Self-adaptive single-diode model parameter identification under small mismatching conditions

Luis Garcia-Marrero, Carlos Pavon-Vargas, Juan Bastidas-Rodriguez, Eric

Monmasson, Giovanni Petrone

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Luis Garcia-Marrero, Carlos Pavon-Vargas, Juan Bastidas-Rodriguez, Eric Monmasson, Giovanni Petrone. Self-adaptive single-diode model parameter identification under small mismatching conditions. Renewable Energy, 2025, 245, pp.122735. 10.1016/j.renene.2025.122735. hal-05069010

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Submitted on 23 Jun 2025

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# Self-adaptive single-diode model parameter identification under small mismatching conditions

Luis E. Garcia-Marrero<sup>a,b,cID*</sup>, Carlos I. Pavon-Vargas<sup>aID</sup>, Juan D. Bastidas-Rodriguez<sup>a,cID</sup>, Eric Monmasson<sup>bID</sup>, Giovanni Petrone<sup>aID</sup>

## ARTICLE INFO

### Keywords:

- Single-diode model

- Parameter identification

- Mismatching conditions

- PV diagnosis

- Evolutionary algorithms

## ABSTRACT

Current model-based methods for monitoring photovoltaic (PV) modules typically rely on the single-diode model (SDM) or its variants, assuming uniform operating conditions across the module. However, these ideal conditions are difficult to realize in real-world applications due to partial shading, soiling, degradation, and other phenomena. This paper proposes a 7-parameter self-adapting Double SDM model (D-SDM) to enhance the accuracy and reliability of parameter identification in PV modules under real operating conditions. A robust methodology based on evolutionary algorithms is proposed to estimate the parameters of the D-SDM, directly from the I-V characteristic of a PV module, applicable in both uniform and mismatched scenarios. The proposed methodology also includes a robust fitting error calculation that only considers the section of the I-V curve where all the cells operate with positive voltage. The methodology is validated using experimental and simulated I-V curves across various mismatching patterns, demonstrating the superior stability and reliability of the proposed method, which can be used for PV system monitoring and diagnosis in complex conditions.

## 1. Introduction

In real-world applications, monitoring the condition of photovoltaic (PV) modules, quantifying degradation, and diagnosis are often evaluated using model-based parameter identification methods [1,2]. These approaches involve fitting the electrical behavior of the module to an equivalent electric circuit, or model, which requires identifying a set of parameters. By analyzing the behavior of these parameters over time, it is possible to detect performance degradation or faults [3]. In addition to PV modules, PV submodules can be used for a more detailed diagnosis, representing a set of PV cells connected in series with a bypass diode in antiparallel [4].

The single-diode model (SDM) is the most widely used model for PV submodules, modules, or strings operating under uniform conditions since it provides a trade-off between accuracy and complexity. Moreover, this model has shown to be effective in modeling PV generators with crystalline cells, which is the most widely adopted technology [4]. Extensive research has been conducted on methods for identifying the parameters of the SDM. These methods can generally be classified into two categories: analytical and numerical approaches [5,6]. Analytical methods simplify the PV model by focusing on key points of its current

vs. voltage (I-V) characteristic [5]. In contrast, numerical methods utilize the entire I-V characteristic, or a portion of it, to determine a set of parameters that provides an optimal fitting [7].

The study in [8] proposes an iterative method based on the Levenberg-Marquardt algorithm and simultaneous determination of PV module temperature and irradiance, eliminating explicit equations. A numerical method is introduced in [9], employing a genetic algorithm (GA) that relies on only five I-V curve points. The work in [10] presents an analytical method using explicit equations based on the Lambert W function for parameter identification from datasheet values. Similarly, [11] describes an integral-based analytical approach that employs linear least squares to enhance accuracy. In [12], a metaheuristic numerical method is presented, using Particle Swarm Optimization (PSO) and incorporating temperature-dependent series resistance.

A hybrid analytical-numerical technique is detailed in [13], combining PSO with temperature modeling for improved results. In [14], a method combining analytical and iterative techniques is described through explicit equations and step-based refinement. The work in [15] proposes a multi-objective evolutionary algorithm designed to optimize

<footnote><sup>a</sup> Dipartimento di Ingegneria dell'Informazione ed Elettrica e Matematica Applicata/DIEM, University of Salerno, Fisciano, 84084, Italy</footnote>

<footnote><sup>b</sup> Systèmes et Applications des Technologies de l'Information et de l'Energie laboratory, CY Cergy Paris University, Cergy-Pontoise, 95031, France</footnote>

<footnote><sup>c</sup> Facultad de Ingeniería y Arquitectura, Universidad Nacional de Colombia, Manizales, 170003, Colombia</footnote>

<footnote>* Corresponding author at: Dipartimento di Ingegneria dell'Informazione ed Elettrica e Matematica Applicata/DIEM, University of Salerno, Fisciano, 84084, Italy.</footnote>

<footnote>E-mail address: lgarciamarrero@unisa.it (L.E. Garcia-Marrero).</footnote>
static and dynamic model parameters simultaneously to achieve consistency across environmental conditions. The study in [16] introduces a deterministic numerical approach using derivative-based optimization, improving accuracy with minimal computation. Lastly, [17] presents a metaheuristic method based on Diversity Improvement-Oriented Differential Evolution (DIODE), which enhances population diversity to prevent premature convergence.

However, all the previous approaches assume that the cells within a PV module are identical and operate under uniform irradiance and temperature conditions. As a result, the model produces I-V curves with the same shape and curvature as those of a single cell, scaled according to the number of series-connected ($N_s$) and parallel-connected ($N_{sh}$) cells. This strong assumption is difficult to fulfill in real applications since the cells in a PV module or string, mismatch due to different causes. Some of the most common causes are partial shading due to surrounding objects (temporary or permanent), soiling, uneven aging of the PV cells, early degradation, and differences in the manufacturing process [18,19]. The same assumption is adopted when using other models in the literature, such as the double-diode model (DDM) and triple-diode (TDM) model, which are applied to model PV generators with cells from various technologies [20].

When a PV module or string is under non-uniform conditions, estimating the SDM parameters from an experimental I-V curve will lead to unrealistic values. Non-uniform conditions produce distorted I-V curves that generate unrealistic parameters even if the distortion is difficult to identify with the naked eye [18,19]. For example, some authors have shown that mismatching conditions generate an I-V curve with a sharper curvature around the maximum power point [21]. In that case, the uniformity assumption provides unrealistic parameters for the SDM due to the lack of the necessary degrees of freedom to account for the underlying physical structure. Moreover, the unrealistic SDM parameters may lead to wrong diagnostics or degradation quantification of the monitored system.

One approach to consider realistic mismatching conditions of the cells forming a PV module or string is to increase the model granularity by representing each cell with the Bishop equivalent circuit [22], which requires eight parameters. This approach has been used to simulate different PV generators operating under mismatching conditions assuming as known the parameters of the cells [23]. Nevertheless, the parameter identification of a model with this granularity significantly increases the complexity and computational burden of the model, since in a module or string with $N_s$ cells, the number of parameters to identify would be $8N_s$. Additionally, in a PV module, measuring the voltage of each cell is not possible, making it even more difficult to identify the parameters of the Bishop model.

