Real-Time Temperature Prediction of Partially Shaded PV Modules

Abstract

Temperature prediction for partially shaded photovoltaic (PV) modules is essential for ensuring the stability and safety of PV systems. However, existing methods suffer from high computational complexity, limiting their applicability in engineering practice. Aimed at a real-time and portable algorithm that can be embedded in mobile devices for intelligent monitoring of PV stations, a simple and fast method is designed in this work for estimating the thermal behavior of PV modules under partial shading conditions. To the best of our knowledge, this is the first work in this field that achieves computational simplicity without relying on professional commercial software. The experimental results validate the accuracy of the proposed method in comparison with the multiphysics model (which is widely regarded as the benchmark in this field) while significantly improving computational efficiency. Simulations are conducted to explore the effects of shading proportions and environmental conditions. Shading proportions ranging from 6% to 90% are prone to promoting the development of hotspots under conditions that involve partial shading of an individual cell. Higher irradiance, a higher ambient temperature and a lower wind speed result in a higher temperature of the PV module.

1. Introduction

Renewable energy has received increasing attention worldwide [1]. Among renewable sources, solar energy stands out for its economic viability, environmental benefits, and widespread availability [2]. Driven by continuous improvements in efficiency and reduced prices, the deployment of solar photovoltaic (PV) modules has accelerated considerably in recent years [3]. The International Renewable Energy Agency reported comprehensive statistics showing that installed PV capacity had reached 1858 GW by 2025 [4].

Partial shading represents a prevalent issue within PV systems due to bird droppings, soil, leaves and shadows [5,6,7,8]. A partially shaded cell may consume electricity [9,10,11] and develop hotspots [12,13]. Under certain working points, shading proportions, and environmental conditions, the maximum temperature of a partially shaded cell can reach 140 °C. An increase in temperature not only affects the electrical properties [14,15,16,17,18] but also accelerates the degradation of PV modules [19,20,21]. Worst of all, flammable material coming into contact with the overheating region might result in potential fire hazards. In order to avoid these problems, it is necessary to predict the temperature of partially shaded PV modules.

The cell-level temperature distribution within a PV module depends on its thermal power and heat transfer [22]. Numerous studies [23,24] have estimated the thermal power based on the absorbed solar power and electrical efficiency of solar cells. With regard to heat transfer, the finite element method (FEM) has been widely used [25,26,27,28,29,30,31]. The multiphysics model is a benchmark that comprehensively takes into account both thermal power and heat transfer. However, the multiphysics model is highly computationally complex and relies on professional software, such as ANSYS [32,33] and COMSOL [34]. It is usually used only for scientific analysis.

For emerging applications such as intelligent monitoring and digital twins of PV stations [35,36,37,38], the requirements shift from ultimate accuracy to real-time capability, computational efficiency, and embeddability. The high complexity and software dependence of conventional multiphysics models pose significant barriers to meeting these requirements. Addressing this specific limitation, this study introduces a streamlined modeling approach that sacrifices non-essential physical details to achieve real-time temperature prediction of partially shaded PV modules. The proposed method is designed to be simple and fast, forming the core of a portable algorithm suitable for deployment in resource-constrained edge devices for real-time field diagnostics. The contributions of our work are summarized as follows:

• Inspired by the FEM, the partially shaded cell is divided into finite layers, and the heat transfer of each layer is analyzed. An iterative algorithm is proposed to calculate the temperature of each layer. To the best of our knowledge, this is the first work in this field that achieves computational simplicity without relying on professional commercial software.
• The proposed method enables real-time temperature prediction for the partially shaded PV module. Experimental results validate that the proposed method achieves accuracy comparable to that of the multiphysics model (which is widely regarded as the benchmark in this field) while significantly improving computational efficiency.
• Simulations are conducted to explore the effects of shading proportions and environmental conditions. Shading proportions ranging from 6% to 90% are prone to promoting the development of hotspots under conditions that involve partial shading of an individual cell. Higher irradiance, a higher ambient temperature and a lower wind speed result in higher temperatures of the PV module.

This paper is structured as follows. The methodology of real-time temperature prediction of partially shaded PV modules is described in Section 2, and experimental results are reported to show the performance in Section 3. Subsequently, simulations reveal the effects of shading proportions and environmental conditions in Section 4. The paper concludes with Section 5, which summarizes the findings and their implications.

2. Materials and Methods

The temperature of a PV module depends on its thermal power and heat transfer. This section begins with the basic concept of the equivalent circuit, proceeds to estimate the cell-level thermal power, analyzes the heat transfer, and culminates in predicting the temperature of a partially shaded PV module. Based on this progression, the overall framework is outlined in Figure 1.

2.1. Modeling of the Equivalent Circuit

Generally, there are three types of silicon cells in a partially shaded PV module, as shown in Figure 2: the partially shaded cell, the homogeneous cell in the partially shaded sub-module, and the homogeneous cell in the normal sub-module. To simplify the description, homogeneous cells in the normal sub-module and in the partially shaded sub-module are called homogeneous cells.

2.1.1. The Single-Diode Model

We employ the single-diode model (SDM) [39] to characterize the I-V behavior of homogeneous cells, as depicted in Figure 3. Its parameters include the photocurrent ($I_{ph}$), the ideality factor (n) and the reverse saturation current ($I_s$) of the diode; the shunt resistance ($R_h$); and the series resistance ($R_s$). These Parameters are extracted using the dual-iteration algorithm [40].

2.1.2. The Parallel Model

The parallel model [41] is applied to represent the partially shaded cell, as illustrated in Figure 4a; here, the unshaded and shaded regions are indicated by gray and green boxes, respectively. The corresponding equivalent parallel model merging two diodes into one is shown in Figure 4b. The simplified parallel model merging the photocurrents ($I_{ph1}$ and $I_{ph2}$) and shunt resistances ($R_{h1}$ and $R_{h2}$) is shown in Figure 4c.

