A novel analytical approach for accurate photovoltaic model parameters identification

Abstract

Precise assessment of photovoltaic (PV) parameters is crucial for enhancing solar system performance, ensuring reliable energy predictions, and improving efficiency. Estimating the PV model parameters remains fraught with theoretical and practical issues, as evidenced by the continuous emergence of various optimization techniques to address the shortcomings of traditional approaches. This paper propels this domain forward by representing an analytical-derived solution that concurrently eradicates iteration-dependent convergence problems associated with optimization techniques with enhanced estimation accuracy. The presented analytical technique depends on data derived from the I–V characteristics, circumventing the need for approximations. The approach methodically divided the I–V curve into three separate operating zones: the high-current zone, the maximum power zone, and the high-voltage zone. Each zone yields distinct equations using derivative-based analysis, clumping in a fully specified system of five equations. This technique facilitates the accurate extraction of the five essential single-diode model (SDM) PV parameters by generating five equations for these unknowns. The suggested analytical approach was validated using experimental data for three different PV panels/cells: R.T.C France solar cell, PWP201, and ET-M53690W PV module. The efficacy of this methodology was rigorously assessed, exhibiting substantial concordance with empirical I–V characteristics and being compared to seven modern optimization strategies. The findings demonstrate the enhanced precision of the proposed analytical method compared to optimization-based techniques.

Introduction

The increasing reliance on renewable energy sources to satisfy global energy demands is driven by their cost-effectiveness, widespread availability, and minimal adverse environmental effects (Osman et al., 2023). Within this framework, solar photovoltaic (PV) energy has emerged as a pivotal research area focused on enhancing efficiency (Al-Shahri et al., 2021; El Hammoumi et al., 2022). The performance of PV systems is highly sensitive to environmental factors, including solar irradiance and temperature. This dependence underscores the need to develop precise modeling frameworks that maximize energy yield and system reliability (Bupi et al., 2021).

Accurate modeling of PV model parameters is essential for guaranteeing the reliability and efficiency of PV system design, operation, and performance evaluation. Precise modeling of essential electrical characteristics, namely, photocurrent $I_{ph}$ , saturation current $I_S$ , ideality factor n, series resistance $R_S$ , and shunt resistance $R_{Sh}$ allows PV models to depict real-world performance authentically under various operating situations. This precision is essential for maximizing energy output, augmenting the efficacy of maximum power point tracking (MPPT), and refining system-level simulations.

Due to the nonlinear nature of the equations, determining the unidentified parameters of photovoltaic models presents significant challenges. This complexity has garnered considerable interest from researchers (Abdel-Basset et al., 2021; Ebrahimi et al., 2019). Various solutions have been suggested in the literature to determine these unknown parameters (Zhang et al., 2023). These methodologies can be categorized into four main types: analytical, numerical, optimization, and hybrid methods.

Analytical methods for PV modeling typically derive I–V characteristics using key parameters, including short-circuit current, open-circuit voltage, maximum power point current, and voltage. These approaches are computationally efficient, relying on the solution of explicit mathematical equations. For instance Sevillano-Bendezú et al. (2024) assess the effectiveness of analytical techniques for extracting unknown parameters in the SDM. However, these methods often rely on simplifying assumptions, such as uniform operating conditions and idealized parasitic resistance losses, which can compromise accuracy under variable environmental conditions. To mitigate these constraints, Syed et al. (2024) introduce an enhanced analytical technique for the DDM model, necessitating seven parameters obtained from three separate I–V curve regions: the linear segment and two exponential segments. Although analytical approaches provide simplicity, their dependence on approximations or predetermined I–V data may limit the precision of parameter extraction. Consequently, numerical approaches are sometimes used as a more reliable option for resolving PV model equations.

In Elhammoudy et al. (2023b), an efficient numerical approach based on dichotomy is employed to determine unknown parameters for the SDM model. This method achieves optimal parameters through an iterative process. Similarly, Newton-Raphson techniques are utilized to ascertain unknown parameters for the SDM (Adak et al., 2022). In Gao et al. (2016), a Lambert W-function-based exact representation for a physics-based DDM model of solar cells employs a restart-based bound-constrained Nelder-Mead (rbcNM) algorithm, alongside the reported Rcr-IJADE algorithm to get optimal parameter values for DDM. While these techniques are computationally intensive, they often yield higher precision by progressively minimizing error with each iteration.

Alternative approaches utilize optimization techniques that reformulate the parameter identification problem as an optimization problem, employing metaheuristic algorithms. These algorithms are user-friendly due to their applicability in numerous complex optimization problems. In most studies, the root mean square error (RMSE) is used as the fitness function to resolve the issue. Various meta-heuristics have been successfully applied to identify unknown parameters of three types of PV cells and PV modules. For accurately determining the parameters of different PV cells and modules, an advanced spherical evolution method based on a novel dynamic sine-cosine mechanism (DSCSE) was proposed in Zhou et al. (2021). Other notable algorithms include the enhanced multistrategy sine–cosine algorithm (ESCA) in Zhou and Shang( 2024), a Whippy Harris Hawks Optimization (WHHO) (Naeijian et al., 2021), meta-heuristic heap-based optimizer (HBO) (AbdElminaam et al., 2022), Drunken Adaptive Walking Chaotic Wolf Swarm (DCWPA) (Wu et al., 2022), particle swarm optimization (PSO) (Ahmed et al., 2022), Ranking Teaching-Learning-Based Optimization (RTLBO) (Yu et al., 2023), Comprehensive Learning Rao-1 Optimization (CLRao-1) (Farah et al., 2022), Improved Arithmetic Optimization (IAOA) algorithm in (Abbassi et al., 2022), a Bio-Dynamics Grasshopper Optimization Algorithm (BDGOA) (Jabari et al., 2024), Whale Optimization Algorithm (WOA)-based meta-heuristic (Yang et al., 2024), Enhanced Prairie Dog Optimizer (En-PDO) (Izci et al., 2024), Drone Squadron Optimization (DSO) (Kumari et al., 2024), and Improved Incorporating the Subtraction-Average Snake Optimization (ISASO) algorithm in Mai et al. (2024).

These meta-heuristic algorithms enhance accuracy in solving the PV parameter estimation optimization issue, even in the face of changes in meteorological conditions. For instance, in Zhang et al. (2024), a novel algorithm was suggested to improve accuracy under changing weather conditions. For instance, in Smaili et al. (2024), a novel, innovative version of the enhanced artificial rabbit optimization (EARO) algorithm was designed to investigate the issue of identifying PV module characteristics for the TDM model, while considering three distinct real-world PV modules. However, these meta-heuristic algorithms necessitate significant computational effort. That is why in Aoufi et al. (2023), the authors develop the nested loop biogeography-based optimization—differential evolution optimizer (NLBBODE) to identify PV parameters with reasonable computational effort and minimum execution time, despite the nonlinearity of PV system dynamics and the insufficiency of data. These algorithms are based on random searching of model parameters. The primary issue is the arbitrary parameter initialization. An incorrect choice of parameter initialization can degrade model performance or prevent convergence.

