An adaptable method for efficient modeling of photovoltaic generators’ performance based on the double-diode model

Abstract

The aim of this study is to establish an effective modeling technique for simulating the performance of photovoltaic modules by calculating their electrical parameters based on the two-diode model. The suggested methodology involves reducing the scope of the study from seven unknown parameters to only three, and that without resorting to any approximations. The first four parameters are calculated analytically based on the data representing the crucial positions on the current-voltage graph and using a new expression of the fill-factor derived from the two-diode equivalent circuit. The remaining parameters are established numerically based on a simple iterative technique adaptable with two sites of data availability. The photovoltaic modeling begins by utilizing the values of key-points. Subsequently, to ensure the proposed approach's adaptability to various scenarios of available information about PV generators, it is invested and applied for an optimization process. The accuracy is evaluated for diverse types of photovoltaic modules, and the results are weighed against widely reviewed numerical methods and evolutionary optimization algorithms in the literature. As a result, the new method demonstrates superior performance, yielding the smallest values for the utilized statistical indicators and reducing compilation time. These findings underscore its flexibility and high efficiency in simulating photovoltaic devices.

Highlights

• •Hybrid extraction of PV modules' seven parameters based on the datasheet.
• •Accuracy is tested for PV panels of various technologies.
• •The new method is invested and adapted for an optimization process.
• •Performance is tested against recent meta-heuristic algorithms.
• •Best accuracy and execution time are provided by the proposed method.

Nomenclature

I_L

Light-generated current (A)

I₀₁

Saturation current of diode D₁ (A)

R_s

Series resistance (Ω)

n₁

Ideality factor of diode D₁

I₀₂

Saturation current of diode D₂ (A)

n₂

Ideality factor of diode D₂

R_p

Parallel resistance (Ω)

G_p

Parallel admittance (Ω⁻¹)

I

PV generator's output current (A)

V

PV generator's output voltage (V)

I_sc

Short-circuit current (A)

I_m

Peak power current (A)

V_m

Peak power voltage (V)

V_oc

Open-circuit voltage (V)

I_Me

Vector of measured current (A)

V_M

Vector of measured voltage (V)

I_C

Calculated current (A)

N

Number of measured data

AM

Air Mass equals to 1.5

G

Operating irradiance (W/m²)

T

Operating temperature (°C)

N_s

Number of cells in series of a PV module

K_B

Boltzmann's constant (K_B = 1.38064852 × 10⁻²³J/K)

q

Electron charge (q = 1.60217646 × 10⁻¹⁹C)

V_th

Thermal voltage (V)

K_isc

Temperature response factor of I_sc (A/°C)

K_im

Temperature response factor of I_m (A/°C)

K_Vm

Temperature response factor of V_m (V/°C)

K_Voc

Temperature response factor of V_oc (V/°C)

α_Vm and α_Voc

Two adjusted parameters

Abbreviations

ABC-DE

Artificial Bee Colony-Differential Evolution

BMA

Barnacles Mating Algorithm

DDM

Double-Diode Model

FF

Fill-Factor

IAE

Individual Absolute Error

IJAYA

Improved JAYA

IGSK

Improved Gaining-Sharing Knowledge Algorithm

IMPA

Improved Marine Predators Algorithm

ITLBO

Improved Teaching Learning-Based Optimization

GCPSO

Guaranteed Convergence Particle Swarm Optimization

GWO

Grey Wolf Optimizer

MLBSA

Multiple Learning Backtracking Search Algorithm

MPPT

Maximum Power Point Traking

NRMSE

Normalized Root Mean Square Error

RMSE

Root Mean Square Error

SR

Squared Error

STC

Standard Test Conditions

TLABC

Teaching–Learning–Based Artificial Bee Colony

WHHO

Whippy Harris Hawks Optimization Algorithm

WDO

Wind Driven Optimization

1. Introduction

At present, humanity is undergoing rapid technological advancements, coinciding with the substantial growth of the world's population. This swift development is aimed at meeting the evolving needs and luxuries of the citizens, thereby sustaining their continual increase [1]. Historically, the energy requirements accompanying the countries' industrialization were responded using fossil fuels, such as natural coal, gas, and oil [[2], [3], [4]]. Moreover, the use of these energy sources persists to this day, with the vast majority of energy needs still being met by fossil fuels. However, the gigantic utilization of these kinds of sources has led to their fast drain threatening the future generations at different levels: climate, health, and economy [[5], [6], [7]]. In recent decades, humans have begun to recognize the risks associated with the overexploitation of fossil fuels and have sought safer and renewable alternatives. Hence, It is expected that renewable energies will cover 49–67 % of primary energy by the year 2050 [3,8].

Since 1990, the increased implementation of photovoltaic systems has positioned sunlight as an extremely promising alternative [9]. Solar energy has then spurred various research avenues, drawing the attention of researchers worldwide [3]. These include the development of advanced materials with several applications that includes PV modules improvements and applications [10,11], as well as the creation of efficient PV simulation tools [2]. Given the costs associated with PV installations, modeling PV modules is an essential task before their integration into PV systems, particularly in large installations. This modeling is necessary to simulate their performance and predict their interactions with the installation environment by estimating the influence of weather conditions on their efficiency. However, predicting the electrical production of PV modules is not an obvious task [12]. It requires tribal knowledge about the internal electrical parameters of the PV generator influencing its performance. Indeed, in the present time, the demand for photovoltaic modeling methods has grown simultaneously with the great interest given to renewable energies [13].

Diverse electrical models were introduced in the literature to describe the physical phenomenon occurring inside PV generators. The one-diode model with five electrical parameters and the two-diode model with seven parameters are the most adopted by the authors [9,[12], [13], [14], [15], [16]]. To reduce the complexity and the nonlinearity degree of the current equation, the single-diode model is extensively employed in academic research [[12], [13], [14], [15], [16], [17], [18]]. However, since it ignores the impact of the recombination process within the depletion area, the single-diode model does not give the correct description of the physical effects inside real PV cells, and it is considered as a non-effective model [19]. To address this problem and improve precision, a second diode was introduced into the circuit to represent the recombination losses occurring within the space charge region. Consequently, another model with dual diodes and seven parameters was introduced [[19], [20], [21], [22], [23]].

The works in this paper are built on the two-diode circuit as the basic model to evaluate the electrical characteristics and the maximum power anticipated to be provided by PV modules under real environmental conditions. The model contains seven unknown parameters; the photo-current (I_L), the shunt parasitic resistance (R_p), the series resistance (R_s), two ideality factors (n₁ and n₂), and two reverse saturation currents (I₀₁ and I₀₂) [23]. Taking into account the nonlinearity nature of the current equation, calculating all the previous seven parameters is considered as a great challenge. Accordingly, the techniques presented in prior studies to address this problem can be categorized into three major groups. The first set includes the techniques based on resolution of a set of analytical equations. By employing series of approximations, analytical techniques seem viable when dealing with the single-diode model. But, for the double-diode model, additional approximations will be required, and this large number of assumptions leads to the loss of the method's accuracy [16,25,26]. The second group contains numerical methods divided into two approaches: the numerical solving of a group of nonlinear equations using numerical algorithms. This first kind requires good initialization for all seven unknowns to guarantee the attainment of accurate solutions [24]. Moreover, obtaining all seven nonlinear equations without using any approximations is not a simple task [24]. The numerical resolution also involves curve adjustment based on the entire measured data curve, as well as the utilization of meta-heuristic algorithms. However, these algorithms have major disadvantages, such as requiring significant computational time due to the arbitrary initialization of populations and the need for preliminary delimitation of variation limits for each parameter [[27], [28], [29]]. The hybrid methods form the third group of identification techniques, this kind of methods is based on the combination of more than one approach to deal with the disadvantages of the previous approaches [25,30,31].

Lately, various optimization methods based on meta-heuristic principles have been proposed to determine the values of the seven parameters in the two-diode model. These methods include the BMA (barnacles mating algorithm) [32], the MLBSA (multiple learning backtracking search algorithm) [33], the IJAYA (improved JAYA algorithm) [34], the FPSO (Flexible Particle Swarm Optimization Algorithm) [35], the ITLBO (improved teaching learning-based optimization) [36], the IGSK (improved gaining-sharing knowledge algorithm) [37], the WHHO (whippy Harris Hawks optimization algorithm) [38], the IMPA (improved marine predators algorithm) [39], the TLABC (Teaching–learning–based artificial bee colony) [40], and the WDO (wind driven optimization) [41]. These methods have the advantage of avoiding mathematical complications since they are based on optimization algorithms already ready to be used for diversified real-world problems. However, their performance still suffers from various challenges, including the selection of specific algorithm parameters such as initial populations, crossover rate, mutation probability, number of generations, cognitive learning factor, and others [39,42]. Incorrect choice of the appropriate values of these optimization parameters can significantly impact the convergence speed or directs the algorithm to achieve the local best solution, which can have no physical significance. Moreover, for every PV generator, the use of evolutionary algorithms requires prior knowledge of the interval of variation of each of its internal physical parameters. According to the literature, it can be noticed that the variation ranges of the parameters differ from one work to another for the same PV generator without any justification [34,38]. Indeed, a wrong determination of the variations interval can either lead to the convergence toward local optimums or increase the execution time.