Submodule-level simulations, where each submodule includes a group of cells protected by a single bypass diode, reduce computational complexity. In [24], a submodule-based modeling approach is proposed for series-parallel PV arrays under mismatching conditions, using the bisection method instead of Jacobian-based methods. In [25], a model is introduced to predict the performance of large PV systems, focusing on submodule-level mismatch calculations and explicit I-V curve evaluation. In [26], submodule-level simulations are compared to finer cell granularity methods, such as the cell-by-cell method, achieving similar accuracy, especially when the submodule is uniformly shaded. Although these approaches are sufficiently accurate for simulation studies, they all assume uniform operating conditions across the submodule. This assumption limits their applicability for reliable parameter identification from experimental I-V curves, as demonstrated in [19].

Another approach to model PV strings operating under mismatching conditions is to represent each module with the SDM [5,27], DDM [28], or TDM [29]. Then, the modules are connected in series to form an equivalent circuit that represents the string. A methodology proposed in [27] focuses on detecting voltage inflection points and knees in the I-V curve. In this approach, the SDM is first fitted under various uniform

operating conditions to derive a functional relationship between the parameters and the operating conditions. However, these approaches also assume that all the cells in each module are equal and operate under uniform conditions, which is not realistic as explained before.

A model-based online diagnostic tool may rely on mismatch detection algorithms to run parameter identification procedures when operating conditions are near uniform. However, in the presence of small mismatches, such as minor soiling or manufacturing differences, uniformity may be incorrectly assumed, resulting in significant errors in parameter estimation. These errors can obscure the SDM parameters trends, complicating the identification of early signs of malfunctions or degradation. Relying solely on mismatch detection to decide when to perform parameter identification, assumes that uniform conditions are always necessary, which is unrealistic in real-world PV systems. Many small mismatches occur frequently that can distort critical parameters if not accounted for. To fill this gap, this paper proposes a novel parameter identification procedure that remains robust to non-ideal operating conditions. Consequently, this work seeks to improve the practical applicability of parameter identification methods, enabling their use in online diagnostic tasks under real operating conditions.

This paper is an extension of the conference paper [19], where a procedure to estimate the SDM parameters under a small mismatch produced by irregular artificial illumination is proposed. This new paper improves the model by reducing the number of parameters to identify. Moreover, a new procedure is proposed, that can be applied to I-V curves with diverse mismatching conditions common in real applications, including a robust error function that automatically selects the part of the I-V curve where all the cells operate in positive voltages. The self-adaptive nature of the procedure allows it to adjust its parameters in response to varying shading patterns, ensuring consistent performance across different scenarios, and making it applicable in both uniform and mismatched conditions. Finally, an experimental validation in outdoor conditions under different mismatching patterns is proposed in this new paper.

In summary, the main contributions of this work are:

* A 7-parameter self-adapting SDM-based model, denominated Double Single-Diode Model (D-SDM), is proposed to enhance the accuracy and reliability of parameter identification in PV modules under mismatched conditions

* A robust methodology based on evolutionary algorithms is introduced to estimate the parameters of the proposed D-SDM model

* The advantages of the proposed model and methodology are demonstrated using experimental and simulated I-V curves across various real partial shading patterns

The rest of the paper is organized as follows. Section 2 introduces the SDM model for PV cells and modules highlighting its limitations for mismatching conditions. Section 3 outlines the foundations of the proposed D-SDM model, including the circuit design, underlying assumptions, and relevant equations. Section 4 describes the methodology to extract the parameters of the D-SDM model. Section 5 validates the model and methods by using experimental and simulated I-V curves, and Section 6 closes the paper with the conclusions.

## 2. Model of a PV module based on SDM

The SDM is widely used to represent a PV module consisting of $N_s$ series-connected cells and protected by bypass diodes connected in antiparallel [4]. However, this approach assumes that all the cells are the same and operate under the same irradiance and temperature conditions, which is difficult to guarantee in real applications. This section introduces the SDM for the cell and the module, highlighting the limitations of the uniformity assumption when the PV module is under mismatching conditions.
<figure><img src="image_4.png" /><figcaption>Fig. 1. Single diode model for a single solar cell.</figcaption></figure>

## 2.1. SDM of a PV cell and module

The basic constructive element of a PV module is a cell, which can be represented by the SDM shown in Fig. 1. The relation between the cell's current ($I_c$) and voltage ($V_c$) is introduced in (1), where $I_{ph-c}$ is the photo-induced current, $I_{s-c}$ is the inverse saturation current, $\eta_c$ is the diode ideality factor, $R_{s-c}$ is the series resistance, and $R_{sh-c}$ is the parallel resistance. Moreover, $V_{th-c}$ is the thermal voltage, which can be defined as a function of the cell temperature ($T_c$), the Boltzmann constant $k$, and the electron charge $q$ as follows: $V_{th-c} = kT/q$.

$$I_c = I_{ph-c} - I_{s-c} \left( \exp\left(\frac{V_c + I_c R_{s-c}}{\eta_c V_{th-c}}\right) - 1 \right) - \frac{V_c + I_c R_{s-c}}{R_{sh-c}} \quad (1)$$

The SDM can be scaled to represent a PV module consisting of $N_s$ cells connected in series, provided that all $N_s$ cells are identical and operate under uniform conditions. In that case, it is possible to use the same structure of (1) but scaling the parameters as shown in (2)–(7), where subindex $m$ is used to indicate the parameters of the entire module.

$$V_{th-m} = N_s \cdot V_{th-c} \quad (2)$$

$$I_{ph-m} = I_{ph-c} \quad (3)$$

$$I_{s-m} = I_{s-c} \quad (4)$$

$$\eta_m = \eta_c \quad (5)$$

$$R_{s-m} = N_s \cdot R_{s-c} \quad (6)$$

$$R_{sh-m} = N_s \cdot R_{sh-c} \quad (7)$$

In realistic scenarios, each PV cell operates under unique irradiance and temperature conditions, influenced by factors such as shading or manufacturing differences, which can result in mismatches among the cells. Even when the mismatch level is small enough that the deformation of the I-V curve is not visually noticeable, fitting the SDM to such a curve will fail to identify the parameters accurately. The five parameters adjusted during the curve-fitting process often reach unrealistic values.