In addition to these methods, hybrid approaches have also been explored for PV parameter identification of different PV cells and modules. Authors in Choulli et al. (2023), Li et al. (2023), and Qais et al. (2019) use hybrid methods through a combination of analytical techniques and optimization algorithms, such as sunflower optimization (SFO), PSO, and artificial hummingbird metaheuristic algorithms, respectively. These methods extract certain parameters of each model using analytical methods, while the remaining parameters are determined using optimization methods. In Elhammoudy et al. (2023a), other hybrid methods are presented, combining a Dandelion Optimizer (DO) metaheuristic algorithm with a numerical method, Newton-Raphson (NR). In Tifidat et al. (2023), a combination of numerical and analytical methods is applied to reduce the number of unknown parameters from five to only two. In Taleshian et al. (2023), hybrid methods between Flexible Improved PSO (FIPSO) and Sequential Quadratic Programming (SQP). Despite their effectiveness, many existing methods exhibit limitations in accuracy due to inherent assumptions or reliance on optimization algorithms. Table 1 presents a comparative analysis of the advantages and limitations of the four PV parameter estimation techniques, highlighting the enhanced efficacy of the proposed method over existing approaches.

Table 1.

PV parameters estimation types review summary.

Type	Accuracy	Time convergence	Computational efficiency	Implementation complexity	Convergence reliability	Needed for initial conditions
Analytical methods (Sevillano-Bendezú et al., 2024; Syed et al., 2024)	Moderate	Very low	High	Low	Medium	No
Numerical methods (Adak et al., 2022; Elhammoudy et al., 2023b; Gao et al., 2016)	Medium	Moderate	Medium	Medium	low	Yes
Optimization methods (Mai et al., 2024; Singla et al., 2024; Zhou and Shang, 2024)	High	High	Moderate	High	Medium	Yes
Hybrid methods (Choulli et al., 2023; Lidaighbi et al., 2022; Premkumar et al., 2021)	High	High	Low	Very High	high	Yes
Proposed method	High	Very low	Very High	Low	High	No

This paper proposes a novel analytical approach designed explicitly to determine the parameters in a SDM of a PV system. Unlike conventional analytical methods, which rely on various approximations, this approach relies solely on empirical data measurements while achieving high accuracy in determining the five key parameters. The proposed method categorizes the I–V curve characteristics into three distinct operational intervals: (a) the current-saturation region, characterized by a nearly constant current; (b) the nonlinear region surrounding the maximum power point (MPP); and (c) the high-voltage linear region nearing open-circuit conditions, as shown in Figure 1.

Figure 1.

Equivalent circuit of SDM PV cell.

This methodology ensures:

High accuracy in parameter estimation through direct utilization of characteristic points along the I–V curve.

Simplicity of the computational technique.

Intrinsic robustness, independent of initial conditions or convergence challenges commonly faced by existing algorithms.

Modeling of a photovoltaic module

SDM model

This section aims to address the PV modeling problem. Then, an electrical model will be used to describe the photovoltaic module. In this respect, the SDM is considered.

The SDM is often exploited to describe PV behavior because of its balance of simplicity and precision. The equivalent circuit of the SDM model is given in Figure 2. This model consists of a current source, a diode, $R_S$ and $R_{Sh}$ .

Figure 2.

Typical $I-V$ characteristic of a SDM PV cell and its repartition into three operational zones.

Unknown parameter description

According to KCL, the equation of the SDM model is defined as:

$$I=I_{Ph}-I_S\left[exp\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}}\right)-1\right]-{\displaystyle \frac{V+R_{S}I}{R_{Sh}}}$$

(1)

where $I_{Ph}$ represents the photocurrent, $I_S$ represents the reverse saturation current, n is the ideality factor of the diode, and $V_T$ stands for the thermal voltage, which can be expressed as:

$$V_T={\displaystyle \frac{KT}{q}}$$

(2)

Where K is the Boltzmann constant (1.3806503 × 10^–23 J/K), q represents the electric charge of the electron (1.60217646 × 10⁻¹⁹C), and T represents the temperature of the solar cell.

Generally, the SDM model includes five undefined parameters (Lidaighbi et al., 2022; Premkumar et al., 2021), namely $I_{Ph}$ , $I_S$ , n, $R_S$ , and $R_{Sh}$ .

Method description

The PV model is known for its complex nonlinear behavior. Then, several previous works have been established based on problematic assumptions and approximations. Presently, a multimodel approach is proposed to address the parameter identification issue of a PV system. Specifically, the I–V nonlinear curve can be divided into several intervals. A model can be involved in each interval. In this respect, it is seen that the I–V characteristic can be divided into three intervals (Figure 1):

First interval $V\in [0,V_1]$ , which corresponds to zone 1, where $V_1$ is the upper voltage limit of zone 1.

In this interval, it is seen that the $I-V$ characteristic can be approximated by a linear curve of negative slope (denoted $-g_1$ ( ${\mathrm{\Omega }}^{-1}$ )). Accordingly, the current I is related to the voltage V by the following expression:

$$I=-g_{1}V+I_0$$

(3)

where $I_0(A)$ is the value of I for $V=0V$ , that is the short-circuit current value $I_{SC}$ (Figure 1).

• Second interval $V\in [V_2,V_{OC}]$ , which corresponds to zone 3, where $V_2$ is the lower voltage limit of zone 3:

The value of the upper bound of this second interval corresponds to the open-circuit voltage $V_{OC}$ , that is the voltage value for $I=0A$ . In this interval, the curve of current I with respect to the voltage V can be approximated by a linear segment of slope $-g_2$ . This means that the relationship between I and V can be expressed as:

$$I=-g_{2}V+I_0^{\mathrm{\prime }}$$

(4)

It is readily seen that equation (4) can be rewritten as:

$$I=-g_2(V-V_{OC})$$

(5)

Using equations (4) and (5), one immediately has $I_0^{\mathrm{\prime }}=g_2{V}_{OC}$ .

Third interval $V\in [V_1,V_2]$ , which corresponds to zone 2:

Assumptions

For any given temperature, the curve of the current I versus the voltage V is assumed to be continuous. During the experiment time, the irradiance and temperature are considered constant. These assumptions are generally satisfied. The shape of the curve in this interval is nonlinear. Therefore, according to the Weierstrass theorem, the $I-V$ curve can be described by a polynomial function (see, e.g., Brouri, 2022), where the current I is related to the voltage V as:

$$I=\sum _{l=0}^m{a}_l{V}^l$$

(6)

• A majorant of Taylor expansion truncation m, given in equation (6), is supposed to be known.

• The number of measurement points in the interval $[V_1,V_2]$ is greater than $m+1$ .

At this stage, the Taylor expansion truncation m is considered known. Practically, the setting issue of the truncation order m will be discussed later.

The objective of the proposed approach is to establish some equations describing the PV behavior within any interval using the analytical expression between the current I and voltage V as in equation (1). Then, based on the set of obtained equations and using data acquisition (set of measurement points $\{(V(K),I(K))\}$ , the model's unknown parameters can be identified.

PV model (Figure 2), or equation (1), there are five unknown parameters: $I_{ph}$ , $I_S$ , n, $R_S$ , and $R_{Sh}$ . The determination of these parameters requires a minimum of five specified equations. One or more equations will be derived from any interval in this context. In this respect, the Taylor series expansion of the exponential function $exp(x)$ in equation (1), where $x=\frac{V+R_{S}I}{n{V}_T}$ , around any value $X_0$ will be developed. One thus has: where $o\left({\left(\frac{V+R_{S}I}{n{V}_T}-X_0\right)}^m\right)$ represents the order of the remainder term in the Taylor expansion. The expansion order m and the point $X_0$ vary concerning the chosen interval.

The value of $X_0$ is assumed to be constant within each given interval ( $[0,V_1]$ , $[V_2,V_{OC}]$ , $[V_1,V_2]$ ) but can vary from one interval to another.

Let N denote the number of measurements and $\{(V(k),I(k));k=1\dots N\}$ the set of measurement points. For clarity, the main steps of the algorithm for selecting the voltages $V_1$ and $V_2$ from the acquired data are summarized in Table 2.

