A Sensitivity-Inspired Parameter Identification Method for the Single-Diode Model of Photovoltaic Modules

Abstract

Parametrization of photovoltaic (PV) modules makes an important foundation for monitoring and fault diagnosis. This work focus on the sensitivity of parameters for the single-diode model (SDM), which fills the gap in existing research. The sensitivity analysis in this work provides a fundamentally new perspective on understanding parameter robustness as well as the prior knowledge for the parameter identification method. Based on insights into the sensitivity analysis, a novel parameter identification method is proposed, which combines analytical expressions with the grid search algorithm. The proposed method reduces the relative error of the extracted parameters in the simulated dataset, and the quantitative improvement of the reverse saturation current is significant (12.6% average reduction). This method achieves the state-of-the-art overall performance in the measured dataset, and the Friedman test confirms that this improvement is statistically significant (p < 0.05). The transition capability of the proposed method is excellent under varying operating conditions, which implies that it has the potential to be applied to the intelligent operation and maintenance of photovoltaic systems.

1. Introduction

In recent years, the share of fossil fuels in the global energy mix has gradually reduced. According to World Energy Outlook (2024), more than 150 countries have policies to expand the utilization of renewable energy, and the supply of renewable energy has increased by 5% year-on-year [1]. According to World Energy Transitions Outlook (2023), the transformation of the global energy sector from fossil-based to zero-carbon sources is expected to be achieved by 2050 [2]. As one of the most profitable renewable energy resources, solar energy is clean and widely available [3]. Over the last decade, the solar photovoltaic (PV) capacity has increased from 220 GW to 1858 GW, according to Renewable Energy Statistics (2025) reported by International Renewable Energy Agency [4]. With the rapid development of the PV industry, the efficiency, stability, and safety of PV systems have attracted widespread attention of researchers [5]. In fact, the need to mitigate potential adverse grid impacts of PV integration motivates the use of condition monitoring and fault diagnosis, which in turn relies on accurate parameterization of PV modules [6,7].

In order to simulate the I–V curves of PV modules [8], different equivalent circuits have been studied. The ideal single-diode model (SDM) is the simplest equivalent circuit [9]. The double-diode model (DDM) [10] and the triple-diode model (TDM) [11] achieve better precision at the cost of increasing computational burden. Among the existing equivalent circuits, the improved SDM with five parameters is widely used, due to its good compromise between accuracy and complexity [12].

Parametrization of PV modules make an important foundation for monitoring and fault diagnosis [13,14]. Analytical and numerical methods have been developed to identify the unknown parameters of equivalent circuits [15]. Analytical approaches [16] simplify nonlinear equations of equivalent circuits via remarkable points, including the short-circuit point, the open-circuit point, and the maximum power point. Using the current, the voltage, and the slope of remarkable points, explicit expressions for estimating the parameters have been provided [17,18]. Analytical methods are simple and fast, but the corresponding assumptions and approximations may generate significant errors of some parameters [19].

Numerical approaches minimize the overall error between the simulated and measured I–V curves. In the literature, numerical optimization methods are classified into deterministic and meta-heuristic branches [20]. The representative of deterministic approaches is the iterative algorithm, which sets the initial guess values of parameters, and then updates these values gradually to solve nonlinear and multivariable functions [21]. Conventional iterative methods, such as Newton–Raphson [22] and Levenberg–Marquardt [23], are sensitive to the initial guess values. Sometimes, these techniques encounter the challenge of slow convergence [24].

Meta-heuristic methods search the solution of the optimization problem, inspired by natural phenomena [25]. The genetic algorithm [26] and the differential evolution [27] are population-based methods. Combined with social group [28], multimutation [29], dynamic diversity capture [30], and self-adaptive [31,32] algorithms, the performance of differential evolution has been improved. Bio-inspired methods [33,34] and physics-based methods [35] have also been investigated. These meta-heuristic methods rely on the reasonable range of each parameter. The objective function may become stuck at local optimal values with stochastic initialization, resulting in uncertainty of the extracted parameters [36].

Considering the advantages and limitations of the above methods, hybrid approaches have been developed. The study [37] finds the ideality factor and the series resistance using the dual-iteration algorithm, while analytically calculating other parameters. Some researchers construct two lower-dimensional spaces using parameter separation techniques [38,39]. In one sub-space, the ideality factor and the series resistance are extracted using optimization algorithms; in another sub-space, other parameters are calculated analytically [40,41]. However, the same parameter can be calculated using different analytical expressions. These methods do not specify which analytical expression achieves better performance.

Existing approaches have not taken into account the sensitivity of parameters for the SDM, which lead to errors of parameters, remarkable points, and the overall I–V curve. The early studies on parameter identification of photovoltaic modules have primarily focused on verifying the effectiveness of the equivalent circuit models and their corresponding mathematical models, with the aim of simulating the I–V curves. However, as the research progresses and develops, extracting accurate parameters has become as important as validating the models and simulating the I–V curves. Therefore, one of the motivations of this study is to promote the rigor of the methodology, transforming the sensitivity analysis into the prior knowledge for the parameter identification method. Sensitivity analysis provides a fundamentally new perspective on understanding parameter robustness beyond traditional methods. Sensitivity analysis can explain which parameter is suitable for calculation using the analytical expression and which analytical expression achieves better performance.

