# Analytical versus Metaheuristic Methods to Extract the Photovoltaic Cells and Panel Parameters

Authors: Daniel T. Cotfas; Petru A. Cotfas; Mihai P. Oproiu; Paul A. Ostafe
Journal: International Journal of Photoenergy
 (2021)

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## Abstract

The parameters of the photovoltaic cells and panels are very important to forecast the power generated. There are a lot of methods to extract the parameters using analytical, metaheuristic, and hybrid algorithms. The comparison between the widely used analytical method and some of the best metaheuristic algorithms from the algorithm families is made for datasets from the specialized literature, using the following statistical tests: absolute error, root mean square error, and the coefficient of determination. The equivalent circuit and mathematical model considered is the single diode model. The result comparison shows that the metaheuristic algorithms have the best performance in almost all cases, and only for the genetic algorithm, there are poorer results for one chosen photovoltaic cell. The parameters of the photovoltaic cells and panels and also the current-voltage characteristic for real outdoor weather conditions are forecasted using the parameters calculated with the best method: one for analytical—the five-parameter analytical method—and one for the metaheuristic algorithms—hybrid successive discretization algorithm. Additionally, the genetic algorithm is used. The forecast current-voltage characteristic is compared with the one measured in real sunlight conditions, and the best results are obtained in the case of a hybrid successive discretization algorithm. The maximum power forecast using the calculated parameters with the five-parameter method is the best, and the error in comparison with the measured ones is 0.48%.


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## Body

## 1. Introduction

Nowadays, the power forecasting for the photovoltaic panels and systems plays a very important role for the investors to increase the investments having a realistic scenario. One of the steps to achieve this goal is to accurately and quickly determine the parameters of the photovoltaic cells and panels.The extraction of the photovoltaic cell parameters is a widely studied issue [1, 2], but it remains current due to its importance and the new possibilities created by the metaheuristic algorithms and artificial intelligence [3].The parameter extraction is possible if there is a dataset which consists of voltage-current pairs (V,I) for the photovoltaic panel, or if the current-voltage characteristic (I-V) is measured. The parameters and dataset can be obtained using the photovoltaic panel datasheet given by the producer [4].The most commonly used mathematical model to characterize the photovoltaic cells and panels is the single diode model (SD) [1], followed by the double diode model (DD) [5] and rarely three diode model (TD) [6]. The number of the parameters which have to be extracted varies, being five for SD, seven for DD, and nine for TD. There are a lot of methods to extract these parameters, their complexity growing with the increasing number of parameters.The methods used to extract the parameters of the photovoltaic cells or panels can be classified into analytical, metaheuristic, and hybrid methods [7]. Each of these methods has both advantages and disadvantages.The contributions and novelty of this paper are as follows:(i)
The main analytical methods and metaheuristic algorithms grouped on families are briefly presented(ii)
The performance of the methods is analyzed in function of the accuracy with which the parameters are extracted analyzing the absolute error, the root mean square error, and the coefficient of determination(iii)
Choosing the best analytical method considering the following: simplicity of application, the execution time, and the accuracy(iv)
Choosing the metaheuristic algorithm with the smallest root mean square error (RMSE) for different photovoltaic cells and panels from all algorithms considered(v)
Comparing for the first time the analytical method (modified five parameters) and metaheuristic algorithm (hybrid successive discretization algorithm) to forecast theI-V characteristic and the maximum power generated by the commercial monocrystalline photovoltaic panel, giving the manufacturers a tool to choose the best option to characterize the PV for their applications. Additionally, the genetic algorithm is considered in the comparisonThe rest of the paper is organized as follows: the equivalent circuits and diode models, statistical tests used for comparison, and the mathematical formulas for calculating the photovoltaic cells and panel parameters at different temperatures and irradiances in the function of their values at the standard test conditions (STC-irradiance 1000 W/m2, temperature 25°C, and air mass 1.5) are described in Section 2. A brief presentation of the used methods is made in Section 3. The results and discussions are presented in Section 4, and the last section is dedicated to conclusions and future works.


## 2. Methods

### 2.1. Photovoltaic Cells and Panel Diode Models

The mathematical model which describes the dependence between the current and the voltage generated by the photovoltaic cells and panels depends on the mechanisms which are taken into account and consequently on the equivalent circuits, Figure1. The simplest model is the ideal one. The most commonly used model is single diode, Figure 1(a), due to its simplicity but also because it manages to describe the behaviour of most types of photovoltaic cells and panels very well. Equation (1) is the mathematical relation for one diode model:(1)I=Iph−IoeV+IRs/nVT−1−V+IRsRsh,where Iph is the photogenerated current, Io is the reverse saturation current, Rs is the series resistance, Rsh is the shunt resistance, n is the ideality factor of diode, and VT is the thermal voltage, VT=kT/q. k is the Boltzmann constant, T is the temperature, and q is the elementary electrical charge.Figure 1
The equivalent circuit of photovoltaic cell: (a) one diode model; (b) two diode model.
(a)(b)The double diode model is described by(2)I=Iph−Io1eV+IRs/n1VT−1−Io2eV+IRs/n2VT−1−V+IRsRsh,where index 1 relates to the diffusion mechanism and 2 the generation-recombination mechanism. The accuracy to determine the parameters of the photovoltaic cell increases especially at low solar radiation when the two diode model is used [8].The mathematical model for the photovoltaic panel is described by(3)I=NpIph−NpIoeNpV+NsIRs/nNpNsVT−1−NpV+NsIRsNsRsh,where Ns represents the number of the photovoltaic cells connected in series and Np represents the number of the photovoltaic cells connected in parallel.


### 2.2. Statistical Test

The comparison between analytical and metaheuristic algorithms is achieved using different statistical error tests, such as absolute error (AE) Equation (4), the root mean square error Equation (5), and the coefficient of determination R2 Equation (6).(4)AE=∑i=1nIic−Iim,(5)RMSE=∑i=1nIic−Iim2n,(6)R2=1−∑i=1nIic−Iim2∑i=1nIim−I¯im2,whereIic and Iim represent the calculated and the measured current, respectively, and n is the total number of measurements.


### 2.3. Irradiance and Temperature Dependence of the PV Parameters

The irradiance and temperature influence more or less the parameters of the photovoltaic cells and panels. The power generated is also dependent on these two factors. So, the relation for photogenerated current, function of the irradiance, and temperature is the following [9]:(7)Iph=GGrefIph,ref+αscT,Tref,where G is irradiance, T represent the temperature, and αsc is the temperature coefficient of the current. The index ref is for the parameters at STC. The reverse saturation current can be calculated with Equation (8) [9, 10]:(8)Io=Io,refTTref3eq/KnEg,ref/Tref−Eg/Tr,where Eg is the energy bandgap. This value depends slightly on temperature [11, 12]:(9)Eg=Eg,ref1−0.0002677T−Tref.The ideality factor of diode depends slightly on the irradiance [13]. The temperature dependence can be written as Equation (10) [9]:(10)n=nrefTTref.The behaviour of the shunt resistance is inversely proportional to that of irradiance, the irradiance increasing as the shunt resistance decreases:(11)Rsh=Rsh,refGrefG.The dependence of the series resistance on temperature and irradiance is described by Equation (12). It decreases linearly with the increase in temperature and increases with the increase of irradiance; β is constant and is considered equal to 0.217 [11, 14].(12)Rsh=Rsh,refTTref1−βlnGGref.


## 2.1. Photovoltaic Cells and Panel Diode Models

The mathematical model which describes the dependence between the current and the voltage generated by the photovoltaic cells and panels depends on the mechanisms which are taken into account and consequently on the equivalent circuits, Figure1. The simplest model is the ideal one. The most commonly used model is single diode, Figure 1(a), due to its simplicity but also because it manages to describe the behaviour of most types of photovoltaic cells and panels very well. Equation (1) is the mathematical relation for one diode model:(1)I=Iph−IoeV+IRs/nVT−1−V+IRsRsh,where Iph is the photogenerated current, Io is the reverse saturation current, Rs is the series resistance, Rsh is the shunt resistance, n is the ideality factor of diode, and VT is the thermal voltage, VT=kT/q. k is the Boltzmann constant, T is the temperature, and q is the elementary electrical charge.Figure 1
The equivalent circuit of photovoltaic cell: (a) one diode model; (b) two diode model.
(a)(b)The double diode model is described by(2)I=Iph−Io1eV+IRs/n1VT−1−Io2eV+IRs/n2VT−1−V+IRsRsh,where index 1 relates to the diffusion mechanism and 2 the generation-recombination mechanism. The accuracy to determine the parameters of the photovoltaic cell increases especially at low solar radiation when the two diode model is used [8].The mathematical model for the photovoltaic panel is described by(3)I=NpIph−NpIoeNpV+NsIRs/nNpNsVT−1−NpV+NsIRsNsRsh,where Ns represents the number of the photovoltaic cells connected in series and Np represents the number of the photovoltaic cells connected in parallel.