In order to identify the values of the unknown parameters of the double-diode model, the authors in Ref. [19] use several assumptions to reduce the nonlinearity of the electrical equation. In fact, the search space's dimension was condensed from seven to just four parameters, based on these assumptions. The first approximation used by the authors is the assignment of 1 as a value of the first ideality factor (n₁) and assigning a value greater than 1.2 to the second ideality factor (n₂). The latter assumption was also adopted later by the authors of [20]. However, this last assumption is not fair in all cases, and an incorrect choice of the values of these two parameters may involve a significant waste of accuracy [24,29]. The second hypothesis used by the authors in Ref. [19] is the equality of both saturation currents (I₀₁ and I₀₂). The same assumption was used in several research papers [20,26,43], yet this has no physical significance [24]. The recombination current, which models the charge carrier loss caused by electron-hole recombination, is always far higher compared to the saturation current. On the other hand, the saturation current, models the leak current in the semiconductor and is mainly influenced by temperature variations. Thus, the adopted approximation in the literature does not match the physical significance of these two parameters, consequently adversely affecting the model's precision [23]. In addition to the two previous assumptions, the authors in Refs. [19,20], and [21] assume I_L=I_sc equality, further worsening the accuracy of the results. The series resistance was determined by the authors of [19,20] using slow increments built on the error minimization at the peak power point, the technique which was first used for the single diode circuit by Villalva et al. [44]. Unfortunately, the use of a large number of approximations has a ruinous effect on the method's accuracy and deviates the modeling of its physical meaning.

With the purpose of reducing the computational time and the sophistication of the current model, Chitti Babu et al. [21] neglect the shunt and series resistances. Thus, all resistive effects are ignored inside the solar cells, which is not physically true since the resistive effects always exist and are influenced by temperature and irradiance's levels. Moreover, intending to calculate the values of both reverse saturation currents, the first one was calculated using an approximate analytical equation. The second saturation current was deduced analytically using the value of the first one, according to a predefined equation derived from previous works, which was also adopted by the authors in Refs. [29,30]. However, this equation lacks a physical interpretation or explanation. Subsequently, the magnitudes of the ideality factors were extracted in Ref. [21] using an iterative process. In addition to the used approximations for I_L and I₀₂ in Ref. [30], the values of the two ideality factors were hypothesized as fixed and taken directly from two different works [42,45]. Considering that the works in Refs. [42,45] are based on evolutionary algorithms, which require a large computational time to extract all seven parameters at once and not just n₁ and n₂, the execution time of the technique described in Ref. [30] must be further increased due to the iterative process used to extract R_s.

In an attempt to mitigate the non-linearity resulting from the tedious calculations used, the authors in Ref. [26] assumed the n₁ = n₂ equality and found that both ideality factors are close to 1. However, these results lack physical evidence. Indeed, the ideality factor describes the mechanism of recombination within the P–N junction. When recombination is low, this factor takes values between 1 and 2, and near 2 in cases of high recombination [25]. Hence, the authors in Refs. [23,24] mentioned that, generally, n₁+n₂ is greater than 3, but in Ref. [24], it was assumed that n₁+n₂ = 3 to complete a system of seven equations to be solved numerically through the use of the 'fsolve' function of MATLAB. However, this assumption is not valid. Moreover, as it revolves around solving a set of seven equations, the proposed technique in Ref. [24] requires accurate initialization for every single parameter, the thing which increases the number of approximations, risks its convergence, and leads to tedious calculations.

Recently, the authors in Refs. [25,43] have attempted to circumvent some of the previous assumptions regarding n₁, n₂, and R_s by increasing the number of iterative processes employed. In fact [25], proposes two methods that differ in the techniques used to extract the two ideality factors. Initially, n₁ was assumed to be equal to 1, and n₂ was presumed to be equal to 2. These approximations were also utilized previously in Ref. [31] yet, not always true. Next, to avoid the last two assumptions, the authors in Ref. [25] used an iterative process for calculating both ideality factors simultaneously. This process involved slow increments and assumed that an increase in n₁ implies the increase in n₂, according to a predefined relationship between the two parameters. In addition to the previous iterative process, an additional loop was used to find the value of R_s that minimizes the prediction error at the peak power point. To avoid the unproven relationships between the two ideality factors, the authors in Ref. [43] chose to use three slow sequential iterative processes. However, each additional separate iterative technique further increases the execution time.

Some prior numerical approaches utilized an enormous number of approximations, which contradicts the physical relevance of the seven parameters. Moreover, these methods often rely on multiple iterative loops, significantly impacting the modeling's convergence speed. Such challenges are particularly problematic for developing simulation tools for PV systems and dynamic applications like maximum power point tracking (MPPT), necessitating further improvement and simplification. Consequently, the current work aims to address these issues encountered by previous modeling techniques, which may hinder their efficiency. By minimizing approximations and streamlining the iterative process, the proposed approach seeks to ensure greater accuracy and efficiency in determining PV module parameters. This advancement is of significant importance for various PV module simulation applications.

This work proposes a novel reduced-form modeling technique that combines both analytical and numerical approaches. On one side, the number of unknowns in the problem is reduced. Specifically, the four parameters I_L, I₀₁, I₀₂, and R_p are expressed as functions of the remaining parameters (n₁, n₂, and R_s), without relying on any assumptions or simplifications. Consequently, by extracting these last three parameters, it becomes possible to analytically calculate the first four parameters. On the other side, by eliminating the four parameters from the current equation by replacing them with their expressions as functions of n₁, n₂, and R_s, these three parameters are extracted at once. This is achieved using a numerical method based on the Levenberg-Merquardt algorithm and employing a single fast iterative approach. At first, only the available values of the key points of PV panels from their datasheets are used for parameter extraction and for predicting the I–V characteristics of six PV panels of different technologies. Then, the performance of these predictions is tested against the most widely reviewed numerical methods in the literature using various statistical indicators. Next, the proposed algorithm is adapted to an optimization process. Thus, for a fair comparison, the proposed method is then used for optimization by exploiting the experimental I–V characteristic of the RTC France solar cell. The speed of convergence of the method, as well as its accuracy, is validated against several of the most advanced meta-heuristic optimization algorithms. Section 2 introduces the double-diode model. The proposed reduced form in this work is detailed in Section 3. Section 4 of the current paper analyzes the results and presents the performance of the method under different weather conditions, comparing it with other numerical methods and some meta-heuristic optimization algorithms from the literature.

2. Double-diode electrical model

The double-diode circuit model of PV generators is illustrated in Fig. 1. The model contains a resistance (R_s) representing losses due to the Joule effect within a solar cell, a parallel resistance (R_p) accounting for losses caused by manufacturing defects, the light-generated current (I_L), and two diodes (D₁ and D₂) [25,46].

Fig. 1.

Double-diode model.

For a PV generator, the characteristic equation linking its output current to the output voltage is given in equation (1), as follows:

$$\mathrm{I}={\mathrm{I}}_{\mathrm{L}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{\mathrm{V}+\mathrm{I}\times {\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_1{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{\mathrm{V}+\mathrm{I}\times {\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_2{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-{\mathrm{G}}_{\mathrm{p}}\left(\mathrm{V}+\mathrm{I}\times {\mathrm{R}}_{\mathrm{s}}\right)$$ (1)

Where V_th refers to the thermal voltage given as ${\mathrm{V}}_{\text{th}}=\frac{\mathrm{T}\times {\mathrm{K}}_{\mathrm{b}}}{\mathrm{q}}$, K_B is the Boltzmann's constant, T is the operating temperature of the PV generator, q is the electron charge, and G_p is the parallel admittance equals 1/R_p.

3. Theoritical model

This section is dedicated to detailing the steps followed to establish the values of the electrical parameters of the Double-Diode Model (DDM). The section is split into two subsections. The first details the methodology used to derive analytical expressions for calculating the parameters I_L, I₀₁, I₀₂, and R_p as functions of R_s, n₁, and n₂. This proposed procedure relies on four equations relating I_L, I₀₁, I₀₂, and R_p as unknown parameters. The four equations are derived from the current equation evaluated at the three specific points of the electrical I–V curve, along with a new expression for the fill factor based on the DDM. The second subsection introduces the numerical procedure used to find out the values of R_s, n₁, and n₂. This procedure entails the numerical resolution of a system comprising three nonlinear equations.

