Revealing the Intrinsic Mechanisms of Hot and Cold Spots within a Locally Shaded Photovoltaic Module Based on Micro-Electrical Characteristics

Abstract

Hot-spot generation is critical to the performance and lifespan of photovoltaic (PV) modules; however, the underlying mechanisms of hot-spot formation have not been fully elucidated. This work conducted a localized shading test on a PV module, measured the micro-electrical characteristics and temperature distributions of both the shaded and unshaded cells, calculated the heat-source power densities, and then predicted the occurrence and locations of hot and cold spots via numerical simulations. It was found that, under an irradiance of 750 W/m², when one cell in a PV module is shaded by 1/2, the unshaded area within the shaded cell exhibited a hot spot, with the temperature reaching up to 77.66 °C, approximately 22.5 °C higher than the surrounding cells. The intrinsic mechanism for the occurrence of the hot spot is that, compared with the unshaded cells, the unshaded portion of the shaded cell can generate an extra significantly large Joule heat power density, about 1079.62 W/m². The reason for generating such a large Joule heat power density is that this portion is in a reverse-bias state with a high current density flowing through it, according to our measurements. In contrast, the shaded portion forms a cold spot, about 7.5 °C cooler than the surrounding cells. This is because the shaded portion can only generate a Joule heat power density of about 46.98 W/m² due to the small reverse-bias current density flowing through it and fails to absorb heat from solar irradiance, which is about 645 W/m². Moreover, this work demonstrates that the hot-spot temperature initially rises and then decreases with increasing shading ratio, with the highest temperatures and the most pronounced temperature changes occurring around a shading ratio of 1/2. The presented method can be also used to evaluate the performance and reliability of various other PV modules under local shading conditions.

1. Introduction

As of 2023, China’s cumulative photovoltaic (PV) installed capacity has reached 610 GW, accounting for 20.27% of the total power generation capacity [1]. Among this, distributed photovoltaics account for 43.27%. The main installation scenarios for distributed photovoltaics include rooftop PV systems [2], PV curtain walls [3], and PV roads [4], among others. Distributed photovoltaics are highly susceptible to local shading from nearby buildings [5], plants [6], fallen leaves [7], bird droppings [8], etc. When a PV module is locally shaded, the shaded cell’s current decreases, subsequently suppressing the module’s overall current due to the series mismatch effect. The reduced current causes the shaded cell to become reverse biased, leading to power dissipation [9]. This results in an increase in temperature and the formation of a “hot spot” [10]. The hot-spot temperature on the locally shaded cell can reach 101.4 °C [11], which will significantly impact the lifespan [12] and safety of the PV module [13]. Bypass diodes are commonly used in PV modules to mitigate the effects of hot spots, by conducting a part of the current in the PV string. However, they cannot prevent the “energy” generated by the module itself from being lost on the shaded cell [14]. Given the prevalence and damage of hot-spots under local shading [15], studying their impact on the power output and temperature distribution of PV modules is highly significant.

Several studies have explored the impacts of local shading on the power output and hot spots of PV modules. For example, Fodah et al. found that the temperature of PV modules shaded by bird droppings increased by approximately 5% compared to clean modules [16]. However, they did not analyze the temperature distribution across different areas within the PV modules. García et al. observed that the temperature of a shaded cell in a PV module could be 70–80 °C higher than that of the unshaded cells [17]. They mainly attributed the hot spot to the cell quantity (leakage current) and the number of series-connected cells. They did not differentiate between the contributions of the shaded and unshaded areas within a cell nor explore the relationship between the temperature distribution in the PV module and the shading ratio. Consequently, the understanding of the influence of the local shading on PV modules remains incomplete. This study aims to reveal the mechanisms behind hot and cold spots based on the micro-electrical characteristics, which distinguish between the shaded and unshaded portions within the same cell.

This study first measured the micro-electrical characteristics of both shaded and unshaded cells within a PV module. It then analyzed and calculated the heat sources at different positions within the module and carried out numerical simulations to predict temperatures at various positions, comparing these predictions with experimentally measured results. Based on these findings, the intrinsic mechanisms behind localized hot and cold spots were revealed and the impacts of shading ratio on the temperature distribution of PV modules were discussed. This research is important for predicting high-temperature warnings and the power-generation capabilities of PV modules.