The errors of the extracted parameters are derived from measurement error and optimization error. On the one hand, the minor errors of the measured I–V curve are prone to be magnified, as the mathematical model of the equivalent circuit contains the exponential term. On the other hand, the optimization problem yields sub-optimal solutions, which implies that the extracted parameters do not exactly match the true values. Therefore, the parametrization procedure should avoid the magnification of measurement errors, and make the loss function as low as possible.

In light of the above, the sensitivity of parameters to the current, the voltage, and the I–V gradient are analyzed, which reveals that the analytical calculation of the photoinduced current using the short-circuit current does not magnify the measurement error. The sensitivity of the loss function to parameters is analyzed, which demonstrates that the diode ideality factor has the most significant impact on the loss function. Based on insights into the sensitivity, a novel parameter identification method for the SDM is proposed. The contributions of this article are summarized as follows:

• To the best of our knowledge, this is the first work that focuses on the sensitivity of parameters for the SDM. The sensitivity analysis in this work explains which parameter is suitable for calculation using the analytical expression and which analytical expression achieves better performance, providing a fundamentally new perspective on understanding parameter robustness as well as the prior knowledge for the parameter identification method.
• Based on insights into the sensitivity, a novel parameter identification method for the SDM is proposed, which combines analytical expressions with the grid search algorithm. The proposed method reduces the relative error of the extracted parameters in the simulated dataset, and the quantitative improvement of the reverse saturation current is significant (12.6% average reduction). This method achieves the state-of-the-art overall performance in the measured dataset, and the Friedman test confirms that this improvement is statistically significant (p < 0.05).
• When the parameters are transferred to varying operating conditions, the proposed method can simulated the I–V curve and the P–V curve accurately. The excellent transition capability of this method implies that it has the potential to be applied to the intelligent operation and maintenance of photovoltaic systems. Furthermore, this work provides an important foundation for monitoring and fault diagnosis.

The rest of this article is organized as follows. The sensitivity analysis and the proposed parameter identification method are described in Section 2. Subsequently, experimental results are presented in Section 3. Next, parameters are interpreted and applications of the proposed method are discussed in Section 4. Finally, conclusions are drawn in Section 5.

2. Materials and Methods

In this section, a sensitivity-inspired parameter identification method for the SDM is proposed. Firstly, the SDM and its parameters are described in Section 2.1. Secondly, the sensitivity of these parameters is analyzed in Section 2.2. Based on insights into the sensitivity, the parameter identification algorithm is designed in Section 2.3.

2.1. Parameters of the SDM

The improved SDM is adopted in this work due to its good compromise between accuracy and complexity for PV modeling. The equivalent circuit of the SDM is shown in Figure 1. Parameters of the SDM consist of the photoinduced current $I_{ph}$, the shunt resistance $R_h$, the series resistance $R_s$, the reverse saturation current $I_s$, and the ideality factor n of the diode.

The relationship between the current I and voltage V is shown by the nonlinear equation of Equation (1):

$$I=I_{ph}-I_s\left[exp\left({\displaystyle \frac{V+I·R_s}{n·V_T}}\right)-1\right]-{\displaystyle \frac{V+I·R_s}{R_h}}$$

(1)

where the thermal voltage $V_T$ is defined in Equation (2),

$$V_T={\displaystyle \frac{k·T_c}{q}}$$

(2)

where k is the Boltzmann constant ($1.38\times {10}^{-23}\mathrm{J}/\mathrm{K}$), q is the electron charge ($1.60\times {10}^{-19}\mathrm{C}$), and $T_c$ is the cell temperature.

2.2. Sensitivity Analysis of Parameters

Sensitivity analysis is beneficial for understanding the relationship among measurement errors (involving the current/voltage/gradient), parameters, and the loss function.

2.2.1. Sensitivity of Parameters to the Current

According to Equation (1), the partial derivatives of parameters with respect to the current are calculated, respectively, and the sensitivity of parameters to the current is obtained using Equations (3)–(7).

$$\begin{array}{c}S\left(n,I\right)={\displaystyle \frac{I}{I_s}}·{\displaystyle \frac{1+\frac{R_s}{R_h}}{exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{V+I{R}_s}{n{V}_T}}}+{\displaystyle \frac{I{R}_s}{V+I{R}_s}}\end{array}$$

(3)

$$\begin{array}{c}S\left(I_{ph},I\right)={\displaystyle \frac{I·\left[1+\frac{R_s}{R_h}+exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{I_s{R}_s}{n{V}_T}\right]}{I_{ph}}}\end{array}$$

(4)

$$\begin{array}{c}S\left(I_s,I\right)={\displaystyle \frac{I}{I_s}}·{\displaystyle \frac{1+\frac{R_s}{R_h}+exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{I_s{R}_s}{n{V}_T}}{1-exp\left(\frac{V+I{R}_s}{n{V}_T}\right)}}\end{array}$$

(5)

$$\begin{array}{c}S\left(R_s,I\right)=-{\displaystyle \frac{1}{exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{I_s{R}_s}{n{V}_T}+\frac{R_s}{R_h}}}-1\end{array}$$