The shading proportion ($p\%$) is defined as the ratio of the partial shading area to the cell area, typically ranging from 0 (fully illuminated) to 1 (fully shaded). $tr$ represents the transmittance of the shading material. The photocurrents ($I_{ph,1}$ and $I_{ph,2}$) are calculated, and the equivalent photocurrent ($I_{ph}^{*}$ is) calculated in (3).

$$I_{ph,1}=I_{ph}·(1-p\%)$$

(1)

$$I_{ph,2}=I_{ph}·tr·p\%$$

(2)

$$I_{ph}^{*}=I_{ph,1}+I_{ph,2}$$

(3)

The shunt resistances ($R_{h,1}$ and $R_{h,2}$) are calculated, and the equivalent shunt resistance ($R_h^{*}$) is calculated in (6).

$$R_{h,1}={\displaystyle \frac{R_h}{(1-p\%)}}$$

(4)

$$R_{h,2}={\displaystyle \frac{R_{h,shade}}{p\%}}$$

(5)

$$R_h^{*}={\displaystyle \frac{R_{h,1}·R_{h,2}}{R_{h,1}+R_{h,2}}}$$

(6)

where $R_{h,shade}$ represents the shunt resistance under the shading condition, as calculated in (7).

$$R_{h,shade}={\displaystyle \frac{1}{tr}}·R_h$$

(7)

2.1.3. Electric Power of Each Cell

For homogeneous cells, the electrical power is calculated directly in (8). For the partially shaded cell, the calculations are performed separately for the unshaded and shaded regions, as shown in (9) and (10).

$$P_{electric}=I_{cell}·V_{cell}$$

(8)

$$P_{electric,1}=\left\{I_{ph,1}-\left(1-p\%\right)·I_d-{\displaystyle \frac{V_{cell}+I_{cell}·R_s}{R_{h,1}}}\right\}·V_{cell}$$

(9)

$$P_{electric,2}=\left\{I_{ph,2}-p\%·I_d-{\displaystyle \frac{V_{cell}+I_{cell}·R_s}{R_{h,2}}}\right\}·V_{cell}$$

(10)

where $I_{cell}$ and $V_{cell}$ denote the cell-level current and voltage, respectively, and $I_d$ denotes the current of the diode. The detailed calculations of $I_{cell}$, $V_{cell}$ and $I_d$ can be found in [31].

2.2. Thermal Power Estimation

A portion of the solar energy reaching the photovoltaic modules is reflected, while the majority is absorbed. The absorbed solar energy is partitioned between electricity and heat. The PV module is composed of several PV cells encapsulated by the ethylene vinyl acetate (EVA), the glass cover, and the Tedlar/PET/Tedlar (TPT) backsheet. The schematic diagram of the PV module from the side view is shown in Figure 5.

The solar energy absorbed by these materials in the related region is calculated in (11)–(14):

$$P_{glass}=G·S_{cell}·{\alpha }_g$$

(11)

$$P_{eva}=G·S_{cell}·{\beta }_g·{\alpha }_e$$

(12)

$$P_{silicon}=G·S_{cell}·{\beta }_g·{\beta }_e·{\alpha }_s$$

(13)

$$P_{tpt}=G·S_{cell}·{\beta }_g·{\beta }_e·{\beta }_s·{\alpha }_t$$

(14)

where G denotes the irradiance on the surface of the solar panel; $S_{cell}$ is the area of the silicon cell; ${\beta }_g$, ${\beta }_e$ and ${\beta }_s$ represent the transmissivity of the glass cover, EVA, and silicon cell; and ${\alpha }_g$, ${\alpha }_e$, ${\alpha }_s$ and ${\alpha }_t$ denote the absorptivity of the glass cover, EVA, silicon cell, and TPT backsheet. Standard optical properties of typical materials in the PV module are adopted from [24], as shown in

Table 1. Standard optical properties of typical materials in the PV module.

Material	Emissivity	Reflectivity	Absorptivity	Transmissivity
Glass	0.85	4%	4%	92%
EVA	-	2%	8%	90%
Silicon Cell	-	8%	90%	2%
TPT (Polymer)	0.9	86%	12.8%	1.2%

.

The total absorbed solar energy in the related region is calculated in (15). The thermal power of the homogeneous cell is derived from the calculated total absorbed solar energy and the fraction converted to electricity, as shown in (16).

$$P_{solar}=P_{glass}+P_{eva}+P_{silicon}+P_{tpt}$$

(15)

$$P_{thermal}=P_{solar}-P_{electric}$$

(16)

For the partially shaded cell, the thermal power of the unshaded region is given by (17), and that of the shaded region is given by (18).

$$P_{thermal,1}=P_{solar}·\left(1-p\%\right)-P_{electric,1}$$

(17)

$$P_{thermal,2}=P_{solar}·p\%-P_{electric,2}$$

(18)

2.3. Heat Transfer

Generally, heat transfer occurs through three primary mechanisms: conduction, convection, and radiation.

2.3.1. Heat Conduction

Heat conduction along x is defined by Fourier’s law, as shown in (19).

$${\mathsf{\Phi }}_{cond}=-\lambda ·A·{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}x}}$$

(19)

where $\lambda$ represents the thermal conductivity, A is the cross-sectional area and T represents the temperature.

Standard thermal properties of typical materials in the PV module are adopted from [28], as listed in

Table 2. Standard thermal properties of typical materials in the PV module.

Material	Thermal Conductivity [W/(m·K)]
Glass	2.00
EVA	0.31
Silicon Cell	130.00
TPT (Polymer)	0.15

. The silicon cell contributes to most of the heat conduction due to its high thermal conductivity. To simplify the calculation, heat conduction within the glass, EVA, and TPT layers is not considered, as justified by their in-plane thermal conductivity being 2–3 orders of the magnitude less than that of the silicon cell.

2.3.2. Heat Convection

Newton’s law of cooling is applied to describe heat convection, as expressed in (20). The convective heat transfer coefficients [42] are calculated in (21) and (22).

$${\mathsf{\Phi }}_{conv}=h·S·\mathsf{\Delta }T$$

(20)

$$h_f=5.82+4.07·v$$

(21)

$$h_b=0.6·h_f$$

(22)

where h represents the convective heat transfer coefficient, S represents the area of the surface, $\mathsf{\Delta }T$ is the temperature difference, $h_f$ is the convective heat transfer coefficient at the front surface of the PV module, $h_b$ is the convective heat transfer coefficient at the back surface of the PV module and v is the wind speed.