3.1. Analytical extraction of I_L, I₀₁, I₀₂, and R_p

The aim of this subsection is to analytically compute the four parameters I_L, I₀₁, I₀₂, and R_p as functions of n₁, n₂, and R_s. To achieve this, first, the current equation of equation (1) is computed at the three key points as follows:

At the point of the short circuit (I=I_sc, V = 0), equation (2) is obtained as:

$${\mathrm{I}}_{\text{sc}}={\mathrm{I}}_{\mathrm{L}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{{\mathrm{I}}_{\text{sc}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_1{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{{\mathrm{I}}_{\text{sc}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_2{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-{\mathrm{G}}_{\mathrm{p}}{\mathrm{I}}_{\text{sc}}{\mathrm{R}}_{\mathrm{s}}$$ (2)

At the point of the peak power (I=I_m, V=V_m), equation (3) is obtained as:

$${\mathrm{I}}_{\mathrm{m}}={\mathrm{I}}_{\mathrm{L}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{{\mathrm{V}}_{\mathrm{m}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_1{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{{\mathrm{V}}_{\mathrm{m}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_2{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-{\mathrm{G}}_{\mathrm{p}}\left({\mathrm{V}}_{\mathrm{m}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}\right)$$ (3)

At the point of the open circuit (I = 0, V=Voc), equation (4) is obtained as:

$${\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{01}\left[\exp \left(\frac{{\mathrm{V}}_{\text{oc}}}{{\mathrm{n}}_1{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]+{\mathrm{I}}_{02}\left[\exp \left(\frac{{\mathrm{V}}_{\text{oc}}}{{\mathrm{n}}_2{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]+{\mathrm{V}}_{\text{oc}}{\mathrm{G}}_{\mathrm{p}}$$ (4)

To streamline the notation, the symbols of equation (5) below are employed.

$$\{\begin{array}{l}{\mathrm{A}}_{1,2}=\exp \left(\frac{{\mathrm{I}}_{\text{sc}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_{1,2}{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)\\ {\mathrm{B}}_{1,2}=\exp \left(\frac{{\mathrm{V}}_{\mathrm{m}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_{1,2}{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)\\ {\mathrm{C}}_{1,2}=\exp \left(\frac{{\mathrm{V}}_{\text{oc}}}{{\mathrm{n}}_{1,2}{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)\end{array}$$ (5)

To get rid of I_L of equation (3), it is replaced by its expression of equation (4). Then, an expression for I₀₁ as a function of I₀₂, G_p, n₁, n₂, and R_s can be derived from equation (3) as follows:

$${\mathrm{I}}_{01}\left({\mathrm{I}}_{02},{\mathrm{R}}_{\mathrm{p}},{\mathrm{n}}_1,{\mathrm{n}}_2,{\mathrm{R}}_{\mathrm{s}}\right)=-\frac{{\mathrm{I}}_{\mathrm{m}}+{\mathrm{I}}_{02}\left[{\mathrm{B}}_2-{\mathrm{C}}_2\right]+{\mathrm{G}}_{\mathrm{p}}\left({\mathrm{V}}_{\mathrm{m}}-{\mathrm{V}}_{\text{oc}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}\right)}{\left[{\mathrm{B}}_1-{\mathrm{C}}_1\right]}$$ (6)

From equation (2), I_L can also be expressed as:

$${\mathrm{I}}_{\mathrm{L}}\left({\mathrm{I}}_{01},{\mathrm{I}}_{02},{\mathrm{R}}_{\mathrm{p}},{\mathrm{n}}_1,{\mathrm{n}}_2,{\mathrm{R}}_{\mathrm{s}}\right)={\mathrm{I}}_{01}\left[{\mathrm{A}}_1-1\right]+{\mathrm{I}}_{02}\left[{\mathrm{A}}_2-1\right]+{\mathrm{G}}_{\mathrm{p}}{\mathrm{I}}_{\text{sc}}{\mathrm{R}}_{\mathrm{s}}-{\mathrm{I}}_{\text{sc}}$$ (7)

By substituting I_L from equation (7) into equation (3), and then replacing I₀₁ in the same equation (3) with its expression from equation (6), the following expression for I₀₂ as a function of G_p, n₁, n₂, and R_s is obtained:

$${\mathrm{I}}_{02}=\frac{\left[\left({\mathrm{I}}_{\text{sc}}-{\mathrm{I}}_{\mathrm{m}}\right)\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)+{\mathrm{I}}_{\mathrm{m}}\left({\mathrm{B}}_1-{\mathrm{A}}_1\right)\right]+{\mathrm{G}}_{\mathrm{p}}\left[\left({\mathrm{V}}_{\mathrm{m}}-{\mathrm{V}}_{\text{oc}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}\right)\left({\mathrm{B}}_1-{\mathrm{A}}_1\right)-\left({\mathrm{V}}_{\mathrm{m}}+{\mathrm{R}}_{\mathrm{s}}\left({\mathrm{I}}_{\mathrm{m}}-{\mathrm{I}}_{\text{sc}}\right)\right)\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)\right]}{\left({\mathrm{B}}_2-{\mathrm{A}}_2\right)\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)-\left({\mathrm{B}}_2-{\mathrm{C}}_2\right)\left({\mathrm{B}}_1-{\mathrm{A}}_1\right)}$$ (8)

To simplify the writing of equation (8), the following notations are adopted:

$$\{\begin{array}{l}{\mathrm{d}}_1=\frac{\left[\left({\mathrm{I}}_{\text{sc}}-{\mathrm{I}}_{\mathrm{m}}\right)\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)+{\mathrm{I}}_{\mathrm{m}}\left({\mathrm{B}}_1-{\mathrm{A}}_1\right)\right]}{\left({\mathrm{B}}_2-{\mathrm{A}}_2\right)\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)-\left({\mathrm{B}}_2-{\mathrm{C}}_2\right)\left({\mathrm{B}}_1-{\mathrm{A}}_1\right)}\\ {\mathrm{d}}_2=\frac{\left[\left({\mathrm{V}}_{\mathrm{m}}-{\mathrm{V}}_{\text{oc}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}\right)\left({\mathrm{B}}_1-{\mathrm{A}}_1\right)-\left({\mathrm{V}}_{\mathrm{m}}+{\mathrm{R}}_{\mathrm{s}}\left({\mathrm{I}}_{\mathrm{m}}-{\mathrm{I}}_{\text{sc}}\right)\right)\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)\right]}{\left({\mathrm{B}}_2-{\mathrm{A}}_2\right)\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)-\left({\mathrm{B}}_2-{\mathrm{C}}_2\right)\left({\mathrm{B}}_1-{\mathrm{A}}_1\right)}\end{array}$$ (9)

The expression of I₀₂ of equation (8) can be rewritten using the notations of equation (9) to obtain the form shown in equation (10):

$${\mathrm{I}}_{02}\left({\mathrm{n}}_1,{\mathrm{n}}_2,{\mathrm{R}}_{\mathrm{s}},{\mathrm{R}}_{\mathrm{p}}\right)={\mathrm{d}}_1+{\mathrm{G}}_{\mathrm{p}}{\mathrm{d}}_2$$ (10)

To derive the expression for calculating G_p only as a function of n₁, n₂, and R_s, the procedure starts from the expression of the output power of PV generators defined as follows:

$$\mathrm{P}=\mathrm{I}\times \mathrm{V}$$ (11)

A new power expression, denoted $\hat{\mathrm{P}}$ is obtained by combining equation (11) and equation (1) using the method of undetermined multipliers of Lagrange. This method was first employed by the authors in Ref. [47] to derive the expression of the fill factor in the case of the SDM. In this work, the technique is adapted for the DDM to obtain $\hat{\mathrm{P}}$ as:

$$\hat{\mathrm{P}}=\mathrm{I}\times \mathrm{V}-\lambda \times \left\{-\mathrm{I}+{\mathrm{I}}_{\mathrm{L}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{\mathrm{V}+{\mathrm{R}}_{\mathrm{s}}\mathrm{I}}{{\mathrm{n}}_1{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{\mathrm{V}+{\mathrm{R}}_{\mathrm{s}}\mathrm{I}}{{\mathrm{n}}_2{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\right)-1\right]-\left(\frac{\mathrm{V}+{\mathrm{R}}_{\mathrm{s}}\mathrm{I}}{{\mathrm{R}}_{\mathrm{p}}}\right)\right\}$$ (12)