2. Experimental Methods and Calculation Settings

2.1. Experimental Methods

Measuring the micro-electrical characteristics is the first step in the research process of this paper. In this step, the current and voltage of different cells within the locally shaded PV module were measured. By analyzing these micro-electrical characteristics, it was determined whether each cell, as well as the shaded and unshaded areas within the same cell, consumed electrical energy and generate heat, as detailed in Section 3.1.

The schematic diagram of the experimental setup is shown in Figure 1. The PV module used in this study is a crystalline silicon PERC module with 36 cells connected in series and a bypass diode connected in parallel. Its standard parameters under an irradiance of 1000 W/m² are P_max = 100 W, V_oc = 18 V, and I_sc = 6.66 A, respectively. A shading material with a transmittance of 0 shades a single cell in the module, with shading ratios R = 0, 1/4, 1/2, 3/4, and 1.

The experiments took place on 16 October 2023, from 10:30 AM to 2:00 PM in Changping District, Beijing. The ambient temperature during the experiments, measured with a thermometer, was 20 °C ± 1 °C. In the experiment, the PV module was constantly maintained at the maximum power point using an MPPT controller from Yike Solar Technology Co., Ltd, Dongguan, China.

The actual solar irradiance received by the PV module was measured using a solar irradiance meter from Lailx Technology, Beijing, China, mounted on the module. By adjusting the tilt angle, the solar irradiance received by the PV module was controlled at 250 W/m², 500 W/m², and 750 W/m². The experiments at each irradiance level were completed within 10 min. During this time, the variation in the actual measured solar irradiance was less than 4 W/m².

The micro-electrical characteristics of the shaded and unshaded cells were measured sequentially using a Rigol DM 3058 multimeter from RIGOL Technologies Co., Ltd., Suzhou, China. A hole was drilled in the backsheet of the PV module to expose the wires at one end of the shaded cell. The voltage of the shaded cell, V_shaded, was directly measured using the digital multimeter, while the voltage of the unshaded cells, V_unshaded, was calculated using the formula: (V_module − V_shaded)/35, where V_module is the whole voltage across the PV module. This method assumes that all cells in the photovoltaic module are identical. The temperature distribution within the PV module was measured using a Fluke F 563 infrared thermometer from Fluke corporation, Washington, DC, USA.

2.2. Calculation Settings

Next, the temperature distribution within the PV module was simulated using COMSOL 6.2 [18], with ultra-fine mesh division accuracy. The thermal model of the PV module consists of a glass cover plate, EVA layers, crystalline silicon solar cells, TPT layer, and an aluminum frame, with material parameters detailed in

Table 1. Material parameters.

Materials	Glass	EVA	Si	TPT	Al
Thermal conductivity (W·m−1·K−1)	6.17	0.34	148.0	0.24	230.0
Density (kg·m−3)	2508.0	951.0	2400.0	2200.0	2700.0
Constant pressure heat capacity (J·kg−1·K−1)	830.0	1400.0	124.0	1050.0	900.0

. The module dimensions are 119 cm in length and 54 cm in width.

The heat-transfer process between the PV module and the environment includes heat source input, solid heat transfer, surface radiation, and forced convection. The heat-source input is derived from the calculated heat source distribution at different positions within the PV module. This distribution was quantitatively determined based on the micro-electrical data of the cells and the incident irradiance power. Details of the heat-source distribution calculation are provided in Section 3.2. The heat-source distribution was applied to the upper surface of the photovoltaic cells as the surface heat-source power density Q_b (W/m²). This is because sunlight is incident on the top surface of the cells and crystalline silicon exhibits an excellent thermal conductivity.