## 2.2. Statistical Test

The comparison between analytical and metaheuristic algorithms is achieved using different statistical error tests, such as absolute error (AE) Equation (4), the root mean square error Equation (5), and the coefficient of determination R2 Equation (6).(4)AE=∑i=1nIic−Iim,(5)RMSE=∑i=1nIic−Iim2n,(6)R2=1−∑i=1nIic−Iim2∑i=1nIim−I¯im2,whereIic and Iim represent the calculated and the measured current, respectively, and n is the total number of measurements.


## 2.3. Irradiance and Temperature Dependence of the PV Parameters

The irradiance and temperature influence more or less the parameters of the photovoltaic cells and panels. The power generated is also dependent on these two factors. So, the relation for photogenerated current, function of the irradiance, and temperature is the following [9]:(7)Iph=GGrefIph,ref+αscT,Tref,where G is irradiance, T represent the temperature, and αsc is the temperature coefficient of the current. The index ref is for the parameters at STC. The reverse saturation current can be calculated with Equation (8) [9, 10]:(8)Io=Io,refTTref3eq/KnEg,ref/Tref−Eg/Tr,where Eg is the energy bandgap. This value depends slightly on temperature [11, 12]:(9)Eg=Eg,ref1−0.0002677T−Tref.The ideality factor of diode depends slightly on the irradiance [13]. The temperature dependence can be written as Equation (10) [9]:(10)n=nrefTTref.The behaviour of the shunt resistance is inversely proportional to that of irradiance, the irradiance increasing as the shunt resistance decreases:(11)Rsh=Rsh,refGrefG.The dependence of the series resistance on temperature and irradiance is described by Equation (12). It decreases linearly with the increase in temperature and increases with the increase of irradiance; β is constant and is considered equal to 0.217 [11, 14].(12)Rsh=Rsh,refTTref1−βlnGGref.


## 3. Analytical and Metaheuristic Methods

Pillai and Rajasekar classified the methods to extract the parameters of the photovoltaic cells and panels in analytical, metaheuristic, and hybrid (consisting of those mentioned before) methods [8]. The analytical methods are based on formulas obtained using approximation and/or particular points on the I-V characteristic and some parts of the I-V characteristics. Multiobjective optimization problems were tough issues, but the development and use of metaheuristic algorithms in the last years led to solutions with a very good accuracy [15]. These metaheuristic algorithms were quickly adapted and used to solve the multimodal problem of the current-voltage dependence of the photovoltaic devices.


### 3.1. Analytical Methods

These methods were used to calculate the parameters of the photovoltaic cells since the 60s [13]. A lot of methods have been developed, especially for the SD model, but in the last years, they were developed for DD and even TD models. They can calculate one, more than one, or all parameters of the photovoltaic cells and panels. The several analytical methods are presented in Table 1.Table 1
The analytical methods.
MethodsParametersModelsRemarksComplexity of usageAccuracyRef.Analytical five-parameter methodIph,Io,n,Rs,RshSDUsing part ofI-V characteristic to determine the Rs and RshMediumHigh[16]TivanovIph,Io,n,Rs,RshSDUsing part ofI-V characteristic to determine the Rs and Rsh and Iph~IscMediumMedium[17]Ortiz-CondeIph,Io,n,Rs,RshSDUsing the CC function to calculate the equation coefficientsCV1,CI1,CV2andCI2LowMedium[18]Garrido-AlzarIph,Iod,Ior,nr,Rs,RshDDnr is considered 1, and four points (V,I) are usingMediumLow[19]Generalized arean,Rs,RshSDUsingIph~Isc, and the subgraphic area for three I-V characteristicsHighLow[20]AreaRsSDUsing subgraphic area forI-V characteristic and n=1LowLow[21]KaminskiIod,Ior,nr,Rs,RshDDThe parameters are determined in dark conditionsLowLow[22]Rs modelIod,Ior,nr,Rs,SDRsh is considered ∞MediumLow[23]L functionRs,RshSDUsing Lambert W functionHighMedium[24]Explicit method for the five parametersIph,Io,n,Rs,RshSDUsing two points forIsc zone and Voc zone to calculate the slopesMediumMedium[9]CotfasRs and sSDUsingI-V characteristic measured and the ideal oneMediumMedium[1]Modified five parametersIph,Io,n,Rs,RshSDUsing two empirically equations to calculateRs and RshLowHigh[25]ElkholyIph,Io,n,Rs,RshSDUsing a method based on the nonlinear least-squares algorithm; the parameters are calculated at different environmental conditionsMediumMedium[7]NdegwaIph,Io,n,Rs,RshSDn and Io are calculated firstly using Isc,Im,Vm,Voc, and Rs=0, Rsh≈∞; Rs and Rsh are then evaluated for different values of ns in the neighborhood of no (1≤n≤no)MediumMedium[26]TRDLAIph,Io,n,Rs,RshSDn is calculated firstly using the data provided by the manufacturer’s datasheet; the other four parameters are calculated using the trust-region-dogleg algorithmMediumMedium[27]BranoIph,Io,n,Rs,RshSDUsing five equations derived from Equation (1)MediumMedium[28]The complexity of usage and the accuracy of the method to extract the parameters of the photovoltaic cells or panels are two key indicators. Three-level ratings are used for each of them: low, medium, and high. For the complexity of usage, they mean as follows: low: simple formulas are used; medium: complex formulas, fitting and iterative procedure are necessary; high: the analytical method needs dedicated computational software [29]. For the accuracy, the level rating is the function of the statistical test [29, 30]. The rating for each method is shown in Table 1. Their results can be used by the manufacturers to choose the optimum method to characterize the photovoltaic cells.The analytical five-parameter method, 5P, is the most widely used of the analytical ones to extract the parameters of the photovoltaic cells. The first step is to calculate the slope around the open-circuit voltage to calculateRso, Equation (12), and the short circuit current to calculate Rsho, Equation (13). The slope calculation requires having many points on the I-V characteristic in the two regions. In case this is not possible and to reduce the complexity of the method, Rso and Rsho can be calculated using Equations (15) and (16), empirically obtained in modified five parameters 5Pm [25].(13)Rso=−dVdIV=Voc,(14)Rsh=Rsho=−dVdII=Isc,(15)Rso=0.002102+0.318070Rsm,Rsm=Voc−VmIm,(16)Rsho=−0.051914+2.505219Rsm,Rshm=VmIsc−Im,where Im and Vm represent the maximum power point coordinates.(17)n=Vm+RsoIm−VocVtlnIsc−Vm/Rsh−Im−lnIsc−Voc/Rsh+Im/Isc−Voc/Rsh,Io=Isc−VocRshexp−VocnVT,Rs=Rso−nVtIoexp−VocnVT,Iph=Isc1+RsRsh+IoexpIscRsnVT−1.