Table 2.

Algorithm for determining the voltages ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ .

Step 1: Initialization step and data acquisition. Collect the measurement data $\{(V(k),I(k));k=1\dots N\}$ . Set $I_{SC}=I(1)$ , $V_{OC}=V(N)$ , Initialize $k=1$ and the slope $g(k)=0(A/V)$ . Set a tolerance D (e.g., 0.05, 0.03).

Step 2: Dynamic slope calculation in zone 1. Increase k ( $k=k+1$ ) and compute the slope $g(k)$ as: $g(k)=\frac{I(k)-I(k-1)}{V(k)-V(k-1)}$

Step 3: Determination of $V_1$ and the slope of zone 1. If $|g(k)|>(1+D)|g(k-1)|$ , then set $V_1=V(k)$ , $g_1=|g(k)|$ , $k_1=k$ , and go to Step 4. Else, return to Step 2.

Step 4: Dynamic slope calculation of zone 2. Increase k and update the slope $g(k)$ : $g(k)=\frac{I(k)-I(k-1)}{V(k)-V(k-1)}$

Step 5: Determination of $V_2$ . Return to Step 4 if the following condition is not satisfied: $(1-D)|g(k-1)|<|g(k)|<(1+D)|g(k-1)|$ Else, set $V_2=V(k)$ , $k_2=k$ , and go to Step 6.

Step 6: Dynamic slope calculation of zone 3. Increase k and compute the slope $g(k)$ as follows: $g(k)=\frac{I(k)-I(k-1)}{V(k)-V(k-1)}$

Step 7: Determination of the slope $g_2$ of zone 3. If $k=N$ , then set the slope $g_2=|g(k)|$ and go to End Step. Else, return to Step 6.

Step 8: End of algorithm.

Identification method of the PV model

Description of the PV behavior in the first interval

Initially, voltage V is regarded as belonging to the interval [0, $V_1$ ], referred to as the first interval. It is seen that when $V\in [0,V_1]$ , the $I-V$ characteristic is a linear curve having a slight negative slope (Figure 1). Furthermore, the value of the term $\frac{V+R_{S}I}{n{V}_T}$ , for $V\in [0,V_1]$ , remains small, i.e., $X_0\approx 0$ in equation (7). For convenience, the order m in equation (7) can be set to 0. One thus gets:

$$exp\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}}\right)=1$$

(8)

By replacing the term $exp\left(\frac{V+R_{S}I}{n{V}_T}\right)$ in equation (1) with its expression in equation (8), the current can be expressed as:

$$I=I_{Ph}-{\displaystyle \frac{V}{R_{Sh}}-{\displaystyle \frac{R_S}{R_{Sh}}I}}$$

(9)

It follows from equation (9) that the expression of the current I with respect $I_{ph}$ and V can be written as:

$$I=-{\displaystyle \frac{1}{R_S+R_{Sh}}V+{\displaystyle \frac{R_{sh}}{R_S+R_{Sh}}{I}_{Ph}}}$$

(10)

The I–V characteristic (Figure 1) indicates that the current $I$ as a function of voltage V, for $V\in [0,V_1]$ , approximates a linear relationship with a negative slope of $-g_1$ , as specified in equation (7). This outcome is corroborated by equation (10), where:

$$g_1={\displaystyle \frac{1}{R_S+R_{Sh}}}$$

(11)

and:

$$I_0=I_{SC}={\displaystyle \frac{R_{sh}}{R_S+R_{Sh}}{I}_{Ph}}$$

(12)

Let us consider $I_1$ , that denotes the current value for $V=V_1$ (Figure 1). Using the $I-V$ characteristic, the values of the slope $g_1$ and short-circuit current $I_{SC}$ can be obtained. Specifically, $g_1$ can be determined as follows:

$$g_1={\displaystyle \frac{I_{SC}-I_1}{V_1}}$$

(13)

and $I_{SC}$ is the current value for $V=0$ . It follows equations (11) and (12) and their estimated $g_1$ and $I_{SC}$ that are given by the $I-V$ characteristic (or data acquisition), two equations can be involved given as follows:

$$\{\begin{array}{c}{\displaystyle \frac{1}{R_S+R_{Sh}}=g_1={\displaystyle \frac{I_{SC}-I_1}{V_1}}}\\ {\displaystyle \frac{R_{Sh}}{R_S+R_{Sh}}{I}_{Ph}=I_{SC}}\end{array}$$

(14)

It is important to observe that the two equations in equation (14) encompass three unknown parameters, $I_{Ph}$ , $R_S$ and $R_{Sh}$ . The two developed equations are inadequate for estimating the unknown parameters. This identification issue involves five unknown parameters (i.e., $I_{Ph}$ , $R_S$ , $R_{Sh}$ , $I_S$ and $n$ ). Consequently, a minimum of five equations is necessary to estimate these parameters.

Description in the second interval

In this part, the voltage V is considered to belong to the interval $[V_2,V_{oc}]$ (Figure 1), denoted the second interval. It follows from the $I-V$ characteristic in the second interval ( $V\in [V_2,V_{oc}]$ ) that the current I decreases proportionally as the voltage V increases (Figure 1). The aim at this stage is to choose the expansion order $m$ in equation (7). It is observed that the $I-V$ characteristic in this interval can be approximated by a linear curve. Then, the expansion order $m$ in equation (7) can be set either to $0$ or $1$ . For convenience, the order m is set to 1. Furthermore, the value of $X_0$ in equation (7) for a high value of voltage, V is not necessarily fixed to 0.

Remark 1.

If the expansion order m is taken more than 1, the curve of current I concerning the voltage V will take a nonlinear form. However, it is well known that the $I-V$ characteristics of the PV module exhibit a linear curve during this second interval (Figure 1). This confirms the choice of expansion order $m=1$ in equation (7).

Following Remark 1, the Taylor expansion in equation (7) around $X_0$ becomes:

$$exp\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}}\right)=exp(X_0)\left(1+\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}-X_0}\right)\right)$$

(15)

where $X_0\ne 0$ and it is noted in this interval $X_{02}$ . By replacing the $exp(.)$ in equation (1) with its expression given in equation (15), the current I in equation (1) becomes as:

$$I=I_{ph}-I_S\left[exp(X_{02})\left(1+{\displaystyle \frac{V+R_{S}I}{n{V}_T}-X_{02}}\right)-1\right]-{\displaystyle \frac{V+R_{s}I}{R_{Sh}}}$$

(16)

Generally, $exp(X_{02})(1-X_{02})$ is much greater than $1$ . Then, the expressions of the current I become:

$$I=I_{ph}-I_{S}exp(X_{02})[1-X_{02}]-I_{S}exp(X_{02}){\displaystyle \frac{V+R_{S}I}{n{V}_T}-{\displaystyle \frac{V}{R_{Sh}}-{\displaystyle \frac{R_s}{R_{Sh}}I}}}$$

(17)

If follows from equation (17) that the current I is correlated with the voltage V as:

$$I=-{\displaystyle \frac{{\displaystyle \frac{1}{R_{Sh}}+{\displaystyle \frac{I_S}{n{V}_T}exp(X_{02})}}}{1+R_S\left({\displaystyle \frac{1}{R_{Sh}}+{\displaystyle \frac{I_S}{n{V}_T}exp(X_{02})}}\right)}\left(V-{\displaystyle \frac{I_{Ph}-I_{S}exp(X_{02})(1-X_{02})}{{\displaystyle \frac{1}{R_{Sh}}+{\displaystyle \frac{I_S}{n{V}_T}exp(X_{02})}}}}\right)}$$

(18)