−n·q_b = Q_b

(1)

In Equation (1), n represents the normal vector of the upper surface of the photovoltaic cell and q_b denotes the heat flux vector resulting from Q_b.

q_T = −k∇T

(2)

In Equation (2), q_T represents the conductive heat flux and k is the thermal conductivity of the solid. The radiant heat flux between the solid surface and the environment is described by Equation (3):

−n·q_r = εσ(T⁴_amb − T⁴)

(3)

In Equation (3), q_r represents the radiation heat flux, ε is the surface emissivity, σ is the Stefan–Boltzmann constant, 5.67 × 10⁻⁸ W/(m²·K⁴), and T_amb is the ambient temperature, set at 20 °C. Forced convection is calculated using empirical Equations (4)–(8):

−n·q_f = q₀

(4)

q₀ = h(T_amb − T)

(5)

$$\mathrm{h}=\left\{\begin{array}{c}2{\displaystyle \frac{K}{L}}{\displaystyle \frac{0.3387{P_r}^{1/3}{R_{e_L}}^{1/2}}{{[1+{(0.0468/P_r)}^{2/3}]}^{1/4}}} (R_{e_L}\le {5·10}^5)\\ 2{\displaystyle \frac{K}{L}}{{\mathrm{P}}_{\mathrm{r}}}^{1/3}(0.037{R_{e_L}}^{4/5}-871) (R_{e_L}>{5·10}^5)\end{array}\right.$$

(6)

$$R_{e_L}={\displaystyle \frac{\mathrm{U}\mathrm{L}}{v}}$$

(7)

$$P_r={\displaystyle \frac{\mu C_p}{\lambda }}$$

(8)

Here, q_f represents the convective heat flux, h is the convective heat transfer coefficient for the module surface (W·m⁻²·K⁻¹), and L is the characteristic length (1.19 m, the length of the module). ReL is the Reynolds number based on the characteristic length, P_r is the Prandtl number for air, v is the kinematic viscosity of air (m²/s), μ is the dynamic viscosity of air (Pa·s), C_p is the specific heat capacity of air at constant pressure (J·kg⁻¹·K⁻¹), λ is the thermal conductivity of air (W·m⁻¹·K⁻¹), and U is the environmental wind speed, set at 1 m/s.

3. Results and Discussion

3.1. Micro-Electrical Characteristics of Different Solar Cells within a Locally Shaded PV Module

Table 2. The micro-electrical characteristics of different solar cells in the PV module.

Irradiance (W/m2)	Shading Ratio	Voltage of Module (V)	Voltage of Shaded Cell (V)	Voltage of Unshaded Cell (V)	Current (A)	Reverse-Bias Current Density (A/m2)
250	0	16.488	0.458	0.458	1.681	-
	1/4	15.970	−0.130	0.460	1.310	10.717
	1/2	14.120	−3.800	0.512	0.950	11.944
	3/4	13.380	−5.240	0.532	0.560	13.235
	1	13.195	−5.600	0.537	0.210	13.558
500	0	16.524	0.459	0.459	2.952	-
	1/4	15.950	−0.220	0.462	2.280	10.330
	1/2	14.745	−2.580	0.495	1.610	11.686
	3/4	14.100	−3.890	0.514	0.950	12.331
	1	13.100	−5.870	0.542	0.220	14.204
750	0	16.560	0.460	0.460	3.770	-
	1/4	15.925	−0.350	0.465	3.350	10.653
	1/2	14.180	−3.810	0.514	2.290	12.331
	3/4	13.105	−5.900	0.543	1.200	13.816
	1	12.055	−7.580	0.561	0.250	16.141

shows the measured micro-electrical characteristics of the shaded and unshaded cells under various shading ratios and solar irradiance levels. It is evident that the voltage across the shaded cell is negative, indicating a reverse-bias state. This negative voltage implies that the shaded cell consumes electrical energy, generating Joule heat.

The values of these negative voltages are significantly higher than the positive voltage of the unshaded cells. This is because the reverse-bias voltage across the shaded cell results from the combined effect of all the unshaded cells, as given by V_shaded = V_module − 35 × V_unshaded. As the shading ratio increases, the output voltage of the unshaded cells also rises. This is due to the rightward shift in the operating point of the unshaded cells, as their current decreases with the increasing shading ratio. Additionally, for a constant shading ratio, higher solar irradiance leads to a greater reverse-bias voltage on the shaded cell for similar reasons.

