 09
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.In addition, Figure4 gives all the above algorithms’ convergence performance. It shows that DE_WOA converges the fastest in the early stage (before about 800 evaluations). However, after that, DPGSK exceeds DE_WOA, and it can find a better global optimal value instead of falling into a local optimal value. Furthermore, compared with GSK, the convergence speed of DPGSK is significantly improved, which demonstrates that the proposed dual-population evolution strategy indeed boosts the convergence of GSK considerably. In short, the DPGSK algorithm has excellent convergence in the SDM.Figure 4
Convergence curves in the SDM.


## 5.2. Results of the DDM

The identification results of the DDM are provided in Table8. Table 9 and Figures 5(a) and 5(b) show the IAE results and characteristic curves achieved by DPGSK, respectively. We can see that the IAE value of DPGSK (0.01759995) is the smallest among all algorithms, indicating DPGSK is more competitive in the parameter identification accuracy of the DDM. Table 10 shows the RMSE values in the DDM, and DPGSK gets the best values in all the indexes (9.82484859E-04, 9.87885272E-04, 9.84815569E-04, and 1.68677548E-06, respectively). Table 11 shows the R+, R−, and p values of Wilcoxon’s rank-sum. Obviously, the values of R+ are higher than those of R− consistently for all algorithms, and the p values are all lower than 0.05 except for GSK. It indicates that DPGSK is significantly better than other algorithms and comparable to GSK. Therefore, it can conclude that DPGSK has a stronger search ability and robustness in the DDM. Moreover, the convergence curves in the DDM are presented in Figure 6. DE-WOA has the fastest convergence in the early stage but falls into local optimum quickly. DPGSK surpasses DE-WOA after about 2000 evaluations. The original GSK converges relatively slowly in the first half of the evolution, resulting in inaccurate values for the unknown parameters. In contrast, DPGSK has better overall convergence performance than other algorithms and searches for the global optimum more effectively.Table 8
Parameter identification results of the DDM.
Algorithm
Parameter
I
ph (A)
I
sd1 (μA)
I
sd2 (μA)
R
s (Ω)
R
sh (Ω)
n
1
n
2
DPGSK
0.76077168
0.25583590
0.11052916
0.03644723
54.01092626
1.46639593
1.64126723
GSK
0.76077847
0.17092238
0.26394496
0.03648583
54.28926304
1.76856035
1.46597779
ITLBO
0.76077553
0.14239269
0.18110179
0.03637850
53.72272176
1.48849728
1.47607649
SATLBO
0.76096271
0.40876719
0.16662876
0.03648058
54.25291092
1.70980158
1.43554851
TLABC
0.76084150
0.14106070
0.39227701
0.03679306
54.15110002
1.42305668
1.68539145
IJAYA
0.76079609
0.23469918
0.21099112
0.03651187
54.73346624
1.68421377
1.45209458
PPSO
0.76083767
0.10308949
0.60945205
0.03760498
52.22118743
1.39259550
1.75664548
CLPSO
0.76077382
0.04056545
0.16670401
0.04103186
51.35048578
1.31412732
1.58710667
AGDE
0.76084759
0.11467499
0.21581341
0.03643901
52.88962572
1.53128949
1.46569019
DE_WOA
0.76078731
0.38618205
0.10301104
0.03677098
54.69665473
1.62937670
1.40773189Table 9
IAE results in the DDM.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
−0.2057
0.7640
0.76404493
0.76405437
0.76408737
0.76412147
0.76404506
0.76422705
0.76421064
2
−0.1291
0.7620
0.76264663
0.76264432
0.76266247
0.76270783
0.76264646
0.76276121
0.76276331
3
−0.0588
0.7605
0.76134678
0.76135011
0.76135466
0.76141028
0.76136273
0.76141570
0.76143492
4
0.0057
0.7605
0.76015271
0.76016206
0.76015425
0.76021906
0.76018428
0.76018028
0.76021561
5
0.0646
0.7600
0.75906006
0.75907473
0.75905596
0.75912848
0.75910566
0.75904889
0.75910005
6
0.1185
0.7590
0.75805209
0.75807121
0.75804319
0.75812109
0.75810989
0.75800317
0.75807141
7
0.1678
0.7570
0.75710434
0.75712665
0.75709182
0.75717100
0.75717191
0.75701636
0.75710543
8
0.2132
0.7570
0.75615666
0.75618033
0.75614233
0.75621582
0.75623053
0.75602526
0.75614234
9
0.2545
0.7555
0.75510107
0.75512352
0.75508757
0.75514495
0.75517617
0.75492077
0.75507511
10
0.2924
0.7540
0.75366391
0.75369196
0.75366466
0.75369265
0.75374364
0.75344150
0.75364091
11
0.3269
0.7505
0.75138959
0.75140000
0.75138812
0.75137426
0.75144661
0.75111331
0.75135496
12
0.3585
0.7465
0.74733934
0.74733975
0.74734825
0.74728651
0.74737732
0.74704911
0.74730858
13
0.3873
0.7385
0.74007737
0.74006754
0.74009662
0.73999500
0.74009380
0.73983240
0.74005480
14
0.4137
0.7280
0.72737048
0.72735306
0.72739641
0.72728047
0.72736897
0.72725349
0.72735795
15
0.4373
0.7065
0.70688719
0.70690744
0.70695291
0.70686026
0.70691820
0.70701042
0.70692291
16
0.4590
0.6755
0.67527661
0.67526034
0.67529466
0.67526010
0.67527383
0.67558755
0.67527618
17
0.4784
0.6320
0.63087826
0.63086933
0.63088427
0.63091974
0.63089314
0.63135733
0.63087640
18
0.4960
0.5730
0.57208764
0.57208659
0.57208221
0.57217132
0.57212484
0.57260736
0.57208087
19
0.5119
0.4990
0.49950384
0.49950813
0.49949187
0.49959773
0.49955993
0.49991695
0.49949193
20
0.5265
0.4130
0.41350590
0.41351180
0.41349377
0.41357743
0.41357227
0.41370317
0.41349169
21
0.5398
0.3165
0.31722685
0.31723122
0.31721961
0.31725638
0.31729325
0.31717851
0.31721518
22
0.5521
0.2120
0.21210344
0.21210483
0.21210315
0.21208974
0.21216130
0.21186020
0.21209971
23
0.5633
0.1035
0.10271591
0.10271488
0.10272123
0.10267609
0.10276008
0.10239586
0.10272483
24
0.5736
−0.0100
−0.00925611
−0.00925748
−0.00924897
−0.00923317
−0.00922742
−0.00950125
−0.00923084
25
0.5833
−0.1230
−0.12438481
−0.12438340
−0.12438136
−0.12438285
−0.12437102
−0.12438554
−0.12434028
26
0.5900
−0.2100
−0.20918955
−0.20918379
−0.20919287
−0.20913412
−0.20918506
−0.20890333
−0.20912974
∑
I
A
E
0.01759995
0.01760029
0.01770340
0.01763368
0.01763637
0.01769859
0.01787209
The optimum values are highlighted in bold.Figure 5
Characteristic curves of DPGSK in the DDM: (a) I-V and (b) P-V.
(a)
(b)Table 10
RMSE values in the DDM.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
9.82484859E − 04
9.87885272E − 04
9.84815569E − 04
1.68677548E − 06
GSK
9.82681198E − 04
9.91166708E − 04
9.85246463E − 04
1.78610712E − 06
ITLBO
9.84345253E − 04
4.31097698E − 03
1.75430730E − 03
8.04653832E − 04
SATLBO
9.90006061E − 04
2.98794798E − 03
1.71833393E − 03
5.73888171E − 04
TLABC
9.85002657E − 04
2.31622995E − 03
1.32240610E − 03
3.65025331E − 04
IJAYA
9.86676609E − 04
2.12261011E − 03
1.31023130E − 03
2.60436562E − 04
PPSO
9.87968584E − 04
2.78834805E − 03
1.62052853E − 03
5.06325501E − 04
CLPSO
1.53872824E − 03
4.52312248E − 03
2.99124493E − 03
7.37512494E − 04
AGDE
9.82862593E − 04
1.84667496E − 03
1.07310473E − 03
2.00132037E − 04
DE_WOA
9.83101057E − 04
1.36161947E − 03
1.01475307E − 03
7.93377243E − 05
The optimum values are highlighted in bold.Table 11
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms in the DDM.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
974.00
856.00
3.87099778E − 01
≈
ITLBO
1333.00
497.00
6.72195436E − 10
↑
SATLBO
1365.00
465.00
3.01985936E − 11
↑
TLABC
1344.00
486.00
2.37146943E − 10
↑
IJAYA
1362.00
468.00
4.07716485E − 11
↑
PPSO
1365.00
465.00
3.01985936E − 11
↑
CLPSO
1365.00
465.00
3.01985936E − 11
↑
AGDE
1175.00
655.00
1.24770538E − 04
↑
DE_WOA
1177.00
653.00
1.10577260E − 04
↑
The sign “↑” and “≈” means that DPGSK is significantly superior and similar to the compared competitor.Figure 6
Convergence curves different algorithms in the DDM.