2.3.3. Heat Radiation

The Stefan–Boltzmann law is adopted to describe heat radiation, as given in (23).

$$\mathsf{\Delta }{\mathsf{\Phi }}_{rad}=S·\sigma ·\left({\epsilon }_1·T_1^4-{\epsilon }_2·T_2^4\right)$$

(23)

where S denotes the area of the surface; $\sigma$ denotes the Stefan–Boltzmann constant, where $\sigma =5.67\times {10}^{-8}$ $\mathrm{W}/({\mathrm{m}}^2·{\mathrm{K}}^4)$; $T_1$ represents the temperature of the surface; $T_2$ represents the temperature of the external heat radiation source, which is taken as the ambient temperature in this article; ${\epsilon }_1$ represents the emissivity of the surface, which is shown in Table 1; and ${\epsilon }_2$ represents the emissivity of the external heat radiation source. Assume that the external heat radiation source is a black body and ${\epsilon }_2=1$.

2.3.4. Heat Transfer Analysis of the Partially Shaded Solar Panel

For the homogeneous cell, the temperature is assumed to be uniform. This means that heat conduction of the homogeneous cell can be ignored. Consequently, the heat transfer of the homogeneous cell only consists of heat convection and heat radiation, as illustrated in Figure 6a.

We have observed that in some situations, a significant temperature difference can arise between the unshaded and shaded regions of a partially shaded cell. This means that heat conduction of the partially shaded cell should be taken into consideration, as depicted in Figure 6b.

2.4. Real-Time Temperature Prediction

The real-time temperature prediction of the PV module involves homogeneous cells and the partially shaded cell.

2.4.1. Temperature of Homogeneous Cells

Taking the steady state into consideration, the thermal power of the homogeneous cell is equal to the heat transfer of this cell, as shown in (24).

$$P_{thermal}={\mathsf{\Phi }}_{conv}+\mathsf{\Delta }{\mathsf{\Phi }}_{rad}$$

(24)

where $P_{thermal}$ is the thermal power of the homogeneous cell; ${\mathsf{\Phi }}_{conv}$ is the heat convection of the homogeneous cell, which is calculated in (25); and $\mathsf{\Delta }{\mathsf{\Phi }}_{rad}$ represents the heat radiation between the homogeneous cell and the surrounding environment, which is calculated in (26).

$${\mathsf{\Phi }}_{conv}=(h_f+h_b)·S_{cell}·(T_{cell}-T_a)$$

(25)

$$\mathsf{\Delta }{\mathsf{\Phi }}_{rad}=S_{cell}·\sigma ·\left[({\epsilon }_g+{\epsilon }_p)·T_{cell}^4-2·T_a^4\right]$$

(26)

where $T_{cell}$ denotes the temperature of the homogeneous cell, $T_a$ denotes the ambient temperature, ${\epsilon }_g$ is the emissivity of the glass cover, ${\epsilon }_p$ is the emissivity of the TPT backsheet, and ${\epsilon }_g$ and ${\epsilon }_p$ are shown in Table 1.

The ambient temperature is measured in real time. Given $P_{thermal}$, $h_f$, $h_b$, $S_{cell}$, $T_a$, ${\epsilon }_g$, ${\epsilon }_p$ and $\sigma$, (24) can be considered as a quartic equation with $T_{cell}$. It has been proven that this equation has two real roots (one is positive, and the other is negative), as well as two imaginary roots. The temperature of the homogeneous cell ($T_{cell}$) is obtained by the positive real root.

2.4.2. Temperature Distribution of the Partially Shaded Cell

To analyze heat conduction within the partially shaded cell, we discretize it into multiple layers. If the layer is infinitesimal, as shown in Figure 7a, differential equations of temperature (T) with x can be obtained. Unfortunately, it is hard to find the analytical solution of these differential equations. As a compromise, the partially shaded cell is divided into finite layers, as shown in Figure 7b. Then, the temperature distribution of the partially shaded cell is calculated approximately.

In the steady state, the thermal power of each layer in Figure 7b is equal to its heat transfer, as shown in (27).

$$P_{thermal\mathsf{\Delta }} = \mathsf{\Delta }{\mathsf{\Phi }}_{cond\mathsf{\Delta }} + {\mathsf{\Phi }}_{conv\mathsf{\Delta }} + \mathsf{\Delta }{\mathsf{\Phi }}_{rad\mathsf{\Delta }}$$

(27)

where $P_{thermal\mathsf{\Delta }}$ is the thermal power of this layer, $\mathsf{\Delta }{\mathsf{\Phi }}_{cond\mathsf{\Delta }}$ represents the heat conduction between this layer and its neighboring layers, ${\mathsf{\Phi }}_{conv\mathsf{\Delta }}$ is the heat convection of this layer, and $\mathsf{\Delta }{\mathsf{\Phi }}_{rad\mathsf{\Delta }}$ denotes heat radiation between this layer and the surrounding environment.

If the i-th layer is the unshaded region, its thermal power is calculated in (28). Otherwise, if the i-th layer is the shaded region, its thermal power is calculated in (29).

$$P_{thermal,u,\mathsf{\Delta }}\left(i\right)=P_{thermal,1}·{\displaystyle \frac{L·\mathsf{\Delta }{x}_i}{S_{cell}·(1-p\%)}}$$

(28)

$$P_{thermal,s,\mathsf{\Delta }}\left(i\right)=P_{thermal,2}·{\displaystyle \frac{L·\mathsf{\Delta }{x}_i}{S_{cell}·p\%}}$$

(29)

where $P_{thermal,1}$ is the thermal power of the unshaded region in the partially shaded cell, $P_{thermal,2}$ is the thermal power of the shaded region, L is the length of the silicon cell and $\mathsf{\Delta }{x}_i$ is the width of the i-th layer.