(6)

$$\begin{array}{c}S\left(R_h,I\right)={\displaystyle \frac{1+\frac{R_s}{R_h}+exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{I_s{R}_s}{n{V}_T}}{\frac{V+I{R}_s}{I{R}_h}}}\end{array}$$

(7)

In order to capture the variability of crystalline technology, two types of PV modules are selected to demonstrate the characteristics of the sensitivity. The detailed information of these PV modules is given in Section 3. The logarithm of the absolute sensitivity of parameters to the current is shown in Figure 2. Around the short-circuit point, the logarithm of the absolute sensitivity of $I_{ph}$ to the current tends to zero and stabilizes, which implies that the calculation error of $I_{ph}$ falls within the same order of magnitude as the measurement error of the current.

2.2.2. Sensitivity of Parameters to the Voltage

According to Equation (1), the partial derivatives of parameters with respect to the voltage are calculated, respectively, and the sensitivity of parameters to the voltage is obtained using Equations (8)–(12).

$$\begin{array}{c}S\left(n,V\right)={\displaystyle \frac{V}{V+I{R}_s}}·{\displaystyle \frac{\frac{n{V}_T}{R_h}+I_s·exp\left(\frac{V+I{R}_s}{n{V}_T}\right)}{I_s·exp\left(\frac{V+I{R}_s}{n{V}_T}\right)}}\end{array}$$

(8)

$$\begin{array}{c}S\left(I_{ph},V\right)={\displaystyle \frac{V}{I_{ph}}}·\left[{\displaystyle \frac{1}{R_h}}+exp\left({\displaystyle \frac{V+I{R}_s}{n{V}_T}}\right)·{\displaystyle \frac{I_s}{n{V}_T}}\right]\end{array}$$

(9)

$$\begin{array}{c}S\left(I_s,V\right)={\displaystyle \frac{V}{I_s}}·{\displaystyle \frac{\frac{1}{R_h}+exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{I_s}{n{V}_T}}{1-exp\left(\frac{V+I{R}_s}{n{V}_T}\right)}}\end{array}$$

(10)

$$\begin{array}{c}S\left(R_s,V\right)={\displaystyle \frac{\partial R_s}{\partial V}}·{\displaystyle \frac{V}{R_s}}=-{\displaystyle \frac{V}{I·R_s}}\end{array}$$

(11)

$$\begin{array}{c}S\left(R_h,V\right)={\displaystyle \frac{1+exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{I_s{R}_h}{n{V}_T}}{1+\frac{I{R}_s}{V}}}\end{array}$$

(12)

The logarithm of the absolute sensitivity of these parameters to the voltage is shown in Figure 3. Between the optimum operating voltage and the open-circuit voltage, the sensitivity of n to the voltage tends to zero and stabilizes, which implies that the calculation error of n falls within the same order of magnitude as the measurement error of the voltage.

2.2.3. Sensitivity of Parameters to the I–V Gradient

Taking the derivative of both sides of Equation (1) with respect to the current, and denoting the I–V gradient as $g={\displaystyle \frac{\mathrm{d}V}{\mathrm{d}I}}$, the parameter $I_{ph}$ no longer appears in Equation (13). The partial derivatives of other parameters with respect to g are calculated, respectively, and the sensitivity of these parameters to g is obtained using Equations (14)–(17).

$$1=-I_s·exp\left({\displaystyle \frac{V+I{R}_s}{n{V}_T}}\right)·{\displaystyle \frac{g+R_s}{n{V}_T}}-{\displaystyle \frac{g+R_s}{R_h}}$$

(13)

$$\begin{array}{c}S\left(n,g\right)={\displaystyle \frac{g}{g+R_s}}·{\displaystyle \frac{\frac{n{V}_T}{I_s{R}_h}+exp\left(\frac{V+I{R}_s}{n{V}_T}\right)}{\left(1+\frac{V+I{R}_s}{n{V}_T}\right)·exp\left(\frac{V+I{R}_s}{n{V}_T}\right)}}\end{array}$$

(14)

$$\begin{array}{c}S\left(I_s,g\right)=-{\displaystyle \frac{g}{g+R_s}}·\left[1+{\displaystyle \frac{1}{\frac{I_s{R}_h}{n{V}_T}·exp\left(\frac{V+I{R}_s}{n{V}_T}\right)}}\right]\end{array}$$

(15)

$$\begin{array}{c}S\left(R_s,g\right)={\displaystyle \frac{\partial R_s}{\partial g}}·{\displaystyle \frac{g}{R_s}}=-{\displaystyle \frac{g}{R_s}}\end{array}$$

(16)

$$\begin{array}{c}S\left(R_h,g\right)={\displaystyle \frac{g·\left[1+exp\left(\frac{V+I{R}_s}{n{V}_T}\right)·\frac{I_s{R}_h}{n{V}_T}\right]}{g+R_s}}\end{array}$$

(17)

The logarithm of the absolute sensitivity of these parameters to the I–V gradient is shown in Figure 4. Around the short-circuit point, the sensitivity of $R_h$ to the I–V gradient tends to zero and stabilizes, which implies that the calculation error of $R_h$ falls within the same order of magnitude as the calculation error of the I–V gradient.