## 5.3. Results of the PV Modules

The parameters identified for the Photowatt-PW201, STM6-40/36, and STP6-120/36 are listed in Tables12–14, respectively. The IAE values are shown in Tables 15–17, and the characteristic curves are presented in Figures 7(a)–7(c), respectively. Moreover, Tables 18–20 show different algorithms’ RMSE values, and Tables 21–23 show the R+, R−, and p values of Wilcoxon’s rank-sum test. Figures 8(a)–8(c) provide the convergence curves for these three PV modules, respectively.Table 12
Parameter identification results of the Photowatt-PW201 module.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
1.03051430
3.48226306
1.20127100
981.98224632
48.64283503
GSK
1.03051431
3.48226099
1.20127104
981.98120353
48.64283278
ITLBO
1.03044768
3.43239211
1.20299680
985.81188317
48.58697983
SATLBO
1.03066993
2.61812262
1.23945153
952.61293357
47.56009632
TLABC
1.03059985
3.41535302
1.20283782
957.61637732
48.56901287
IJAYA
1.03067358
3.32748556
1.20534996
923.06854887
48.47141869
PPSO
1.03066227
3.40455344
1.20359921
956.17388108
48.55670692
CLPSO
1.03194503
4.80166061
1.14178462
753.68578258
49.92457911
AGDE
1.03051430
3.48226269
1.20127102
981.98235599
48.64283461
DE_WOA
1.03051401
3.48247551
1.20126216
982.02716996
48.64306864Table 13
Parameter identification results of the STM6-40/36 module.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
1.66390478
1.73865689
0.00427377
15.92829400
1.52030292
GSK
1.66390491
1.73865480
0.00427377
15.92830421
1.52030275
ITLBO
1.66291802
2.32922408
0.00329262
17.67901268
1.55312223
SATLBO
1.66242121
3.24072656
0.00219364
19.81988076
1.59198324
TLABC
1.66496850
1.49591379
0.00474385
14.81882861
1.50398507
IJAYA
1.66566937
1.31619537
0.00500220
13.97331288
1.49035945
PPSO
1.66400834
1.66355701
0.00440936
15.70905258
1.51546361
CLPSO
1.68515659
1.33316783
0.00238545
14.63260110
1.78305059
AGDE
1.66390479
1.73866072
0.00427376
15.92829744
1.52030316
DE_WOA
1.66390477
1.73865738
0.00427377
15.92829743
1.52030295Table 14
Parameter identification results of the STP6-120/36 module.
Algorithm
Parameter
I
ph (A)
I
sd (μA)
R
s (Ω)
R
sh (Ω)
n
DPGSK
7.47252993
2.33499503
0.00459463
22.21989009
1.26010347
GSK
7.47252810
2.33498605
0.00459464
22.22282108
1.26010320
ITLBO
7.47867495
2.48328081
0.00456420
16.93389869
1.26533610
SATLBO
7.55010206
33.20410083
0.00293965
17.73907128
1.52961944
TLABC
7.47266894
2.33243162
0.00459544
22.08141044
1.26001308
IJAYA
7.51783186
5.12186124
0.00417303
12.29917328
1.32886699
PPSO
7.50376792
2.31232586
0.00317927
20.02371401
1.48672583
CLPSO
7.53505228
4.16487988
0.00278201
15.23020319
1.55760973
AGDE
7.47254343
2.33103110
0.00459542
22.14767462
1.25996110
DE_WOA
7.47253093
2.33497159
0.00459464
22.21826350
1.26010264Table 15
IAE results in the Photowatt-PW201 module.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
0.1248
1.0315
1.02919209
1.02919208
1.02905920
1.02917072
1.02918849
1.02923011
1.02923011
2
1.8093
1.0300
1.02738435
1.02738436
1.02732835
1.02738979
1.02734233
1.02744658
1.02744658
3
3.3511
1.0260
1.02574214
1.02574214
1.02569285
1.02570863
1.02560225
1.02576310
1.02576310
4
4.7622
1.0220
1.02405399
1.02406101
1.02406176
1.02403615
1.02387760
1.02408867
1.02408867
5
6.0538
1.0180
1.02228341
1.02228341
1.02224939
1.02218680
1.02198398
1.02223779
1.02223779
6
7.2364
1.0155
1.01988740
1.01990740
1.01989398
1.01979923
1.01956222
1.01984925
1.01984925
7
8.3189
1.0140
1.01635081
1.01635081
1.01634184
1.01622066
1.01596271
1.01627036
1.01627036
8
9.3097
1.0100
1.01049143
1.01049143
1.01050210
1.01036154
1.01009951
1.01041151
1.01041151
9
10.2163
1.0035
1.00067876
1.00067876
1.00071461
1.00056341
1.00031699
1.00061378
1.00061378
10
11.0449
0.9880
0.98465335
0.98465335
0.98471834
0.98456668
0.98435574
0.98461667
0.98461667
11
11.8018
0.9630
0.95969741
0.95969741
0.95979128
0.95964995
0.95949003
0.95969740
0.95969740
12
12.4929
0.9255
0.92304876
0.92304876
0.92316502
0.92304371
0.92294136
0.92308508
0.92308508
13
13.1231
0.8725
0.87258816
0.87258817
0.87271422
0.87261959
0.87257006
0.87265070
0.87265070
14
13.6983
0.8075
0.80739213
0.80739013
0.80743031
0.80736470
0.80735418
0.80738205
0.80738205
15
14.2221
0.7265
0.72795782
0.72795783
0.72805795
0.72801921
0.72803047
0.72802144
0.72802144
16
14.6995
0.6345
0.63646618
0.63646619
0.63653731
0.63652005
0.63653864
0.63650849
0.63650849
17
15.1346
0.5345
0.53569607
0.53569608
0.53573580
0.53573308
0.53575082
0.53571142
0.53571142
18
15.5311
0.4275
0.42881615
0.42881616
0.42882779
0.42883256
0.42884801
0.42880600
0.42880600
19
15.8929
0.3185
0.31866266
0.31866267
0.31865925
0.31866529
0.31868219
0.31863952
0.31863952
20
16.2229
0.2085
0.20785712
0.20785712
0.20783547
0.20783778
0.20786289
0.20781823
0.20781823
21
16.5241
0.1010
0.09835421
0.09835421
0.09832938
0.09832420
0.09836545
0.09831560
0.09831560
22