The heat conduction values of the first layer ($i=1$), the last layer ($i=n$) and inner layers ($1<i<n$) are calculated in (30)–(32), respectively.

$$\mathsf{\Delta }{\mathsf{\Phi }}_{cond\mathsf{\Delta }}\left(1\right)=-{\lambda }_s{\delta }_{s}L·{\displaystyle \frac{T\left(2\right)-T\left(1\right)}{(\mathsf{\Delta }{x}_2+\mathsf{\Delta }{x}_1)/2}}$$

(30)

$$\mathsf{\Delta }{\mathsf{\Phi }}_{cond\mathsf{\Delta }}\left(n\right)=-{\lambda }_s{\delta }_{s}L\left[-{\displaystyle \frac{T\left(n\right)-T(n-1)}{(\mathsf{\Delta }{x}_n+\mathsf{\Delta }{x}_{n-1})/2}}\right]$$

(31)

$$\mathsf{\Delta }{\mathsf{\Phi }}_{cond\mathsf{\Delta }}\left(i\right)=-{\lambda }_s{\delta }_{s}L\left[{\displaystyle \frac{T(i+1)-T\left(i\right)}{(\mathsf{\Delta }{x}_{i+1}+\mathsf{\Delta }{x}_i)/2}}-{\displaystyle \frac{T\left(i\right)-T(i-1)}{(\mathsf{\Delta }{x}_i+\mathsf{\Delta }{x}_{i-1})/2}}\right]$$

(32)

where ${\lambda }_s$ is the thermal conductivity of the silicon cell, ${\delta }_s$ is the thickness of the silicon cell and $T\left(i\right)$ denotes the temperature of the i-th layer.

The heat convection and heat radiation of the i-th layer are given in (33) and (34) respectively.

$${\mathsf{\Phi }}_{conv\mathsf{\Delta }}\left(i\right)=(h_f+h_b)·(L·\mathsf{\Delta }{x}_i)·\left[T\left(i\right)-T_a\right]$$

(33)

$$\mathsf{\Delta }{\mathsf{\Phi }}_{rad\mathsf{\Delta }}\left(i\right)=(L·\mathsf{\Delta }{x}_i)·\sigma ·\left[({\epsilon }_g+{\epsilon }_p)·T^4\left(i\right)-2·T_a^4\right]$$

(34)

Now, we calculate the temperature of each layer using the iterative method, as shown in the following steps:

Step 1: Temporarily ignoring heat conduction, the partially shaded cell can be simplified as two layers, as illustrated in Figure 7c. The left half is unshaded, and the right half is shaded. Solving the quartic equations with $T_u$ and $T_s$ in (35) and (36) yields the temperature distribution across these layers.

$$\begin{array}{cc}{P}_{thermal,1}\hfill & =(h_f+h_b)·S_{cell}·(1-p\%)·(T_u-T_a)\hfill \\ & + S_{cell}·(1-p\%)·\sigma ·\left[({\epsilon }_g+{\epsilon }_p)·T_u^4-2·T_a^4\right]\hfill \end{array}$$

(35)

$$\begin{array}{cc}{P}_{thermal,2}\hfill & =(h_f+h_b)·S_{cell}·p\%·(T_s-T_a)\hfill \\ & + S_{cell}·p\%·\sigma ·\left[({\epsilon }_g+{\epsilon }_p)·T_s^4-2·T_a^4\right]\hfill \end{array}$$

(36)

Step 2: Reconsidering heat conduction, the partially shaded cell is divided as finite layers, as shown in Figure 7b. The number of layers is set to m. Generally, a larger m means more accurate results and more computational cost.

Step 3: Set the width of each layer. To simplify the calculation, we set $\mathsf{\Delta }{x}_1=\mathsf{\Delta }{x}_2=\cdots =\mathsf{\Delta }{x}_m=\mathsf{\Delta }x=L/m$.

Step 4: Set the initial temperature of each layer in (37).

$$T{\left(i\right)}_{\left(0\right)}=\left\{\begin{array}{c}{T}_u,\mathrm{if} \mathrm{the}i\text{-}\mathrm{th} \mathrm{layer} \mathrm{is} \mathrm{unshaded}\hfill \\ T_s,\mathrm{if} \mathrm{the}i\text{-}\mathrm{th} \mathrm{layer} \mathrm{is} \mathrm{shaded}\hfill \end{array}\right.$$

(37)

Step 5: If $T_u>T_s$, skip to Step 6; otherwise, skip to Step 12.

Step 6: Set $k=1$ and $flag=0$.

Step 7: The temperature of each layer is calculated iteratively. The first layer and the second layer are calculated by (38) and (39) respectively. The $(i+1)$-th layer ($1<i<m$) is calculated in (40).

$$T{\left(1\right)}_{\left(k\right)}=T{\left(1\right)}_{(k-1)}-\tau$$

(38)

$$\begin{array}{cc}T{\left(2\right)}_{\left(k\right)}\hfill & ={\displaystyle \frac{{\left(\mathsf{\Delta }x\right)}^2}{{\lambda }_s{\delta }_s}}·(h_f+h_b)·\left[T{\left(1\right)}_{\left(k\right)}-T_a\right]\hfill \\ & + {\displaystyle \frac{{\left(\mathsf{\Delta }x\right)}^2}{{\lambda }_s{\delta }_s}}·\sigma ·\left[({\epsilon }_g+{\epsilon }_p)·T{\left(1\right)}_{\left(k\right)}^4-2·T_a^4\right]\hfill \\ & - {\displaystyle \frac{\mathsf{\Delta }x}{{\lambda }_s{\delta }_{s}L}}·P_{thermal,\mathsf{\Delta }}\left(1\right)+T{\left(1\right)}_{\left(k\right)}\hfill \end{array}$$

(39)

$$\begin{array}{cc}T{(i+1)}_{\left(k\right)}\hfill & ={\displaystyle \frac{{\left(\mathsf{\Delta }x\right)}^2}{{\lambda }_s{\delta }_s}}·(h_f+h_b)·\left[T{\left(i\right)}_{\left(k\right)}-T_a\right]\hfill \\ & + {\displaystyle \frac{{\left(\mathsf{\Delta }x\right)}^2}{{\lambda }_s{\delta }_s}}·\sigma ·\left[({\epsilon }_g+{\epsilon }_p)·T{\left(i\right)}_{\left(k\right)}^4-2·T_a^4\right]\hfill \\ & - {\displaystyle \frac{\mathsf{\Delta }x}{{\lambda }_s{\delta }_{s}L}}·P_{thermal,\mathsf{\Delta }}\left(i\right)+2·T{\left(i\right)}_{\left(k\right)}-T{(i-1)}_{\left(k\right)}\hfill \end{array}$$

(40)

where $\tau$ is a parameter that needs to be set manually.