2.2.4. Sensitivity of the Loss Function to Parameters

According to Equation (1), the loss function is defined using Equation (18),

$$\begin{array}{c}\hfill f=I_{ph}-I_s\left[exp\left({\displaystyle \frac{V+I{R}_s}{n{V}_T}}\right)-1\right]-{\displaystyle \frac{V+I{R}_s}{R_h}}-I\end{array}$$

(18)

where the value of f depends on the current, the voltage, and the parameters.

The partial derivatives of f with respect to parameters are calculated, respectively, and the sensitivity of f to parameters is obtained using Equations (19)–(23).

$$\begin{array}{c}S\left(f,n\right)=I_s·exp\left({\displaystyle \frac{V+I{R}_s}{n{V}_T}}\right)·{\displaystyle \frac{V+I{R}_s}{n·V_T·f}}\end{array}$$

(19)

$$\begin{array}{c}S\left(f,I_{ph}\right)={\displaystyle \frac{\partial f}{\partial I_{ph}}}·{\displaystyle \frac{I_{ph}}{f}}={\displaystyle \frac{I_{ph}}{f}}\end{array}$$

(20)

$$\begin{array}{c}S\left(f,I_s\right)=\left[1-exp\left({\displaystyle \frac{V+I{R}_s}{n{V}_T}}\right)\right]·{\displaystyle \frac{I_s}{f}}\end{array}$$

(21)

$$\begin{array}{c}S\left(f,R_s\right)=-{\displaystyle \frac{I}{f}}·\left[exp\left({\displaystyle \frac{V+I{R}_s}{n{V}_T}}\right)·{\displaystyle \frac{I_s{R}_s}{n{V}_T}}+{\displaystyle \frac{R_s}{R_h}}\right]\end{array}$$

(22)

$$\begin{array}{c}S\left(f,R_h\right)={\displaystyle \frac{\partial f}{\partial R_h}}·{\displaystyle \frac{R_h}{f}}={\displaystyle \frac{V+I{R}_s}{R_h·f}}\end{array}$$

(23)

The root mean square error (RMSE) of the relative sensitivity of the loss function with respect to parameters is shown in Figure 5. Obviously, the loss function is more sensitive to n.

2.3. The Sensitivity-Inspired Parameter Identification Method

The identification of parameters should avoid the magnification of measurement errors, and make the loss function as low as possible. Based on insights into the sensitivity, a novel parametrization algorithm is designed in this section.

2.3.1. Insights into the Sensitivity

Understanding the characteristics of the sensitivity contributes to more accurate and efficient identification of parameters. Insights into the sensitivity are explored as follows:

n: The RMSE of the relative sensitivity of the loss function to n is much higher in Figure 5, which implies that n has the most significant impact on the loss function. In order to reduce the calculation error, numerical methods are more suitable than analytical methods for n [37].

$I_{ph}$: When the logarithm of the absolute sensitivity is higher in Figure 2, the measurement error of the current will lead to more significant calculation errors of the parameters. Around the short-circuit point, the logarithm of the absolute sensitivity of $I_{ph}$ to the current approaches zero, which means that the short-circuit current can be used to analytically calculate $I_{ph}$ without the magnification of the measurement error.

$R_h$: When the sensitivity of the parameter to the I–V gradient stabilizes within a certain range in Figure 4, the slope of the I–V curve within this range has a relatively consistent influence on the parameter. Around the short-circuit point, the logarithm of the absolute sensitivity of $R_h$ to the I–V gradient approaches zero and stabilizes while the slope of the I–V curve remains almost constant, which means that the I–V gradient around the short-circuit point can be used to analytically calculate $R_h$.

$I_s$: Around the maximum power point, the logarithm of the absolute sensitivity of $I_s$ to the I–V gradient approaches zero and changes slowly, as shown in Figure 4. Considering that the variation of the I–V gradient around the maximum power point would introduce additional errors, it is a better choice to analytically calculate $I_s$ using multiple current and voltage pairs rather than the slope of the I–V curve.

$R_s$: Around the open-circuit point, the sensitivity of $R_s$ is extremely high in Figure 3, which implies that the measurement errors are prone to magnification during the calculation of $R_s$. Therefore, numerical methods are more suitable than analytical methods for $R_s$.

2.3.2. Analytical Expression of Parameters

Based on insights into the sensitivity, $I_{ph}$ and $R_h$ can be calculated using the current and the I–V gradient at the short-circuit point, respectively. When $V=0$, the current through the diode in Figure 1 is infinitesimal and the magnitude of the diode current is typically less than ${10}^{-6}$ compared to the total current. Ignoring the relevant term of the diode, Equation (13) is approximately simplified to Equation (24),

$$R_h=-\left(g_{sc}+R_s\right)$$

(24)

where $g_{sc}$ represents the I–V gradient around the short-circuit point. Similarly, Equation (1) is simplified to Equation (25),

$$I_{ph}=I_{sc}·\left(1+{\displaystyle \frac{R_s}{R_h}}\right)$$

(25)

where $I_{sc}$ represents the short-circuit current.