Since the value of $X_{02}$ is significant in this interval ( $V\in [V_2,V_{oc}]$ ), the value 1 can be neglected in the denominator of I in equation (18). Then, the current can be rewritten as follows:

$$I=-{\displaystyle \frac{1}{R_S}\left(V-{\displaystyle \frac{I_{Ph}-I_{S}exp(X_{02})(1-X_{02})}{{\displaystyle \frac{I_S}{n{V}_T}exp(X_{02})+{\displaystyle \frac{1}{R_{Sh}}}}}}\right)}$$

(19)

The result given by equation (19) shows that the current I versus the voltage V is a linear curve of slope $-\frac{1}{R_S}$ , which confirms the expression of I given in equation (9), where

$$g_2={\displaystyle \frac{1}{R_S}}$$

(20)

and:

$$V_{OC}={\displaystyle \frac{I_{Ph}-I_{S}exp(X_{02})(1-X_{02})}{{\displaystyle \frac{I_S}{n{V}_T}exp(X_{02})+{\displaystyle \frac{1}{R_{Sh}}}}}}$$

(21)

Using the $I-V$ characteristic, the slope $-g_2$ and $V_{OC}$ can be estimated (Figure 1) as follows:

$$g_2={\displaystyle \frac{I_2}{V_{OC}-V_2}}$$

(22)

The value of V is equal to $V_{OC}$ when $I=0$ (Figure 1). Following equations (20) and (21) and their estimations, two equations can be obtained given as:

$$\{\begin{array}{c}{\displaystyle \frac{1}{{\mathrm{R}}_{\mathrm{S}}}={\displaystyle \frac{{\mathrm{I}}_2}{{\mathrm{V}}_{\mathrm{OC}}-{\mathrm{V}}_2}}}\\ {\displaystyle \frac{{\mathrm{I}}_{\mathrm{Ph}}-{\mathrm{I}}_{\mathrm{S}}\exp ({\mathrm{X}}_{02})(1-{\mathrm{X}}_{02})}{{\displaystyle \frac{{\mathrm{I}}_{\mathrm{S}}}{\mathrm{n}{\mathrm{V}}_{\mathrm{T}}}\exp ({\mathrm{X}}_{02})+{\displaystyle \frac{1}{{\mathrm{R}}_{\mathrm{Sh}}}}}}={\mathrm{V}}_{\mathrm{OC}}}\end{array}$$

(23)

By combining the first equation of (23) with the two equations obtained in the first interval, given in equation (14), a system of three equations having three unknown parameters (namely, $R_S$ , $R_{Sh}$ and $I_{Ph}$ ) is obtained. Accordingly, these three parameters can be determined. At this stage, the remaining unknown parameters are $I_S$ and $n$ . It is essential to highlight that there exists just one equation pertaining to these parameters, namely the second equation in equation (23). The former is inadequate for estimating $I_S$ and $n$ . Consequently, an additional equation including $I_S$ and $n$ is necessary to evaluate these values.

Remark 2.

Presently, the PV parameters will be identified for any given temperature (i.e., T is constant). Then, the thermal voltage, $V_T$ is considered constant. Furthermore, it is interesting to note that the same identification approach can be applied to other temperature values.

Description in the third interval

Let us consider that $V\in [V_1,V_2]$ , called the third interval (Figure 1). In this interval, the $I-V$ characteristic cannot be defined by a linear curve. Indeed, it is seen that for $V\in [V_1,V_2]$ the $I-V$ curve is nonlinear (Figure 1). In this respect and following Remark 1, the expansion order $m$ of equation (7) should be taken greater than $1$ (i.e., $m>1$ ). For instance, let us set the order m in equation (7) to 2, one has:

$$exp\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}}\right)=exp(X_{03})\left[1+\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}-X_{03}}\right)+{\displaystyle \frac{1}{2!}{\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}-X_{03}}\right)}^2}\right]$$

(24)

where $X_0$ is noted $X_{03}$ in this interval. By replacing the $exp(.)$ term in equation (1) with its expression in equation (24), the expression of current I in equation (1) can be reformulated as:

$$I=I_{ph}-I_S\left[exp(X_{03})\left(1+{\displaystyle \frac{V+R_{S}I}{n{V}_T}-X_{03}+{\displaystyle \frac{1}{2}{\left({\displaystyle \frac{V+R_{S}I}{n{V}_T}-X_{03}}\right)}^2}}\right)-1\right]-{\displaystyle \frac{V}{R_{Sh}}-{\displaystyle \frac{R_S}{R_{Sh}}I}}$$

(25)

Generally, the value of $exp(X_{03})$ is significantly greater than $1$ , the value $1$ may be neglected in equation (25). By doing this and rearranging equation (25), one gets:

Let us introduce the new variables y, ${\alpha }_1$ , and ${\alpha }_2$ , defined respectively as:

$$y=I_{Ph}-\left(1+{\displaystyle \frac{R_S}{R_{Sh}}}\right)I-{\displaystyle \frac{V}{R_{Sh}},{\alpha }_1={\displaystyle \frac{V+R_{S}I}{V_T},{\alpha }_2={\displaystyle \frac{V^2+2R_{S}IV+R_S^2{I}^2}{2V_T^2}}}}$$

(27)

At this stage, the only unknown parameters are n and $I_S$ while ( $R_S$ , $R_{Sh}$ and $I_{Ph}$ are already obtained using the first and second intervals). It is worth noting that for any point $(V,I)$ for $V\in [V_1,V_2]$ , the parameter ${\alpha }_1$ and ${\alpha }_2$ can be obtained using the estimates of $R_S$ , $R_{Sh}$ and $I_{Ph}$ . Similarly, for any measurement point $(V,I)$ in this interval, the value of y can be determined. Then, it is readily seen that using equations (27) and (26), y can be expressed as:

$$y=I_{S}exp(X_0)\left[1-X_{03}+{\displaystyle \frac{1}{2}{X}_{03}^2+{\displaystyle \frac{{\alpha }_1}{n}(1-X_{03})+{\displaystyle \frac{{\alpha }_2}{n^2}}}}\right]$$

(28)

where the unknown parameters are n, $I_S$ and $X_{03}$ . Therefore, for an expansion order $m=2$ , three points should be taken in this interval, $(V_1,I_1)$ , $(V_2,I_2)$ and any other point $(V_3,I_3)$ such that $V_3\in [V_1,V_2]$ . For any measurement point $(V_j,I_j)$ ( $j=1\dots 3$ ), there corresponds to a value of $(y,{\alpha }_1,{\alpha }_2)$ , noted respectively $(y_j,{\alpha }_{1,j},{\alpha }_{2,j})$ , for $j=1\dots 3$ . Accordingly, by using these measurements and equation (25), three equations can be obtained in this interval, summarized as follows:

$$\{\begin{array}{c}{y}_1=I_{S}exp(X_{03})\left(1-X_{03}+{\displaystyle \frac{1}{2}{X}_{03}^2+{\displaystyle \frac{{\alpha }_{1,1}}{n}(1-X_{03})+{\displaystyle \frac{{\alpha }_{2,1}}{n^2}}}}\right)\\ y_2=I_{S}exp(X_{03})\left(1-X_{03}+{\displaystyle \frac{1}{2}{X}_{03}^2+{\displaystyle \frac{{\alpha }_{1,2}}{n}(1-X_{03})+{\displaystyle \frac{{\alpha }_{2,2}}{n^2}}}}\right)\\ y_3=I_{S}exp(X_{03})\left(1-X_{03}+{\displaystyle \frac{1}{2}{X}_{03}^2+{\displaystyle \frac{{\alpha }_{1,3}}{n}(1-X_{03})+{\displaystyle \frac{{\alpha }_{2,3}}{n^2}}}}\right)\end{array}$$

(29)

At this stage, the parameters of the PV model that remain unknown are n and $I_S$ . However, the second equation of equations (23) and (29) generate two new unknowns, which are $X_{02}$ and $X_{03}$ , respectively. Then, four unknown parameters ( $n$ , $I_S$ , $X_{02}$ , and $X_{03}$ ) should be determined using four equations (i.e., the second equation of equations (23) and (29)). Accordingly, these equations are sufficient to estimate the unknown parameters.