### 3.2. Metaheuristic Methods

The metaheuristic algorithms have been used to extract the parameters of the PV since the 2000s, when Jervase et al. used the genetic algorithms and the DD model to extract the seven parameters of the photovoltaic cell [31]. By using metaheuristic algorithms, all parameters of the photovoltaic cells and panels can be calculated. There are a lot of metaheuristic algorithms applied to extract the parameters of the photovoltaic cells and panels. The lower and upper values for the photovoltaic cells or panel parameters are necessary to be considered for the limitation of the global optimum search. Table 2 presents some of them, classified on family and on whether they are simple or hybrid [8]. The families of the algorithms presented are genetic algorithms (GA) [31–34], differential evolution (DE) [35–38], particle swarm optimization (PSO) [39–43], discretization [3, 44, 45], artificial bee colony (ABC) [46, 47], shuffled complex evolution [48, 49], simulated annealing (SA) [50, 51], flower pollination algorithm (FPA) [52, 53], harmony search (HS) [54], JAYA algorithm [55–57], teaching–learning-based optimization algorithm [58, 59], whale optimization algorithm [60, 61], and backtracking search algorithm [62, 63]. Additionally, the diode model is shown, computational time and the iteration number when these are given.Table 2
The metaheuristic algorithms.
Family algorithmsTypeModelsPVRange setComputational time/iterationsReferenceGeneticSimpleSD50 W panelPartially-/50[31]GeneticSimpleSD57 mm RTC France solar cellYes-/-[32]GA-RSimpleSD57 mm RTC France solar cell, mSi commercial photovoltaic cellPartially56 s/5000[33]Genetic GA-LSHybridSD57 mm RTC France solar cellYes-/-[34]Differential evolution DESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes12 s and 16 s/10000 and 20000[35]Differential evolution Rcr-IJADESimpleSD, DD57 mm RTC France solar cellYes33 s and 58 s/10000 and 20000[36]Penalty differential evolution P-DESimpleDDpSi-S75 and S115 mSi-SM55 and SQ150PC tin film-ST36 and ST40Partially42 s/500[37]Differential evolution with an individual-dependent mechanism IDESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/10000 and 20000[35]Linear population success-history-based adaptive DE L-SHADESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes35.4 and 62.66 s/10000 and 20000[35]Differential evolution DEIMHybridSD, DDKC120 PV moduleYes32 s/10000 and 20000[38]Particle swarm optimization CPSOSimpleSD57 mm RTC France solar cellYes-/4500[39]Particle swarm optimization VCPSOSimpleDD—No-/-[40]Particle swarm optimization NM-MPSOHybridSD, DD57 mm RTC France solar cellYes-/350000[41]Fractional chaotic ensemble particle swarm optimizer FC-EPSOHybridSD, DD57 mm RTC France solar cellYes11.5 s and 12 s/200[42]Chaotic heterogeneous comprehensive learning PSO C-HCLPSOHybridSD, DD57 mm RTC France solar cellYes204 s and 225 s/2000[43]Hybrid successive discretization algorithm HSDAHybridSD, DD57 mm RTC France solar cell,3×3 cm monocrystalline silicon photovoltaic cell, PWP201 photovoltaic panel, STP6-120/36, STM6-40/36, etc.Yes28 s and 46 s/4[44]Discretization SDASimpleSD57 mm RTC France solar cell,3×3 cm monocrystalline silicon photovoltaic cell, PWP201 photovoltaic panelYes142 s and 266 s/4[3]Discretization PSDASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel, Kyocera KC200GT photovoltaic panelYes28 s and 46 s/4[45]Artificial bee colony optimization ABCOSimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel, STM6-40/36Yes-/10000[46]Artificial bee colony optimization ABC-NMSHybridSD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/5000[47]Shuffled complex evolution ISCESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/5000 and 10000[48]Shuffled complex evolution-opposition-based learning ESCE-OBLHybridSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/5000 and 10000[49]Simulated annealing SASimpleSD, DD57 mm R.T.C France solar cell, Photowatt-PWP201—-/-[64]Simulated annealing LM-SAHybridSD57 mm RTC France solar cell—-/2050[50]Flower pollination FPASimpleSD, DD57 mm RTC France solar cell, Photowatt-PWP201Yes-/25000[51]Flower pollination BPFPAHybridSD, DD57 mm RTC France solar cellPartially-/20000[52]Harmony search HSSimpleSD, DD57 mm RTC France solar cell—-/5000[53]Innovative global harmony search IGHSSimpleSD, DD57 mm RTC France solar cell—-/5000[53]Pattern search PSSimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/-[54]JAYA algorithm IJAYASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/50000[55]Performance-guided JAYA algorithm PGJAYASimpleSD, DD57 mm R.T.C France solar cell, Photowatt-PWP201 KC200GT, SM55, thin-film Shell ST40Yes-/50000[56]Comprehensive learning JAYA algorithm CJAYASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/20000 and 48000[57]Teaching–learning-based optimization TLBOSimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/20000[58]Improved TLBO ITLBOSimpleSD, DD57 mm R.T.C France solar cell, PWP201 photovoltaic panel, STP6-120/36, STM6-40/36—5.95 s (30) and 6.60s (30)/50000[59]Whale optimization algorithm WOASimpleSD, DDKC200GTYes-/45000[60]Improved version of WOA IWOASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel, JAM6-60-295W-4BB, CS6U-320PYes-/100000[61]Multiple learning backtracking search algorithm MLBSASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes39 and 44 s/50000[62]BSA-Lévy flight (LFBSA)SimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/50000[63]One of the new and the best algorithms, HSDA [44], is used against the modified analytical method to forecast the I-V characteristic and maximum power generated. The HSDA algorithm is an improved version of the SDA algorithm [3]. It is a hybrid one. The first algorithm used is one of the existent algorithms and gives a solution for SDA. A vicinity is considered around it, and the parameters can be extracted with very good accuracy using SDA for this vicinity. The flow chart of the HSDA algorithm is presented in Figure 2.Figure 2
HSDA algorithm flow chart [44].


## 3.1. Analytical Methods

These methods were used to calculate the parameters of the photovoltaic cells since the 60s [13]. A lot of methods have been developed, especially for the SD model, but in the last years, they were developed for DD and even TD models. They can calculate one, more than one, or all parameters of the photovoltaic cells and panels. The several analytical methods are presented in Table 1.Table 1
The analytical methods.
MethodsParametersModelsRemarksComplexity of usageAccuracyRef.Analytical five-parameter methodIph,Io,n,Rs,RshSDUsing part ofI-V characteristic to determine the Rs and RshMediumHigh[16]TivanovIph,Io,n,Rs,RshSDUsing part ofI-V characteristic to determine the Rs and Rsh and Iph~IscMediumMedium[17]Ortiz-CondeIph,Io,n,Rs,RshSDUsing the CC function to calculate the equation coefficientsCV1,CI1,CV2andCI2LowMedium[18]Garrido-AlzarIph,Iod,Ior,nr,Rs,RshDDnr is considered 1, and four points (V,I) are usingMediumLow[19]Generalized arean,Rs,RshSDUsingIph~Isc, and the subgraphic area for three I-V characteristicsHighLow[20]AreaRsSDUsing subgraphic area forI-V characteristic and n=1LowLow[21]KaminskiIod,Ior,nr,Rs,RshDDThe parameters are determined in dark conditionsLowLow[22]Rs modelIod,Ior,nr,Rs,SDRsh is considered ∞MediumLow[23]L functionRs,RshSDUsing Lambert W functionHighMedium[24]Explicit method for the five parametersIph,Io,n,Rs,RshSDUsing two points forIsc zone and Voc zone to calculate the slopesMediumMedium[9]CotfasRs and sSDUsingI-V characteristic measured and the ideal oneMediumMedium[1]Modified five parametersIph,Io,n,Rs,RshSDUsing two empirically equations to calculateRs and RshLowHigh[25]ElkholyIph,Io,n,Rs,RshSDUsing a method based on the nonlinear least-squares algorithm; the parameters are calculated at different environmental conditionsMediumMedium[7]NdegwaIph,Io,n,Rs,RshSDn and Io are calculated firstly using Isc,Im,Vm,Voc, and Rs=0, Rsh≈∞; Rs and Rsh are then evaluated for different values of ns in the neighborhood of no (1≤n≤no)MediumMedium[26]TRDLAIph,Io,n,Rs,RshSDn is calculated firstly using the data provided by the manufacturer’s datasheet; the other four parameters are calculated using the trust-region-dogleg algorithmMediumMedium[27]BranoIph,Io,n,Rs,RshSDUsing five equations derived from Equation (1)MediumMedium[28]The complexity of usage and the accuracy of the method to extract the parameters of the photovoltaic cells or panels are two key indicators. Three-level ratings are used for each of them: low, medium, and high. For the complexity of usage, they mean as follows: low: simple formulas are used; medium: complex formulas, fitting and iterative procedure are necessary; high: the analytical method needs dedicated computational software [29]. For the accuracy, the level rating is the function of the statistical test [29, 30]. The rating for each method is shown in Table 1. Their results can be used by the manufacturers to choose the optimum method to characterize the photovoltaic cells.The analytical five-parameter method, 5P, is the most widely used of the analytical ones to extract the parameters of the photovoltaic cells. The first step is to calculate the slope around the open-circuit voltage to calculateRso, Equation (12), and the short circuit current to calculate Rsho, Equation (13). The slope calculation requires having many points on the I-V characteristic in the two regions. In case this is not possible and to reduce the complexity of the method, Rso and Rsho can be calculated using Equations (15) and (16), empirically obtained in modified five parameters 5Pm [25].(13)Rso=−dVdIV=Voc,(14)Rsh=Rsho=−dVdII=Isc,(15)Rso=0.002102+0.318070Rsm,Rsm=Voc−VmIm,(16)Rsho=−0.051914+2.505219Rsm,Rshm=VmIsc−Im,where Im and Vm represent the maximum power point coordinates.(17)n=Vm+RsoIm−VocVtlnIsc−Vm/Rsh−Im−lnIsc−Voc/Rsh+Im/Isc−Voc/Rsh,Io=Isc−VocRshexp−VocnVT,Rs=Rso−nVtIoexp−VocnVT,Iph=Isc1+RsRsh+IoexpIscRsnVT−1.