Given multiple current and voltage pairs, $I_s$ can be calculated by the loss function. $I_x$ and $V_x$ represent the current and voltage at the x-th point on the I–V curve, respectively. $f_x$ represents the corresponding loss. In order to obtain the minimum of $\sum f_x^2$ when $x\in \mathsf{\Phi }$, the partial derivatives of $I_s$ are shown in Equation (26). By combining Equations (18), (21) and (26), $I_s$ can be calculated using Equation (27),

$$\sum _{x\in \mathsf{\Phi }}{\displaystyle \frac{\partial f_x^2}{\partial I_s}}=\sum _{x\in \mathsf{\Phi }}\left(2f_x·{\displaystyle \frac{\partial f_x}{\partial I_s}}\right)=0$$

(26)

$$I_s={\displaystyle \frac{{\displaystyle \sum _{x\in \mathsf{\Phi }}}\left\{\left(I_{ph}-I_x-\frac{V_x+I_x{R}_s}{R_h}\right)\left[exp\left(\frac{V_x+I_x{R}_s}{n{V}_T}\right)-1\right]\right\}}{{\displaystyle \sum _{x\in \mathsf{\Phi }}}{\left[exp\left(\frac{V_x+I_x·R_s}{n{V}_T}\right)-1\right]}^2}}$$

(27)

where $\mathsf{\Phi }$ represents the set of data indices around the maximum power point.

2.3.3. Algorithm for Parameter Identification

The flowchart of the proposed algorithm for parametrization is shown in Figure 6. The input of the algorithm includes the short-circuit current $I_{sc}$, the open-circuit voltage $V_{oc}$, the optimum operating current $I_m$, the optimum operating voltage $V_m$, and the thermal voltage $V_T$. As the ranges of n and $R_s$ can be determined based on the literature and experience, n and $R_s$ are suitable for iteration. Given n and $R_s$, the other three parameters are calculated by Equation (24), Equation (25), and Equation (27), respectively.

The overall loss of the I–V curve $f_{all}$ is defined using Equation (28). As the local characteristics around the short-circuit point and the maximum power point have already been taken into account in the calculation of $R_h$, $I_{ph}$, and $I_s$, it is reasonable to consider the loss of the open-circuit point in addition to the overall loss. The loss of the open-circuit point $f_{oc}$ is shown in Equation (29), and the mixed loss function $f_{mix}$ is defined using Equation (30),

$$f_{all}=\sqrt{{\displaystyle \frac{1}{M}}\sum _{x\in \mathsf{\Psi }}{f}_x^2}$$

(28)

$$f_{oc}=I_{ph}-I_s\left[exp\left({\displaystyle \frac{V_{oc}}{n{V}_T}}\right)-1\right]-{\displaystyle \frac{V_{oc}}{R_h}}$$

(29)

$$f_{mix}=w_1·f_{all}+w_2·f_{oc}$$

(30)

where $\mathsf{\Psi }$ represents the set of data indices for the entire I–V curve; $V_{oc}$ represents the open-circuit voltage; $w_1$ and $w_2$ denote the weights of $f_{all}$ and $f_{oc}$, respectively. The values of $w_1$ and $w_2$ are set within the range of [0, 1], while $w_1+w_2=1$. The larger $w_1$ generates the better overall performance, while the larger $w_2$ leads to the better accuracy at the open-circuit point. The selection of $w_1$ and $w_2$ depends on the balance between overall performance and the accuracy of the remarkable point. In this work, $w_1=0.95$ and $w_2=0.05$ are selected.

The outer iteration of n and the inner iteration of $R_s$ search for the minimum $f_{mix}$, which takes into account the global trend of the I–V curve as well as the local characteristics. The output of the algorithm includes five parameters of the SDM with the minimum $f_{mix}$. The initial guesses for n and $R_s$ depend on their empirical ranges. The parameter ranges adopted for the simulated and measured datasets in this work are described in the next section. In order to avoid becoming stuck in the local optimal solution, this work adopts the classical grid search method within the predefined ranges. The grid search method is simple and effective for low-dimensional problems and will not encounter the issue of nonconvergence. The time complexity is $O({\lambda }_1·{\lambda }_2)$, and the space complexity is $O\left(1\right)$, where ${\lambda }_1$ and ${\lambda }_2$ represent the number of discrete points for the ranges of n and $R_s$, respectively.

3. Results

Experiments are conducted on the dataset derived from Simulink to evaluate the accuracy of extracted parameters, shown in Section 3.1. The performance of the SDM with the proposed parameter identification method is evaluated on the dataset provided by National Renewable Energy Laboratories (NREL), shown in Section 3.2.

3.1. Validation of the Parameters

In order to evaluate the performance of the proposed parameter identification method, two types of PV modules are selected for experiments, including monocrystalline (LR6-60-285M) and multicrystalline (CS6X-310P). The I–V curves are derived from Simulink under standard testing condition (STC). The characteristics of these PV modules are shown in

Table 1. Characteristics of PV modules in the simulated dataset.

Characteristics	LR6-60-285M	CS6X-310P
Short-circuit current ($I_{sc}$)	9.59 A	9.08 A
Open-circuit voltage ($V_{oc}$)	38.6 V	44.9 V
Optimum operating current ($I_m$)	9.05 A	8.52 A
Optimum operating voltage ($V_m$)	31.5 V	36.4 V
Maximum power ($P_m$)	285.075 W	310.128 W
Number of cells ($N_c$)	60	72

.