16.7987
−0.0080
−0.00816934
−0.00816935
−0.00818905
−0.00820433
−0.00819923
−0.00819817
−0.00819817
23
17.0499
−0.1110
−0.11096845
−0.11096845
−0.11097591
−0.11103032
−0.11090721
−0.11097918
−0.11097918
24
17.2793
−0.2090
−0.20910752
−0.20910754
−0.20910699
−0.20914695
−0.20910487
−0.20910337
−0.20910337
25
17.4885
−0.3030
−0.30202238
−0.30202236
−0.30198950
−0.30204274
−0.30187032
−0.30197816
−0.30197816
∑
I
A
E
—
0.04153981
0.04156890
0.04184719
0.04171346
0.04161306
0.04157073
0.04157073
The optimum values are highlighted in bold.Table 16
IAE results in the STM6-40/36 module.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
0.0000
1.6630
1.66345813
1.66345826
1.66260805
1.66443537
1.66507301
1.66354109
1.66345815
2
0.1180
1.6630
1.66325224
1.66325237
1.66242248
1.66421410
1.66483839
1.66333233
1.66325226
3
2.2370
1.6610
1.65955120
1.65955133
1.65908545
1.66023736
1.66062210
1.65957999
1.65955121
4
5.4340
1.6530
1.65391446
1.65391460
1.65398860
1.65418895
1.65421589
1.65386715
1.65391448
5
7.2600
1.6500
1.65056580
1.65056595
1.65093009
1.65061376
1.65044382
1.65047762
1.65056582
6
9.6800
1.6450
1.64543044
1.64543058
1.64610666
1.64521365
1.64481498
1.64529874
1.64543045
7
11.5900
1.6400
1.63923405
1.63923420
1.63999761
1.63888800
1.63838498
1.63909183
1.63923407
8
12.6000
1.6360
1.63371510
1.63371525
1.63440865
1.63336092
1.63286405
1.63358524
1.63371511
9
13.3700
1.6290
1.62728848
1.62728863
1.62784771
1.62697130
1.62652647
1.62718088
1.62728850
10
14.0900
1.6190
1.61831518
1.61831533
1.61867332
1.61807425
1.61772795
1.61824074
1.61831519
11
14.8800
1.5970
1.60306738
1.60306753
1.60312097
1.60295914
1.60278898
1.60304385
1.60306739
12
15.5900
1.5810
1.58158500
1.58158515
1.58131434
1.58163074
1.58168525
1.58161748
1.58158501
13
16.4000
1.5420
1.54232746
1.54232760
1.54171614
1.54254961
1.54292624
1.54242399
1.54232746
14
16.7100
1.5240
1.52122498
1.52122512
1.52052881
1.52149721
1.52200675
1.52134082
1.52122499
15
16.9800
1.5000
1.49920573
1.49920587
1.49847526
1.49950443
1.50012831
1.49933327
1.49920574
16
17.1300
1.4850
1.48527115
1.48527129
1.48454267
1.48557516
1.48626024
1.48540239
1.48527116
17
17.3200
1.4650
1.46564322
1.46564336
1.46494367
1.46594192
1.46670030
1.46577552
1.46564323
18
17.9100
1.3880
1.38759935
1.38759947
1.38723486
1.38776734
1.38869950
1.38770062
1.38759937
19
19.0800
1.1180
1.11837212
1.11837216
1.12017631
1.11756378
1.11833822
1.11819130
1.11837216
20
21.0200
0.0000
−0.00002133
−0.00002166
−0.00039888
0.00020655
0.00005339
−0.00002328
−0.00002136
∑
I
A
E
—
0.06429210
0.06432185
0.06740885
0.06733509
0.07094699
0.06453392
0.06436733
The optimum values are highlighted in bold.Table 17
IAE results in the STP6-120/36 module.
Item
V
L (V)
I
Lmeasured (A)
I
Lcalculated (A)
DPGSK
GSK
ITLBO
TLABC
IJAYA
PPSO
AGDE
1
0.0000
7.4800
7.47698179
7.47698019
7.47665638
7.47111081
7.51527601
7.50310413
7.47098999
2
9.0600
7.4500
7.45253756
7.45253743
7.45451827
7.45259805
7.48530660
7.49450105
7.45251333
3
9.4700
7.4200
7.44934550
7.44934543
7.45111948
7.44940349
7.48101491
7.48368056
7.44932094
4
10.3200
7.4400
7.43909225
7.43909232
7.44039457
7.43914567
7.46785784
7.47142399
7.43906828
5
11.1700
7.4100
7.42026504
7.42026524
7.42101522
7.42031503
7.44495300
7.44796198
7.42024411
6
11.8100
7.3800
7.39587321
7.39587350
7.39613741
7.39592154
7.41644421
7.40403264
7.39585681
7
12.3600
7.3700
7.36326491
7.36326527
7.36305632
7.36331247
7.37943806
7.36924502
7.36325428
8
12.7400
7.3400
7.33148326
7.33148365
7.33091921
7.33153057
7.34413104
7.33486735
7.33147771
9
13.1600
7.2900
7.28413020
7.28413063
7.28315199
7.28417735
7.29245408
7.28686643
7.28413125
10
13.5900
7.2300
7.21776129
7.21776173
7.21634963
7.21780812
7.22130851
7.21603695
7.21776995
11
14.1700
7.1000
7.08813761
7.08813803
7.08617271
7.08818289
7.08511764
7.08444912
7.08815694
12
14.5800
6.9700
6.95844943
6.95844981
6.95616503
6.95849205
6.95118750
6.95847554
6.95847554
13
14.9300
6.8300
6.81486058
6.81486089
6.81239172
6.81489929
6.80465661
6.74814273
6.81489101
14
15.3900
6.5800
6.56792998
6.56793018
6.56540330
6.56796057
6.55542580
6.50237572
6.56796260
15
15.7100
6.3600
6.34872814
6.34872825
6.34632033
6.34875084
6.33602884
6.28894274
6.34875907
16
16.0800
6.0000
6.03749323
6.03749322
6.03541018
6.03750464
6.02623996
5.99024242
6.03751827
17
16.3400
5.7500
5.77681472
5.77681465
5.77508163
5.77681719
5.76764593
5.74212427
5.77683304
18
16.7600
5.2700
5.27376618
5.27376604
5.27278276
5.27375386
5.26946723
5.26494555
5.27376968
19
16.9000
5.0700
5.08193494
5.08193479
5.08123945
5.08191802
5.07951765
5.08277576
5.08193271
20
17.1000
4.7900
4.78583409
4.78583395
4.78556827
4.78581135
4.78615088
4.80074242
4.78582332
21
17.2500
4.5600
4.54629057
4.54629048
4.54635205
4.54626429
4.54858123
4.57184242
4.54627337
22
17.4100
4.2900
4.27393013
4.27393010
4.27433549