As an example, consider the experimental curve shown in Fig. 2. It refers to the PV string shown in Fig. 7, Case I, where two modules connected in series seem under the same irradiance conditions, but there is a small mismatch possibly caused by the reflexes on the surface of the module. In these conditions, bypass diodes are not active and the I-V curve is expected to be well fitted with the SDM. Nevertheless, when attempting to fit the SDM to the experimental curve from 0 to $V_{OC}$, the optimization algorithm tends to prioritize solutions that fit a larger number of points, which compromises the accuracy near the MPP (as seen in Fig. 2, curve B). One potential solution to improve the fitting around the MPP is to focus on a portion of the curve near this point. Fig. 2, curve C illustrates the fitting when using data points from voltages slightly below the MPP to $V_{OC}$. As seen in Table 1, this modification results in significantly different estimated parameters. The series resistance $R_s$ decreases notably, while the parallel resistance $R_{sh}$ increases dramatically, converging to the upper bound of the search interval. These inconsistencies between the two fittings, when similar values might be expected, highlight the limitations of the uniformity assumption, even when the mismatching conditions are not immediately visible.

<figure><img src="image_5.png" /><figcaption>Fig. 2. Uniform assumption limitations. A. Experimental data; B. SDM fitting using the full experimental points, C. SDM fitting using only the points with voltages between $V_{mpp}$ and $-0.2V_{mpp}$ to $V_{oc}$.</figcaption></figure>

<table><caption>Table 1
Uniform assumption limitations. A. Experimental data; B. SDM fitting using the full experimental points, C. SDM fitting using only the points with voltages between $V_{mpp}$ and $-0.2V_{mpp}$ to $V_{oc}$.</caption><thead><tr><th>Fitting</th><th><i>I</i><sub>ph</sub>[A]</th><th><i>I</i><sub>s</sub>[A]</th><th><i>η</i></th><th><i>R</i><sub>s</sub>[Ω]</th><th><i>R</i><sub>sh</sub>[Ω]</th></tr></thead><tbody><tr><td>B</td><td>5.6029</td><td>4.8474e-06</td><td>1.4934</td><td>0.17562</td><td>93.121</td></tr><tr><td>C</td><td>5.6089</td><td>9.5054e-05</td><td>1.8901</td><td>0.090265</td><td>∞</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Two-irradiance levels assumption for a PV module with $N_s$ cells in series.</figcaption></figure>

## 3. Adaptive double-SDM of a PV module

To consider the mismatching conditions in a PV module, it is proposed to divide the $N_s$ module cells into two groups, i.e. group a and group b, considering that the module cells operate under two main irradiance levels: shaded and unshaded. This approach increments the freedom degrees without significantly increasing the model complexity, the number of parameters to identify, and the computational burden, as occurs in the case of using a dedicated model for each cell. In realistic scenarios, each PV cell operates under unique irradiance and temperature conditions. However, in cases of small mismatches, where the difference in irradiance is not significant, the conditions of the affected cells are often similar. As a result, the two-irradiance levels assumption becomes a reasonable approximation. Fig. 3 presents a schema of the two-irradiance levels assumption for a PV module with $N_s$ cells in series, where $N_{s-a}$ and $N_{s-b}$ are the number of cells in each group and $N_s = N_{s-a} + N_{s-b}$.

Small mismatch scenarios are of particular interest, as these are more challenging to detect from I-V curve analysis and are highly relevant for robust parameter identification. In this case, the operating conditions differ only slightly, making the deviations in the I-V curve less pronounced and more difficult to identify. Significant mismatch scenarios, in contrast, typically result in clearly noticeable deviations in the I-V curve. In such cases, running a parameter identification method may not be meaningful, as these mismatches are often transient conditions caused by factors such as clouds.

In the proposed approach, each group is represented by an SDM as illustrated in Fig. 4. It is important to note that the modeling
<figure><img src="image_7.png" /><figcaption>Fig. 4. D-SDM model representing groups a and b in series.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Simulink block of the schema shown in Fig. 3.</figcaption></figure>

strategy focuses on estimating the number of affected cells within the module rather than identifying their exact locations. By categorizing cells into shaded and unshaded groups, the model achieves a more accurate representation of the electrical behavior of the module under non-uniform conditions, without the complexity of modeling each cell individually. The parameters $N_{s-a}$ and $N_{s-b}$ dynamically adjust themselves to the operating conditions, allowing the model to adapt itself to various scenarios. This adaptability makes the model more general, as it can handle uniform and non-uniform operating conditions. Hence, the circuit model shown in Fig. 4 is denominated as Double SDM (D-SDM).

This modeling strategy aims to maintain the critical parameters of the SDM ($I_{s-c}$, $\eta_c$, $R_{s-c}$, $R_{sh-c}$) independent from mismatching patterns. This approach prevents distortions in the I-V curve caused by mismatch issues from resulting in incorrect parameter estimation intended to fit the curve. Instead, any distortion in the I-V curve is expected to be captured by the introduced parameters $N_{s-a}$, $N_{s-b}$, $I_{ph-a}$, and $I_{ph-b}$.

Assuming the cells of groups a and b operate under irradiances $G_a$ and $G_b$, respectively, and $G_b > G_a$ without loss of generality, it is possible to analyze the section of the I-V curve where all the cells operate at positive voltages. The validity of the SDM is limited to positive voltages, so it is important to consider only the part of the I-V curve where all cells are operating within this range. As widely discussed in the literature, shaded cells in a PV string may operate in

reversed mode depending on the number of shaded cells, the shading effect level, and the PV module operating voltage. In the presence of a significant mismatch level, as for the example discussed in [30], the shading effect introduces a visible deformation of the I-V curve by reducing the delivered power and with the risk of triggering hot-spots in the cells operating with negative voltage. On the other side, if the level of mismatch is small, the effect is not visible on the global I-V curve, nevertheless it may introduce a difference in the operating voltage of each cell.

To explain this effect, a Simulink block reproducing the scheme of Fig. 3 has been simulated to evaluate the voltage across the shaded and unshaded cells as a function of the PV output voltage for three cases: 1 shaded cell, 18 shaded cells, and 35 shaded cells in a PV module composed of 36 series connected cells. A 2% difference in irradiance conditions between the cells in each block ($G_a = 0.98G_b$) is analyzed for each simulation. The Solar Cell Simulink embedded block [31] is used with its default parameter values, as shown in Fig. 5. Details on the Solar Cell block and its integration with a variable electric load for I-V curve tracing are available on the MATLAB webpage [31]. As it is evident from the plots on the left side of Fig. 6 the voltage of each cell changes significantly as a function of the number of shaded cells even if the I-V curve of the PV module remains almost unchanged (right plots of Fig. 6). Hence, the key idea of having a self-adapting number of shaded/unshaded cells helps to tune the D-SDM also in the presence of small mismatching conditions, when the I-V curves seem almost uniform; moreover, this model automatically converges to the SDM in the presence of really uniform conditions.

It is worth noting that $R_{sh-b} \approx \infty \Omega$ in the model presented in Fig. 4 for the group of unshaded cells. This approximation is performed considering that $R_{sh-b}$ has a negligible impact on the output current of group b (i.e. in the I-V curve of group b) for high voltages; since in this section the output current principally depends on the current of the diode, which increases exponentially and is much larger than $R_{sh-b}$ current. Moreover, the section of the module I-V curve where shaded and unshaded cells operate at positive voltages corresponds to high voltages for the I-V curve of group b (unshaded cells). Numerical validation of this simplification is provided in Section 5.4.