Remark 3.

To determine the parameters of the PV model, some equations have been derived in each interval. Two equations have been derived for the first and second intervals, given by equations (14) and (23), respectively. To explain how equations can be deduced in the third interval, the expansion order m in equation (3) is chosen to be equal to 2. Roughly, the expansion order $m$ is not necessarily fixed to 2. The choice of order m can be made by balancing simplicity and accuracy.

Let ${\epsilon }_m$ denote the Root Sum of Squares (RSS) error (or the estimation error) in the third interval, defined as follows:

$${\epsilon }_m=\sqrt{\sum _{j=1}^{m+1}{(I_j-{\hat{I}}_j)}^2}$$

where $I_j$ and ${\hat{I}}_j$ , for $j=1\dots m+1$ , denote the true current and its estimate, respectively.

The following algorithm for setting the order m is given below:

To address accuracy issues related to segmenting the characteristic into sub-intervals, a validation mechanism is implemented and is given in Figure 3. Using the true data and the estimated data, the $RMSE$ is computed and compared to a given threshold $RMS{E}_M$ .

Figure 3.

Adaptive validation of the $I-V$ curve segmentation.

Remark 4.

For a given threshold $RMS{E}_M$ set by the designer and for the chosen segmentation, the $RMSE$ is computed and compared with $RMS{E}_M$ . If $RMSE<RMS{E}_M$ , he chosen segmentation is considered valid. Otherwise, the boundary voltages $V_1$ and $V_2$ are adjusted ( $V_1$ can be decreased, $V_2$ can be increased), or the degree of approximation is increased. Furthermore, if necessary, the data length N is also modified.

Results and discussion

To validate the effectiveness of the proposed analytical approach for identifying photovoltaic panel parameters using a SDM, the proposed method is applied to three different PV modules/cells based on experimental data. Initially, to facilitate comparison with other methodologies, the well-recognized RTC France solar cell and PWP 201 PV panel are evaluated using the experimental data from (Gao et al., 2018). Furthermore, the experimental results for the ET-M53690, obtained directly from the laboratory, were conducted under varying temperature conditions. To assess the effectiveness of the presented methodology, two performance metrics are employed to analyze the discrepancy between the estimated and actual PV parameters: the RMSE and Absolute Error (AE).

$$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{(I_{exp}(i)-I_{model}(i))}^2}{N}}}$$

(30)

$$AE=|y_{exp}(i)-y_{model}(i)|$$

(31)

where $y_{model}$ and $y_{exp}$ are the currents obtained using the proposed analytical method and the experimental data, respectively, and N is the number of points. The value of $y_{model}$ is obtained using an SDM which involves three models applied over different intervals.

Validation through benchmark data

In this section, the experimental data of the R.T.C solar cell and PWP201 are used to evaluate the proposed approach. The electrical characteristics of these PV modules at the tested temperature are given in Table A1 (see the Appendix).

Implementation of analytical approach on benchmark solar cell: R.T.C. France solar cell

In this case, the R.T.C solar cell is utilized to validate the accuracy of the proposed SDM five parameter estimation. The experimental data were collected at $33\circ \mathrm{C}$ and $1000\mathrm{W}/{\mathrm{m}}^2$ of irradiance. To determine the appropriate expansion order over the voltage interval $V\in [V_1,V_2]$ , the procedure described in Table 3 is applied. The process starts by fixing the initial expansion order to $m=2$ and setting an admissible error threshold (Step 1 of Table 3). In this work, we set ${\epsilon }_1=0.08$ .

Table 3.

Choice of the expansion order.

Step 1: Initialization step. Set an error threshold ${\epsilon }_{Max}$ and initialize the expansion order m (Preferably, $\ge 2$ ).

Step 2: Data acquisition. Let $\{(V_{3,j},I_{3,j});j=1,\dots m+1\}$ denoting a set of measurement points (i.e., set of $m+1$ voltage values and their corresponding current values), where $V_{3,j}\in [V_1,V_2]$ for $j=1\dots m+1$ .

Step 3: Estimation of the nonlinearity coefficients describing the $I-V$ curve. Using data acquisition (Step 2), give the coefficient estimate of $a_l$ , for $l=0\dots m$ , of nonlinearity describing the $I-V$ characteristic within $[V_1,V_2]$ , satisfying: $I_{3,j}=\sum _{l=0}^m{a}_l{V}_{3,j}^l,forj=1\dots m+1$ or its equivalent: $\left(\begin{array}{c}{I}_{3,1}\\ I_{3,2}\\ ⋮\\ I_{3,m+1}\end{array}\right)=\left(\begin{array}{ccccc}{V}_{3,1}^m& V_{3,1}^{m-1}& \dots & V_{3,1}& 1\\ V_{3,2}^m& V_{3,2}^{m-1}& \dots & V_{3,2}& 1\\ ⋮& ⋮& \dots & ⋮& ⋮\\ V_{3,m+1}^m& V_{3,m+1}^{m-1}& \dots & V_{3,m+1}& 1\end{array}\right)\left(\begin{array}{c}{a}_m\\ a_{m-1}\\ ⋮\\ a_0\end{array}\right)$

Step 4: Calculation of error estimation. Using the nonlinearity estimate describing the $I-V$ curve in the interval $[V_1,V_2]$ (Step 3), compute the estimated current values ${\hat{I}}_{3,j}$ , for $j=1\dots m+1$ , corresponding to the voltages $V_{3,j}$ . Then, using the estimated current ${\hat{I}}_{3,j}$ , for $j=1\dots m+1$ , compute the RSS error as follows: ${\epsilon }_m=\sqrt{\sum _{j=1}^{m+1}{(I_{3,j}-{\hat{I}}_{3,j})}^2}$

Step 5: Order verification. Using the error estimation (Step 4), check if the condition ${\epsilon }_m<{\epsilon }_{Max}$ is satisfied. Then, the expansion order m can be retained and go to Step 6. Otherwise, increase m ( $m=m+1$ ) and go to Step 2.

Step 6: End of algorithm.

Using the set of experimentally acquired data points within the interval $[V_1,V_2]$ , the approximation error (RSS) is computed as given in Step 4 of Table 3: ${\epsilon }_m=\sqrt{\sum _{j=1}^{m+1}{(I_{3,j}-{\hat{I}}_{3,j})}^2}$ where $I_{3,j}$ (for $j=1\dots m+1$ ) denotes the measured current, and ${\hat{I}}_{3,j}$ the approximated current.

Check if the RSS error ${\epsilon }_2$ exceeds the predefined threshold $0.08$ (Step 5 of Table 3). If this condition is not satisfied, increase the expansion order to $m=3$ and evaluate the error criterion again. Otherwise, set to $m=2$ .

Using the obtained experimental data, it was observed that the RSS error satisfies the condition ${\epsilon }_2<0.08$ . Therefore, the expansion order is retained for the voltage interval $[V_1,V_2]$ is $m=2$ .

The same selection procedure is applied to the other dataset in the latter.