λ is the undetermined multiplier of Lagrange optimizing $\hat{\mathrm{P}}$ with respect to the current (I) and the voltage (V). Indeed, λ satisfies the following system of equations:

$$\{\begin{array}{l}{\frac{\widehat{\partial \mathrm{P}}}{\partial \mathrm{I}}|}_{\left({\mathrm{V}}_{\mathrm{m}},{\mathrm{I}}_{\mathrm{m}}\right)}=0\\ {\frac{\widehat{\partial \mathrm{P}}}{\partial \mathrm{V}}|}_{\left({\mathrm{V}}_{\mathrm{m}},{\mathrm{I}}_{\mathrm{m}}\right)}=0\end{array}$$ (13)

Replacing equation (12) in equation (13), the voltage at the peak power point (V_m) and the current at the peak power point (I_m) are obtained as:

$$\{\begin{array}{l}{\mathrm{V}}_{\mathrm{m}}=\mathrm{\lambda }\left[-1-{\mathrm{I}}_{01}\frac{{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{n}}_1{\mathrm{V}}_{\text{th}}}{\mathrm{B}}_1-{\mathrm{I}}_{02}\frac{{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{n}}_2{\mathrm{V}}_{\text{th}}}{\mathrm{B}}_2-{\mathrm{R}}_{\mathrm{s}}{\mathrm{G}}_{\mathrm{p}}\right]\\ {\mathrm{I}}_{\mathrm{m}}=\mathrm{\lambda }\left[-{\mathrm{I}}_{01}\frac{1}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{n}}_1{\mathrm{V}}_{\text{th}}}{\mathrm{B}}_1-{\mathrm{I}}_{02}\frac{1}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{n}}_2{\mathrm{V}}_{\text{th}}}{\mathrm{B}}_2-{\mathrm{G}}_{\mathrm{p}}\right]\end{array}$$ (14)

Given that the computation of the PV generator's maximum power can be derived using the equation below [47]:

$${\mathrm{P}}_{\mathrm{m}}={\mathrm{I}}_{\mathrm{m}}^2\times {\mathrm{R}}_{\mathrm{m}}$$ (15)

R_m can be calculated using the system of equations in equation (14) and can be expressed as shown in equation (16) below:

$${\mathrm{R}}_{\mathrm{m}}=\frac{{\mathrm{V}}_{\mathrm{m}}}{{\mathrm{I}}_{\mathrm{m}}}={\mathrm{R}}_{\mathrm{s}}+{\left[1+\frac{{\mathrm{R}}_{\mathrm{p}}}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\left(\frac{{\mathrm{I}}_{01}}{{\mathrm{n}}_1}\mathrm{B}{}_1+\frac{{\mathrm{I}}_{02}}{{\mathrm{n}}_2}{\mathrm{B}}_2\right)\right]}^{-1}$$ (16)

Equations (15), (16) are combined to obtain the expression of the Fill-Factor (FF), which represents the squareness of the PV generator's characteristics according to the two-diode model. The expression is as follows:

$$\text{FF}=\frac{{\mathrm{I}}_{\mathrm{m}}^2}{{\mathrm{V}}_{\text{oc}}{\mathrm{I}}_{\text{sc}}}\times \left({\mathrm{R}}_{\mathrm{s}}+\frac{{\mathrm{R}}_{\mathrm{p}}}{1+\frac{{\mathrm{R}}_{\mathrm{p}}}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\left(\frac{{\mathrm{I}}_{01}}{{\mathrm{n}}_1}\mathrm{B}{}_1+\frac{{\mathrm{I}}_{02}}{{\mathrm{n}}_2}{\mathrm{B}}_2\right)}\right)$$ (17)

The expression for FF obtained in equation (17) is written as a function of I₀₁, I₀₂, R_s, n₁, and n₂. By replacing I₀₁ and I₀₂ in equation (17) with their expressions from equations (6), (10), respectively, the parallel resistance R_p is deduced uniquely as a function of n₁, n₂, and R_s, as shown in equation (18):

$${\mathrm{R}}_{\mathrm{p}}\left({\mathrm{n}}_1,{\mathrm{n}}_2,{\mathrm{R}}_{\mathrm{s}}\right)=\frac{\left(\frac{{\text{FFV}}_{\text{oc}}{\mathrm{I}}_{\text{sc}}}{{\mathrm{I}}_{\mathrm{m}}^2}-{\mathrm{R}}_{\mathrm{s}}\right)\times \left({\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}-{\mathrm{E}}_1+{\mathrm{d}}_2\times {\mathrm{E}}_2\right)}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}-\left(\frac{{\text{FFV}}_{\text{oc}}{\mathrm{I}}_{\text{sc}}}{{\mathrm{I}}_{\mathrm{m}}^2}-{\mathrm{R}}_{\mathrm{s}}\right)\times \left({\mathrm{d}}_1\times {\mathrm{E}}_2-\frac{\mathrm{I}{}_{\mathrm{m}}{\mathrm{B}}_1}{{\mathrm{n}}_1\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)}\right)}$$ (18)

Where E₁ and E₂ are defined in the system of equation (19) below:

$$\{\begin{array}{l}{\mathrm{E}}_1=\frac{{\mathrm{V}}_{\mathrm{m}}-{\mathrm{V}}_{\text{oc}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{n}}_1\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)}{\mathrm{B}}_1\\ {\mathrm{E}}_2=\frac{{\mathrm{B}}_2}{{\mathrm{n}}_2}-\frac{{\mathrm{B}}_1\left({\mathrm{B}}_2-{\mathrm{C}}_2\right)}{{\mathrm{n}}_1\left({\mathrm{B}}_1-{\mathrm{C}}_1\right)}\end{array}$$ (19)

3.2. Determination of n₁, n₂, and R_s numerically

In this subsection, two different techniques are employed for computing the values of the three parameters. The first technique relies solely on the available values of I_sc, I_m, V_m, and V_oc from the PV module's datasheet. The second technique involves utilizing an optimization algorithm, which is based on measured data.

3.2.1. Identification based on remarkable-points’ values (prediction)

This subsection outlines the steps for extracting the parameters n₁, n₂, and R_s based on the values of key points. Numerically calculating the values of these parameters requires a system of three equations linking them. To that end, the first equation of this system is derived by substituting I_L from equation (4) into equation (2), resulting in equation (20), as:

$${\mathrm{I}}_{01}\left[{\mathrm{C}}_1-{\mathrm{A}}_1\right]+{\mathrm{I}}_{02}\left[{\mathrm{C}}_2-{\mathrm{A}}_2\right]+{\mathrm{G}}_{\mathrm{p}}\left({\mathrm{V}}_{\text{oc}}-{\mathrm{I}}_{\text{sc}}{\mathrm{R}}_{\mathrm{s}}\right)-{\mathrm{I}}_{\text{sc}}=0$$ (20)

By substituting the expressions for I₀₁, I₀₂, and R_p from equations (6), (10), (18) respectively into equation (17), it can be used as the second equation in the system of three equations. Finally, by substituting the expressions for I_L, I₀₁, I₀₂, and R_p from equations (7), (6), (10), (18), respectively into equation (3), it becomes the last equation in the system.

The system regrouping the set of the three equations to be solved numerically is as follows:

$$\{\begin{array}{l}{\mathrm{I}}_{01}\left[{\mathrm{C}}_1-{\mathrm{A}}_1\right]+{\mathrm{I}}_{02}\left[{\mathrm{C}}_2-{\mathrm{A}}_2\right]+{\mathrm{G}}_{\mathrm{p}}\left({\mathrm{V}}_{\text{oc}}-{\mathrm{I}}_{\text{sc}}{\mathrm{R}}_{\mathrm{s}}\right)-{\mathrm{I}}_{\text{sc}}=0\\ \text{FF}-\frac{{\mathrm{I}}_{\mathrm{m}}^2}{{\mathrm{V}}_{\text{oc}}{\mathrm{I}}_{\text{sc}}}\times \left({\mathrm{R}}_{\mathrm{s}}+{\mathrm{R}}_{\mathrm{p}}{\left(1+\frac{{\mathrm{R}}_{\mathrm{p}}}{{\mathrm{N}}_{\mathrm{s}}{\mathrm{V}}_{\text{th}}}\left(\frac{{\mathrm{I}}_{01}}{{\mathrm{n}}_1}\mathrm{B}{}_1+\frac{{\mathrm{I}}_{02}}{{\mathrm{n}}_2}{\mathrm{B}}_2\right)\right)}^{-1}\right)=0\\ {\mathrm{I}}_{\mathrm{L}}-{\mathrm{I}}_{01}\left[{\mathrm{B}}_1-1\right]-{\mathrm{I}}_{02}\left[{\mathrm{B}}_2-1\right]-{\mathrm{G}}_{\mathrm{p}}\left({\mathrm{V}}_{\mathrm{m}}+{\mathrm{I}}_{\mathrm{m}}{\mathrm{R}}_{\mathrm{s}}\right)-{\mathrm{I}}_{\mathrm{m}}=0\end{array}$$ (21)

Where I_L, I₀₁, I₀₂, and R_p are replaced, respectively, by their expressions of equations (7), (6), (10), (18) gotten as functions of n₁, n₂, and R_s.