4.27390109
4.27813192
4.31024324
4.27390624
23
17.6500
3.8300
3.83228334
3.83228345
3.83317162
3.83225270
3.83869284
3.88291542
3.83225035
24
19.2100
0.0000
0.00116402
0.00116712
0.00184104
0.00124967
-0.02787477
0.00762152
0.00115712
∑
I
A
E
—
0.27460349
0.27460538
0.28986916
0.28038683
0.48079071
0.64413128
0.28045679
The optimum values are highlighted in bold.Figure 7
Characteristic curves of DPGSK: (a) Photowatt-PW201 module, (b) STM6-40/36 module, and (c) STP6-120/36 module.
(a)
(b)
(c)Table 18
RMSE values in the Photowatt-PW201 module.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
2.42507487E − 03
2.42507487E − 03
2.42507487E − 03
4.75324029E − 17
GSK
2.42507487E − 03
2.43124074E − 03
2.42529952E − 03
1.12527868E − 06
ITLBO
2.42613506E − 03
6.78078310E − 03
2.83290424E − 03
8.11128363E − 04
SATLBO
2.43587695E − 03
6.03229203E − 03
3.04534170E − 03
8.69364652E − 04
TLABC
2.42618247E − 03
4.05234846E − 03
2.67703585E − 03
3.83080237E − 04
IJAYA
2.42566534E − 03
2.95825827E − 03
2.53272931E − 03
1.44953051E − 04
PPSO
2.42508316E − 03
8.83755053E − 03
2.94600003E − 03
1.24746322E − 03
CLPSO
2.83408533E − 03
6.75827193E − 03
4.01222813E − 03
9.56232818E − 04
AGDE
2.42507487E − 03
2.42507628E − 03
2.42507492E − 03
2.56868962E − 10
DE_WOA
2.42507489E − 03
3.54660632E − 03
2.56291129E − 03
2.64120863E − 04
The optimum values are highlighted in bold.Table 19
RMSE values in the STM6-40/36 module.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
6.69268329E − 18
GSK
1.72981371E − 03
1.72981372E − 03
1.72981371E − 03
1.50670485E − 12
ITLBO
1.80302370E − 03
5.04532930E − 02
5.08181795E − 03
8.64964315E − 03
SATLBO
2.00426673E − 03
2.98051206E − 02
8.68645469E − 03
7.72801077E − 03
TLABC
1.73011828E − 03
6.71432512E − 02
5.15375645E − 03
1.18096851E − 02
IJAYA
1.80656350E − 03
3.38695808E − 03
2.69248270E − 03
4.76741812E − 04
PPSO
1.73296413E − 03
3.08607469E − 02
4.78248606E − 03
5.06632794E − 03
CLPSO
5.51270979E − 03
2.82327010E − 02
1.59673399E − 02
7.07623601E − 03
AGDE
1.72981371E − 03
1.90393613E − 03
1.73951550E − 03
3.26588994E − 05
DE_WOA
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
7.83732302E − 13
The optimum values are highlighted in bold.Table 20
RMSE values in the STP6-120/36 module.
Algorithm
RMSE
Best
Worst
Mean
Std
DPGSK
1.66006031E − 02
1.66006031E − 02
1.66006031E − 02
3.64215013E − 16
GSK
1.66006031E − 02
1.66006379E − 02
1.66006043E − 02
6.34321846E − 09
ITLBO
1.66026542E − 02
1.82932081E − 01
4.09561836E − 02
3.36881282E − 02
SATLBO
1.67283597E − 02
5.94301196E − 02
4.09933264E − 02
1.18675783E − 02
TLABC
1.66007077E − 02
2.27284953E − 01
3.51585815E − 02
4.39960971E − 02
IJAYA
1.73920763E − 02
2.74919656E − 02
2.09806168E − 02
3.00496502E − 03
PPSO
1.66391981E − 02
1.14874892E − 01
4.06962594E − 02
1.85185406E − 02
CLPSO
3.52931825E − 02
8.67923901E − 01
1.77656627E − 01
1.93537380E − 01
AGDE
1.66006031E − 02
1.34812942E − 01
2.06460413E − 02
2.15653571E − 02
DE_WOA
1.66006031E − 02
1.66007829E − 02
1.66006139E − 02
4.10680224E − 08Table 21
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms for the Photowatt-PW201 module.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1365.00
465.00
2.96913841E − 11
↑
ITLBO
1365.00
465.00
2.96913841E − 11
↑
SATLBO
1365.00
465.00
2.96913841E − 11
↑
TLABC
1365.00
465.00
2.96913841E − 11
↑
IJAYA
1365.00
465.00
2.96913841E − 11
↑
PPSO
1365.00
465.00
2.96913841E − 11
↑
CLPSO
1365.00
465.00
2.96913841E − 11
↑
AGDE
1327.50
502.50
1.09035598E − 09
↑
DE_WOA
1365.00
465.00
2.96913841E − 11
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.Table 22
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms for the STM6-40/36 module.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1356.50
473.50
4.42956883E − 11
↑
ITLBO
1365.00
465.00
1.98787859E − 11
↑
SATLBO
1365.00
465.00
1.98787859E − 11
↑
TLABC
1365.00
465.00
1.98787859E − 11
↑
IJAYA
1365.00
465.00
1.98787859E − 11
↑
PPSO
1365.00
465.00
1.98787859E − 11
↑
CLPSO
1365.00
465.00
1.98787859E − 11
↑
AGDE
1365.00
465.00
1.98787859E − 11
↑
DE_WOA
1341.00
489.00
2.03566531E − 10
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.Table 23
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. other algorithms for the STP6-120/36 module.
DPGSK vs.
R
+
R
−
p value
Sig.
GSK
1363.00
467.00
3.29327551E − 11
↑
ITLBO
1365.00
465.00
2.69269726E − 11
↑
SATLBO
1365.00
465.00
2.69269726E − 11
↑
TLABC
1365.00
465.00
2.69269726E − 11
↑
IJAYA
1365.00
465.00
2.69269726E − 11
↑
PPSO
1365.00
465.00
2.69269726E − 11
↑
CLPSO
1365.00
465.00
2.69269726E − 11
↑
AGDE
1365.00
465.00
2.69269726E − 11
↑
DE_WOA
1354.50
475.50
7.59190233E − 11
↑
The sign “↑” means that DPGSK is significantly superior to the compared competitor.Figure 8
Convergence curves: (a) Photowatt-PW201 module, (b) STM6-40/36 module, and (c) STP6-120/36 module.
(a)
(b)
(c)