The optimal five parameters of the model, along with their corresponding RMSE, are compared with other approaches at the same temperature, demonstrating that the proposed method achieves higher accuracy than previous studies, as shown in Table 4. From this table, it is clear that the small values of RMSE are achieved by the proposed approach, which is $6.7910\times {10}^{-4}$ . Moreover, the I–V curve of the SDM model and the experimental data of the R.T.C solar cell are highly confused, as shown in Figure 4.

Figure 4.

$I-V$ characteristics of the R.T.C. France solar cell: comparison between experimental data and estimated data obtained using the proposed method.

Table 4.

Comparison of the suggested technique with other ways to R.T.C France solar cells.

Method	$I_{Ph}(A)$	$I_S(\mu A)$	$n$	$R_S(\mathrm{\Omega })$	$R_{Sh}(\mathrm{\Omega })$	$RMSE\times {10}^{-4}$
Ana-PSO (Choulli et al., 2023)	0.7608	0.3119	1.4777	0.0365	52.9953	7.8093
ISCE (Gao et al., 2018)	0.7607	0.3230	1.4811	0.0363	53.7185	9.8602
IWOA (Xiong et al., 2018)	0.7608	0.3232	1.4812	0.0364	53.7317	9.8602
ITLBO (Li et al., 2019)	0.7608	0.3230	1.4812	0.0364	53.7185	9.8602
SDA (Cotfas et al., 2019)	0.7607	0.3244	1.4816	0.0363	53.8427	9.8598
LFBSA (Zhang et al., 2020)	0.7607	0.2259	1.4510	0.0367	55.4854	9.8248
MDE-NR (Chermite and Douiri, 2025)	0.7607	0.3230	1.4811	0.0363	53.7185	7.7537
HFGD (Sharma et al., 2023)	0.7607	0.3106	1.4772	0.0365	52.8899	7.72
Hybrid (ANA-Iterative) (Lidaighbi et al., 2022)	0.7610	0.2933	1.4713	0.0367	49.9807	8.86
WSO-MTBO (Jeridi, 2024)	0.7607	0.3230	1.4811	0.0363	53.7185	7.7298
WSO-HO (Jeridi et al., 2025)	0.7607	0.3106	1.4773	0.0365	52.8899	7.7298
Proposed	0.7666	0.2777	1.5445	0.0537	59.2642	6.7910

Furthermore, this figure reveals excellent agreement between the simulated I–V characteristics and experimental measurements, with near-perfect overlap of the curves. This close match confirms the high performance of the proposed method in extracting the five SDM parameters.

As shown in Table 4, some parameters obtained using the proposed method differ from those reported in other studies. This is mainly due to the nonuniqueness of PV model parameters, where different parameter combinations may yield the same $I-V$ curve. Despite these differences, the extracted parameters remain physically plausible, ensuring a fair comparison based on both accuracy and physical consistency.

Figure 5 illustrates the absolute error for each voltage value. This figure shows that the absolute error remains consistently low (<0.0005) across the entire voltage range. These results confirm that the parameters extracted through the proposed analytical method accurately reproduce the $I-V$ characteristics of the R.T.C France solar cell, with exceptional precision.

Figure 5.

Absolute error of the R.T.C solar cell using the suggested method.

Implementation of analytical approach on benchmark PV module: Photowatt PWP201

In this case, the experimental data of the PWP201 PV panel, which is composed of 36 polycrystalline cells connected in series at irradiation of $1000\mathrm{W}/{\mathrm{m}}^2$ and a temperature of $45\circ \mathrm{C}$ are used to extract the five parameters of SDM. The unknown parameters and the corresponding RMSE value of the proposed model for the PV module are compared with the parameters obtained using various approaches in seven studies, as presented in Table 5. This table shows that the suggested analytical method obtains the best RMSE value, which is $2.0953\times {10}^{-3}$ . Furthermore, Figure 6 illustrates a robust correlation between the $I-V$ curve derived from both estimated and experimental data, validating the high precision of the suggested parameter extraction approach. The association is also corroborated by the performance metric in Table 5, which affirms the method's efficacy.

Figure 6.

$I-V$ characteristics of the PWP201 PV panel: comparison between experimental data and estimated data obtained using the proposed method.

Table 5.

Comparison between the proposed method and the other approaches of PWP201 PV panel.

Method	$I_{Ph}(A)$	$I_S(\mu A)$	$n$	$R_S(\mathrm{\Omega })$	$R_{Sh}(\mathrm{\Omega })$	$RMSE\times {10}^{-3}$
Ana-PSO (Choulli et al., 2023)	1.0335	2.8255	1.3295	1.2240	689.3294	2.1547
ISCE (Gao et al., 2018)	1.0305	3.4822	1,3511	1.2012	981.9822	2.4250
IWOA (Xiong et al., 2018)	1.0305	3.4717	1,3508	1.2016	978.6771	2.4251
ITLBO (Li et al., 2019)	1.0305	3.4823	1,3511	1.2013	981.9823	2.4251
SDA (Cotfas et al., 2019)	1.0305	3.4816	1,3499	1.2012	981.5996	2.4250
LFBSA (Zhang et al., 2020)	1.0305	3.4822	1.3511	1.2012	981.9823	2.4250
DE (Yang et al., 2019)	1.0305	3.4823	1,3511	1.2013	981.9819	2.4251
EHA-NMS (Chen et al., 2016)	1.03051	3.4822	1.3511	1.2012	981.9822	2.425
Hybrid (ANA-Iterative) (Lidaighbi et al., 2022)	1.0334	2.9484	1.3339	1.2217	716.5402	2.164
BLPSO (Wang et al., 2024)	1.0304	3.4477	1.3500	1.2027	997.118	2.4266
Proposed	1.0326	7.4506	1.4401	1.2102	1246.820	2.0953

It can be seen from this table that some parameters identified by the proposed method differ from those obtained in the literature. This behavior can be explained by the fact that the parameters of PV models are not unique. Several sets of parameters might lead to an equivalent adjustment of the I–V curves. However, the parameters extracted using the suggested method remain physically feasible and fall into categories that are consistent with the physical characteristics of PV modules. Therefore, the comparison is considered fair when the evaluation is based primarily on the accuracy of adjustment and the physical plausibility of the parameters.

Figure 7 illustrates the absolute error in each point measurement between the estimated and experimental current.

Figure 7.

Absolute error of the PWP201 PV panel using the proposed method.

From Figure 7 we can remark that the AE value stays below 0.001 for the majority of the points of the $I-V$ characteristic and reaches 0.024 for a point close to 13.69 V. This demonstrates that the SDM model using the proposed method can highly simulate the real behavior of the PWP201 PV module.

Experimental validation: Implementation of analytical approach ET-M53690W at different temperatures: $25\circ \mathbf{C}$ , $35\circ \mathbf{C}$ and $45\circ \mathbf{C}$

The experimental setup used to acquire the I–V characteristics of the PV modules is presented in Figure 8. It consists of a PV module connected to a load (variable resistive). To acquire the current and voltage measurements, the load variation is performed within a very short time interval, enabling rapid sweeping of the operating points over the entire I–V curve. The current and voltage are measured using a digital acquisition system, while temperature and irradiance are measured using the temperature and irradiance sensors. This configuration enables accurate data collection under various environmental conditions, which is crucial for the reliable identification of the SDM parameters. The $I-V$ characteristics were acquired at different temperatures. The tests were conducted at other times in Morocco, allowing for natural temperature variations.

Figure 8.

Experimental setup for PV panel characterization.

To ensure a uniform temperature across the entire surface of the panel, the zone where the experiment is carried out is judiciously chosen so that the temperature across the whole surface is almost the same.