## 3.2. Metaheuristic Methods

The metaheuristic algorithms have been used to extract the parameters of the PV since the 2000s, when Jervase et al. used the genetic algorithms and the DD model to extract the seven parameters of the photovoltaic cell [31]. By using metaheuristic algorithms, all parameters of the photovoltaic cells and panels can be calculated. There are a lot of metaheuristic algorithms applied to extract the parameters of the photovoltaic cells and panels. The lower and upper values for the photovoltaic cells or panel parameters are necessary to be considered for the limitation of the global optimum search. Table 2 presents some of them, classified on family and on whether they are simple or hybrid [8]. The families of the algorithms presented are genetic algorithms (GA) [31–34], differential evolution (DE) [35–38], particle swarm optimization (PSO) [39–43], discretization [3, 44, 45], artificial bee colony (ABC) [46, 47], shuffled complex evolution [48, 49], simulated annealing (SA) [50, 51], flower pollination algorithm (FPA) [52, 53], harmony search (HS) [54], JAYA algorithm [55–57], teaching–learning-based optimization algorithm [58, 59], whale optimization algorithm [60, 61], and backtracking search algorithm [62, 63]. Additionally, the diode model is shown, computational time and the iteration number when these are given.Table 2
The metaheuristic algorithms.
Family algorithmsTypeModelsPVRange setComputational time/iterationsReferenceGeneticSimpleSD50 W panelPartially-/50[31]GeneticSimpleSD57 mm RTC France solar cellYes-/-[32]GA-RSimpleSD57 mm RTC France solar cell, mSi commercial photovoltaic cellPartially56 s/5000[33]Genetic GA-LSHybridSD57 mm RTC France solar cellYes-/-[34]Differential evolution DESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes12 s and 16 s/10000 and 20000[35]Differential evolution Rcr-IJADESimpleSD, DD57 mm RTC France solar cellYes33 s and 58 s/10000 and 20000[36]Penalty differential evolution P-DESimpleDDpSi-S75 and S115 mSi-SM55 and SQ150PC tin film-ST36 and ST40Partially42 s/500[37]Differential evolution with an individual-dependent mechanism IDESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/10000 and 20000[35]Linear population success-history-based adaptive DE L-SHADESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes35.4 and 62.66 s/10000 and 20000[35]Differential evolution DEIMHybridSD, DDKC120 PV moduleYes32 s/10000 and 20000[38]Particle swarm optimization CPSOSimpleSD57 mm RTC France solar cellYes-/4500[39]Particle swarm optimization VCPSOSimpleDD—No-/-[40]Particle swarm optimization NM-MPSOHybridSD, DD57 mm RTC France solar cellYes-/350000[41]Fractional chaotic ensemble particle swarm optimizer FC-EPSOHybridSD, DD57 mm RTC France solar cellYes11.5 s and 12 s/200[42]Chaotic heterogeneous comprehensive learning PSO C-HCLPSOHybridSD, DD57 mm RTC France solar cellYes204 s and 225 s/2000[43]Hybrid successive discretization algorithm HSDAHybridSD, DD57 mm RTC France solar cell,3×3 cm monocrystalline silicon photovoltaic cell, PWP201 photovoltaic panel, STP6-120/36, STM6-40/36, etc.Yes28 s and 46 s/4[44]Discretization SDASimpleSD57 mm RTC France solar cell,3×3 cm monocrystalline silicon photovoltaic cell, PWP201 photovoltaic panelYes142 s and 266 s/4[3]Discretization PSDASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel, Kyocera KC200GT photovoltaic panelYes28 s and 46 s/4[45]Artificial bee colony optimization ABCOSimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel, STM6-40/36Yes-/10000[46]Artificial bee colony optimization ABC-NMSHybridSD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/5000[47]Shuffled complex evolution ISCESimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/5000 and 10000[48]Shuffled complex evolution-opposition-based learning ESCE-OBLHybridSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/5000 and 10000[49]Simulated annealing SASimpleSD, DD57 mm R.T.C France solar cell, Photowatt-PWP201—-/-[64]Simulated annealing LM-SAHybridSD57 mm RTC France solar cell—-/2050[50]Flower pollination FPASimpleSD, DD57 mm RTC France solar cell, Photowatt-PWP201Yes-/25000[51]Flower pollination BPFPAHybridSD, DD57 mm RTC France solar cellPartially-/20000[52]Harmony search HSSimpleSD, DD57 mm RTC France solar cell—-/5000[53]Innovative global harmony search IGHSSimpleSD, DD57 mm RTC France solar cell—-/5000[53]Pattern search PSSimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/-[54]JAYA algorithm IJAYASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/50000[55]Performance-guided JAYA algorithm PGJAYASimpleSD, DD57 mm R.T.C France solar cell, Photowatt-PWP201 KC200GT, SM55, thin-film Shell ST40Yes-/50000[56]Comprehensive learning JAYA algorithm CJAYASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/20000 and 48000[57]Teaching–learning-based optimization TLBOSimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel—-/20000[58]Improved TLBO ITLBOSimpleSD, DD57 mm R.T.C France solar cell, PWP201 photovoltaic panel, STP6-120/36, STM6-40/36—5.95 s (30) and 6.60s (30)/50000[59]Whale optimization algorithm WOASimpleSD, DDKC200GTYes-/45000[60]Improved version of WOA IWOASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panel, JAM6-60-295W-4BB, CS6U-320PYes-/100000[61]Multiple learning backtracking search algorithm MLBSASimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes39 and 44 s/50000[62]BSA-Lévy flight (LFBSA)SimpleSD, DD57 mm RTC France solar cell, PWP201 photovoltaic panelYes-/50000[63]One of the new and the best algorithms, HSDA [44], is used against the modified analytical method to forecast the I-V characteristic and maximum power generated. The HSDA algorithm is an improved version of the SDA algorithm [3]. It is a hybrid one. The first algorithm used is one of the existent algorithms and gives a solution for SDA. A vicinity is considered around it, and the parameters can be extracted with very good accuracy using SDA for this vicinity. The flow chart of the HSDA algorithm is presented in Figure 2.Figure 2
HSDA algorithm flow chart [44].


## 4. Results and Discussion

### 4.1. Analytical vs. Metaheuristic

The representative analytical method—modified five parameters—which has low difficulty to use and high accuracy, for SD model, it is compared with the best metaheuristic algorithms for each family, for several photovoltaic cells and panels when the results for the parameters are available. The parameter values for the SD model given in the references are used to calculate the statistical tests AE, RMSE, andR2 used for comparison. Additionally, the GA which will be used for forecast is considered.


#### 4.1.1. RTC France Solar Cell

The first comparison is made for the RTC France solar cell, one of the widely used by researchers to prove the performance of the developed algorithms to extract the parameters of the photovoltaic cells and panels. The result of the parameters and the statistical tests, RMSE, AE, andR2, are presented in Table 3.Table 3
The parameters and statistical tests for RTC cell.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]0.76077580.3230165321.481182320.0363770853.7145208859.8602E−040.02152770.9999893Rcr-IJADE [36]0.760775530.32302081.48118360.0363770953.7185259.86021E−40.02152710.9999893C-HCLPSO [43]0.760790.310621.47710.03654852.8851.1201E−030.02091150.999986ABC–NMS [47]0.7607760.3230211.4811840.03637753.7185219.86023E−040,0215330.9999893ESCE-OBL [49]0.760780.323021.481180.0363853.71859.8602E−040.02152690.9999893LM-SA [50]0.760780.318491.479760.0364353.326449.8646E−040.02151040.9999892FPA [51]0.760790.310621.477070.0365552.87711.214E−030.02167880.9999837IGHS [53]0.760770.343511.487400.0361353.28451.033E−030.02120250.9999882CLJAYA [57]0.760780.32302081.4811840.036377153.7185219.8603E−040.02154150.9999892ITLBO [59]0.76080.32301.48120.036453.71859.9161E−040.0218090.9999891IWOA [61]0.76080.32321.48120.036453.73179.9486E−040.0211310.9999891MLBSA [62]0.76080.323021.48120.036453.71859.8969E−040.02172160.99998925Pm [3]0.76120.19661.430.04295.288.674E−030.1596980.999086GA [33]0.76190.80871.57510.029942.37290.019080.2776730.995997In the case of the RTC France solar cell, the 5Pm method has the RMSE, which is widely used to measure the performance of the methods, higher than the ones obtained for the metaheuristic algorithms with the exception of the GA. Also, for the AE andR2, the values are higher.To have a complete image of the results obtained using different methods, Figure3(a) presents the absolute current error. The 5Pm method and GA algorithm overestimate or underestimate the current around the open-circuit voltage point, where the other algorithms calculate the current better. The reverse saturation current extracted with the GA algorithm is more than two times higher than that calculated with HSDA. The parasitic resistances, the series resistance, and the shunt resistance present also a high variation. Figure 3(b) shows the behaviour of the AE around the open-circuit voltage region. The AE values for the C-HCLPSO and FPA algorithms alternate around the AE average of the other algorithms considered, having high values for some regions and very small for other regions.Figure 3
(a) The absolute current error (AE) for RTC France solar cell; (b) AE in open-circuit voltage region.
(a)(b)