Four existing parameter identification methods are adopted for comparison, including the analytical method [20], the meta-heuristic method [28], the deterministic method [37], and the hybrid method [42]. The analytical method does not require the range of parameters. The deterministic method requires the range of n and $R_s$. The hybrid method requires the range of n, $R_s$, and $R_h$. The meta-heuristic method requires the range of each parameter. The empirical range of parameters at STC is shown in

Table 2. Range of parameters in the simulated dataset.

Parameters	LR6-60-285M	CS6X-310P
$n_c$	[0.9, 1]	[0.9, 1]
$I_{ph}$ (A)	[0, 12]	[0, 12]
$I_s$ (A)	[0, ${10}^{-8}$]	[0, ${10}^{-8}$]
$R_s$ ($\mathsf{\Omega }$)	[0, 0.5]	[0, 0.5]
$R_h$ ($\mathsf{\Omega }$)	[0, 500]	[0, 500]

. The parameters of these PV modules at STC are extracted. The equivalent ideality factor n of the PV module is calculated using Equation (31), and the relative error of the extracted parameter $r_p$ is calculated using Equation (32),

$$n=n_c·N_c$$

(31)

$$r_p={\displaystyle \frac{|e_p-s_p|}{s_p}}\times 100\%$$

(32)

where $n_c$ represents the ideality factor of each cell, $e_p$ represents the extracted parameter, and $s_p$ represents the parameter provided by Simulink.

As shown in Figure 7, the relative errors of $I_s$ using existing methods are obviously higher compared to other parameters. This problem is avoided using the proposed method. The relative error of each parameter using the proposed method is extremely low and almost imperceptible, indicating that the proposed parametrization method performs well.

3.2. Validation of the Model

The measured dataset is provided by NREL [43] for flat-plate PV modules, including current–voltage curves and associated meteorological data for approximately one-year periods. Although SDM achieves a good balance between accuracy and complexity for PV modeling, it is more rigorous to conduct experiments on the measured dataset in addition to the simulated dataset.

The measured I–V curves are selected at conditions around $T=25$ °C, allowing a deviation within ±5 °C. Irradiance levels including 200, 400, 600, 800, and 1000 W/m² are selected, allowing a deviation within ±10 W/m². For each irradiance level, ten set of I–V curves are randomly collected.

In order to evaluate the performance of SDM with the proposed parametrization method, two types of PV modules are selected for experiments, including monocrystalline (xSi12922) and multicrystalline (mSi0188). The characteristics of these PV modules at STC are shown in

Table 3. Characteristics of PV modules in the measured dataset.

Characteristics	xSi12922	mSi0188
Short-circuit current ($I_{sc}$)	5.127 A	2.73 A
Open-circuit voltage ($V_{oc}$)	22.06 V	22.07 V
Optimum operating current ($I_m$)	4.724 A	2.522 A
Optimum operating voltage ($V_m$)	17.58 V	18.16 V
Maximum power ($P_m$)	83.04 W	45.8 W
Number of cells ($N_c$)	36	36

. The empirical range of parameters is shown in

Table 4. Range of parameters in the measured dataset.

Parameters	xSi12922	mSi0188
$n_c$	[1, 1.5]	[1, 1.5]
$I_{ph}$ (A)	[0, 6]	[0, 3]
$I_s$ (A)	[0, ${10}^{-6}$]	[0, ${10}^{-6}$]
$R_s$ ($\mathsf{\Omega }$)	[0, 0.5]	[0, 1]
$R_h$ ($\mathsf{\Omega }$)	[0, 1500]	[0, 3000]

.

I–V curves are simulated by calculating the current based on the voltage using the Lambert W function [20]. The simulated I–V curves using existing and proposed parameter identification methods are shown in Figure 8. The relative error of current is calculated using Equation (33),

$$r_{c,j}={\displaystyle \frac{|I_{sim,j}-I_{msd,j}|}{I_{sc}}}\times 100\%$$

(33)

where $r_{c,j}$ represents the relative error of current of the j-th point; $I_{sim,j}$ and $I_{msd,j}$ represent the simulated current and the measured current of the j-th point, respectively.

The relative errors of current are shown in Figure 9. When the SDM can accurately describe the I–V curve, the errors of the proposed method are obviously lower than those of the existing methods, as shown in Figure 9a. The errors are prone to becoming higher at low irradiance levels, as shown in Figure 9b.

The overall performance is evaluated by the normalized root mean square error (nRMSE), the mean relative error (MRE), and the coefficient of determination (R²), shown as Equations (34)–(36),

$$\mathrm{nRMSE}={\displaystyle \frac{\sqrt{\frac{1}{m}·{\sum }_{j=1}^m{\left(I_{sim,j}-I_{msd,j}\right)}^2}}{I_{sc}}}\times 100\%$$

(34)

$$\mathrm{MRE}={\displaystyle \frac{1}{m}}·\sum _{j=1}^m\left|{\displaystyle \frac{I_{sim,j}-I_{msd,j}}{I_{msd,j}}}\right|\times 100\%$$

(35)

$$R^2=1-{\displaystyle \frac{{\sum }_{j=1}^m{\left(I_{sim,j}-I_{msd,j}\right)}^2}{{\sum }_{j=1}^m{\left(I_{sim,j}-\overline{I}\right)}^2}}\times 100\%$$

(36)

where $\overline{I}$ represents the average value of the measured current.