It is worth noting that the cell temperature is assumed to be the same across the module $T_a = T_b = T_c$. In steady-state conditions, the cell temperature $T_c$ can be estimated as a function of the ambient temperature $T_{amb}$ and the incident irradiance $G_T$ through (8) [32], where $T_{NOCT}$ is the Nominal Operating Cell Temperature (NOCT), typically 45 °C; $T_{amb,NOCT}$ is the ambient temperature under NOCT conditions, typically 20 °C; and $G_{NOCT}$ is the solar irradiance under NOCT conditions, typically 800 W/m².

$$T_c = T_{amb} + \left( \frac{T_{NOCT} - T_{amb,NOCT}}{G_{NOCT}} \right) G_T \quad (8)$$

The difference in cell temperature $\Delta T_c = T_b - T_a$ due to a difference in irradiance $\Delta G = G_b - G_a$ in a PV module can be derived by manipulating (8) as:

$$\Delta T_c = \left( \frac{T_{NOCT} - T_{amb,NOCT}}{G_{NOCT}} \right) \Delta G \quad (9)$$

For typical values ($T_{NOCT}$ = 45 °C, $T_{amb,NOCT}$ = 20 °C, $G_{NOCT}$ = 800 W/m²), the proportionality constant is approximately 0.03. Thus, a difference in irradiance of 100 W/m² corresponds to a cell temperature variation of approximately 3 °C. Given that the proposed approach focuses on small mismatches and the minimal effect of irradiance variation on temperature changes, temperature variations are not included in the model proposed in this paper to improve the computational efficiency. However, it is important to note that such variations can be included in the D-SDM model by increasing the number of parameters to identify, which would consequently raise the computational complexity.

In real applications, measuring the temperatures of all the cells is not practical. Therefore, it is assumed that the temperature can be
<figure><img src="image_9.png" /><figcaption>Fig. 6. Operating voltage of Cell versus PV module voltage in different mismatched scenarios for a 2% difference in irradiance conditions ($G_a = 0.98G_b$).</figcaption></figure>

approximated either as the average of the available temperature sensors in the module [33] or, as discussed earlier, estimated using ambient temperature and irradiance data [32,34].

From the circuit shown in Fig. 4, it is possible to write (10) and (11), where the voltage and the current of the module are represented by $V_m$ and $I_m$ respectively. In those equations, the SDM parameters of each group are identified by subscripts a and b: $I_{ph-a}$, $I_{s-a}$, $\eta_a$, $R_{s-a}$, $R_{sh-a}$ for group a, and $I_{ph-b}$, $I_{s-b}$, $\eta_b$, and $R_{s-b}$ for group b. It can be observed that the currents of groups a and b are equal to the module current, i.e. $I_m = I_a = I_b$; while the relation between $V_m$ and the voltages of groups a ($V_a$) and b ($V_b$) is given in (12).

$$I_a = I_{ph-a} - I_{s-a} \left( \exp\left(\frac{V_a + I_a R_{s-a}}{\eta_a V_{th-a}}\right) - 1 \right) - \frac{V_a + I_a R_{s-a}}{R_{sh-a}} \quad (10)$$

$$I_b = I_{ph-b} - I_{s-b} \left( \exp\left(\frac{V_b + I_b R_{s-b}}{\eta_b V_{th-b}}\right) - 1 \right) \quad (11)$$

$$V_m = V_a + V_b \quad (12)$$

The inverse saturation current and ideality factor do not depend on the number of cells in each group, according to (4) and (5); hence, they have the same values for groups a and b (see (13) and (14)). The inverse saturation current is well known to depend on the temperature, but the assumption of the same temperature for all the cells justifies the use of (13). Furthermore, the series resistances, parallel resistance, and the thermal voltages of groups a and b depend on the number of cells in each group as introduced in (15), (16), and (17) since those parameters are directly proportional to the number of cells connected in series in each group.

$$I_{s-c} = I_{s-a} = I_{s-b} \quad (13)$$

$$\eta_c = \eta_a = \eta_b \quad (14)$$

$$R_{s-c} = \frac{R_{s-a}}{N_{s-a}} = \frac{R_{s-b}}{N_{s-b}} \quad (15)$$

$$R_{sh-c} = \frac{R_{sh-a}}{N_{s-a}} \quad (16)$$

$$V_{th-a} = N_{s-a} \cdot \frac{k T_c}{q} \qquad V_{th-b} = N_{s-b} \cdot \frac{k T_c}{q} \quad (17)$$

Eqs. (13)–(16) incorporate certain assumptions regarding the dependence of parameters on irradiance and temperature. Although experimental studies in the literature do not provide a consensus on the behavior of each model parameter under varying temperature or irradiance levels [35], some assumptions are widely accepted. The saturation current is typically assumed to depend only on temperature [36,37], while the parallel resistance is generally considered to depend only on irradiance [37,38]. The ideality factor is assumed to be independent of irradiance, while some studies suggest it may depend on temperature [36,38]. The series resistance is often treated as independent of both irradiance and temperature [37], although some research reports a linear variation of $R_s$ with temperature [39].

Consistent with these prior studies, this work assumes that the saturation current, the ideality factor, and the series resistance are independent of irradiance, a widely supported assumption in the literature [40]. Although these assumptions are adopted here with good results, the proposed D-SDM model could incorporate alternative dependency relationships between the two blocks, which, depending on the PV technology, may lead to improved accuracy. Finally, the parameters $I_{ph-a}$ and $I_{ph-b}$ depend on $G_a$ and $G_b$, respectively. In summary, the D-SDM is represented in Fig. 4 and described by (10)–(12). Considering Eqs. (13)–(17), the parameters to identify can be deduced:

* Replacing $N_{s-b} = N_s - N_{s-a}$ in (17)

* Replacing (13)–(17) in (10) and (11)

Performing these steps, the D-SDM has seven (7) parameters to be identified that can be defined as follows:

$$\bar{P} = [N_{s-a}, I_{ph-a}, I_{ph-b}, I_{s-c}, \eta_c, R_{s-c}, R_{sh-c}] \quad (18)$$

It is important to note that the identified parameters correspond to one cell; therefore, they can be scaled to represent groups a and b by using (2)–(7) but replacing $N_s$ with $N_{s-a}$ and $N_{s-b}$, respectively.

## 4. Proposed method to identify the D-SDM parameters

The identification of $\bar{P}$ can be formulated as an optimization problem by fitting the Eqs. (10)–(12) to the experimental I-V curves. As
widely discussed in the literature, different approaches can be considered for fitting the SDM parameters to experimental data [17,19]. One way is to use the experimental voltage values ($V_m = V_{exp}$) in the model and determine the parameters $\vec{P}$ that best fit the calculated current ($I_m$) with the experimental current values ($I_{exp}$), thus requiring the current to be expressed as a function of voltage. A complementary approach is given if the experimental current values are used in the model ($I_m = I_{exp}$) and parameters are optimized to fit the calculated voltage ($V_m$) with the corresponding experimental values ($V_{exp}$). It should be noted that Eqs. (10)–(11) have an implicit form; therefore, they cannot be directly used to determine the module current ($I_m$) for a given voltage ($V_m$) or vice versa, which is a key step in calculating the error between the model and the experimental I-V curve.