#### 4.1.2. Commercial Monocrystalline Silicon Photovoltaic Cell

There are three methods to extract parameters of mSi commercial photovoltaic cell. The 5Pm analytical method gives the weakest results for all statistical tests, Table4. If in the case of the RTC photovoltaic cell, the RMSE obtained using the 5Pm method is almost ten times higher; for the mSi photovoltaic cell, the RMSE is 4.5 times higher, but the GA algorithm significantly improves its performance.Table 4
The parameters and statistical tests for mSi commercial photovoltaic cell.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]0.425753160.5162416131.679334060.0913289899.0759801765.63098E−040.01426480.99998535Pm [3]0.42550.306455671.6183110.10352224145.2222.25639E−030.0474920.99976GA [33]0.42568820.83833111.739260.0859435123.36596.9741E−040.01633540.99997These changes in the performance of the methods can be easily observed in Figure4. The performance of the GA algorithm is substantially improved for the region around the open circuit point, while the 5Pm method shows weakness in this region.Figure 4
The absolute current error for mSi commercial photovoltaic cell.


#### 4.1.3. PWP201 Photovoltaic Panel

Analyzing the results obtained for RMSE in the case of the PWP201 photovoltaic panel, RMSE and AE obtained with the 5Pm method are 1.65 times higher than the ones obtained with the HSDA algorithm. There are three algorithms with the best values for all three statistical tests, HSDA, Rcr-IJADE, and CLJAYA, Table 5.Table 5
The parameters and statistical tests for PWP201 photovoltaic panel.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]1.03051430.34822630448.6428351.201271981.2012712.42507E−030.04892370.99997Rcr-IJADE [36]1.03051433.482262948.6428351.201271981.982162.42507E−030.04892370.99997ABC–NMS [47]1.030513.4822648.6431.20127981.9822.42518E−030.04894540.99997FPA [51]1.0320913.04753848.131281.217583811.37212.57361E–030.05337460.99996CLJAYA [57]1.0305143.482262848.642831.201271981.9822792.42507E−030.04892270.99997ITLBO [59]1.03053.482348.64281.2013981.98232.42519E−030.04888780.99997IWOA [61]1.03053.471748.63131.2016978.67712.42523E−030.04889330.99997MLBSA [62]1.03053.482348.64281.20163981.98232.42561E−030.048780.999975Pm [3]1.0343.57148.711.2061123.004.019E–030.08335220.99991The absolute current errors for PWP201 photovoltaic panel are under 0.01 (A), having a uniform distribution, but keeping the high values in the open-circuit voltage region, Figure5(a). Although, in this case, the methods estimate the current without high difference for certain voltages in comparison with the ones measured, the RMSE and AE have high values. The AE for the PWP201 photovoltaic panel around the open-circuit voltage is higher for the FPA and 5Pm methods. The other algorithms considered have the same behaviour, Figure 5(b).Figure 5
(a) The absolute current error (AE) for PWP201 photovoltaic panel; (b) AE in open-circuit voltage region.
(a)(b)


#### 4.1.4. STM6-40 Photovoltaic Panel

The 5Pm method has the statistical test high values, Table6. The (V,I) pairs of the STM6-40 photovoltaic panel are not uniformly distributed. There are very few points in the open-circuit voltage region [46], which leads to poorer results in this case. The value of the coefficient of determination confirms this issue.Table 6
The parameters and statistical tests for the STM6-40 photovoltaic panel.
AlgorithmIph (A)Io (μA)nRs (Ω)RshΩRMSEAER2HSDA [44]1.66390477991.738654397854.7308990.153855932543.418349851.72981E−030.02190350.99997731Rcr-IJADE [36]1.66391.738754.73080.1548573.41881.73428E−030.02161480.99997719ITLBO [59]1.66391.738754.73080.1548573.41881.73428E−030.02161480.999977195Pm [3]1.66362.6541E−433.35340.9121898.163.53507E–020.5409480.98999793The best results are obtained for the HSDA algorithm for RSME andR2. The plot of the absolute current errors, Figure 6(a), shows the weakness of the 5Pm method in the region around the open-circuit voltage. For some points, the current calculated with the 5Pm method is twenty times higher than the ones calculated with the HSDA algorithm.Figure 6
(a) The absolute current error (AE) for STM6-40 photovoltaic panel; (b) AE in open-circuit voltage region.
(a)(b)The behaviour of the AE around the open-circuit voltage is similar for the ITLBO and the Rcr-IJADE algorithms, Figure 6(b). The AE for the HSDA algorithm has some small variations.


#### 4.1.5. Commercial Monocrystalline Silicon Photovoltaic Panel

The statistical tests for commercial mSi photovoltaic panel are presented in Table7.Table 7
The parameters and statistical tests for mSi photovoltaic panel.
AlgorithmIph (A)Io (μA)nRs (Ω)Rsh (Ω)RMSEAER2HSDA [44]1.2242064.677E−418.3649940.144071544.3617242.77734E−031.939260.99956777GA [33]1.2230824.988143E−320.52890.021472921765.3884.96271E−032.614560.99845Pm [3]1.2240.334E−318.020.1341242.916.21773E–032.424160.9978The shape of the absolute current error curves is the same in the case of the mSi photovoltaic panel. The highest value for the AE is again in the region of the open-circuit voltage, Figure7.Figure 7
The absolute current error for mSi photovoltaic panel.The coefficient of determination for all photovoltaic cells and panels is very good, less for the GA algorithm in case of RTC photovoltaic cell and STM6-40 photovoltaic panel.


### 4.2. Forecast Comparison

TwoI-V characteristics are measured for the commercial monocrystalline photovoltaic panel for the irradiance of 700 W/m2 and 983 W/m2. The measurements were made on the roof of the university building [3].TheI-V characteristic measured at 700 W/m2 was used to extract the parameters of the mSi photovoltaic panel with three methods HSDA algorithm, GA algorithm, and 5Pm analytical method. These parameters are considered as Iph,ref,Io,ref,nref,Rs,ref,andRsh,ref. The parameters of the mSi photovoltaic panel are calculated for 983 W/m2 irradiance, using Equations (7)–(11). These values are used to compare the measured I-V characteristic for 983 W/m2 in real conditions with the ones forecasted with the five parameters calculated with Equations (7)–(11). The temperature coefficient of the current was determined, and it is 15.76 mA/°C.The root mean square error for the forecastedI-V characteristic of the mSi photovoltaic panel is presented in Table 8 in comparison with one for the I-V characteristic measured. The RMSE f obtained when the parameters are calculated with Equations (7)–(11) is worse in the case of the HSDA and GA algorithms than the RMSE m obtained using the extracted parameters with those algorithms for the measured characteristic. In case of the 5Pm methods, the results are improved.Table 8
The statistical tests for the forecastI-V characteristic.
AlgorithmRMSEfRMSEmHSDA [44]5.30433E–033.234E–03GA [33]1.63618E–026.707E–035Pm [3]6.23956E–032.86E–02The absolute current error calculated for the forecastI-V characteristic in comparison with the ones measured is presented in Figure 8(a). The highest values for AE are in the region of the open-circuit voltage. The behaviour of the curves obtained with parameters calculated with Equations (7)–(11) for HSDA and 5Pm methods is very similar, while the ones obtained with GA algorithms have an accentuated increase in the open-circuit voltage region.Figure 8
(a) The absolute current error (AE) for the forecastedI-V characteristic for photovoltaic panel; (b) AE in the voltage which corresponds to maximum power region.
(a)(b)Figure8(b) shows the behaviour of the absolute current error around the maximum voltage, Vm, which is the voltage coordinate for the maximum power point. The best results are obtained for the 5Pm method for all regions considered.The comparison between maximum power generated,Pmax,m, the mSi panel calculated from real measurements and ones forecasted, Pmax,f, using the extracted parameters with the three methods, Table 9, shows that the best estimation is for the maximum power estimated with parameters extracted with the 5Pm method.Table 9
ThePmax forecasted and measured.
AlgorithmHSDA [44]GA [33]5Pm [3]MeasuredPmax (W)13.80713.79513.80913.875Pmax,f/Pmax,m (%)99.50998.4599.52By analyzing the results obtained through comparison between the analytic method and metaheuristic algorithms, it can be concluded that the 5Pm method can be used to extract the parameters of the photovoltaic cells and panels. Additionally, the analytical method can be used to estimate theI-V characteristics and the power generated using the parameters given by the producers. It can be used due to the advantages which are presented in Table 10.Table 10
Advantages and disadvantages.
MethodAdvantagesDisadvantagesAnalytical(i) Using the 5Pm method is very easy, theRso and Rsh are calculated with the empirical equations [25](ii) Very easy to implement on the computer for all software which allows calculation(iii) The necessary time to extract the parameters is very short for the 5Pm method (duration is under ms)(iv) Does not require a powerful or dedicated PC, which leads to the small cost of the PC, and the used software can be a common one(i) The accuracy of the parameters extracted decreases due to approximations used [8](ii) If some points of the I-V characteristic are used, the equation system obtained is nonlinear and it is necessary to use software that allows solving this (this can be costly)(iii) The complexity increases when the two diode model is used (seven equations are needed)Metaheuristic(i) The accuracy is higher than that of the analytical methods(ii) The number of iterations can be reduced (for example, for the ISCE algorithm, the iteration number is 5000)(iii) The algorithms can be easily adapted for the two diode model, but the computational time will increase(iv) Using algorithms as grey wolf, prey predator, and fire fly optimization, the performance of the parameters extraction can be improved [8](v) GA algorithm family can easy be part of the hybrid algorithm [65](vi) DE algorithm family has a high convergence(vii) PSO can be improved if it is used together with Nelder-Mead methods, this hybridisation can reduce the computational resources [65](viii) SDA family has very good performance, and it can be easily used with other algorithms in hybrid structure(ix) Improved version of SCE has performance comparable with HSDA(i) The computational time can be high, it depends on the PC power and the metaheuristic algorithm (the number of the iterations and the complexity), there are algorithms with over 100000 iterations or even 350000-NM-MPSO algorithm [41](ii) The cost of the PC can be high and the dedicated software must be used(iii) The performance of the most metaheuristic algorithms depend by the initial parameter range(iv) Requires very good knowledge for implementation(v) GA, SCE, TLBO, WOA, and BSA algorithms have a slow convergence(vi) DE and PSO algorithms can converge prematurely(vii) The computational time is high in case of the PSO algorithms(viii) The computational time is relatively high, but using parallelization, it can be reduced; in the case presented, the time was reduced 6 times