### 5.3.1. Analysis of the IAE Values

Figures7(a)–7(c) illustrate that the currents calculated by DPGSK fit the measured currents to a high degree. For the IAE results, DPGSK reaches the smallest IAE values in these modules. To be specific, the IAE values are 0.04153981, 0.06429210, and 0.27460349, respectively, demonstrating that DPGSK has higher accuracy in the parameter identification of these three PV modules than other algorithms.


### 5.3.2. Analysis of the RMSE Values

For the best RMSE index, in the Photowatt-PW201 module, DPGSK, GSK and AGDE yield the optimum result (2.42507487E-03). In the other two modules, four algorithms including DPGSK, GSK, AGDE, and DE_WOA, obtain the optimum values, which are 1.72981371E-03 and 1.66006031E-02, respectively.For the worst RMSE index, DPGSK gets the optimum values in the Photowatt-PW201 and STP6-120/36, and the values are 2.42507487E-03 and 1.66006031E-02, respectively. For the STM6-40/36, both DPGSK and DE_WOA achieve the same optimum value (1.72981371E-03).For the mean RMSE index, DPGSK yields the optimum values, i.e., 2.42507487E-03 and 1.66006031E-02, respectively, in the Photowatt-PW201 and STP6-120/36 modules. For the STM6-40/36 module, DPGSK, GSK, and DE_WOA reach the same optimum value (1.72981371E-03).For the standard deviation, only DPGSK provides the best performance in all the three PV modules. The values are 4.75324029E-17, 6.69268329E-18, and 3.64215013E-16, respectively. Particularly, the optimal standard deviation values of DPGSK are significantly better than that of others.The above comparisons fully indicate that DPGSK obtains the most accurate parameters’ values for the three PV modules and shows the strongest robustness in searching for the global optimal solutions.


### 5.3.3. Analysis of the Convergence Performance

In the Photowatt-PW201 module, DPGSK and DE-WOA converge faster than other algorithms during the initial stage (before about 1000 evaluations). Moreover, compared with GSK, DPGSK improves the convergence performance effectively. In the last two modules, DE-WOA converges fastest, and DPGSK converges similarly to GSK. In the STM6-40/36 module, DPGSK surpasses DE-WOA at approximately 4000 evaluations. In the STP6-120/36 module, DPGSK completely surpasses GSK at approximately 3000 evaluations and surpasses DE-WOA at approximately 6000 evaluations. DE-WOA falls into local convergence in all the PV modules, while DPGSK can avoid the local optimum effectively and surpass GSK to search for the global optimum. Therefore, it proves again that DPGSK has stronger competitiveness in convergence performance.


### 5.3.4. Analysis of the Test Results

The test outcomes provided in Tables21–23 reveal that all the R+ values are clearly higher than the R− values. For most methods, the R+ is 1365, and only the R+ of DE-WOA is 1327.5 in the Photowatt-PW201 module. The R+ of GSK and DE-WOA in the STM6-40/36 module are 1356.5 and 1341, respectively. In the STP6-120/36 module, GSK and DE-WOA have an R+ of 1356.5 and 1341, respectively. Obviously, in these three modules, the p values are considerably below 0.05, indicating that DPGSK significantly outperforms all the other methods. It concludes that the suggested DPGSK is statistically more reliable than other competitors.


## 5.3.1. Analysis of the IAE Values

Figures7(a)–7(c) illustrate that the currents calculated by DPGSK fit the measured currents to a high degree. For the IAE results, DPGSK reaches the smallest IAE values in these modules. To be specific, the IAE values are 0.04153981, 0.06429210, and 0.27460349, respectively, demonstrating that DPGSK has higher accuracy in the parameter identification of these three PV modules than other algorithms.