As a consequence, the first step is to express the implicit Eq. (10) as an explicit equation by using the Lambert-W function [41]. That explicit equation is introduced in (19), where $W(\cdot)$ indicates the principal branch of the Lambert-W function. In contrast, $V_b$ can be expressed explicitly from (11), as shown in (22), thanks to the parallel resistance simplification of group b.

$$V_a = -I_m (R_{s-a} + R_{sh-a}) + R_{sh-a} (I_{ph-a} + I_{s-a}) - \eta_a V_{th-a} L_a \quad (19)$$

$$L_a = W \left\{ \frac{I_{s-a} R_{sh-a} \exp(S_a)}{\eta_a V_{th-a}} \right\} \quad (20)$$

$$S_a = \frac{R_{sh-a}(-I_m + I_{ph-a} + I_{s-a})}{\eta_a V_{th-a}} \quad (21)$$

$$V_b = \eta_b V_{th-b} \ln \left( \frac{I_{ph-b} - I_m}{I_{s-b}} + 1 \right) - I_m R_{s-b} \quad (22)$$

Using Eqs. (19) and (22), $V_a$ and $V_b$ can be calculated for a given $I_m$, followed by the calculation of $V_m$ using (12). Hence, it could be possible to fit an I-V curve by defining the fitness function of the optimization problem as a function of the module voltage; in other words, using $I_m$ as the independent variable and $V_m$ as the dependent one. Nevertheless, this fitting approach tends to produce more errors near the short-circuit current of any subblock (i.e., at low voltages) because small errors or variations in $I_m$ lead to significant variations in $V_m$.

To increase the robustness of the fitting process, a binary search procedure is proposed to determine $I_m$ for a given $V_m$, with a defined accuracy $\epsilon$. For each experimental voltage point, Algorithm 1 estimates the corresponding current using an iterative approach that leverages the monotonicity of the I-V curve. The inputs of this algorithm are $V_{exp}(i)$, $I_{exp}(i)$, $\epsilon$, $\tau$, and the parameters of the D-SDM model ($\vec{P}$), where $V_{exp}(i)$ and $I_{exp}(i)$ describe the ith experimental point of the I-V curve, and $\tau$ is a constant greater than 1 that defines the maximum value of $I_m$ for the initial search range. The output of Algorithm 1 is $I_m(i)$, which is the calculated current of the PV module corresponding to $V_m(i) = V_{exp}(i)$; a brief description of how it works is reported below.

The algorithm begins by defining an initial search range, i.e. $I_{min} < I_m < I_{max}$, where $I_{max} = \tau I_{exp}$. Then, it calculates the midpoint between $I_{min}$ and $I_{max}$ ($I_{mid}$), which is used to determine $V_a$ and $V_b$ for $I_m = I_{mid}$ by using (19) and (22) as well as $V_m$ by evaluating (12). After, $V_m > V_{exp}(i)$ means that $I_{mid} > I_{exp}(i)$; then, the search range of $I_m(i)$ can be reduced by setting $I_{min} = I_{mid}$; otherwise, the search range can be restricted by making $I_{max} = I_{mid}$. It is important to note that this algorithm is valid because the I-V curve is always decreasing.

## 4.1. Error calculation

At this point, it is possible to calculate the currents of an experimental I-V curve with N points by using D-SDM for a given set of parameters $\vec{P}$. The experimental I-V curve is described by $V_{exp}(i)$ and $I_{exp}(i)$ with $i \in [1, ..., N]$; then, the D-SDM currents ($I_m(i)$) are calculated by executing Algorithm 1 for each value of $V_{exp}(i)$.

To enhance the robustness of the error calculation for a given set of parameters $\vec{P}$, only experimental points where the validity of the D-SDM is satisfied are considered; i.e. all the cells operate at positive

### Algorithm 1 Binary search for finding $I_m(i)$

Inputs: $V_{exp}(i)$, $I_{exp}(i)$, $\epsilon$, $\tau$, and $\vec{P}$

Output: $I_m(i)$

for i=1:N do

$$I_{min} = 0, I_{max} = \tau I_{exp}[i] \\ \text{while } I_{max} - I_{min} > \epsilon \text{ do}$$

$$I_{mid} = \frac{I_{max} + I_{min}}{2}$$

Calculate $\vec{V}_a$ and $V_b$ for $I_{mid}$ using (19) and (22)

$$V_m = V_a + V_b$$

if $V_m > V_{exp}(i)$ then

$$I_{min} = I_{mid}$$

else

$$I_{max} = I_{mid}$$

end if

end while

$$I_m(i) = \frac{I_{max} + I_{min}}{2}$$

end for

voltages. In other words, the error calculation only considers the section of the I-V curve where $V_a$ and $V_b$ are positive. More formally, if the experimental I-V points are sorted in decreasing order by their voltage values:

$$V_{exp}(i) > V_{exp}(i + 1) \quad \text{for } i \in [1, ..., N - 1],$$

then, the error calculation considers M points, where M is the maximum index such that the point $V_{exp}(M + 1)$ fails to meet the validity criteria $V_a > 0$ V and $V_b > 0$ V.

Given this rule for selecting valid points on the I-V curve, the error function must be carefully defined. It should prioritize including as many points as possible from the I-V curve while minimizing the discrepancies between the experimental and calculated curves. Then, the error function is defined based on the relative area between these two curves. The error is computed using the trapezoidal rule, where the voltage difference between consecutive points is defined as:

$$\Delta V(i) = V_{exp}(i + 1) - V_{exp}(i)$$

The error, denoted by $\epsilon$, is then calculated as the ratio of the area between the experimental and estimated curves to the area under the experimental curve:

$$\epsilon = \frac{\sum_{i=1}^{M-1} \left[ \frac{|I_{exp}(i) - I_m(i)| + |I_{exp}(i+1) - I_m(i+1)|}{2} \right]}{\sum_{i=1}^{M-1} \left[ \frac{|I_{exp}(i) + I_{exp}(i+1)|}{2} \right]} \Delta V(i) \quad (23)$$

It is worth noting that M must be identified each time (23) is evaluated since it depends on the parameters defined in $\vec{P}$ (18).

## 4.2. Optimization method

Differential evolution algorithm (DE) [42] is selected to find the values of $\vec{P}$ that minimizes $\epsilon$. Particularly, the implemented variant is DE/best/1/bin using the Python framework Pymoo [43]. Moreover, the algorithm parameters are selected by attending the guidelines given in [44]: scale factor $F = 0.8$, and crossover rate $Cr = 0.9$, which helps to optimize non-separable functions.