In this case, the proposed analytical approach is implemented to extract the five parameters of the SDM based on experimental data from the ET-M53690W PV module, which consists of 36 monocrystalline cells. Three different temperatures were considered to demonstrate that the proposed approach maintains its effectiveness despite temperature changes. The electrical characteristics of this PV panel at each tested temperature are presented in Table A2 (see the Appendix).

The experimental data are obtained from the laboratory at different temperatures such as $25\circ \mathrm{C}$ , $35\circ \mathrm{C}$ , and $45\circ \mathrm{C}$ and an irradiance of $1000\mathrm{W}/{\mathrm{m}}^2$ . Figure 9 illustrates the $I-V$ characteristics of the measured data and the proposed analytical approach at different temperatures. This result shows that the proposed method performs well despite temperature variations, indicating its applicability even when the temperature changes. The five parameters obtained at each tested temperature, along with the resulting RMSE error and AE, are presented in Table 6.

Figure 9.

$I-V$ characteristics of the ET-M53690W PV panel: comparison between experimental data and estimated data obtained using the proposed method at different temperature.

Table 6.

Optimal parameters of the proposed analytical method of ET-M53690W PV panel at three different temperatures.

Parameters	Temperature $(\circ \mathrm{C})$
	25	35	45
${\mathrm{I}}_{\mathrm{Ph}}(\mathrm{A})$	5.549	5.588	5.601
${\mathrm{I}}_{\mathrm{S}}(\mathrm{A})$	2.426 $\times$ 10−4	0.002	0.008
$\mathrm{n}$	1.000	1.000	1.019
${\mathrm{R}}_{\mathrm{S}}(\mathbf{\Omega })$	0.1922	0.1904	0.174
${\mathrm{R}}_{\mathrm{Sh}}(\mathbf{\Omega })$	50.716	50.307	42.76
$\mathrm{RMSE}\times {10}^{-3}$	1.620	2.796	0.571
$\mathrm{MAE}\times {10}^{-3}$	1.312	1.127	0.443

Figure 10 presents the absolute error in different temperatures ( $T=25\circ \mathrm{C}$ , $T=35\circ \mathrm{C}$ , and $T=45\circ \mathrm{C}$ ) for each measurement point. Examining Figure 10 reveals that the absolute error in the majority of the voltage range is very low at the three temperatures. It is seen that the variation in temperature does not affect the performance of the proposed method. From this figure, it is clear that the absolute error of $T=25\circ \mathrm{C}$ is superior in the majority of point measurements compared with other temperatures. The absolute error in the voltage, close to 21 V, reaches its maximum value.

Figure 10.

Absolute error of the ET-M53690W PV panel using the proposed method at different temperature conditions.

To evaluate the proposed model, a set of statistical indicators, namely the mean absolute error (MAE), minimum (Min), maximum (Max), mean (ME), and standard deviation (SD) has been calculated and summarized in Table 7. These metrics provide a more detailed and reliable assessment of the model's accuracy as well as its consistency under the different temperature conditions considered.

Table 7.

Performance metrics of the ET-M53690W PV using the proposed method at three different temperatures.

Tested temperature	MAE	$Min$	$Max$	$ME$	SD
$T=25\circ C$	0.001312	0.0018253	0.0036528	0.0031816	0.0028327
$\mathrm{T}=35\circ C$	0.001127	0.0060823	0.013941	0.0079457	0.0037871
$T=45\circ C$	0.000443	0.00116965	0.002306	0.0026591	0.0031401

The results demonstrate that the proposed approach maintains a low estimation error across all tested temperature conditions. A slight increase in the error metrics is observed as the temperature increases. Nevertheless, the overall stability of the MAE and SD values confirms the robustness and reliability of the method under varying meteorological conditions.

In addition, since the proposed method relies exclusively on analytical expressions and based on experimental data, avoiding iterative numerical solvers. As a result, the computational burden is very low, making the method suitable for fast simulations.

Remark 5.

To identify the PV parameters for other temperatures, the same approach can be carried out. When atmospheric conditions change, some PV parameters vary, while others remain constant. For instance, in the case when the temperature varies and the irradiance is constant, it is seen that the short circuit current $I_{SC}$ (for $V\in [0,V_1]$ ) is almost constant (Figure 9).

Furthermore, it is shown that the open circuit voltage $V_{OC}$ does not keep the same voltage (Figure 9). For $V\ge V_2$ , it is seen that the obtained straight lines, for different temperatures, are almost parallel. Then, they have the same slope, which is given analytically by equation (22). This means that the expression of slope equation (22) remains unchanged even if the temperature varies.

Generalization and robustness analysis

Validation using a number of PV cells/modules

To address the generalizability of the proposed analytical parameter estimation method, an extensive validation was carried out using multiple photovoltaic (PV) modules including monocrystalline, polycrystalline, and thin-film technologies, as summarized in Table 8. Specifically, the validation dataset includes six different PV modules, namely STM6-40/36, Ultra85-P, LSM 20, STP6-120/36, STE 4/100, and PVM-752GaAs, which differ in terms of cell technology, number of series-connected cells. This selection enables the evaluation of the proposed method across diverse PV architectures, rather than being limited to a single panel configuration.

Table 8.

Validation results of the proposed method for different PV technologies.

PV parameters	Monocrystalline			Polycristalline		Thin film
	STM6-40/36	Ultra85-P	LSM 20	STP6-120/36	STE 4/100	PVM_752GaAs
Temperature	51 °C	48.7 °C	24 °C	55 °C	22 °C	25 °C
Number of cells	36	36	20	36	4	1
$I_{Ph}(A)$	1.663	5.227	0.155	7.475	0.02661	0.0998
$I_S(\mu A)$	1.743	10.458	1.731	1.930	0.00784	0.000385
$n$	1.520	1.568	2	1.244	1.312058	2
$R_S(\mathrm{\Omega })$	0.153	0.397	0.525	0.168	0.537	0.512
$R_{Sh}(\mathrm{\Omega })$	573.655	136.888	1033.686	569.581	1500	1468.511
$RMSE\times {10}^{-3}$	1.72	2.56	3.74	14.23	0.33	0.047

The obtained results demonstrate that the proposed method consistently achieves low RMSE values across all tested PV technologies, confirming its high accuracy and numerical stability regardless of material type, or number of cells. This indicates that the method does not rely on technology-specific assumptions and can be effectively applied to both crystalline silicon and thin-film PV modules.

Sensitivity analysis

To evaluate the robustness of the PV model under parameter uncertainty, a sensitivity analysis was conducted for the five key parameters of the single-diode model: photocurrent $I_{Ph}$ , saturation current $I_S$ , ideality factor n, series resistance $R_S$ , and shunt resistance $R_{Sh}$ . The sensitivity coefficients were derived analytically, and the impact of parameter variations on model accuracy was assessed using RMSE and MAE metrics.

Sensitivity analysis with respect to ${\mathit{I}}_{\mathit{P}\mathit{h}}$

This sensibility will noted $S_{I_{Ph}}$ which is given as:

$$S_{I_{Ph}}={\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }{I}_{Ph}}={\displaystyle \frac{1}{1+{\displaystyle \frac{I_s{R}_s}{n{V}_T}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)+{\displaystyle \frac{R_s}{R_{sh}}}}}}}$$

(32)

This expression can be simplified to:

$$S_{I_{Ph}}={\displaystyle \frac{n{V}_T{R}_{sh}}{n{V}_T(R_{sh}+R_s)+I_s{R}_s{R}_{sh}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)}}$$

(33)

The sensitivity $S_{I_{Ph}}$ is always positive and less than 1. Then, it depends on the operating point $(V,I)$ .