## 5.3.2. Analysis of the RMSE Values

For the best RMSE index, in the Photowatt-PW201 module, DPGSK, GSK and AGDE yield the optimum result (2.42507487E-03). In the other two modules, four algorithms including DPGSK, GSK, AGDE, and DE_WOA, obtain the optimum values, which are 1.72981371E-03 and 1.66006031E-02, respectively.For the worst RMSE index, DPGSK gets the optimum values in the Photowatt-PW201 and STP6-120/36, and the values are 2.42507487E-03 and 1.66006031E-02, respectively. For the STM6-40/36, both DPGSK and DE_WOA achieve the same optimum value (1.72981371E-03).For the mean RMSE index, DPGSK yields the optimum values, i.e., 2.42507487E-03 and 1.66006031E-02, respectively, in the Photowatt-PW201 and STP6-120/36 modules. For the STM6-40/36 module, DPGSK, GSK, and DE_WOA reach the same optimum value (1.72981371E-03).For the standard deviation, only DPGSK provides the best performance in all the three PV modules. The values are 4.75324029E-17, 6.69268329E-18, and 3.64215013E-16, respectively. Particularly, the optimal standard deviation values of DPGSK are significantly better than that of others.The above comparisons fully indicate that DPGSK obtains the most accurate parameters’ values for the three PV modules and shows the strongest robustness in searching for the global optimal solutions.


## 5.3.3. Analysis of the Convergence Performance

In the Photowatt-PW201 module, DPGSK and DE-WOA converge faster than other algorithms during the initial stage (before about 1000 evaluations). Moreover, compared with GSK, DPGSK improves the convergence performance effectively. In the last two modules, DE-WOA converges fastest, and DPGSK converges similarly to GSK. In the STM6-40/36 module, DPGSK surpasses DE-WOA at approximately 4000 evaluations. In the STP6-120/36 module, DPGSK completely surpasses GSK at approximately 3000 evaluations and surpasses DE-WOA at approximately 6000 evaluations. DE-WOA falls into local convergence in all the PV modules, while DPGSK can avoid the local optimum effectively and surpass GSK to search for the global optimum. Therefore, it proves again that DPGSK has stronger competitiveness in convergence performance.


## 5.3.4. Analysis of the Test Results

The test outcomes provided in Tables21–23 reveal that all the R+ values are clearly higher than the R− values. For most methods, the R+ is 1365, and only the R+ of DE-WOA is 1327.5 in the Photowatt-PW201 module. The R+ of GSK and DE-WOA in the STM6-40/36 module are 1356.5 and 1341, respectively. In the STP6-120/36 module, GSK and DE-WOA have an R+ of 1356.5 and 1341, respectively. Obviously, in these three modules, the p values are considerably below 0.05, indicating that DPGSK significantly outperforms all the other methods. It concludes that the suggested DPGSK is statistically more reliable than other competitors.


## 5.4. Whole Performance

The DPGSK algorithm is compared with other algorithms in different models separately above. However, its performance could not be verified comprehensively by the experimental results in a single model. Therefore, the Friedman test is employed here for the statistical analysis of multiple models simultaneously at a confidence level of 0.05. The test results, as displayed in Figure9, demonstrate that DPGSK ranks first with an average ranking of 1.40, followed by GSK, DE_WOA, AGDE, IJAYA, TLABC, PPSO, ITLBO, SATLBO, and CLPSO, which further indicates the superior status of the presented algorithm over others.Figure 9
Friedman test result.


## 5.5. Analysis of the Components

It is known that GSK contains two phases, i.e., the junior and the senior. In the proposed DPGSK, it adopts the dual-population evolution strategy to divide the senior and junior phases into two random subpopulations of equal size. In this subsection, the effect of each phase on DPGSK is evaluated. Two variants are considered. One variant that only uses the junior phase is IGSK1, and the other variant that only performs the senior phase is IGSK2.


### 5.5.1. Analysis of the RMSE Values

For the best RMSE indicator, all four algorithms get the optimum values 9.86021878E-04 and 2.42507487E-03 in the SDM and the Photowatt-PW201 module, respectively, as shown in Table24. In the DDM, IGSK2 gets the best value (9.82484852E-04), and DPGSK’s value (9.82484859E-04) stays close behind IGSK2. In the STM6-40/36 and STP6-120/36 modules, DPGSK, GSK, and IGSK2 obtain the same optimum values, which are 1.72981371E-03 and 1.66006031E-02, respectively.Table 24
RMSE values for different variants of DPGSK.
Model
Algorithm
RMSE
Best
Worst
Mean
Std
SDM
DPGSK
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
5.01330110E − 17
GSK
9.86021878E − 04
9.86021881E − 04
9.86021878E − 04
5.77198278E − 13
IGSK1
9.86021878E − 04
1.14970749E − 03
1.00334452E − 03
3.09917403E − 05
IGSK2
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
4.95508657E − 17
DDM
DPGSK
9.82484859E − 04
9.87885272E − 04
9.84815569E − 04
1.68677548E − 06
GSK
9.82681198E − 04
9.91166708E − 04
9.85246463E − 04
1.78610712E − 06
IGSK1
9.83129226E − 04
3.35318293E − 03
1.69916657E − 03
5.92923212E − 04
IGSK2
9.82484852E − 04
9.90885053E − 04
9.85386108E − 04
1.82893145E − 06
Photowatt-PW201
DPGSK
2.42507487E − 03
2.42507487E − 03
2.42507487E − 03
4.75324029E − 17
GSK
2.42507487E − 03
2.43124074E − 03
2.42529952E − 03
1.12527868E − 06
IGSK1
2.42507487E − 03
2.43852468E − 03
2.42617462E − 03
2.72385502E − 06
IGSK2
2.42507487E − 03
2.42774735E − 03
2.42516515E − 03
4.87738798E − 07
STM6-40/36
DPGSK
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
6.69268329E − 18
GSK
1.72981371E − 03
1.72981372E − 03
1.72981371E − 03
1.50670485E − 12
IGSK1
1.73191719E − 03
7.07659606E − 03
3.38453446E − 03
1.41010667E − 03
IGSK2
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
2.49962266E − 15
STP6-120/36
DPGSK
1.66006031E − 02
1.66006031E − 02
1.66006031E − 02
3.64215013E − 16
GSK
1.66006031E − 02
1.66006379E − 02
1.66006043E − 02
6.34321846E − 09
IGSK1
1.66333141E − 02
3.40593689E − 02
2.35744620E − 02
6.13100930E − 03
IGSK2
1.66006031E − 02
1.66026785E − 02
1.66006723E − 02
3.78914430E − 07
The optimum values are highlighted in bold.For the worst RMSE indicator, DPGSK and IGSK2 get the optimum values in the SDM and STM6-40/36 module, which are 9.86021878E-04 and 1.72981371E-03, respectively. In the DDM, Photowatt-PW201 module, and STP6-120/36 module, only DPGSK achieves the optimum values, i.e., 9.87885272E-04, 2.42507487E-03, and 1.66006031E-02, respectively.For the mean RMSE indicator, DPGSK, GSK, and IGSK2 get the optimum values in the SDM and STM6-40/36 module and the values are 9.86021878E-04 and 1.72981371E-03, respectively. In the DDM, Photowatt-PW201 module, and STP6-120/36 module, only DPGSK achieves the optimum values, i.e., 9.84815569E-04, 2.42507487E-03, and 1.66006031E-02, respectively.For the standard deviation, DPGSK achieves the optimum values in the DDM, Photowatt-PW201, STM6-40/36, and STP6-120/36 modules, which are 1.68677548E-06, 4.75324029E-17, 6.69268329E-18, and 3.64215013E-16, respectively. Although IGSK2 gets the optimum value of 4.95508657E-17 in the SDM, DPGSK’s value (5.01330110E-17) is only a little short of IGSK2.Summarizing the above comparisons, we can see that DPGSK obtains the optimal values the most often. When it does not, the difference between its value and the best value is very slight. IGSK2 obtains the optimal value the second most often, GSK follows, and IGSK1 is at the end. Overall, DPGSK proves to be more effective in identifying accurate and reliable parameters for these PV models.