Three remarkable points on the I–V curve involve the short-circuit current $I_{sc}$, the open-circuit voltage $V_{oc}$, and the maximum power $P_m$. The relative errors of remarkable points and the performance evaluation of the overall I–V curve are shown in Figure 10. It can be observed that the proposed method achieves a good compromise between remarkable points and the overall performance.

The average computational time is shown in

Table 5. Computational time.

	Analytical	Deterministic	Meta-Heuristic	Hybrid	Proposed
Monocrystalline	0.0001 s	0.0064 s	0.0922 s	0.0401 s	0.0067 s
Multicrystalline	0.0001 s	0.0112 s	0.0912 s	0.0409 s	0.0129 s

. The analytical method is the fastest, and the meta-heuristic method requires more time. The computational time of the proposed method is moderate compared to existing methods.

Considering that the measurement error of current is usually less than 0.001 A, the Gaussian white noise (with a standard deviation of 0.001 A) is added to the current of the measured dataset to analyze the robustness.

Table 6. Robustness under noisy data of monocrystalline.

	Analytical	Deterministic	Meta-Heuristic	Hybrid	Proposed
$I_{sc}$	0.060%	0.064%	0.048%	0.057%	0.060%
$V_{oc}$	0.452%	0.456%	0.576%	0.459%	0.453%
$P_m$	0.074%	0.033%	0.084%	0.001%	0.024%
nRMSE	1.168%	0.836%	0.348%	0.134%	0.097%
MRE	0.579%	0.470%	0.162%	0.098%	0.065%
$1-{\mathrm{R}}^2$	0.382%	0.166%	0.058%	0.005%	0.002%

and

Table 7. Robustness under noisy data of multicrystalline.

	Analytical	Deterministic	Meta-Heuristic	Hybrid	Proposed
$I_{sc}$	0.150%	0.182%	0.351%	0.155%	0.169%
$V_{oc}$	0.450%	0.451%	0.621%	0.454%	0.419%
$P_m$	0.448%	0.041%	0.495%	0.001%	0.171%
nRMSE	1.919%	1.234%	0.698%	0.800%	0.529%
MRE	1.042%	0.822%	0.367%	0.503%	0.318%
$1-{\mathrm{R}}^2$	1.009%	0.478%	0.333%	0.194%	0.141%

show the average performance with varying irradiance levels under noisy data of monocrystalline and multicrystalline, respectively. The proposed method is robust and exhibits the best overall performance as well as moderate accuracy at remarkable points.

The statistical significance of performance differences between methods is analyzed using the Friedman test. As shown in

Table 8. Statistical significance.

	p-Value	Significance	Best Method
$I_{sc}$	0.6339	No	–
$V_{oc}$	0.4510	No	–
$P_m$	0.0021	Yes	Hybrid
nRMSE	0.0093	Yes	Proposed
MRE	0.0040	Yes	Proposed
$1-{\mathrm{R}}^2$	0.0093	Yes	Proposed

, the proposed method achieves the best overall performance, including nRMSE, MRE, and $1-{\mathrm{R}}^2$.

4. Discussion

In order to explain how the sensitivity-inspired parameter identification method works, monocrystalline (LR6-60-285M) is taken as an example in this section. The effect of parameters is analyzed, and the correlation between parameters is discussed in Section 4.1. Subsequently, applications of the proposed method are shown in Section 4.2. Next, the comparison of different methods is summarized in Section 4.3.

4.1. Interpretation of Parameters

The individual effect of parameters on the I–V curve due to variation of n, $I_{ph}$, $I_s$, $R_s$, and $R_h$ is shown in Figure 11. The photoinduced current $I_{ph}$ determines the short-circuit current, while the shunt resistance $R_h$ determines the slope around the short-circuit point. The influence of these two parameters on the I–V curve appears to be relatively independent. The saturation current $I_s$ and the ideality factor n have a highly correlated influence on the the open-circuit voltage. The series resistance $R_s$ affects the slope around the open-circuit point.

If $I_s$ changes along with n to keep the open-circuit voltage constant, while other parameters remain unchanged, the collaborative effect of n and $I_s$ on the I–V curve is shown in Figure 12a. It can be observed that the slope around the open-circuit point changes slightly, and the maximum power point changes obviously. In order to adjust the slope around the open-circuit point, $R_s$ can be found using the minimum overall loss of the I–V curve, as shown in Equation (28). The collaborative effect of n, $I_s$, and $R_s$ on the I–V curve is shown in Figure 12b. It indicates that when $I_s$ and $R_s$ change along with n, the I–V curve may remain approximately consistent.

Based on the above analysis, it can be inferred that the parameters that can roughly simulate the I–V curve are not unique. Within a certain margin of error, the correlation between n, $I_s$, and $R_s$ is shown in Figure 13. It can be observed that $I_s$ is positively correlated with n, and $R_s$ is negatively correlated with n. It is particularly worth noting that a slight change in n will cause a significant change in $I_s$, which explains why the relative error of $I_s$ is the largest among all the parameters in Figure 7.