Step 8: Given the threshold ($th$), evaluate the error ($Er\left(k\right)$) using (41). If $\left|Er\right(k\left)\right|>th$, skip to Step 9; otherwise, skip to Step 12.

$$\begin{array}{cc}Er\left(k\right)\hfill & ={\displaystyle \frac{{\left(\mathsf{\Delta }x\right)}^2}{{\lambda }_s{\delta }_s}}·(h_f+h_b)·\left[T{\left(m\right)}_{\left(k\right)}-T_a\right]\hfill \\ & + {\displaystyle \frac{{\left(\mathsf{\Delta }x\right)}^2}{{\lambda }_s{\delta }_s}}·\sigma ·\left[({\epsilon }_g+{\epsilon }_p)·T{\left(m\right)}_{\left(k\right)}^4-2·T_a^4\right]\hfill \\ & - {\displaystyle \frac{\mathsf{\Delta }x}{{\lambda }_s{\delta }_{s}L}}·P_{thermal,\mathsf{\Delta }}\left(m\right)+T{\left(m\right)}_{\left(k\right)}-T{(m-1)}_{\left(k\right)}\hfill \end{array}$$

(41)

Step 9: If $Er\left(k\right)<0$ or $minT\left(i\right)<T_s$, set $flag=1$ and $\tau =-\left|\tau \right|$; if $Er\left(k\right)>0$ and $minT\left(i\right)>T_s$, set $\tau =\left|\tau \right|$.

Step 10: If $flag=1$, set $\tau =-0.5\tau$. This step can ensure the convergence of the algorithm.

Step 11: Update $k=k+1$; then, skip to Step 7.

Step 12: The temperature at x ($0\le x\le L$), as shown in Figure 7a, is calculated by linear interpolation in (42) and (43).

$$i=\lfloor {\displaystyle \frac{x}{\mathsf{\Delta }x}}+{\displaystyle \frac{1}{2}}\rfloor$$

(42)

$$T\left(x\right)=T\left(i\right)+\left({\displaystyle \frac{x}{\mathsf{\Delta }x}}-{\displaystyle \frac{2i-1}{2}}\right)·\left[T(i+1)-T\left(i\right)\right]$$

(43)

where $\lfloor ·\rfloor$ represents the floor function and $i=0,1,\cdots ,m$. Here, we define $T\left(0\right)$ and $T(m+1)$ as the temperatures of adjacent cells at both sides of the partially shaded cell.

Method for selecting the initial $\mathbf{\tau }$: The $\tau$ parameter is not chosen arbitrarily. Its initial value is derived directly from the physical setup and discretization of the problem to ensure a meaningful and stable starting point for the iteration. We set the initial value as follows: $\tau = |T_u-T_s|/m$, where $T_u$ and $T_s$ are the initial temperatures of unshaded and shaded layers, respectively and m is the number of discretized layers. The initial value of $\tau$ provides a balanced starting point. In practice, this adaptive method avoids poor manual guesses and leads to reliable performance.

Influence of $\mathbf{\tau }$ on Convergence and Stability: The initial selection of $\tau$ directly controls the convergence iteration count. The role of $\tau$ is to serve as a step-size controller in our iterative scheme. Its behavior directly governs convergence. Since the absolute value of $\tau$ only remains constant or is halved, the adjustment of the solution in each step is strictly controlled or reduced. This prevents oscillations or divergence, ensuring numerical stability. The halving mechanism is triggered when the current solution does not meet the convergence criterion. Because the absolute value of $\tau$ is bounded from below (approaching zero in the limit), this process guarantees that the algorithm will eventually satisfy any finite precision requirement, leading to asymptotic convergence.

3. Results

Experimental studies are carried out on monocrystalline silicon solar panels (STP 310S-20/Wfw). This section begins with a description of the data collection process, followed by an evaluation of the performance.

3.1. Data Collection

Mono-crystalline silicon solar panels (STP 310S-20/Wfw) are subjected to shading experiments, as shown in Figure 8a. The environmental monitoring station (TRM-ZSA) is presented in Figure 8b. The current and voltage of each solar panel are measured by the monitoring devices (OPT700-RS) presented in Figure 8c. The temperature distribution of each solar panel is measured by the infrared thermal imager (FLIR ONE pro) shown in Figure 8d. Characteristics of the solar panel under the standard testing condition (STC) and at the nominal operation cell temperature (NOCT) are presented in

Table 3. Characteristics of the PV module (STP 310S-20/Wfw).

Characteristics	STC	NOCT
Maximum Power ($P_m$)	310 W	228.7 W
Optimum Operating Voltage ($V_m$)	33.4 V	30.6 V
Optimum Operating Current ($I_m$)	9.29 A	7.47 A
Open-circuit Voltage ($V_{oc}$)	40.2 V	37.0 V
Short-circuit Current ($I_{sc}$)	9.77 A	7.91 A
Temperature Coefficient of $V_{oc}$	−0.34$\%/$°C
Temperature Coefficient of $I_{sc}$	0.060$\%/$°C
Number of Cells	60 (6 × 10)
Number of bypass diodes	3
Cell dimensions	156 × 156 mm

.

The solar panels used in our experiment were installed in 2020, with an expected service life of 25 years. The data analyzed in this work were collected in early 2021, meaning the PV module had experienced minimal degradation at the time of measurement. Prior to data collection, we carefully inspected the module to ensure no visible defects—such as cracks or physical abnormalities—were present, thereby minimizing interference from such factors.

The experimental dataset used to validate the proposed method was collected in January 2021. This historical dataset remains fully applicable, as it was acquired under well-documented conditions, providing a consistent benchmark. While our subsequent research has focused on real-time prediction, this high-quality foundational data is sufficient to verify the core methodology presented in this work.