Given the integer nature of the parameter $N_{s-a}$, this paper evaluates two variants of the optimization method. The first, named D-SDM, treats $N_{s-a}$ as a real parameter, while the second, named D-SDM-I considers it as an integer value. DE is inherently a global optimization algorithm for real-encoded variables [42]; therefore, additional customization is required to handle integer variables. In the first methodology, where $N_{s-a}$ is treated as a real variable, the standard DE is employed with a population size of NP = 40, running the algorithm until convergence.
In the second methodology, where $N_{s-a}$ is treated as an integer parameter, a population of $NP = 40$ individuals evolves until convergence, with $N_{s-a}$ considered as a real parameter, as described previously. After convergence, the final population is expanded by adding 5 random individuals, resulting in a new population of $NP = 45$. This expanded population then evolves again until convergence, but this time an integer repair operator is applied to each individual in every generation before evaluation. The idea behind this approach is to enhance exploration in the early stages of evolution by allowing $N_{s-a}$ to vary without constraints. Once the initial population has converged, diversity is increased by introducing random solutions. The subsequent evolution with the integer constraint on $N_{s-a}$ ensures that the final optimal solution includes $N_{s-a}$ as an integer.

## 5. Validation results

The validation of the proposed identification procedure considers the two variants introduced before **D-SDM** and **D-SDM-I**. The traditional **SDM** under uniform assumption is used as a baseline comparison model (see (1)–(7)). From this point forward, this parameter identification approach will be referred to as **SDM**. For a fair comparison with **SDM**, it is necessary to calculate the minimum voltage $V_{low}$, where $V_{low} < V_{mpp}$, in such a way that the portion of the experimental curve from $V_{low}$ to $V_{oc}$ does not contain any activation of the bypass diode. Such activation is reflected in a drastic change in the first derivative of the I-V curve, which would lead to larger errors in the **SDM** parameters estimation. Then, the experimental points are considered from $V_{low}$ to $V_{oc}$ used for the fittings.

In summary, for each experimental I-V curve, the best-fitting parameters are determined using the compared procedures (**D-SDM**, **D-SDM-I**, and **SDM**). Section 4 provides details on the implementation of the **D-SDM** and **D-SDM-I** parameter identification procedures. The traditional **SDM** procedure follows the model Eqs. (1)–(7) and employs the same optimization algorithm to compute the best-fitting parameters. The estimated parameters are then compared based on metrics such as stability under similar operating conditions to assess the practical relevance of each procedure. Additionally, the **D-SDM** procedure is evaluated for its accuracy in identifying the true model parameters using artificially generated I-V curves designed to represent a wide range of PV technologies.

### 5.1. Experimental campaign

The experimental campaign was designed to test the identification procedures under common mismatching phenomena that may occur in real applications in outdoor conditions. All experimental I-V curves were measured on a commercial PV module manufactured by Solbian with reference *Flex-SP50L* [45] (its specifications under STC are given in [19]). It comprises 16 pseudo square back-contact mono-crystalline Silicon cells, all connected in series, and a bypass diode. A configuration of two *Flex-SP50L* series-connected modules is considered in our experimental campaign with three Pt100 RTD sensors attached to each module to measure the average temperature. A solarimeter LM1-10V PRO [46] was used for measuring the irradiance $G[W/m^2]$ on the plane of each solar module. The temperature and irradiance sensors were connected to ADAM 6015 and ADAM 6017 [47] to save the data in a computer.

Four common mismatch cases are analyzed as illustrated in Fig. 7, which are described below:

* Case (I) closely resembles a scenario with uniform conditions. In this case, the primary cause of mismatch is the non-uniform radiation caused by the reflexes on the surface of the module. While the I-V curve shows only minor deviations from the uniform condition, these deviations impact the accuracy of parameter identification in models that assume fully uniform conditions (see Figs. 2 and 8 (Case I)).

* Case (II) represents partial shading caused by surrounding objects. In this scenario, the deformation of the I-V curve is noticeable, as shown in Fig. 8 (Case II). Fitting only the rightmost portion of the curve from $V_{low}$ to $V_{oc}$ (see definition of $V_{low}$ at the beginning of Section 5), using a model that assumes uniform conditions will not be accurate. This is due to the increased curvature around $V_{mpp}$, which requires the model to have more degrees of freedom to capture the shape effectively [21].

* Case (III) simulates the mismatch effect caused by bird droppings. A small deviation from the uniform condition is evident in the I-V curve, as shown in Fig. 8 (Case III), which is sufficient to challenge traditional parameter identification procedures.

* Case (IV) represents scenarios where dust accumulates and rain redistributes it across the module surface, creating a non-uniform shading pattern. The deformation in the I-V curve is evident, as shown in Fig. 8 (Case IV), but may not be easily noticeable if only the rightmost portion of the curve is considered.

Several I-V curves were measured for each representative mismatch scenario under outdoor conditions. The objective is to assess the capabilities of the proposed identification procedures to obtain consistent parameter estimates under various mismatching conditions, which are introduced in Table 2. This table also includes the fitting error for each case obtained with (23).

### 5.2. Overview of D-SDM parameter estimates

Section 2 highlighted the limitations of the uniformity assumption to fit an I-V curve of a PV module under mismatching conditions. Fig. 9 extends Fig. 2 by including the fitting of the **D-SDM** for the I-V curve under study and considering two cases for the **SDM** (under uniformity assumptions), the first one using all the experimental data (line B in Fig. 9) and the second one considering the experimental points from $V_{low} = V_{mpp} - 0.2V_{mpp}$ to $V_{oc}$ (line C in Fig. 9). In the first case, the **SDM** parameters fail to fit the I-V curve around the MPP, while in the second case, the parameters fail to fit the I-V curve for low voltages. In contrast, the **D-SDM** effectively follows the I-V curve around the MPP and for high voltages.

Regarding the results shown in Fig. 8 and Table 2, it can be observed that the two variants proposed fit significantly better than the **SDM**. The fitting quality is further supported by the data in Table 2, where the **D-SDM**-based identification procedures consistently exhibit errors with one or two orders of magnitudes lower than the ones of the **SDM** for all the mismatching cases. Additionally, the proposed identification procedure automatically identifies the range where all the cells operate at positive voltages for the fitting process.