## 4.1. Analytical vs. Metaheuristic

The representative analytical method—modified five parameters—which has low difficulty to use and high accuracy, for SD model, it is compared with the best metaheuristic algorithms for each family, for several photovoltaic cells and panels when the results for the parameters are available. The parameter values for the SD model given in the references are used to calculate the statistical tests AE, RMSE, andR2 used for comparison. Additionally, the GA which will be used for forecast is considered.


### 4.1.1. RTC France Solar Cell

The first comparison is made for the RTC France solar cell, one of the widely used by researchers to prove the performance of the developed algorithms to extract the parameters of the photovoltaic cells and panels. The result of the parameters and the statistical tests, RMSE, AE, andR2, are presented in Table 3.Table 3
The parameters and statistical tests for RTC cell.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]0.76077580.3230165321.481182320.0363770853.7145208859.8602E−040.02152770.9999893Rcr-IJADE [36]0.760775530.32302081.48118360.0363770953.7185259.86021E−40.02152710.9999893C-HCLPSO [43]0.760790.310621.47710.03654852.8851.1201E−030.02091150.999986ABC–NMS [47]0.7607760.3230211.4811840.03637753.7185219.86023E−040,0215330.9999893ESCE-OBL [49]0.760780.323021.481180.0363853.71859.8602E−040.02152690.9999893LM-SA [50]0.760780.318491.479760.0364353.326449.8646E−040.02151040.9999892FPA [51]0.760790.310621.477070.0365552.87711.214E−030.02167880.9999837IGHS [53]0.760770.343511.487400.0361353.28451.033E−030.02120250.9999882CLJAYA [57]0.760780.32302081.4811840.036377153.7185219.8603E−040.02154150.9999892ITLBO [59]0.76080.32301.48120.036453.71859.9161E−040.0218090.9999891IWOA [61]0.76080.32321.48120.036453.73179.9486E−040.0211310.9999891MLBSA [62]0.76080.323021.48120.036453.71859.8969E−040.02172160.99998925Pm [3]0.76120.19661.430.04295.288.674E−030.1596980.999086GA [33]0.76190.80871.57510.029942.37290.019080.2776730.995997In the case of the RTC France solar cell, the 5Pm method has the RMSE, which is widely used to measure the performance of the methods, higher than the ones obtained for the metaheuristic algorithms with the exception of the GA. Also, for the AE andR2, the values are higher.To have a complete image of the results obtained using different methods, Figure3(a) presents the absolute current error. The 5Pm method and GA algorithm overestimate or underestimate the current around the open-circuit voltage point, where the other algorithms calculate the current better. The reverse saturation current extracted with the GA algorithm is more than two times higher than that calculated with HSDA. The parasitic resistances, the series resistance, and the shunt resistance present also a high variation. Figure 3(b) shows the behaviour of the AE around the open-circuit voltage region. The AE values for the C-HCLPSO and FPA algorithms alternate around the AE average of the other algorithms considered, having high values for some regions and very small for other regions.Figure 3
(a) The absolute current error (AE) for RTC France solar cell; (b) AE in open-circuit voltage region.
(a)(b)


### 4.1.2. Commercial Monocrystalline Silicon Photovoltaic Cell

There are three methods to extract parameters of mSi commercial photovoltaic cell. The 5Pm analytical method gives the weakest results for all statistical tests, Table4. If in the case of the RTC photovoltaic cell, the RMSE obtained using the 5Pm method is almost ten times higher; for the mSi photovoltaic cell, the RMSE is 4.5 times higher, but the GA algorithm significantly improves its performance.Table 4
The parameters and statistical tests for mSi commercial photovoltaic cell.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]0.425753160.5162416131.679334060.0913289899.0759801765.63098E−040.01426480.99998535Pm [3]0.42550.306455671.6183110.10352224145.2222.25639E−030.0474920.99976GA [33]0.42568820.83833111.739260.0859435123.36596.9741E−040.01633540.99997These changes in the performance of the methods can be easily observed in Figure4. The performance of the GA algorithm is substantially improved for the region around the open circuit point, while the 5Pm method shows weakness in this region.Figure 4
The absolute current error for mSi commercial photovoltaic cell.


### 4.1.3. PWP201 Photovoltaic Panel

Analyzing the results obtained for RMSE in the case of the PWP201 photovoltaic panel, RMSE and AE obtained with the 5Pm method are 1.65 times higher than the ones obtained with the HSDA algorithm. There are three algorithms with the best values for all three statistical tests, HSDA, Rcr-IJADE, and CLJAYA, Table 5.Table 5
The parameters and statistical tests for PWP201 photovoltaic panel.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]1.03051430.34822630448.6428351.201271981.2012712.42507E−030.04892370.99997Rcr-IJADE [36]1.03051433.482262948.6428351.201271981.982162.42507E−030.04892370.99997ABC–NMS [47]1.030513.4822648.6431.20127981.9822.42518E−030.04894540.99997FPA [51]1.0320913.04753848.131281.217583811.37212.57361E–030.05337460.99996CLJAYA [57]1.0305143.482262848.642831.201271981.9822792.42507E−030.04892270.99997ITLBO [59]1.03053.482348.64281.2013981.98232.42519E−030.04888780.99997IWOA [61]1.03053.471748.63131.2016978.67712.42523E−030.04889330.99997MLBSA [62]1.03053.482348.64281.20163981.98232.42561E−030.048780.999975Pm [3]1.0343.57148.711.2061123.004.019E–030.08335220.99991The absolute current errors for PWP201 photovoltaic panel are under 0.01 (A), having a uniform distribution, but keeping the high values in the open-circuit voltage region, Figure5(a). Although, in this case, the methods estimate the current without high difference for certain voltages in comparison with the ones measured, the RMSE and AE have high values. The AE for the PWP201 photovoltaic panel around the open-circuit voltage is higher for the FPA and 5Pm methods. The other algorithms considered have the same behaviour, Figure 5(b).Figure 5
(a) The absolute current error (AE) for PWP201 photovoltaic panel; (b) AE in open-circuit voltage region.
(a)(b)


### 4.1.4. STM6-40 Photovoltaic Panel

The 5Pm method has the statistical test high values, Table6. The (V,I) pairs of the STM6-40 photovoltaic panel are not uniformly distributed. There are very few points in the open-circuit voltage region [46], which leads to poorer results in this case. The value of the coefficient of determination confirms this issue.Table 6
The parameters and statistical tests for the STM6-40 photovoltaic panel.
AlgorithmIph (A)Io (μA)nRs (Ω)RshΩRMSEAER2HSDA [44]1.66390477991.738654397854.7308990.153855932543.418349851.72981E−030.02190350.99997731Rcr-IJADE [36]1.66391.738754.73080.1548573.41881.73428E−030.02161480.99997719ITLBO [59]1.66391.738754.73080.1548573.41881.73428E−030.02161480.999977195Pm [3]1.66362.6541E−433.35340.9121898.163.53507E–020.5409480.98999793The best results are obtained for the HSDA algorithm for RSME andR2. The plot of the absolute current errors, Figure 6(a), shows the weakness of the 5Pm method in the region around the open-circuit voltage. For some points, the current calculated with the 5Pm method is twenty times higher than the ones calculated with the HSDA algorithm.Figure 6
(a) The absolute current error (AE) for STM6-40 photovoltaic panel; (b) AE in open-circuit voltage region.
(a)(b)The behaviour of the AE around the open-circuit voltage is similar for the ITLBO and the Rcr-IJADE algorithms, Figure 6(b). The AE for the HSDA algorithm has some small variations.