The system of equation (21) is solved using the Levenberg-Marquardt algorithm implemented in MATLAB as the “fsolve” function [48]. To solve the system and get the most accurate values of the parameters n₁, n₂, and R_s, a rigorous selection of the initial values of these parameters is an essential task to be done. Precision in the initial values is crucial to ensure both the rapid convergence and the convergence to the correct solutions, while avoiding local minima and maintaining the physical significance of the parameters. Therefore, the initial values for the two ideality factors n₁ and n₂ are set near 1 and 2, respectively, as shown in equation (22) and equation (23) [49]:

$${\mathrm{n}}_{1,\mathrm{i}}=1+{\mathrm{\epsilon }}_{\mathrm{n}1}$$ (22)

$${\mathrm{n}}_{2,\mathrm{i}}=2+{\mathrm{\epsilon }}_{\mathrm{n}2}$$ (23)

Where ε_n1 and ε_n2 are small offsets taken randomly in the interval [−0.1, 0.1] to avoid the local optimums (1 and 2), and to always ensures the rapid convergence toward the accurate solutions [49].

The initial value of R_s is the unique value extracted through a swift iterative process. This process optimizes the root mean square error (RMSE) between calculated and measured vectors of the three remarkable currents of the I–V curve. Indeed, the procedure consists of the incrimination of R_s and minimizing the root mean square error (RMSE), given in equation (24), only at the three key-points.

$$\text{RMSE}=\sqrt{\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{I}}_{\mathrm{i},\text{Me}}-{\mathrm{I}}_{\mathrm{i},\mathrm{C}}\right)}^2}$$ (24)

Where I_i,Me represents the ith measured current of the vector of measured currents at the three remarkable points given as: ${\mathrm{I}}_{\text{Me}}=\left[{\mathrm{I}}_{\text{sc},\text{measured}},{\mathrm{I}}_{\mathrm{m},\text{measured}},{\mathrm{I}}_{\text{oc},\text{measured}}\right]$.

I_i,C represents the ith calculated current of the vector of calculated currents at the three remarkable-points given as: ${\mathrm{I}}_{\mathrm{M}}=\left[{\mathrm{I}}_{\text{sc},\text{calculated}},{\mathrm{I}}_{\mathrm{m},\text{calculated}},{\mathrm{I}}_{\text{oc},\text{calculated}}\right]$.

In this case, N is equal to 3, representing the three remarkable points. However, when evaluating the RMSE value between the predicted and measured electrical curves, N represents the total number of measured data points (N > 3).

The iterative process is exclusively reserved for initializing R_s, with no involvement in its primary prediction, thereby ensuring a swift extraction.

The steps to be followed for the method in the current work are shown in the flowchart of Fig. 2.

Fig. 2.

Suggested method's flow chart.

3.2.2. Identification based on measured data (optimization)

As a second task, to ensure the adaptability of the proposed method to various case studies, it is utilized as an optimization algorithm. Therefore, instead of solving equation (21) to obtain the values of the three parameters, additional measured or experimental data can be incorporated into the vector of measured currents I_Me. Accordingly, the squared error between the calculated and measured currents, given in equation (25) below, can be minimized to further improve the accuracy of estimation of the whole I–V curve [49]. To that end, the 'lsqnonlin' function of the MATLAB Optimization Toolbox [50] is used to minimize the following calculated error:

$$\text{SR}={\sum _{\mathrm{i}=1}^{\mathrm{N}}\left({\mathrm{I}}_{\mathrm{i},\mathrm{C}}\left({\mathrm{V}}_{\mathrm{i},\mathrm{M}},{\mathrm{n}}_1,{\mathrm{n}}_2,{\mathrm{R}}_{\mathrm{s}}\right)-{\mathrm{I}}_{\mathrm{i},\text{Me}}\right)}^2$$ (25)

3.3. Prediction for various weather conditions

Generally, PV generators are placed outdoors and exposed to various conditions of temperature and irradiance (T, G). Hence, the extraction of the seven parameters' values for non-standard test conditions is a requirement of great importance. According to the literature, several authors [25,30,31] use seven equations to calculate the values of parameters at non-STC, exploiting the calculated values at STC. However, some of the seven equations are obtained using approximations and involve additional unstable parameters such as the band gap energy [25,30,51]. This parameter varies across different PV technologies and is highly dependent on temperature levels, which can lead to significant precision losses at higher temperatures [49]. Therefore, the new contribution in this work is the utilization of the three key points' transition from STC to non-STC and the application of the proposed algorithm for external conditions’ prediction. Hence, only four equations are required instead of seven. The used equations to switch to the external conditions are given in the system of equation (26) [52].

$$\{\begin{array}{l}{\mathrm{I}}_{\text{sc}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{I}}_{\text{sc},\text{STC}}\times \left(1+{\mathrm{K}}_{\text{isc}}\left(\mathrm{T}-{\mathrm{T}}_{\text{STC}}\right)\right)\times \left(\frac{\mathrm{G}}{{\mathrm{G}}_{\text{STC}}}\right)\\ {\mathrm{I}}_{\mathrm{m}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{I}}_{\mathrm{m},\text{STC}}\times \left(1+{\mathrm{K}}_{\text{im}}\left(\mathrm{T}-{\mathrm{T}}_{\text{STC}}\right)\right)\times \left(\frac{\mathrm{G}}{{\mathrm{G}}_{\text{STC}}}\right)\\ {\mathrm{V}}_{\mathrm{m}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{V}}_{\mathrm{m},\text{STC}}\times \left(1+{\mathrm{K}}_{\text{Vm}}\left(\mathrm{T}-{\mathrm{T}}_{\text{STC}}\right)\right)\times {\left(\frac{\mathrm{G}}{{\mathrm{G}}_{\text{STC}}}\right)}^{{\mathrm{\alpha }}_{\text{Vm}}}\\ {\mathrm{V}}_{\text{oc}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{V}}_{\text{oc},\text{STC}}\times \left(1+{\mathrm{K}}_{\text{Voc}}\left(\mathrm{T}-{\mathrm{T}}_{\text{STC}}\right)\right)\times {\left(\frac{\mathrm{G}}{{\mathrm{G}}_{\text{STC}}}\right)}^{{\mathrm{\alpha }}_{\text{Voc}}}\end{array}$$ (26)

The significant improvement in parameter extraction for real weather conditions lies in the ability to calculate all characteristics at any temperature and irradiance level without the need for iterative processes. Instead of initializing R_s through iterations, the extracted value of R_s for STC is utilized as the initial value for non-STC calculations. The choice done for the initial value of R_s is not arbitrary. In fact, variations in weather conditions do not significantly affect the values of the series resistance. Therefore, several works in the literature assume that the value of R_s under external conditions is equal to its value at standard test conditions [30,51,53]. However, this assumed equality is not always accurate [25,43]. Therefore, in this work, R_s is initialized by R_s,STC for non-STC conditions.

The same strategy is adopted for initializing the search for the two ideality factors. Referring to the literature, the values of n₁ and n₂ are considered slightly dependent on temperature and irradiance levels and thus they were considered independent of weather conditions [30,51,53]. However, these two last assumptions have no physical proof. In this work, to avoid the used assumption in the literature, the values of n₁ and n₂ are extracted at any weather conditions using the proposed algorithm where n₁ and n₂ are initialized by their values extracted at STC.

Table 1 gives the steeps to be followed for predicting I–V characteristics at external conditions.

Table 1.

Proposed algorithm for external conditions.

	Algorithm for external conditions
1	Inputs: Ns, T, G, Isc,STC, Im,STC, Vm,STC, Voc,STC, Kisc, KVoc, Kim, and KVm
2	Calculate Isc(G,T), Im (G,T), Vm (G,T), and Voc (G,T) using Eq. 26
3	Initialize Rs using Rs = Rs,STC Initialize n1 using n1 = n1,STC and n2 using n2 = n2,STC
4	Solve Eq. (21) using nonlinear equation solver function “fsolve” of MATLAB to get the values of n1, n2, and Rs
5	Calculate the values of Rp, I02, I01, and IL using the analytical expressions Eq. (18), Eq. (10), Eq. (6), and Eq. (7), respectively
6	Calculate ‘I’ using the Newton Raphson method

To assess the accuracy of the proposed switching technique, it is compared against other methods introduced in the literature. These include the equations used by Hejri et al. [53] and Chennoufi et al. [25], the adopted equations by Yahya-Khotbehsara et al. [30], the technique used by Zhuo et al., and the procedure proposed by Orioli et al. [43].