4.2. Applications

The parameters at STC can be transferred to any environmental condition using Equations (37)–(41),

$$n=n_0$$

(37)

$$I_{ph}={\displaystyle \frac{G}{G_0}}·\left[I_{ph,0}+{\alpha }_i·\left(T_c-T_{c,0}\right)\right]$$

(38)

$$I_s=I_{s,0}·{\left({\displaystyle \frac{T_c}{T_{c,0}}}\right)}^3·exp\left[{\displaystyle \frac{1}{k}}·\left({\displaystyle \frac{E_{g,0}}{T_{c,0}}}-{\displaystyle \frac{E_g}{T_c}}\right)\right]$$

(39)

$$R_s=R_{s,0}$$

(40)

$$R_h={\displaystyle \frac{G}{G_0}}·R_{h,0}$$

(41)

where G represents the irradiance; the subscript “0” represents STC; ${\alpha }_i$ represents the temperature coefficient of $I_{sc}$, and ${\alpha }_i=0\mathrm{A}/°\mathrm{C}$ for LR6-60-285M; $E_g$ represents the energy bandgap, calculated in Equation (42). The energy bandgap at STC is shown in

Table 9. Energy bandgap at STC.

	Monocrystalline	Multicrystalline
$E_{g,0}$	$1.794\times {10}^{-19}$ J	$1.826\times {10}^{-19}$ J

.

$$E_g=\left[1-0.0002677\left(T_c-T_0\right)\right]·E_{g,0}$$

(42)

The parameters at STC are transferred to $T_c=85°\mathrm{C}$ and the irradiance remains unchanged. The measured and predicted I–V curves, as well as P–V curves, are shown in Figure 14 and Figure 15, respectively.

When the parameters at STC are transferred to $T_c=85°\mathrm{C}$ and the irradiance remains unchanged, the simulated open-circuit voltage and maximum power using existing methods show significant deviations, while the simulated I–V curve and P–V curve using the proposed method are both consistent with the measured curves. It implies that the proposed method can be applied to varying operating conditions, which provides an important foundation for the monitoring of PV systems. In addition, the variation of parameters is related to degradation and aging of the PV module. The accurate parameter identification method is beneficial for fault diagnosis, which exhibits the potential to be applied to the intelligent operation and maintenance of photovoltaic systems.

4.3. Comparison of Different Methods

The analytical method does not involve complex optimization problems; thus, it has the advantage of high computational efficiency. On the contrary, the numerical method, specifically the meta-heuristic method, requires more computational time because each parameter needs to be optimized.

The hybrid method combines the advantages of the analytical method and the numerical method. A hybrid method combining analytical expressions and the Flow Direction Algorithm is adopted for comparison. The objective function of this method is to minimize the error of remarkable points. As a result, the hybrid method exhibits the best accuracy at the maximum point. Due to the integration of the meta-heuristic method, namely, the Flow Direction Algorithm, the result of this hybrid method is stochastic.

A simple hybrid method, namely, the deterministic method, finds the ideality factor and the series resistance using the dual-iteration algorithm. The proposed method adopts a similar framework, and explores insights into the sensitivity of parameters. The proposed sensitivity-inspired method achieves the best accuracy of parameters in the simulated dataset and the best overall performance (including nRMSE, MRE, and $1-{\mathrm{R}}^2$ in the measured dataset. When applied to varying operating conditions, the transition capability of the proposed method is better than existing methods.

However, there is no single method that can outperform others in all evaluation metrics. Furthermore, the SDM is totally applicable to monocrystalline (xSi12922), and is partly suitable for multicrystalline (mSi0188) at high irradiance levels. The performance could be improved if the proposed method is extended to the DDM.

5. Conclusions

This article proposes a novel parameter identification method for the SDM, inspired by the sensitivity. In order to reveal the influence of measurement errors, the sensitivity of parameters to the current, the voltage, and the I–V gradient were analyzed, and it was found that the calculation of the photoinduced current using the short-circuit current does not magnify the measurement error. The sensitivity of the loss function to parameters was analyzed, and it was demonstrated that the error of the diode ideality factor has the most significant impact on the loss function. The explicit expressions of the photoinduced current, the shunt resistance, and the reverse saturation current were derived, making the calculation errors as low as possible. The diode ideality factor and the series resistance were extracted by iteration, searching for the optimal value of the mixed loss function. Simulation results verify that the extracted parameters were consistent with the true values. Experiments on the measured datasets showed that the proposed method achieves a good compromise between remarkable points and the overall performance. This work provides an important foundation for monitoring and fault diagnosis, which contributes to the intelligent operation and maintenance of PV systems. Furthermore, the SDM is totally applicable to monocrystalline (xSi12922) and is partly suitable for multicrystalline (mSi0188) at high irradiance levels. The performance could be improved if the proposed method is extended to the DDM. The proposed method has not been tested on thin-film modules. In future research work, sensitivity analysis and parametrization based on the DDM will be explored, and experiments on thin-film modules will be conducted.