### 4.1.5. Commercial Monocrystalline Silicon Photovoltaic Panel

The statistical tests for commercial mSi photovoltaic panel are presented in Table7.Table 7
The parameters and statistical tests for mSi photovoltaic panel.
AlgorithmIph (A)Io (μA)nRs (Ω)Rsh (Ω)RMSEAER2HSDA [44]1.2242064.677E−418.3649940.144071544.3617242.77734E−031.939260.99956777GA [33]1.2230824.988143E−320.52890.021472921765.3884.96271E−032.614560.99845Pm [3]1.2240.334E−318.020.1341242.916.21773E–032.424160.9978The shape of the absolute current error curves is the same in the case of the mSi photovoltaic panel. The highest value for the AE is again in the region of the open-circuit voltage, Figure7.Figure 7
The absolute current error for mSi photovoltaic panel.The coefficient of determination for all photovoltaic cells and panels is very good, less for the GA algorithm in case of RTC photovoltaic cell and STM6-40 photovoltaic panel.


## 4.1.1. RTC France Solar Cell

The first comparison is made for the RTC France solar cell, one of the widely used by researchers to prove the performance of the developed algorithms to extract the parameters of the photovoltaic cells and panels. The result of the parameters and the statistical tests, RMSE, AE, andR2, are presented in Table 3.Table 3
The parameters and statistical tests for RTC cell.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]0.76077580.3230165321.481182320.0363770853.7145208859.8602E−040.02152770.9999893Rcr-IJADE [36]0.760775530.32302081.48118360.0363770953.7185259.86021E−40.02152710.9999893C-HCLPSO [43]0.760790.310621.47710.03654852.8851.1201E−030.02091150.999986ABC–NMS [47]0.7607760.3230211.4811840.03637753.7185219.86023E−040,0215330.9999893ESCE-OBL [49]0.760780.323021.481180.0363853.71859.8602E−040.02152690.9999893LM-SA [50]0.760780.318491.479760.0364353.326449.8646E−040.02151040.9999892FPA [51]0.760790.310621.477070.0365552.87711.214E−030.02167880.9999837IGHS [53]0.760770.343511.487400.0361353.28451.033E−030.02120250.9999882CLJAYA [57]0.760780.32302081.4811840.036377153.7185219.8603E−040.02154150.9999892ITLBO [59]0.76080.32301.48120.036453.71859.9161E−040.0218090.9999891IWOA [61]0.76080.32321.48120.036453.73179.9486E−040.0211310.9999891MLBSA [62]0.76080.323021.48120.036453.71859.8969E−040.02172160.99998925Pm [3]0.76120.19661.430.04295.288.674E−030.1596980.999086GA [33]0.76190.80871.57510.029942.37290.019080.2776730.995997In the case of the RTC France solar cell, the 5Pm method has the RMSE, which is widely used to measure the performance of the methods, higher than the ones obtained for the metaheuristic algorithms with the exception of the GA. Also, for the AE andR2, the values are higher.To have a complete image of the results obtained using different methods, Figure3(a) presents the absolute current error. The 5Pm method and GA algorithm overestimate or underestimate the current around the open-circuit voltage point, where the other algorithms calculate the current better. The reverse saturation current extracted with the GA algorithm is more than two times higher than that calculated with HSDA. The parasitic resistances, the series resistance, and the shunt resistance present also a high variation. Figure 3(b) shows the behaviour of the AE around the open-circuit voltage region. The AE values for the C-HCLPSO and FPA algorithms alternate around the AE average of the other algorithms considered, having high values for some regions and very small for other regions.Figure 3
(a) The absolute current error (AE) for RTC France solar cell; (b) AE in open-circuit voltage region.
(a)(b)


## 4.1.2. Commercial Monocrystalline Silicon Photovoltaic Cell

There are three methods to extract parameters of mSi commercial photovoltaic cell. The 5Pm analytical method gives the weakest results for all statistical tests, Table4. If in the case of the RTC photovoltaic cell, the RMSE obtained using the 5Pm method is almost ten times higher; for the mSi photovoltaic cell, the RMSE is 4.5 times higher, but the GA algorithm significantly improves its performance.Table 4
The parameters and statistical tests for mSi commercial photovoltaic cell.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]0.425753160.5162416131.679334060.0913289899.0759801765.63098E−040.01426480.99998535Pm [3]0.42550.306455671.6183110.10352224145.2222.25639E−030.0474920.99976GA [33]0.42568820.83833111.739260.0859435123.36596.9741E−040.01633540.99997These changes in the performance of the methods can be easily observed in Figure4. The performance of the GA algorithm is substantially improved for the region around the open circuit point, while the 5Pm method shows weakness in this region.Figure 4
The absolute current error for mSi commercial photovoltaic cell.


## 4.1.3. PWP201 Photovoltaic Panel

Analyzing the results obtained for RMSE in the case of the PWP201 photovoltaic panel, RMSE and AE obtained with the 5Pm method are 1.65 times higher than the ones obtained with the HSDA algorithm. There are three algorithms with the best values for all three statistical tests, HSDA, Rcr-IJADE, and CLJAYA, Table 5.Table 5
The parameters and statistical tests for PWP201 photovoltaic panel.
AlgorithmIph (A)Io (μA)nRsΩRshΩRMSEAER2HSDA [44]1.03051430.34822630448.6428351.201271981.2012712.42507E−030.04892370.99997Rcr-IJADE [36]1.03051433.482262948.6428351.201271981.982162.42507E−030.04892370.99997ABC–NMS [47]1.030513.4822648.6431.20127981.9822.42518E−030.04894540.99997FPA [51]1.0320913.04753848.131281.217583811.37212.57361E–030.05337460.99996CLJAYA [57]1.0305143.482262848.642831.201271981.9822792.42507E−030.04892270.99997ITLBO [59]1.03053.482348.64281.2013981.98232.42519E−030.04888780.99997IWOA [61]1.03053.471748.63131.2016978.67712.42523E−030.04889330.99997MLBSA [62]1.03053.482348.64281.20163981.98232.42561E−030.048780.999975Pm [3]1.0343.57148.711.2061123.004.019E–030.08335220.99991The absolute current errors for PWP201 photovoltaic panel are under 0.01 (A), having a uniform distribution, but keeping the high values in the open-circuit voltage region, Figure5(a). Although, in this case, the methods estimate the current without high difference for certain voltages in comparison with the ones measured, the RMSE and AE have high values. The AE for the PWP201 photovoltaic panel around the open-circuit voltage is higher for the FPA and 5Pm methods. The other algorithms considered have the same behaviour, Figure 5(b).Figure 5
(a) The absolute current error (AE) for PWP201 photovoltaic panel; (b) AE in open-circuit voltage region.
(a)(b)


## 4.1.4. STM6-40 Photovoltaic Panel

The 5Pm method has the statistical test high values, Table6. The (V,I) pairs of the STM6-40 photovoltaic panel are not uniformly distributed. There are very few points in the open-circuit voltage region [46], which leads to poorer results in this case. The value of the coefficient of determination confirms this issue.Table 6
The parameters and statistical tests for the STM6-40 photovoltaic panel.
AlgorithmIph (A)Io (μA)nRs (Ω)RshΩRMSEAER2HSDA [44]1.66390477991.738654397854.7308990.153855932543.418349851.72981E−030.02190350.99997731Rcr-IJADE [36]1.66391.738754.73080.1548573.41881.73428E−030.02161480.99997719ITLBO [59]1.66391.738754.73080.1548573.41881.73428E−030.02161480.999977195Pm [3]1.66362.6541E−433.35340.9121898.163.53507E–020.5409480.98999793The best results are obtained for the HSDA algorithm for RSME andR2. The plot of the absolute current errors, Figure 6(a), shows the weakness of the 5Pm method in the region around the open-circuit voltage. For some points, the current calculated with the 5Pm method is twenty times higher than the ones calculated with the HSDA algorithm.Figure 6
(a) The absolute current error (AE) for STM6-40 photovoltaic panel; (b) AE in open-circuit voltage region.
(a)(b)The behaviour of the AE around the open-circuit voltage is similar for the ITLBO and the Rcr-IJADE algorithms, Figure 6(b). The AE for the HSDA algorithm has some small variations.