4. Results and discussion

The main purpose of this section is to validate the performance of the new modeling method in this work. To that end, the suggested method is implemented for seven PV generators featuring diverse technologies, by calculating the seven parameters of each one, predicting their electrical current-voltage curves, and subsequently comparing the projected characteristics with the measured and experimental data. These chosen cases of study are widely employed in the literature.

Initially, only the key-points of the I–V characteristics of six PV panels are used to calculate the seven parameters, using the proposed technique under STC, and test its performance against some of the extensively reviewed hybrid algorithms of the literature. Subsequently, for decent comparison, the whole experimental I–V characteristic of the RTC France PV cell is employed to test the method's rapidity and accuracy. Then, the results are compared with ten recent evolutionary optimization algorithms.

Table 2 gives the specific parameters of each PV generator.

Table 2.

PV generators specifications and operating conditions.

	Ns	G (W/m2)	T (°C)	Isc(A)	Vm(V)	Im(A)	Voc(V)
Multi-crystalline RTC France	1	1000	33	0.7605	0.4590	0.6755	0.5727
Multi-crystalline KD245GH-4FB2	60	1000	25	8.91	29.8	8.23	36.9
Multi-crystalline S75	36	1000	25	4.7	17.6	4.26	21.6
Mono-crystalline SQ150PC	72	1000	25	4.80	34.0	4.40	43.4
Mono-crystalline SM55	36	1000	25	3.45	17.4	3.15	21.7
Thin Film ST40	36	1000	25	2.68	16.6	2.41	23.3
Thin Film ST36	42	1000	25	2.68	15.8	2.28	22.9

With the aim of studying the proposed method's accuracy, in addition to the Root Mean Square Error (RMSE), two additional statistical indicators are selected. The Individual Absolute Error (IAE) is the second selected metric, and it is defined as shown in equation (27):

$$\text{IAE}\left(\mathrm{A}\right)=\left|{\mathrm{I}}_{\mathrm{i},\mathrm{M}}-{\mathrm{I}}_{\mathrm{i},\mathrm{C}}\right|$$ (27)

The third indicator is the Normalized Root Mean Square Error (NRMSE), given as shown in equation (28):

$$\text{NRMSE}(\%)=\frac{{\left[\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{I}}_{\mathrm{i},\mathrm{M}}-{\mathrm{I}}_{\mathrm{i},\mathrm{C}}\right)}^2\right]}^{\frac{1}{2}}}{\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{I}}_{\mathrm{i},\mathrm{M}}}\times 100$$ (28)

4.1. Predicting characteristics at standard test conditions based on remarkable points’ values

Intending to validate accuracy at STC (G = 1000 W/m², T = 25 °C, and AM = 1.5), the proposed method is applied to six PV modules of different technologies to extract their seven parameters. Using the proposed method, the calculated values of the seven parameters for the two selected multi-crystalline PV modules, along with their corresponding RMSE and NRMSE, are compared with those introduced in Refs. [19,20,24,25,30], and [43], as shown in Table 3. As it can be viewed from the table, the method in this work provides the minimum values of the two statistical metrics for both PV modules. For the KD245GH-4FB2 PV panel, the NRMSE value provided by the current modeling technique does not exceed 0.64 %, while the method in Ref. [24] yields the highest value, reaching 1.11 %. The approach in Ref. [25] yields 0.95 %; however, the rapidity of convergence in this method is compromised by the use of two successive iterative processes to improve precision. To further improve accuracy and prevent negative values, the authors in Ref. [43] employed three iterative processes. However, the increase in the number of iterative processes results in longer execution times. For the S75 PV panel, the numerous approximations used in Refs. [19,20], and [30] to determine the two reverse saturation currents, the two ideality factors, and the light-generated current values led to a corresponding normalized root mean square error (NRMSE) exceeding 2.58 % and reaching 3.51 %. In contrast, the proposed approach ensures superior accuracy, with an NRMSE value not exceeding 0.79 %, without relying on assumptions.

Table 3.

Extracted parameter values for the selected multi-crystalline PV panels versus values obtained using other numerical methods in the literature.

	Multi-crystalline
Module	Keyocera KD245GH-4FB2				Shell S75
Method	Kumar et al. [24]	Orioli et al. [43]	Chennoufi et al. [25]	Proposed	Ishaque et al. [19]	Khezzar et al. [20]	Yahya-Khotbehsara et al. [30]	Proposed
IL(A)	8.9202	8.9337	8.931164	8.928886	4.7	4.7	4.7	4.708956
I01(A)	2.7637 × 10−10	1.8084  × 10−10	3.906668 × 10−10	6.195240  × 10−10	3.39  × 10−10	3.388  × 10−10	5.68 × 10−9	3.477880  × 10−11
I02(A)	6.4751 × 10−6	1.9062  × 10−10	1.094485 × 10−6	4.632532  × 10−7	3.39  × 10−10	3.388  × 10−10	1.47 × 10−8	1.810921  × 10−6
n1	0.9954	0.9735	1.0059	1.024886	1	1	1.14	0.913813
n2	2.0044	1.5649	1.9941	2.057782	1.3	1.3	2.6	1.986519
Rs(Ω)	0.2834	0.3192	0.2862	0.299567	0.27	0.2541	0.23	0.363663
Rp(Ω)	247.9658	120.1628	120.4951	141.3299	84.38	78.641	103.58	190.8942
RMSE	0.07356	0.04855	0.06272	0.042610	0.089657	0.107319	0.121921	0.027115
NRMSE (%)	1.11719	0.73741	0.95250	0.641409	2.584012	3.093053	3.513894	0.781474

Table 4 summarizes the calculated seven parameters obtained using the proposed modeling approach for the two mono-crystalline PV panels (SQ150-PC and SM55) operating under STC. A comparison is made with methods introduced in Refs. [20,25,26,30,31], and [43]. From the table, it is evident that the new method demonstrates high precision for both mono-crystalline PV modules compared to alternative methods from the literature. The results in Table 4 show that the RMSE values produced by the approach are the lowest for both panels. For the SQ150-PC panel, the method proposed in Ref. [43], in addition to employing three sequential slow iterative processes, yields the highest RMSE value, reaching 0.12 A. Meanwhile, the method introduced in Ref. [25] produces an RMSE value of 0.075 A, while the suggested technique keeps the RMSE value below 0.028 A. Concerning the SM55 PV panel, the method presented in Ref. [30] demonstrates the poorest accuracy among the compared methods, resulting in a normalized root mean square error of 3.11 %. This discrepancy can be attributed to the arbitrary selection of the two ideality factors and the underlying assumptions made in Ref. [30]. Similarly, the assumption of identical ideality factors in Ref. [26] has led to both n₁ and n₂ being approximately equal to 1, thereby influencing the accuracy of the model. However, that contradicts the physical significance of n₂ which models the recombination phenomenon occurring inside the semiconductor materials of PV generators and which tends toward 2 [49]. Moreover, the values of the two reverse saturation currents’ values supplied in Ref. [26] are equal, yet, this goes against the fact that the second reverse saturation current models electron-hole recombination and takes the values considerably high than I₀₁ [25,49]. Consequently, the method proposed in Ref. [26] results in an NRMSE of 2.27 % for the SM55 PV module. Similarly, by making the same assumption regarding the reverse saturation currents and using fixed values of n₁ and n₂ for all PV panel technologies, the method in Ref. [31] yields an NRMSE of 1.88 %. Employing additional approximations, the NRMSE value obtained in Ref. [30] reaches 3.11 %, while the proposed method in this study achieves an NRMSE of no more than 1.18 %.

Table 4.

Extracted seven parameters for the two selected mono-crystalline PV modules versus the extracted values using other numerical methods of literature.