### 5.5.2. Analysis of the Test Results

The test results in Table25 show that the R+ exceeds the R− for all models. In the SDM, the significance indicator exceeds 0.05 only for IGSK1, and thus, the other two algorithms are significantly worse than DPGSK. In the DDM, the p-values of GSK and IGSK2 exceed 0.05, indicating they are statistically comparable to DPGSK. In the other three PV modules, the significance indicators are considerably less than 0.05, meaning that DPGSK is significantly superior to the rest algorithms. Considering the above, DPGSK is statistically better than GSK, IGSK1, and IGSK2.Table 25
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. its variants for the five PV models.
Model
DPGSK vs.
R
+
R
−
p value
Sig.
SDM
GSK
1355.50
474.50
7.74308961E − 11
↑
IGSK1
1365
465
3.01418492E − 11
↑
IGSK2
954
875
5.64171398E − 01
≈
DDM
GSK
974.00
856.00
3.87099778E − 01
≈
IGSK1
1326
504
1.28703830E − 09
↑
IGSK2
948
882
6.30876292E − 01
≈
Photowatt-PW201
GSK
1365.00
465.00
2.96913841E − 11
↑
IGSK1
1346.5
483.5
1.82527310E − 10
↑
IGSK2
1167
663
1.95503446E − 04
↑
STM6-40/36
GSK
1356.50
473.50
4.42956883E − 11
↑
IGSK1
1365
465
1.98787859E − 11
↑
IGSK2
1346
484
8.29091595E − 11
↑
STP6-120/36
GSK
1363.00
467.00
3.29327551E − 11
↑
IGSK1
1365
465
2.69269726E − 11
↑
IGSK2
1245
585
8.06050812E − 07
↑
The signs “↑” and “≈” mean that DPGSK is significantly superior and similar to the compared competitor, respectively.


## 5.5.1. Analysis of the RMSE Values

For the best RMSE indicator, all four algorithms get the optimum values 9.86021878E-04 and 2.42507487E-03 in the SDM and the Photowatt-PW201 module, respectively, as shown in Table24. In the DDM, IGSK2 gets the best value (9.82484852E-04), and DPGSK’s value (9.82484859E-04) stays close behind IGSK2. In the STM6-40/36 and STP6-120/36 modules, DPGSK, GSK, and IGSK2 obtain the same optimum values, which are 1.72981371E-03 and 1.66006031E-02, respectively.Table 24
RMSE values for different variants of DPGSK.
Model
Algorithm
RMSE
Best
Worst
Mean
Std
SDM
DPGSK
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
5.01330110E − 17
GSK
9.86021878E − 04
9.86021881E − 04
9.86021878E − 04
5.77198278E − 13
IGSK1
9.86021878E − 04
1.14970749E − 03
1.00334452E − 03
3.09917403E − 05
IGSK2
9.86021878E − 04
9.86021878E − 04
9.86021878E − 04
4.95508657E − 17
DDM
DPGSK
9.82484859E − 04
9.87885272E − 04
9.84815569E − 04
1.68677548E − 06
GSK
9.82681198E − 04
9.91166708E − 04
9.85246463E − 04
1.78610712E − 06
IGSK1
9.83129226E − 04
3.35318293E − 03
1.69916657E − 03
5.92923212E − 04
IGSK2
9.82484852E − 04
9.90885053E − 04
9.85386108E − 04
1.82893145E − 06
Photowatt-PW201
DPGSK
2.42507487E − 03
2.42507487E − 03
2.42507487E − 03
4.75324029E − 17
GSK
2.42507487E − 03
2.43124074E − 03
2.42529952E − 03
1.12527868E − 06
IGSK1
2.42507487E − 03
2.43852468E − 03
2.42617462E − 03
2.72385502E − 06
IGSK2
2.42507487E − 03
2.42774735E − 03
2.42516515E − 03
4.87738798E − 07
STM6-40/36
DPGSK
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
6.69268329E − 18
GSK
1.72981371E − 03
1.72981372E − 03
1.72981371E − 03
1.50670485E − 12
IGSK1
1.73191719E − 03
7.07659606E − 03
3.38453446E − 03
1.41010667E − 03
IGSK2
1.72981371E − 03
1.72981371E − 03
1.72981371E − 03
2.49962266E − 15
STP6-120/36
DPGSK
1.66006031E − 02
1.66006031E − 02
1.66006031E − 02
3.64215013E − 16
GSK
1.66006031E − 02
1.66006379E − 02
1.66006043E − 02
6.34321846E − 09
IGSK1
1.66333141E − 02
3.40593689E − 02
2.35744620E − 02
6.13100930E − 03
IGSK2
1.66006031E − 02
1.66026785E − 02
1.66006723E − 02
3.78914430E − 07
The optimum values are highlighted in bold.For the worst RMSE indicator, DPGSK and IGSK2 get the optimum values in the SDM and STM6-40/36 module, which are 9.86021878E-04 and 1.72981371E-03, respectively. In the DDM, Photowatt-PW201 module, and STP6-120/36 module, only DPGSK achieves the optimum values, i.e., 9.87885272E-04, 2.42507487E-03, and 1.66006031E-02, respectively.For the mean RMSE indicator, DPGSK, GSK, and IGSK2 get the optimum values in the SDM and STM6-40/36 module and the values are 9.86021878E-04 and 1.72981371E-03, respectively. In the DDM, Photowatt-PW201 module, and STP6-120/36 module, only DPGSK achieves the optimum values, i.e., 9.84815569E-04, 2.42507487E-03, and 1.66006031E-02, respectively.For the standard deviation, DPGSK achieves the optimum values in the DDM, Photowatt-PW201, STM6-40/36, and STP6-120/36 modules, which are 1.68677548E-06, 4.75324029E-17, 6.69268329E-18, and 3.64215013E-16, respectively. Although IGSK2 gets the optimum value of 4.95508657E-17 in the SDM, DPGSK’s value (5.01330110E-17) is only a little short of IGSK2.Summarizing the above comparisons, we can see that DPGSK obtains the optimal values the most often. When it does not, the difference between its value and the best value is very slight. IGSK2 obtains the optimal value the second most often, GSK follows, and IGSK1 is at the end. Overall, DPGSK proves to be more effective in identifying accurate and reliable parameters for these PV models.