## 4.1.5. Commercial Monocrystalline Silicon Photovoltaic Panel

The statistical tests for commercial mSi photovoltaic panel are presented in Table7.Table 7
The parameters and statistical tests for mSi photovoltaic panel.
AlgorithmIph (A)Io (μA)nRs (Ω)Rsh (Ω)RMSEAER2HSDA [44]1.2242064.677E−418.3649940.144071544.3617242.77734E−031.939260.99956777GA [33]1.2230824.988143E−320.52890.021472921765.3884.96271E−032.614560.99845Pm [3]1.2240.334E−318.020.1341242.916.21773E–032.424160.9978The shape of the absolute current error curves is the same in the case of the mSi photovoltaic panel. The highest value for the AE is again in the region of the open-circuit voltage, Figure7.Figure 7
The absolute current error for mSi photovoltaic panel.The coefficient of determination for all photovoltaic cells and panels is very good, less for the GA algorithm in case of RTC photovoltaic cell and STM6-40 photovoltaic panel.


## 4.2. Forecast Comparison

TwoI-V characteristics are measured for the commercial monocrystalline photovoltaic panel for the irradiance of 700 W/m2 and 983 W/m2. The measurements were made on the roof of the university building [3].TheI-V characteristic measured at 700 W/m2 was used to extract the parameters of the mSi photovoltaic panel with three methods HSDA algorithm, GA algorithm, and 5Pm analytical method. These parameters are considered as Iph,ref,Io,ref,nref,Rs,ref,andRsh,ref. The parameters of the mSi photovoltaic panel are calculated for 983 W/m2 irradiance, using Equations (7)–(11). These values are used to compare the measured I-V characteristic for 983 W/m2 in real conditions with the ones forecasted with the five parameters calculated with Equations (7)–(11). The temperature coefficient of the current was determined, and it is 15.76 mA/°C.The root mean square error for the forecastedI-V characteristic of the mSi photovoltaic panel is presented in Table 8 in comparison with one for the I-V characteristic measured. The RMSE f obtained when the parameters are calculated with Equations (7)–(11) is worse in the case of the HSDA and GA algorithms than the RMSE m obtained using the extracted parameters with those algorithms for the measured characteristic. In case of the 5Pm methods, the results are improved.Table 8
The statistical tests for the forecastI-V characteristic.
AlgorithmRMSEfRMSEmHSDA [44]5.30433E–033.234E–03GA [33]1.63618E–026.707E–035Pm [3]6.23956E–032.86E–02The absolute current error calculated for the forecastI-V characteristic in comparison with the ones measured is presented in Figure 8(a). The highest values for AE are in the region of the open-circuit voltage. The behaviour of the curves obtained with parameters calculated with Equations (7)–(11) for HSDA and 5Pm methods is very similar, while the ones obtained with GA algorithms have an accentuated increase in the open-circuit voltage region.Figure 8
(a) The absolute current error (AE) for the forecastedI-V characteristic for photovoltaic panel; (b) AE in the voltage which corresponds to maximum power region.
(a)(b)Figure8(b) shows the behaviour of the absolute current error around the maximum voltage, Vm, which is the voltage coordinate for the maximum power point. The best results are obtained for the 5Pm method for all regions considered.The comparison between maximum power generated,Pmax,m, the mSi panel calculated from real measurements and ones forecasted, Pmax,f, using the extracted parameters with the three methods, Table 9, shows that the best estimation is for the maximum power estimated with parameters extracted with the 5Pm method.Table 9
ThePmax forecasted and measured.
AlgorithmHSDA [44]GA [33]5Pm [3]MeasuredPmax (W)13.80713.79513.80913.875Pmax,f/Pmax,m (%)99.50998.4599.52By analyzing the results obtained through comparison between the analytic method and metaheuristic algorithms, it can be concluded that the 5Pm method can be used to extract the parameters of the photovoltaic cells and panels. Additionally, the analytical method can be used to estimate theI-V characteristics and the power generated using the parameters given by the producers. It can be used due to the advantages which are presented in Table 10.Table 10
Advantages and disadvantages.
MethodAdvantagesDisadvantagesAnalytical(i) Using the 5Pm method is very easy, theRso and Rsh are calculated with the empirical equations [25](ii) Very easy to implement on the computer for all software which allows calculation(iii) The necessary time to extract the parameters is very short for the 5Pm method (duration is under ms)(iv) Does not require a powerful or dedicated PC, which leads to the small cost of the PC, and the used software can be a common one(i) The accuracy of the parameters extracted decreases due to approximations used [8](ii) If some points of the I-V characteristic are used, the equation system obtained is nonlinear and it is necessary to use software that allows solving this (this can be costly)(iii) The complexity increases when the two diode model is used (seven equations are needed)Metaheuristic(i) The accuracy is higher than that of the analytical methods(ii) The number of iterations can be reduced (for example, for the ISCE algorithm, the iteration number is 5000)(iii) The algorithms can be easily adapted for the two diode model, but the computational time will increase(iv) Using algorithms as grey wolf, prey predator, and fire fly optimization, the performance of the parameters extraction can be improved [8](v) GA algorithm family can easy be part of the hybrid algorithm [65](vi) DE algorithm family has a high convergence(vii) PSO can be improved if it is used together with Nelder-Mead methods, this hybridisation can reduce the computational resources [65](viii) SDA family has very good performance, and it can be easily used with other algorithms in hybrid structure(ix) Improved version of SCE has performance comparable with HSDA(i) The computational time can be high, it depends on the PC power and the metaheuristic algorithm (the number of the iterations and the complexity), there are algorithms with over 100000 iterations or even 350000-NM-MPSO algorithm [41](ii) The cost of the PC can be high and the dedicated software must be used(iii) The performance of the most metaheuristic algorithms depend by the initial parameter range(iv) Requires very good knowledge for implementation(v) GA, SCE, TLBO, WOA, and BSA algorithms have a slow convergence(vi) DE and PSO algorithms can converge prematurely(vii) The computational time is high in case of the PSO algorithms(viii) The computational time is relatively high, but using parallelization, it can be reduced; in the case presented, the time was reduced 6 times


## 5. Conclusions

The paper briefly reviews the analytical methods and metaheuristic algorithms used to extract the five or seven parameters for the photovoltaic cells and panels. The 5Pm analytical method and ones of the best metaheuristic algorithms from different families are compared for five datasets, two for photovoltaic cells, and three for photovoltaic panels.By analyzing the results obtained, the supremacy of the metaheuristic algorithms for accuracy is shown. In all cases studied, the algorithms have better results for all statistical tests used. The analytical method has a better performance than the GA algorithm for the RTC photovoltaic cell. The performance of the HSDA algorithm is one of the best for all photovoltaic cells and panels analyzed, and it was chosen to be compared in the forecast process.TwoI-V characteristics, measured for the commercial mSi photovoltaic panel, are used to compare the influence of the extract parameter methods on the forecast of the maximum power and I-V characteristic at different values of the irradiance and temperature. Using the parameters calculated with the HSDA algorithm, the forecast of the I-V characteristic was better than for GA and 5Pm methods. However, the 5Pm method forecasted better the maximum power, only 0.48% less than the real one. These are preliminary results, which will further be developed in future research by analysis for various cases. This analysis will be made on different panels, under various irradiance and illumination conditions.The 5Pm methods are based on several relations which are easy to implement, for the measuredI-V characteristic, datasets, or the datasheet parameters, the latter offered by the producer, and the parameters can be quickly calculated with very good accuracy. The necessary time to calculate the parameters is very small, and it does not require a powerful PC, as for the metaheuristic algorithms. These prove that the 5Pm method is a valuable candidate for photovoltaic cells and panel manufacturers. They can use the 5Pm method to characterize the photovoltaic devices and to obtain the optimum photovoltaic panels using cells with the same values of the parameters. The production time and the costs can be optimized.



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*Source: 3608138-2021-09-18.xml*