Partial shading experiments are conducted on solar panels using a uniform shade cloth with a transmittance of 0.05, as presented in Figure 9a. To ensure the thermal steady state prior to the measurement, a stabilization period of 10 min is observed after each shading configuration. The 10 min stabilization period is based on a careful consideration of existing research, practical constraints and empirical observation: The referenced indoor study [29] (which suggests an 8 min stabilization under constant irradiance and temperature) operates under ideal, controlled conditions. Our work is conducted outdoors with inherent variability in irradiance, ambient temperature, and wind speed. Under such varying conditions, a transient thermal model would indeed provide higher accuracy. However, it also demands significantly higher computational complexity, which is often impractical for real-time prediction applications. Therefore, we adopted a quasi-steady-state modeling approach as a pragmatic and efficient compromise that is sufficient for capturing the dominant temperature trends critical for real-time monitoring and management. The stabilization period was determined through our direct experimental observations. After conducting partial shading of the PV module, we continuously monitored its temperature and observed that the rate of temperature change became negligible within approximately 10 min. This empirical finding, specific to the PV module and typical outdoor conditions, formed the primary basis for selecting the 10 min interval. Figure 9b depicts the back-side temperature distribution of the partially shaded PV module captured by the infrared thermal imager.

In order to assess the measurement uncertainty, specifications of the experimental equipment are listed in

Table 4. Specifications of the equipment.

Equipment	Parameter	Resolution	Range	Accuracy	Calibration Status
Environmental monitoring station (TRM-ZSA)	Irradiance	1 W/m2	0 to 2000 W/m2	<5%	Factory-calibrated
	Ambient temperature	0.1 °C	−40 to 80 °C	±0.1 °C
	Wind direction	1°	0 to 360°	±3°
	Wind speed	0.1 m/s	0 to 70 m/s	±0.3 m/s
I-V monitoring device (OPT700-RS)	Current	–	0 to 12 A	–	Factory-calibrated
	Voltage	–	0 to 50 V	–
Infrared thermal imager (FLIR ONE pro)	PV temperature	160 × 120	−20 to 120 °C	±3 °C	Calibrated following [31]

. For the I-V monitoring device (OPT700-RS), the manufacturer guarantees the overall performance via factory calibration rather than specifying the resolution and accuracy of the measured current and voltage, which is a common practice for this device category.

3.2. Performance Evaluation

The performance of the proposed method is evaluated in terms of various shading proportions, a continuous observation period, and computational cost. As a benchmark for temperature analysis, the multiphysics model [31] is adopted for comparison.

3.2.1. Various Shading Proportions

The selected cell is partially shaded with various proportions ranging from 25% to 75%. Environmental conditions, including irradiance (587 to 748 W/${\mathrm{m}}^2$), temperature (2.8 to 10.2 °C) and wind speed (0.7 to 3.8 m/s), are random and uncontrollable.

The maximum temperature ($T_{max}$) and normal temperature ($T_{normal}$) are measured and predicted using 2 min average wind-speed records, as presented in Figure 10. The predicted NOCT of 45.3 °C exceeds the datasheet specification by 0.3 °C for the PV module. The sample points for both maximum and normal temperatures remain close to the reference line, showing strong consistency between predictions and measurements, thereby validating the accuracy of the proposed method.

The performance is evaluated using the root mean square error (RMSE), mean absolute error (MAE), and mean bias error (MBE), as shown in (44)–(46).

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\left(T_{p,i}-T_{m,i}\right)}^2}$$

(44)

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}\left|T_{p,i}-T_{m,i}\right|$$

(45)

$$\mathrm{MBE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}\left(T_{p,i}-T_{m,i}\right)$$

(46)

where N denotes the number of samples, $T_{p,i}$ denotes the predicted temperature of the i-th sample, and $T_{m,i}$ denotes the measured temperature of the i-th sample.

The performance evaluation of the maximum temperature for various shading proportions is shown in

Table 5. Performance evaluation for various shading proportions.

Method	RMSE (°C)	MAE (°C)	MBE (°C)
Multiphysics	6.14	5.05	−1.18
Proposed	6.08	5.13	1.07

. These results establish the accuracy of the proposed method as compared to the multiphysics model.

3.2.2. Continuous Observation Period

On 1 January 2021, we applied 50% partial shading to a single cell within the solar panel between 11:15 and 12:30. Environmental conditions during this continuous observation period are shown in Figure 11a,b. The current and voltage of the partially shaded PV module are presented in Figure 11c. The maximum and normal measurements of temperature are compared with predictions, which were generated using three types of wind-speed input—instantaneous, 2 min average, and 10 min average records—as presented in Figure 11d.

As presented in

Table 6. Performance evaluation for a continuous observation period.

Method	Wind-Speed Records	RMSE (°C)	MAE (°C)	MBE (°C)
Multiphysics	Instantaneous	7.86	7.29	−0.13
	2 min mean	4.20	3.32	−1.29
	10 min mean	4.39	3.72	3.57
Proposed	Instantaneous	8.28	6.73	2.56
	2 min mean	4.15	3.77	1.19
	10 min mean	6.49	6.12	6.12

for the continuous observation, predictions using 2 min average records of wind speed yielded lower RMSE and MAE values than those based on either instantaneous or 10 min average records. The experimental results validate that the proposed method achieves accuracy comparable to that of the multiphysics model.

Wind speed exhibits both a long-term trend and significant short-term fluctuations. Using the instantaneous wind speed introduces substantial random noise due to these fluctuations, which can lead to greater errors in temperature prediction. While averaging wind speed over time can mitigate such noise, the length of the averaging window must be chosen carefully. Averaging over too long a period—such as 10 min—can introduce undesirable lag, smoothing out meaningful temporal variations and delaying the model’s response to changing conditions. This lag can, in turn, degrade prediction accuracy.

To balance the trade-off between reducing noise and minimizing lag, the 2 min average record of wind speed is recommended. This interval effectively dampens high-frequency fluctuations while preserving the dynamic characteristics of wind speed relevant to real-time temperature prediction. As shown in Table 6 of the manuscript, the model using 2 min averaged wind speed consistently outperforms both the instantaneous and 10 min averaged versions in terms of prediction accuracy, confirming that this intermediate window provides an optimal compromise.