The results show that the model remains stable under perturbations of $I_{Ph}$ , with RMSE and MAE varying only slightly around the nominal value as shown in the Table 9. This indicates low sensitivity to uncertainties in $I_{Ph}$ and confirms the robustness of the parameter identification method.

(b) Sensitivity analysis with respect to ${\mathit{I}}_{\mathit{S}}$

Table 9.

Impact of ${\mathrm{I}}_{\mathrm{Ph}}$ variation on the model performance.

$I_{Ph}(A)$	5.15735	5.19219	5.26189	5.29673
RMSE	0.0593	0.0297	0.0297	0.0593
MAE	0.0578	0.0288	0.0289	0.0577

The sensitivity of I with respect to the saturation current $I_S$ is noted $S_{I_S}$ and is given as follows:

$$S_{I_S}={\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }{I}_S}={\displaystyle \frac{1-\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)}{1+{\displaystyle \frac{R_s{I}_s}{n{V}_T}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)+{\displaystyle \frac{R_s}{R_{sh}}}}}}}$$

(34)

Which simplifies to:

$$S_{I_S}={\displaystyle \frac{n{V}_T{R}_{sh}\left(1-\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)\right)}{n{V}_T(R_{sh}+R_s)+R_s{I}_s{R}_{sh}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)}}$$

(35)

This sensitivity is negative, indicating an inverse relationship with the output current, and its absolute value increases as the operating point approaches open-circuit conditions.

The sensitivity is relatively low. The RMSE and MAE remain below 0.014, indicating that the model is not highly sensitive to uncertainties in $I_S$ (Table 10).

(c) Sensitivity analysis with respect to $\mathit{n}$

Table 10.

Impact of ${\mathrm{I}}_{\mathrm{S}}$ variation on model performance.

$I_S(\mu A)$	10.31942	10.38914	10.38914	10.52860
RMSE	0.013653	0.013653	0.007169	0.007089
MAE	0.009522	0.009522	0.005182	0.005066

The expression of sensitivity $S_{\mathrm{n}}$ is given as:

$$S_{\mathrm{n}}={\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }\mathrm{n}}={\displaystyle \frac{{\displaystyle \frac{I_s(V+R_{s}I)}{n^2{V}_T}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)}}{1+{\displaystyle \frac{R_s{I}_s}{n{V}_T}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)+{\displaystyle \frac{R_s}{R_{sh}}}}}}}$$

(36)

By simplifying the equation (36), the expression of $S_{\mathrm{n}}$ can be expressed as follows:

$$S_{\mathrm{n}}={\displaystyle \frac{R_{sh}{I}_s(V+R_{s}I)\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)}{n^2{V}_T(R_s+R_{sh})+n{R}_s{I}_s{R}_{sh}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)}}$$

(37)

This positive sensitivity reflects the exponential dependence on n in the diode term.

The model exhibits moderate sensitivity to n. The error metrics are higher compared to $I_{Ph}$ and $I_S$ , but remain acceptable around the optimal value as shown in Table 11.

(d) Sensitivity analysis with respect to ${\mathit{R}}_{\mathit{s}}$

Table 11.

Impact of $\mathrm{n}$ variation on model performance.

$n$	1.54769	1.55815	1.57906	1.58952
RMSE	0.173576	0.085409	0.082727	0.162662
MAE	0.112324	0.055239	0.053550	0.104369

This sensitivity is noted $S_{R_S}$ and its expression is given as:

$$S_{R_S}={\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }{R}_s}={\displaystyle \frac{-\left[{\displaystyle \frac{I_s}{n{V}_T}exp\left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)+{\displaystyle \frac{1}{R_{sh}}}}\right]}{1+{\displaystyle \frac{R_s{I}_s}{n{V}_T}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)+{\displaystyle \frac{R_s}{R_{sh}}}}}I}}$$

(38)

The expression of $S_{R_S}$ can be written as:

$$S_{R_S}={\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }{R}_s}=-{\displaystyle \frac{I_s{R}_{sh}exp\left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)+n{V}_T}{n{V}_T(R_{sh}+R_s)+R_s{I}_s{R}_{sh}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)}I}}$$

(39)

The negative sign indicates that increasing $R_S$ always decreases output current.

The model is robust to variations in $R_{sh}$ , with minimal errors observed around the nominal value (Table 12).

(e) Sensitivity analysis with respect to ${\mathit{R}}_{\mathit{s}\mathit{h}}$

Table 12.

Impact of ${\mathrm{R}}_{\mathrm{S}}$ variation on model performance.

$R_S$	0.39208	0.39473	0.40003	0.40268
RMSE	0.007582	0.004399	0.004351	0.007525
MAE	0.005983	0.003504	0.003270	0.005565

Table 13.

Impact of ${\mathrm{R}}_{\mathrm{Sh}}$ variation on model performance.

$R_{Sh}$	135.06356	135.97615	137.80133	138.71392
RMSE	0.002816	0.002627	0.002634	0.002816
MAE	0.002244	0.002089	0.002136	0.002257

This sensibility will noted $S_{R_{Sh}}$ :

$$S_{R_{Sh}}={\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }{R}_{sh}}={\displaystyle \frac{{\displaystyle \frac{V+R_{s}I}{R_{sh}^2}}}{1+{\displaystyle \frac{R_s{I}_s}{n{V}_T}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)+{\displaystyle \frac{R_s}{R_{sh}}}}}}}$$

(40)

Then $S_{R_{Sh}}$ can be expressed as:

$$S_{R_{Sh}}={\displaystyle \frac{n{V}_T(V+R_{s}I)}{R_{sh}\left[n{V}_T(R_s+R_{sh})+R_s{I}_s{R}_{sh}\exp \left({\displaystyle \frac{V+R_{s}I}{n{V}_T}}\right)\right]}}$$

(41)

This positive sensitivity demonstrates the beneficial effect of high shunt resistance on current output.

The model exhibits very low sensitivity to $R_{Sh}$ variations. The error metrics are the lowest among all parameters, confirming that shunt resistance has a negligible impact on model accuracy under normal operating conditions as given in Table 13.

Conclusion

In this study, a novel analytical approach has been developed for accurately estimating the unknown parameters of the SDM for PV modules and cells using experimental data. The validity and effectiveness of the proposed method were first demonstrated through its application to benchmark PV solar cells and modules. The results showed strong agreement with reference parameters found in the literature, while achieving significantly lower AE and RMSE values, thereby confirming the accuracy and superiority of the method.

Moreover, the robustness of the proposed technique was further validated by successfully extracting the five SDM parameters under varying temperature conditions. The method maintained high accuracy across all tested temperatures, demonstrating its reliability and adaptability to environmental changes. A key advantage of the proposed approach lies in its simplicity and computational efficiency, as it is based on explicit analytical expressions and does not require iterative numerical procedures. This makes it highly suitable for integration into real-world PV monitoring, diagnostic, and optimization systems. Furthermore, the proposed method offers direct benefits for photovoltaic applications, such as MPPT improvement, performance prediction, and panel diagnostics. Its analytical nature enables the quick and reliable extraction of PV model parameters, which makes it suitable for real-time implementation in embedded and smart inverter systems.

Overall, the proposed method offers a reliable, efficient, and practical solution for SDM parameter estimation, contributing to the advancement of accurate PV modeling and performance analysis. Compared with iterative and metaheuristic techniques, the approach avoids convergence issues and high computational cost, which enhances its practicality for large-scale PV monitoring and onsite characterization.

To verify the proposed algorithm under realistic operating conditions, future work will focus on analyzing the impact of irradiance variations on the same problem.