	Mono-crystalline
Module	Shell SQ150-PC				Shell SM55
Method	Khezzar et al. [20]	Orioli et al. [43]	Chennoufi et al. [25]	Proposed	Lun et al. [26]	Yahya-Khotbehsara et al. [30]	Zhuo et al. [31]	Proposed
IL(A)	4.8	4.8054	4.813156	4.805540	3.46188	3.45	3.462371	3.457012
I01(A)	3.1068  × 10−10	1.8765 × 10−19	5.016823 × 10−9	8.86314  × 10−11	1.5529  × 10−10	2.75  × 10−8	2.15931  × 10−10	1.761321  × 10−11
I02(A)	3.1068  × 10−10	5.9857 × 10−7	1.975384 × 10−8	6.05987  × 10−6	1.55243  × 10−10	7.11  × 10−8	2.15931  × 10−10	4.77548  × 10−6
n1	1	0.730762	1.144401	0.955328	1.01658	1.26	1	0.904542
n2	1.3	1.477593	1.355599	1.971367	1.01658	2.84	2	2.023558
Rs(Ω)	0.9	0.4981	0.738099	0.841687	0.49696	0.36	0.506391	0.468462
Rp(Ω)	275	444.1415	269.2969	730.5390	144.28908	233.46	141.2190	230.702
RMSE	0.04299	0.120658	0.075005	0.027831	0.057689	0.078845	0.047789	0.029956
NRMSE (%)	1.22475	3.43711	2.136620	0.786560	2.275567	3.110064	1.885064	1.181645

Table 5 gives a comparison between the calculated seven parameters’ values for two thin-film PV modules operating under STC, when employing the novel method and the introduced methods in Refs. [[19], [20], [21],30], and [31]. On one hand, assuming the equalities I_L=I_sc and I₀₁ I₀₂, coupled with the assumption used to calculate n₁ and n₂, the methods in Ref. [20] yield the highest NRMSE value (5.55 %) for the ST40 PV module. Similarly, employing the same approximation to obtain the I_L value and directly using the values of n1 and n2 factors from two different works introduced in Refs. [42,45], the proposed technique in Ref. [30] results in an NRMSE value exceeding 3.23 %. For the same PV panel, the method proposed in Ref. [21] yields an NRMSE of 1.47 %, while the method proposed in the current work provides an NRMSE below 0.6 %. On the other hand, when applying the method from Ref. [30] to calculate the values of the seven parameters for the thin-film ST36 PV panel, the values of n₁ and n₂ were adopted from Refs. [42,45]. Consequently, the technique in Ref. [30] results in the highest RMSE value, exceeding 0.15 A. This indicates that employing certain parameter values already calculated simultaneously with the others, and then using them to recalculate the remaining parameters based on a different method, may not always be practical. In the study conducted in Ref. [21], it assumes that the shunt resistance is infinitely high and neglects the series resistance, considering it to be zero. However, these assumptions transform the model of real PV generators into an idealized one, representing PV generators without any joule losses. Consequently, the resulting model becomes unsuitable for high-accuracy modeling. This inference is supported by the RMSE value provided in Ref. [21], which exceeds 0.1 A. Conversely, by avoiding all previous approximations, the proposed method in the current work achieves the lowest RMSE value, equal to 0.069 A.

Table 5.

Extracted parameter values for the two selected thin-film PV modules versus the extracted values using other numerical methods of literature.

	Thin-film
Module	Shell ST40				Shell ST36
Method	Khezzar et al. [20]	Yahya-Khotbehsara et al. [30]	Zhuo et al. [31]	Proposed	Ishaque et al. [19]	Babu et al. [21]	Yahya-Khotbehsara et al. [30]	Proposed
IL(A)	2.68	2.68	2.707725	2.692126	2.68	2.68	2.68	2.687284
I01(A)	3.0641 × 10−11	1.65 × 10−7	2.97551  × 10−11	1.530462  × 10−12	1.63 × 10−9	6.56  × 10−3	8.3173  × 10−6	1.014083  × 10−12
I02(A)	3.0641 × 10−11	4.28 × 10−7	2.97551  × 10−11	5.552361  × 10−6	1.63 × 10−9	1.70  × 10−2	2.1548  × 10−5	1.061637  × 10−4
n1	1	1.58	1	0.918298	1	3.53	1.65	0.908358
n2	1.2	2.15	2	2.059283	1.3	88.83	2.10	2.097903
Rs(Ω)	1.76	1.40	1.747487	1.344099	1.86	0	1.0940	1.231420
Rp(Ω)	211.7	1350.3	168.91734	297.8378	96.76	∞	158.81	475.8942
RMSE	0.109606	0.06393	0.094321	0.011699	0.069361	0.106207	0.158847	0.013416
NRMSE (%)	5.550113	3.237442	4.775389	0.591853	3.742739	5.730988	8.571439	0.723930

The measured and the camputed I–V characteristics for the multi-crystalline S75 PV module, the mono-crystalline SM55 PV module, and the thin-film ST36 PV module, when employing the extracted seven parameters, under STC, using the new modeling approach evaluated against the other methods of the literature, are represented in Fig. 3(a), Fig. 4(a), and Fig. 5(a), respectively. The estimated curves are visualized with the lines, and the measured data illustrated with the black markers. As it can be seen, around the peak power points, the proposed approach assures the best prediction's accuracy compared to the other methods of literature. Moreover, the new technique guarantees the most accurate estimation of the integral I–V curves. The last observation can be also seen in Figs. 3(b)–.4(b) and 5(b), providing the calculated individual absolute errors for the proposed method and the selected approaches of literature used for the S75, the SM55, and the ST36 PV modules, respectively. As illustrated in Fig. 3(b), the absolute errors resulting from the proposed method consistently exhibit the least values across the majority of the I–V data relative to other modeling methods of the literature. Furthermore, it becomes evident from the three figures that the proposed approach yields the least errors for the three remarkable points. That refers to the employement of error reduction for the whole of the key points. Thus, the method can be practical for dynamic applications such as the maximum power point tracking applications.

Fig. 3.

(a) Stimulated I–V characteristics of the S75 photovoltaic panel relative to the measured data and three other methods. (b) Corresponding absolute errors.

Fig. 4.

(a) Stimulated I–V characteristics of the SM55 PV photovoltaic panel relative to the measured data and three other methods. (b) Corresponding absolute errors.

Fig. 5.

(a) Stimulated I–V characteristics of the ST36 photovoltaic panel relative to the measured data and three other methods. (b) Corresponding absolute errors.

4.2. Prediction of electrical characteristics for various weather conditions

In this subsection, the suggested simulating method is used to estimate the I–V characteristics of the PV models KD245GH-4FB2, SQ150-PC, and ST40 when subjected to external conditions. These estimations are then compared with other techniques presented in the literature. To derive the I–V curves under non-standard test conditions, researchers in Ref. [25] utilized equations from Ref. [53], enabling the calculation of the seven parameters for non-STC based on their values extracted at STC. This approach, employing the band-gap energy value and adjusted parameters, extracted independently of the PV panels' technology and doping, is a widely used technique in the literature to transfer the seven parameters' values and calculate the characteristics at non-STC [30,31]. However, assuming the same dependence of the band gap energy to the temperature variations for the whole PV technologies can highly influence the characteristics' calculation, especially for the highest temperatures [49]. In this study, the I–V characteristics are determined by transferring the three remarkable points to external conditions, followed by the application of the proposed algorithm outlined in subsection (3.2.1) to obtain the parameter values. To that end, Table 6 summarizes the required parameters for the calculation got from the PV modules datasheet. α_Vm and α_Voc are two adjustment parameters derived from available graphs on the manufacturer datasheet. Furthermore, to enhance the accuracy of estimation, the temperature coefficients of the three key points (K_isc, K_Voc, K_im, and K_Vm) are also extracted from the available graphs on the PV panels’ datasheet.

Table 6.

Required parameters for the calculation of the remarkable points at non-standard conditions.

	Kisc (A/°C)	KVoc (V/°C)	Kim (A/°C)	KVm (V/°C)	αVm	αVoc
Multi-crystalline KD245GH-4FB2	0.4669 × 10−3	−3.642 × 10−3	0.2362 × 10−3	−4.897 × 10−3	0.03826	0.04271
Mono-crystalline SQ150-PC	0.4237 × 10−3	−3.319 × 10−3	−0.3149 × 10−3	−4.357 × 10−3	0.02525	0.05512
Thin Film ST40	0.468 × 10−3	−4.274 × 10−3	−0.8534 × 10−3	-5.477 × 10−3	0.0436	0.08798

Based on the newly proposed approach, the electrical output characteristics of three PV modules (KD245GH-4FB2, SQ150-PC, and ST40) are calculated for various irradiance levels at T = 25 °C. These characteristics are then plotted in Fig. 6(a), Fig. 7(a), and Fig. 8(a) using red lines. As it can be observed, the calculated I–V characteristics based on the proposed algorithm cross at all the measured data points for the whole irradiance levels and for the three PV panels. The accuracy of prediction for various levels of irradiance is tested against the measured data by calculating the root mean squared errors of the proposed method and comparing them to other methods of literature. The results for the three PV panels (KD245GH-4FB2, SQ150-PC, and ST40) are depicted in Fig. 9(a), Fig. 10(a), and Fig. 11(a), respectively. As shown in the figures, the proposed approach demonstrates superior predictive accuracy across various irradiance levels, with the lowest RMSE values compared to the methods in Refs. [25,30,31], and [43]. For the multi-crystalline panels, the RMSE values corresponding to the new approach do not exceed 0.084 A in the worst case, while the method in Ref. [25] yields 0.141 A, and the technique introduced in Ref. [43] achieves 0.18 A. For the mono-crystalline panel, the RMSE of the proposed method does not exceed 0.031 A, while for the thin-film module, it remains below 0.021 A. In comparison, methods [25,43] yield RMSE values of 0.121 A and 0.075 A, respectively, for SQ150-PC, and methods [30,31] achieve RMSE values of 0.067 A and 0.071 A for ST40. Similarly, for temperature variations at various levels and G = 1 kW/m², the I–V curves for KD245GH-4FB2, SQ150-PC, and ST40 are predicted using the new method and compared with other methods from the literature, as shown in Figs. 6(b)–7(b) and 8(b), respectively. The corresponding RMSE values calculated using various methods for temperature variations are shown in Figs. 9(b)–10(b) and 11(b) for KD245GH-4FB2, SQ150-PC, and ST40, respectively. As evident from the figures depicting the I–V curves, the current technique ensures the highest accuracy when considering the influence of temperature on the I–V curves. This observation is reinforced by the RMSE values obtained, where the proposed method consistently yields significantly lower values for all three PV modules across various temperature levels. In the worst-case scenario, the RMSE does not exceed 0.097 A, whereas [43] yields 0.186 A and [25] yields 0.44 A.