3.2.3. Computational Cost

The proposed algorithm is programmed with MATLAB (R2024a). Temperature prediction of the partially shaded PV module takes an average of 15.6 ms, achieving real-time performance. According to our experience, when the programming language is switched to C or JAVA, even if the codes are embedded in mobile devices, the proposed algorithm can still meet the requirement of real-time performance. The computational cost of the proposed method compared to the multiphysics model is listed in

Table 7. Computational cost.

Method	Space Complexity	Time Complexity	Software	Prediction Time
Multiphysics	$O(m_1·m_2·m_3)$	$O(k^{\prime }·m_1·m_2·m_3)$	ANSYS (Student 2025 R2)	6.7 s
Proposed	$O\left(m\right)$	$O(k·m)$	MATLAB (R2024a)	15.6 ms

, where $m_1$, $m_2$, and $m_3$ represent the number of elements in each dimension of the PV module according to the multiphysics model and $k^{\prime }$ represents the convergence iteration count of the multiphysics model.

Clarification of Parameters: m is the number of discretized layers according to our method (e.g., the spatial resolution), and k is the corresponding convergence iteration count.

Derivation of Space Complexity: The primary memory overhead of our algorithm comes from storing the state variables (e.g., temperature) for the m layers. This requires storage proportional to m, leading to a space complexity of $O\left(m\right)$.

Derivation of Time Complexity: The time complexity of the proposed method consists of two factors: cost per iteration and number of iterations (k). Each iteration involves updating the state for all m layers. This core operation has a cost proportional to m, i.e., $O\left(m\right)$ per iteration. Therefore, the total time complexity, given by the iteration count multiplied by the cost per iteration, is Time Complexity = $O(k·m)$. The per iteration cost ($O\left(m\right)$) scales linearly with the problem size (m). Consequently, the overall time complexity ($O(k·m)$) holds for any positive integer (m).

The computational costs presented in Table 7 were obtained through our own measurements. We conducted all simulations in consistent hardware environment—namely, on a laptop with Intel Core i5 (2.50 GHz) and Windows 10 (64-bit). For each model, the reported time is the average execution time over 20 independent runs to ensure reliability and minimize random fluctuations. The proposed method offers a substantial reduction in computational complexity compared to the multiphysics model, leading to a significant improvement in computational efficiency.

4. Discussion

In this section, simulations are conducted on mono-crystalline solar panels (STP310S-20/Wfw) to display the effects of shading proportions, operating points, and environmental conditions.

4.1. Effect of Shading Proportions

Generally, the current of the solar panel under normal conditions is near the optimum operating current. To reveal the effect of shading proportions, the NOCT condition is considered as the environmental condition, and the optimum operating current ($I_m$) at the NOCT (as shown in Table 3 can be considered as the output current of the solar panel). A single cell is assumed to be shaded in the proportion of 5–90% on the left side. For comparison, no shading and complete shading of this cell are also simulated. The temperature distributions in different proportions are shown in Figure 12.

It is obvious that the maximum temperature of the partially shaded cell increases when the shading proportion ranges from 0% to 10% and varies slightly when the shading proportion ranges from 10% to 50%, then decreases when the shading proportion ranges from 50% to 100%. In our work, the overheating region, which is 20 °C warmer than the normal region, is regarded as a hotspot. The definition of a hotspot is derived from Chinese National Standard GB/T 36567-2018 (Photovoltaic module inspection and maintenance procedure) [43]. This standard specifies a 20 °C temperature difference as a critical criterion for evaluating hotspot susceptibility and its associated long-term degradation risks. Adopting this standardized and industry-accepted threshold ensures that our analysis targets temperature anomalies of engineering applications. The temperature differences between the overheating region and normal region are more than 20 °C when the shading proportion varies from 6% to 90%, which implies that a shading proportion ranging from 6% to 90% is prone to develop hotspots.

4.2. Effect of Operating Point

In order to explore the effect of the operating point, the shading scenario is defined as a case in which one cell is 50% shaded on the left side. The NOCT condition is considered as the environmental condition. Different operating points on the I-V curve of the PV module are shown in Figure 13. $P_{oc}$ is the open-circuit point, $P_{sc}$ is the short-circuit point and $P_{mp}$ denotes the max power point. $P_0$ represents the operating point characterized by equality between the photocurrent of the shaded cell and the output current of the solar panel. $P_{bypass}$ denotes the point when the voltage of the partially shaded sub-module falls to zero. $P_1$, $P_2$ and $P_3$ are sample points between $P_0$ and $P_{bypass}$.

As the current rises from the open-circuit point ($P_{oc}$) along the I-V curve, the voltage across the shunt resistance decreases. Once the current attains the photocurrent of the partially shaded cell at $P_0$, the voltage of shunt resistance $R_{h1}$ and $R_{h2}$ in Figure 3a,b becomes zero. As the current continues to increase, the voltage of the partially shaded cell becomes reversed, and it consumes electricity. When the voltage of the partially shaded sub-module falls to zero at $P_{bypass}$, the corresponding bypass diode turns on, enabling current to flow through it [44,45,46]. The temperature distributions of PV modules at different operating points are shown in Figure 14.

Combining Figure 13 and Figure 14 at the cell level from $P_0$ to $P_{bypass}$, the voltage and electric power decrease dramatically, while the thermal power and maximum temperature increase rapidly. From $P_{bypass}$ to $P_{sc}$, the voltage, electric power, thermal power and maximum temperature remain almost unchanged. At the sub-cell level, from $P_{bypass}$ to $P_{mp}$, the electric power of unshaded cell in the normal sub-module increases, while its thermal power and maximum temperature decrease. From $P_{mp}$ to $P_{sc}$, the electric power of the unshaded cell in the normal sub-module decreases, while its thermal power and maximum temperature increase. The temperature of the partially shaded cell reaches its highest level from $P_{pypass}$ to $P_{sc}$, when the bypass diode of the corresponding sub-module is forward-biased.