## 5.5.2. Analysis of the Test Results

The test results in Table25 show that the R+ exceeds the R− for all models. In the SDM, the significance indicator exceeds 0.05 only for IGSK1, and thus, the other two algorithms are significantly worse than DPGSK. In the DDM, the p-values of GSK and IGSK2 exceed 0.05, indicating they are statistically comparable to DPGSK. In the other three PV modules, the significance indicators are considerably less than 0.05, meaning that DPGSK is significantly superior to the rest algorithms. Considering the above, DPGSK is statistically better than GSK, IGSK1, and IGSK2.Table 25
R
+, R−, and p values of Wilcoxon’s rank-sum test for the RMSE values of the DPGSK vs. its variants for the five PV models.
Model
DPGSK vs.
R
+
R
−
p value
Sig.
SDM
GSK
1355.50
474.50
7.74308961E − 11
↑
IGSK1
1365
465
3.01418492E − 11
↑
IGSK2
954
875
5.64171398E − 01
≈
DDM
GSK
974.00
856.00
3.87099778E − 01
≈
IGSK1
1326
504
1.28703830E − 09
↑
IGSK2
948
882
6.30876292E − 01
≈
Photowatt-PW201
GSK
1365.00
465.00
2.96913841E − 11
↑
IGSK1
1346.5
483.5
1.82527310E − 10
↑
IGSK2
1167
663
1.95503446E − 04
↑
STM6-40/36
GSK
1356.50
473.50
4.42956883E − 11
↑
IGSK1
1365
465
1.98787859E − 11
↑
IGSK2
1346
484
8.29091595E − 11
↑
STP6-120/36
GSK
1363.00
467.00
3.29327551E − 11
↑
IGSK1
1365
465
2.69269726E − 11
↑
IGSK2
1245
585
8.06050812E − 07
↑
The signs “↑” and “≈” mean that DPGSK is significantly superior and similar to the compared competitor, respectively.


## 5.6. Discussions

In this paper, we boost the efficacy of the original GSK algorithm using a dual-population evolution strategy. The resultant DPGSK algorithm to identify the PV models’ parameters is compared with various excellent metaheuristics in five models. The effect of different components on DPGSK is also investigated. From the results, we have reached the following summations:(1)
The characteristic curves of I-V and P-V show that DPGSK obtains superior accuracy and reliability in parameter identification. The IAE values for DPGSK in the five PV models are 0.01769439, 0.01759995, 0.04153981, 0.06429210, and 0.27460349, respectively, which are all the best compared with other algorithms.(2)
The RMSE values show that DPGSK achieves optimal values in the five models, and in particular, it outperforms others evidently in terms of standard deviation. Besides, the testR+, R−, and p values present that DPGSK gets the most accurate values and shows the strongest robustness in searching for the global optimal solutions. It proves that the dual-population evolution strategy does raise the accuracy of GSK in solving the parameter identification of PV models.(3)
With regard to the convergence curves, DPGSK and DE_WOA have faster convergence at the beginning of the iteration. Although DE_WOA converges fastest in the DDM, STM6-40/36, and STP6-120/36, it tends to fall into local extrema. DPGSK can avoid local convergence and search for the global optimum effectively. Besides, it is worth mentioning that DPGSK can address the issue of slow convergence of the GSK algorithm in early iterations.(4)
DPGSK’s overall performance is verified more comprehensively through the Friedman test. DPGSK ranks first with an average ranking of 1.40, followed by GSK, DE_WOA, AGDE, IJAYA, TLABC, PPSO, ITLBO, SATLBO, and CLPSO. The ranking results demonstrate the marked superiority of DPGSK over other algorithms in tackling this studied problem.(5)
The effect of the junior phase and the senior phase on DPGSK is analyzed. The results show that these two phases do affect the performance of DPGSK. Either component is indispensable to DPGSK. Nevertheless, the proposed dual-population evolution strategy can exhibit the power to coordinate them well. Namely, both exploration and exploitation can be balanced well in DPGSK.


## 6. Conclusions

Parameter identification is one of the most critical problems of PV system modeling and adopting an accurate and effective algorithm to solve it is notably necessary. In this paper, a dual-population GSK (DPGSK) is suggested to tackle it. Based on the GSK algorithm, a dual-population evolution strategy is designed to balance the search between exploration and exploitation. The whole population splits randomly into two even subpopulations, one of which performs the junior phase while the other executes the senior phase. Five typical PV models are used to confirm the effectiveness of DPGSK. The effect of different components including the junior and the senior on the performance of DPGSK is also investigated. Comparative simulation results reveal that with the powerful assistance of the dual-population evolution strategy, DPGSK overcomes the shortcoming of slow convergence of GSK effectively and enhances the population diversity and convergence rate significantly to search for the global optimum. Besides, it achieves the overall best RMSE values and IAE values compared with other peer algorithms. It is further confirmed by both Wilcoxon’s rank-sum test and Friedman test, indicating that DPGSK has strong competitiveness in accuracy and robustness in tackling the PV models’ parameter identification problem. Moreover, either component is indispensable to DPGSK.On the other hand, we also found that DPGSK’s convergence is still not fast enough, as DE_WOA beats it at the beginning of the iteration. Thus, in future work, it is desirable to further heighten its convergence with the help of other strategies such as the population size reduction technique [64], quadratic interpolation [65], and reinforcement learning [66, 67]. In addition, we will test the suggested DPGSK algorithm’s performance in more PV models and further ameliorate it to tackle more power system operation problems.



---

*Source: 3788453-2023-03-03.xml*