Fig. 6.

(a) Stimulated and measured I–V curves for the KD245GH-4FB2 PV module operating under different levels of G and T = 25 °C. (b) Simulated I–V characteristics for the KD245GH-4FB2 operating under diverse T levels and G = 1 kW/m².

Fig. 7.

(a) Measured and stimulated I–V curves for the SQ150-PC PV module operating under different levels of G and T = 25 °C. (b) Simulated I–V characteristics for SQ150-PC operating under diverse T levels and G = 1 kW/m².

Fig. 8.

(a) Measured and stimulated I–V curves for the ST40 PV module operating under different levels of G and T = 25 °C. (b) Simulated I–V characteristics for the ST40 operating under diverse T levels and G = 1 kW/m².

Fig. 9.

(a) Calculated RMSE for the KD245GH-4FB2 PV panel at various levels of G and T = 25 °C using the new method compared with the existing methods in the literature. (b) Calculated RMSE for KD245GH-4FB2 PV module at diverse T levels and G = 1 kW/m.².

Fig. 10.

(a) Calculated RMSE for the SQ150-PC PV panel at various levels of G and T = 25 °C using the new method compared with the existing methods in the literature. (b) Calculated RMSE for SQ150-PC PV module at diverse T and G = 1 kW/m.².

Fig. 11.

(a) Calculated RMSE for the ST40 PV panel at various levels of G and T = 25 °C using the new method compared with the existing methods in the literature. (b) Calculated RMSE for the ST40 PV module at diverse T levels and G = 1 kW/m.².

4.3. Application of the proposed algorithm for optimization

Within this subsection, the effectiveness of the suggested method in this work is evaluated against some of deterministic and heuristic-based optimization algorithms. For adequate comparison, the proposed algorithm is invested and used for the optimization.

To prove the performance of the suggested modeling technique as an optimization algorithm, the investigation is based on the experimental I–V curves of the RTC solar cell. The extracted values of the seven parameters for the RTC solar cell, the corresponding root mean square errors, and the time needed to converge, compared with some of the latest meta-heuristic algorithms are shown in Table 7.

Table 7.

Comparing the seven extracted values of the parameters and the corresponding convergence time for the RTC PV cell with alternative optimization algorithms.

	IL(A)	I01(A)	n1	I02(A)	n2	Rs(Ω)	Rp(Ω)	RMSE (A)	Convergence time (s)
BMA [32]	0.7605	1.0  × 10−9	1.3839	1.0 × 10−5	1.9906	0.0155	100.000	8.3049  × 10−3	1.4608
MLBSA [33]	0.7608	2.2728  × 10−7	1.4515	7.3835  × 10−7	2	0.0367	55.4612	9.8249  × 10−4	401.25
IJAYA [34]	0.7601	05.00  × 10−9	1.2186	07.509  × 10−7	1.6247	0.0376	77.8519	9.8293  × 10−4	393.60
FPSO [35]	0.76078	2.2731  × 10−7	1.45160	7.2786  × 10−7	1.99969	0.036737	55.3923	9.8253  × 10−4	Not reported
WDO [41]	0.7608	2.9901  × 10−7	1.5443	1.2078  × 10−7	1.4551	0.0354	44.6653	1.6812  × 10−3	Not reported
ITLBO [36]	0.7608	2.260  × 10−7	1.4510	7.493  × 10−7	2.0000	0.0367	55.4854	9.8248  × 10−4	6.60
IGSK [37]	0.76078	7.493  × 10−7	1.45102	2.260  × 10−7	2.000	0.03674	55.4854	9.8248  × 10−4	14.08
WHHO [38]	0.76078094	2.28574  × 10−7	1.451895	7.27182  × 10−7	2.0000	0.03672887	55.4264328	9.82487  × 10−4	Not reported
IMPA [39]	0.760781	2.25974  × 10−7	1.45102	7.49348  × 10−7	1. 9999	0.03674	55.48544	9.82485  × 10−4	5.88
TLABC [40]	0.76081	4.2394  × 10−7	1.9075	2.4011  × 10−7	1.4567	0.03667	54.66797	9.8414  × 10−4	28.01
Method in Ref. [49]	0.760997	5.40081 × 10−8	1.33326	5.11208 × 10−6	2.163439	0.038877	59.86981	8.10676 × 10−4	0.343648
Proposed method	0.760929	1.241575  × 10−7	1.394539	4.676349  × 10−5	3.242891	0.038339	70.97126	7.971465  × 10−4	0.243023

Table 7 demonstrates that the proposed optimization technique, based on the double-diode model, ensures the highest level of precision in estimating the I–V characteristics of PV generators. This enhanced precision was confirmed through the computed RMSE values for the RTC solar cell, where the new model yielded the lowest value compared to other advanced optimization algorithms in the literature. Moreover, the high precision of estimation is obtained within 0.243 s, which presents the lowest execution time that reduces the MLBSA and IJAYA's calculation times by almost 99.94 %. The meta-heuristic algorithms, in general, suffer from their high execution time caused by the pure random selection of the initial populations, in addition to the influence of the choice of the required predefined variation intervals and the additional parameters related to these algorithms. From the table, it's evident that BMA offers the highest speed of convergence compared to other chosen methods in the literature, but it still takes six times longer than the proposed method. Furthermore, despite the low calculation cost of BMA, it results in an RMSE value more than ten times higher than that provided by the suggested method. The results indicate that the introduced technique in this work guarantees the highest rapidity of convergence and assures the best precision of prediction without using any additional parameters such as the imposed ones by the meta-heuristic optimization algorithms and without requiring any predefined bounds of the parameters' variations. The proposed technique's rapidity and precision make it very useful for the development of PV simulation tools and dynamic applications such as the MPPT applications.

Fig. 12(a) compares the obtained I–V characteristics using the proposed method with the data collected experimentally for the RTC solar cell. Fig. 12(b) gives the corresponding absolute errors to the proposed method, which does not exceed 2.02 mA.

Fig. 12.

(a) Measured and simulated I–V characteristics for the RTC France solar cell. (b) Corresponding absolute errors.

5. Conclusion

In this paper, an accurate and adaptable method is introduced to establish the seven electrical parameters of the two-diode circuit modeling PV cells and modules. The extraction is done using a hybrid approach; four parameters (I_L, I₀₁, I₀₂, and R_p) are calculated based on analytical equations extracted using a new fill-factor expression. Then, the three parameters (n₁, n₂, and R_s) are calculated in two different ways depending on the available information about PV generators. First, the prediction uses only the accessible information on the modules' manufacturer datasheet and is then evaluated across six PV modules of various technologies. The results have established the high performance of the proposed approach by yielding the minimum values of the selected statistical metrics against the widely analyzed and reviewed numerical methods of the literature. This makes the first proposed technique very useful for performance assessment. Additionally, a new technique is proposed and tested to predict the output electrical characteristics of PV generators under non-standard test conditions. The effectiveness of this technique is demonstrated across various temperature and irradiance levels by comparing its performance to other techniques introduced in the literature. Additionally, the proposed algorithm is invested in and adapted for an optimization process. Its accuracy and speed are measured against some of the latest evolutionary algorithms, which typically rely on additional meta-heuristic parameters related to the optimization algorithms used. As a result, the method presented in this work achieves the lowest RMSE value and significantly reduces the required convergence time. Hence, this approach can prove highly beneficial for developing simulation software for PV generators and enhancing hardware tools.

Data availability statement

Data will be made available on request.

Funding

This project is funded by King Saud University, Riyadh, Saudi Arabia.

CRediT authorship contribution statement

Kawtar Tifidat: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. Noureddine Maouhoub: Validation, Supervision, Investigation, Formal analysis, Conceptualization. Fatima Ezzahra Ait Salah: Visualization, Validation, Investigation. S.S. Askar: Visualization, Validation, Funding acquisition. Mohamed Abouhawwash: Writing – review & editing, Visualization, Validation, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.