4.3. Effect of Environmental Conditions

In order to explore the effect of environmental conditions, one cell in the PV module is assumed to be half shaded on the left side. The optimum operating current ($I_m$) is considered as the output current of the solar panel under different environmental conditions. The effect of environmental conditions is shown in Figure 15.

The effect of solar irradiance with different ambient temperatures is shown in Figure 15a. The thermal power of the partially shaded cell is directly affected by the level of solar irradiance. As a result, the maximum temperature increases when the solar irradiance ranges from 0 W/${\mathrm{m}}^2$ to 1200 W/${\mathrm{m}}^2$. At the low irradiance level, the maximum temperatures of the partially shaded cell vary greatly with different ambient temperatures. However, at the high irradiance level, the maximum temperatures change slightly with different ambient temperatures. The effect of wind speed with different ambient temperatures is shown in Figure 15b. Because the wind speed is related to the convective heat transfer coefficient, the maximum temperature of the partially shaded cell decreases while the wind speed rises. At the low wind-speed level, the maximum temperatures of the partially shaded cell vary slightly with different ambient temperatures. However, at the high wind-speed level, the maximum temperatures change greatly with different ambient temperatures. The effects of ambient temperature with varying irradiance and wind speed are shown in Figure 15c,d, respectively. The maximum temperature of the partially shaded cell increases linearly when the ambient temperature ranges from −10 °C to 50 °C. These simulations imply that higher irradiance, higher ambient temperature, and lower wind speed result in a higher temperature of the partially shaded cell.

4.4. Model Limitations and Error Analysis

The proposed model employs strategic simplifications to achieve computational efficiency, which inevitably introduces certain errors. The primary sources of errors and their expected impacts are analyzed as follows:

Temperature substitution: The proposed model equates the cell temperature to the glass and TPT surface temperatures. Based on [39], this leads to an error within 3 °C in predicting the surface temperature. However, in different regions of the surface, this deviation is widespread, which means it has a relatively small impact on the prediction of the temperature difference of the solar panel or hotspot detection (which is a key indicator for monitoring).

Neglected in-plane conduction: Considering that the thermal conductivity of the glass, EVA and backsheet layers is relatively low (two to three orders of magnitude lower than that of the silicon layer), we assume that the in-plane heat conduction within these layers can be neglected. This simplification primarily affects the spatial gradient of predicted hotspots, potentially making them appear slightly more localized than in reality. Nevertheless, its influence on the peak temperature (the most critical indicator for risk assessment) is secondary, as confirmed by the consistent results between our predictions and experimental measurements.

Wear-out effects: A correlation likely exists between wear-out effects (e.g., cracks) and the hotspot temperature. However, quantitative studies on this specific relationship are still lacking. The primary focuses of the present work are the investigation of the influence of partial shading on temperature distribution and the development of a real-time prediction model. Therefore, the potential coupling effect between partial shading and long-term degradation mechanisms was not included in our current modeling scope.

The above assumptions and simplifications, to some extent, limited the accuracy of the proposed model but significantly improved the computational efficiency. This represents an intentional and acceptable trade-off made for the application targeting real-time and embedded monitoring.

4.5. Comparison with Recent Works

Research Focus: Existing studies can be categorized by their primary motivations, as shown in

Table 8. Comparison of research focus.

Research Focus	Scientific/Engineering Problem	Relationship to the Proposed Method
Analysis of partial shading effects [47,48,49,50,51]	What are the physical mechanisms of partial shading? What are the consequences?	The physical complexity and the hotspot risk establish the theoretical and applied foundation of this work.
Real-time detection & diagnosis of partial shading [52]	Is shading occurring in the system? What is its type and location?	Provides the triggering signal for the prediction model.
Temperature prediction of the PV module [53,54,55,56]	What is the temperature (field) of the PV module?	The proposed method belongs to this broad domain but focuses on its most challenging sub-problem.
Real-time temperature prediction under partial shading conditions (this work)	How does the temperature field evolve in real time under partial shading conditions? What is the hotspot risk?	Contribution: This work fills the gap from “offline analysis” and “fault diagnosis” to “online risk quantification”.

. Our work targets a specific and under-explored problem—namely, real-time temperature prediction under partial shading conditions, which fills the gap from “offline analysis” and “fault diagnosis” to “online risk quantification”.

Methodological Advancement: We established a novel framework based on four requirements for a practical solution: (1) handles partial shading, (2) achieves real-time performance, (3) does not depend on commercial software, and (4) does not depend on the training dataset. A systematic comparison of temperature prediction methods is shown in

Table 9. Performance comparison of temperature prediction methods.

Method	Capable of Handling Partial Shading?	Real-Time Performance	Software Dependency	Dataset Dependency
Multiphysics Model [57]	Yes	No	Relies on commercial software	No
Simplified Physical & Empirical Model [58]	No	Yes	No	No
Data-driven Model [59]	Limited	Depends on the scale of the model	Relies on machine learning frameworks	Yes
The Proposed Method	Yes	Yes	No	No

.

5. Conclusions

A real-time approach inspired by FEM is designed in this paper to predict the temperature of solar panels under partial shading conditions. The shaded cell is divided into finite layers, and the heat transfer of each layer is analyzed. A simple and fast iterative algorithm is proposed to calculate the temperature of each layer. The proposed approach successfully balances the need for computational complexity (essential for real-time use) with acceptable predictive accuracy for thermal monitoring purposes. To the best of our knowledge, this is the first work in this field that achieves computational simplicity without relying on professional commercial software. Experimental results validate that the proposed method achieves accuracy comparable to that of the multiphysics model (which is widely regarded as the benchmark in this field) while significantly improving computational efficiency. Simulations reveal that shading proportions ranging from 6% to 90% are prone to the development of hotspots in cases where a single cell within the solar panel is partially shaded. Higher irradiance, higher ambient temperature and lower wind speed result in a higher temperature of the PV module. Furthermore, the proposed method can serve as a core physics-informed prediction module in digital-twin frameworks, allowing for virtual testing, performance forecasting, and operational optimization. It is important to note that the proposed method does not provide a rigorous analytical proof for the relationship between the shading proportion and the operating point. Formalizing this into a predictive theory constitutes a valuable and necessary direction for future theoretical work.