Table 2 also confirms the stability of the estimated series resistance across different mismatching patterns. It is well-known that the series resistance is a crucial parameter in determining aging and potential faults in the modules; thus, a reliable parameter estimation method should be able to detect stable and accurate, variations in this parameter. In contrast, **SDM** predictions vary significantly from Case I to Case II, III, and IV. Another key parameter representing losses is the parallel resistance. The **SDM** was unable to estimate this parameter in the mismatching cases analyzed, as $R_{sh-c}$ consistently converged to the upper bound of the search interval. In contrast, both **D-SDM** and **D-SDM-I** provided reasonable estimations for this parameter across all mismatching cases. It is important to note that the resistance values in the results are scaled to a single cell. It is also important to note, that the results show minimal differences in general between the parameters estimated by **D-SDM** and **D-SDM-I**.
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Mismatching cases analyzed. (I). Clean panel without shadows but with non-uniform radiation produced by the reflexes on the surface; (II). Partial shading by surrounding objects; (III). Partial shading by bird droppings; (IV). Partial shading by soiling.</figcaption></figure>

<table><caption>**Table 2**

Best fitting parameters of a single cell for D-SDM, D-SDM-I, and SDM.</caption><thead><tr><th rowspan="2">Parameter</th><th colspan="3">Case I</th><th colspan="3">Case II</th><th colspan="3">Case III</th><th colspan="3">Case IV</th></tr><tr><th>D-SDM</th><th>D-SDM-I</th><th>SDM</th><th>D-SDM</th><th>D-SDM-I</th><th>SDM</th><th>D-SDM</th><th>D-SDM-I</th><th>SDM</th><th>D-SDM</th><th>D-SDM-I</th><th>SDM</th></tr></thead><tbody><tr><td>I<sub>ph-a</sub>[A]</td><td>5.44</td><td>5.48</td><td>5.61</td><td>4.31</td><td>4.31</td><td>4.39</td><td>5.22</td><td>5.22</td><td>5.35</td><td>4.88</td><td>4.87</td><td>4.98</td></tr><tr><td>I<sub>x-c</sub>[A]</td><td>1.41 × 10<sup>-4</sup></td><td>1.56 × 10<sup>-4</sup></td><td>9.51 × 10<sup>-5</sup></td><td>8.88 × 10<sup>-5</sup></td><td>5.64 × 10<sup>-5</sup></td><td>2.69 × 10<sup>-9</sup></td><td>1.30 × 10<sup>-4</sup></td><td>1.25 × 10<sup>-4</sup></td><td>3.16 × 10<sup>-6</sup></td><td>5.84 × 10<sup>-5</sup></td><td>5.40 × 10<sup>-5</sup></td><td>1.59 × 10<sup>-7</sup></td></tr><tr><td>η<sub>c</sub>[–]</td><td>1.96</td><td>1.98</td><td>1.89</td><td>1.93</td><td>1.85</td><td>1.00</td><td>1.92</td><td>1.92</td><td>1.43</td><td>1.84</td><td>1.83</td><td>1.22</td></tr><tr><td>R<sub>x-c</sub>[Ω]</td><td>0.0024</td><td>0.0023</td><td>0.0028</td><td>0.0025</td><td>0.0028</td><td>0.0068</td><td>0.0025</td><td>0.0026</td><td>0.0055</td><td>0.0029</td><td>0.0029</td><td>0.0069</td></tr><tr><td>R<sub>x-c</sub>[Ω]</td><td>6.66</td><td>11.40</td><td>∞</td><td>27.81</td><td>30.34</td><td>∞</td><td>16.11</td><td>17.55</td><td>∞</td><td>6.53</td><td>7.06</td><td>∞</td></tr><tr><td>I<sub>ph-b</sub>[A]</td><td>5.65</td><td>5.69</td><td>–</td><td>5.31</td><td>5.26</td><td>–</td><td>5.65</td><td>5.64</td><td>–</td><td>5.36</td><td>5.34</td><td>–</td></tr><tr><td>N<sub>x-a</sub>[–]</td><td>1.00</td><td>4</td><td>–</td><td>1.00</td><td>1</td><td>–</td><td>1.00</td><td>1</td><td>–</td><td>1.18</td><td>1</td><td>–</td></tr><tr><td>Error E</td><td>2.12 × 10<sup>-4</sup></td><td>2.21 × 10<sup>-4</sup></td><td>1.01 × 10<sup>-3</sup></td><td>2.96 × 10<sup>-4</sup></td><td>2.85 × 10<sup>-4</sup></td><td>1.36 × 10<sup>-2</sup></td><td>2.48 × 10<sup>-4</sup></td><td>2.48 × 10<sup>-4</sup></td><td>6.18 × 10<sup>-3</sup></td><td>1.88 × 10<sup>-4</sup></td><td>2.04 × 10<sup>-4</sup></td><td>5.99 × 10<sup>-3</sup></td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Example of fitting results for experimental I-V curves. I. Uniform Conditions; II. Partial shading by surrounding objects; III. Partial shading by bird droppings; I–V. Partial shading by soiling.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Uniform assumption limitations. A. Experimental data; B. SDM fitting using all the experimental points, C. SDM fitting using only the points with voltages between $V_{mpp} - 0.2V_{mpp}$ to $V_{oc}$, D. D-SDM fitting using all the experimental points.</figcaption></figure>

### 5.3. Assessing the stability of parameter estimates

This section evaluates the ability to obtain stable and reliable parameters for different partial shading patterns across the compared procedures. Since the SDM parameters are known to fluctuate based on operating conditions, I-V curves measured under similar conditions for each shading pattern are selected (See Fig. 7). In this study, similar conditions are defined as those where the irradiance varies by no more than 15 W/m² and the temperature by no more than 5 K across all

curves. It is important to note that the I-V curves were measured in outdoor conditions, making this approximation necessary.

Fig. 10 compares the coefficient of variation (CV) for the estimated series resistance $R_s$ and the parallel resistance $R_{sh}$ across compared models and shading patterns. The coefficient of variation is a standardized measure of dispersion, often used to assess the relative variability of data. Defined as the ratio of the standard deviation to the mean value, the CV provides a dimensionless number that allows for comparison across datasets with different units or scales. In this sense, a reliable
<figure><img src="image_13.png" /><figcaption>Fig. 10. Comparison of coefficient of variation for $R_s$ and $R_{sh}$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Comparison of coefficient of variation for $I_s$ and $\eta$.</figcaption></figure>

identification procedure should provide similar parameter estimates in similar operating conditions and, as a consequence, a low CV.

Fig. 10 highlights the advantages of the proposed identification procedures over **SDM**. For the series resistance $R_s$, **SDM** achieves a lower coefficient of variation (CV) only in the first mismatching pattern, which closely resembles uniform conditions. However, **SDM** struggles to estimate the parallel resistance $R_{sh}$ accurately. In all mismatching cases, **SDM** estimation of $R_{sh}$ converges to the boundary of the search interval, which is consistent with the limitations previously illustrated in Fig. 2.

Fig. 11 presents the CV for the saturation current $I_s$ and the ideality factor $\eta$ across various mismatching cases. The proposed identification procedures consistently yield a significantly lower CV for both parameters, particularly in shading scenarios from cases II to IV. Notably, the accuracy of the **SDM** estimations deteriorates for the more complex mismatching patterns, which is evident in cases II through IV. A comparison of the two procedures proposed in this paper shows that **D-SDM-I** generally achieves a slightly lower CV than **D-SDM** (see Figs. 10 and 11). However, the computational cost for obtaining the parameters with **D-SDM-I** is nearly twice higher than for **D-SDM** (see Section 4.2). Therefore, the choice of a procedure for practical applications will depend on the computational resources available on the hardware in use.

<table><caption>Table 3
Used ranges for generating simulated curves.</caption><thead><tr><th>Parameter</th><th>Lower limit</th><th>Upper limit</th></tr></thead><tbody><tr><td>$N_s$ [-]</td><td>16</td><td>120</td></tr><tr><td>$N_{s-a}$ [-]</td><td>1</td><td>$N_s$</td></tr><tr><td>$I_{ph-b}$ [A]</td><td>2</td><td>12</td></tr><tr><td>$I_{ph-a}$ [A]</td><td>2</td><td>$I_{ph-b}$</td></tr><tr><td>$I_{s-c}$ [A]</td><td>10<sup>-12</sup></td><td>10<sup>-5</sup></td></tr><tr><td>$\eta_c$ [-]</td><td>1</td><td>2</td></tr><tr><td>$R_{s-c}$ [Ω]</td><td>0.001</td><td>0.01</td></tr><tr><td>$R_{sh-c}$ [Ω]</td><td>10</td><td>50</td></tr><tr><td>$\nu$ [-]</td><td>0.7</td><td>1.31</td></tr><tr><td>$T_c$ [K]</td><td>273</td><td>343</td></tr><tr><td>$a$ [-]</td><td>2 x 10<sup>-3</sup></td><td>15 x 10<sup>-3</sup></td></tr><tr><td>$m_{Bi}$ [-]</td><td>2</td><td>6</td></tr><tr><td>$V_{br}$ [V]</td><td>-40</td><td>-20</td></tr></tbody></table>

## 5.4. Validation through simulated I-V curves

This section aims to validate the identification procedure by using artificially created I-V curves that are expected to cover a wide range of PV technologies under a wide range of partial shading conditions. To numerically validate the proposed simplification of neglecting the parallel resistance of the group at higher irradiance (group b), the effect of this parallel resistance is incorporated into the simulated curves. This allowed us to evaluate the ability of the model to estimate the parallel resistance for the low irradiance block (group a).

To simulate the effect of partial shading conditions, the Bishop model [22] is used to replicate the behavior of the cells when working in the first and second quadrants of the I-V curve. According to this model, $I_c$ is calculated by using (24), where $a$ and $m_{Bi}$ are fitting parameters and $V_{br}$ is the reverse breakdown voltage of the cell.

$$ 
\begin{aligned}
I_c &= I_{ph-c} - I_d - I_{sh} \\
I_d &= I_{s-c} \left( \exp \left( \frac{V_c + I_c R_{s-c}}{\eta_c V_{th-c}} \right) - 1 \right) \\
I_{sh} &= \frac{V_c + I_c R_{s-c}}{R_{sh}} \left[ 1 + a \left( 1 - \frac{V_c + I_c R_{s-c}}{V_{br}} \right)^{-m_{Bi}} \right]
\end{aligned} 
\quad (24) 
$$

In the simulation, the I-V curve of the cell at low irradiance is calculated from (24), then it is scaled by $N_{s-a}$ to obtain the I-V curve of group a and a similar procedure is performed for group b. Then, the module I-V curve is obtained by calculating $V_a$ and $V_b$ for the same values of $I_m$, and finally, $V_m$ is just the sum of $V_a$ and $V_b$ as defined in (12).

The parallel resistance behavior exhibits an exponential trend as described in (25) [19], which shows that $R_{sh-b}$ can be defined in terms of $R_{sh-a}$ and $\nu$. In a previous work [48], the authors showed that $\nu$ varies from 0.7 to 1.31 for different PV technologies.

$$ R_{sh-b} = R_{sh-a} \cdot \left( \frac{N_{s-b}}{N_{s-a}} \right) \cdot \left( \frac{I_{ph-b}}{I_{ph-a}} \right)^{-\nu} \quad (25) $$

Considering this, a summary of the range defined for each parameter to generate the simulated curves is shown in Table 3. The set of parameters for each curve is generated randomly from a uniform distribution. For each simulated curve, the parameters of the **D-SDM** procedure are estimated and compared with the true parameters used for its generation.

Fig. 12 compares the true series resistance with the estimated values from the proposed procedure on the left and the true parallel resistance with the calculated values on the right. In both comparisons, the high $R^2$ coefficients demonstrate the method's effectiveness in accurately determining the series and parallel resistances. Additionally, these results confirm that simplifying the model by neglecting the parallel resistance at high irradiance does not compromise the accuracy of the estimated parallel resistance at low irradiance. This numerical evidence supports the model simplification, which reduces the number of parameters while maintaining high accuracy.
<figure><img src="image_15.png" /><figcaption>Fig. 12. True value versus estimation for $R_{s-c}$ (left) and $R_{sh-c}$ (right).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. True value versus estimation for $N_{s-a}/N_s$ (left) and $I_{ph-a}/I_{ph-b}$ (right).</figcaption></figure>

Fig. 13 assesses the proposed method's ability to detect mismatch levels. The left panel compares the ratio of shaded cells to the total number of cells. In contrast, the right panel compares the ratio of the photogenerated current at low irradiance to that at high irradiance. In this case, the high $R^2$ coefficient further demonstrates the effectiveness of the model in detecting mismatch levels.

## 6. Conclusion

This study presents a 7-parameter self-adapting Double Single-Diode Model (D-SDM) designed to improve the reliability of PV module parameter identification under mismatching conditions. A methodology for extracting the parameters of the D-SDM using the Differential Evolution Algorithm and Binary Search is also proposed. This approach reformulates the fitting problem as a function of voltage, enhancing the overall accuracy of the parameter estimation process. A robust error function was defined, considering only the experimental points where all cells operate at positive voltages.

Two variants of the D-SDM were validated, the first treating the number of shaded cells as a real parameter, while the second considering it an integer parameter. Experimental and simulated I-V curves under several mismatching patterns were used to evaluate the two proposed variants. The experimental results revealed that the D-SDM-based methods offer greater stability and reliability in parameter estimation compared to the standard SDM under uniform operating conditions assumption. Additionally, simulated I-V curves highlighted the methodology's effectiveness in accurately determining the true values of the model parameters.

## CRediT authorship contribution statement

**Luis E. Garcia-Marrero:** Writing – original draft, Software, Methodology, Conceptualization. **Carlos I. Pavon-Vargas:** Writing – review & editing, Methodology, Conceptualization. **Juan D. Bastidas-Rodriguez:** Writing – review & editing, Methodology, Conceptualization. **Eric Monmasson:** Writing – review & editing, Supervision. **Giovanni Petrone:** Writing – review & editing, Supervision, Conceptualization.

## Declaration of competing interest

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

## Acknowledgments

This project has received funding from the European Union's Horizon 2020 and Horizon Europe research and innovation programs under the Marie Skłodowska-Curie grant agreements No 955614 and No 101109355. This work has been also supported by the National Recovery and Resilience Plan, Mission 4, Component 2, Investment 1.1, funded by the European Union – Next Generation EU – Project Title “Enhanced Device for Online diaGnosis of PHOtovoltaic and Storage Systems (DOGPHOSS)” P20229FWZK-CUP D53D23015980001, and by FARB funds of the Università degli Studi di Salerno.

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