In this work, the PV module was maintained at the MPPT, resulting in relatively high values for V_module, as shown in Table 2. Consequently, V_shaded is quite small, given the relationship V_shaded = V_module – 35 × V_unshaded. If such a locally shaded PV module with a bypass diode is connected in series within a PV string, the bypass diode will conduct, causing V_module to approach zero and the PV module to be in an almost short-circuit condition. In this scenario, V_shaded will be significantly larger (about 20 V), and the current across it will also increase.

3.2. Heat-Source Power Densities for Different Positions within a Locally Shaded PV Module

A diagram of the energy flows at different positions within the locally shaded PV module is provided in Figure 2. For clarity, these positions are marked as ①, ②, and ③. Position ① denotes the shaded area of the shaded cell, ② denotes the unshaded area of the shaded cell, and ③ denotes any unshaded cell.

In calculating the heat-source power densities, four main factors were considered: irradiance power density (P_In), reflectance power density (P_R), transmittance power density (P_T), and power density of electrical output (P_E). According to the law of energy conservation, the heat-source power density (P) for different positions within the locally shaded PV module is as follows:

P = P_In − P_R − P_T − P_E = P_sunlight − P_E

(9)

The power density absorbed by the solar cell from sunlight, denoted as P_sunlight, is defined as P_sunlight = P_In − P_R − P_T. The reflectivity of a silicon PV module to solar irradiance, P_R/P_In, is approximately 11.94% [19]. Meanwhile, the silicon module allows about 5% of the light with wavelengths longer than 1000 nm to pass through [20]. Since the energy of light in this wavelength range accounts for approximately 41.13% of the AM1.5 spectrum, P_T/P_In can be estimated by 41.13% × 5% ≈ 2.06%. Therefore, P_sunlight = P_In − P_R − P_T ≈ 86% × P_In. Note that the transmittance and reflectance values are sourced from PERC modules. When applying this method to other types of modules, these values should be substituted with those specific to the modules being studied.

The calculation of P_E is crucial in determining heat-source power densities, especially in partially shaded scenarios where direct measurement of current densities in the shaded area ① and unshaded area ② separately is not feasible. The general formula for the calculation of P_E is as follows,

P_E = V × J

(10)

where V and J represent the voltage and current density of each cell (or each area of the shaded cell), respectively. In the case of a fully shaded scenario, the current density J of each cell, whether shaded or unshaded, can be measured directly. In the case of a partially shaded scenario, the current density J of the unshaded cells can be measured directly. However, the current densities of the shaded area J_① and unshaded area J_② of the shaded cell cannot be directly measured.

As the irradiance received by the shaded area ① is zero, its current density should depend solely on the magnitude of the applied bias voltage on it. Therefore, the current density of the shaded area J_① can be replaced by the current density of a single cell under the same voltage in a dark environment, which can be easily measured. The current densities of a single cell under various reverse-bias voltages (listed in Column 4 of Table 2) were measured in a dark environment, and the results were given in the last column.

The current density in the unshaded area ② of the shaded cell can be calculated according to a parallel model [21],

$$J_{②}={\displaystyle \frac{I_{shaded}-J_{\mathrm{①}}AR}{A(1-R)}}$$

(11)

where I_shaded represents the total current of the shaded cell, A represents the area of a single cell, and R is the shading ratio. Equation (11) is the key equation for extracting the micro-electrical characteristics of the shaded and unshaded portions within the shaded cell. Based on Equations (9)–(11), the heat-source power densities at different positions within the PV module were calculated.

The obtained heat-source power densities at different positions within the PV module are presented in

Table 3. Heat-source power densities at different positions within the PV module (W/m²), where ① and ② denote the shaded and unshaded area of the shaded cell, respectively; and ③ denotes the unshaded cell.

R	Position	250 W/m2			500 W/m2			750 W/m2
		Psunlight	PE	P	Psunlight	PE	P	Psunlight	PE	P
0	-	215	49.71	165.29	430	87.48	342.52	645	111.96	533.04
1/4	①	0	−1.40	1.39	0	−2.27	2.27	0	−3.73	3.73
	②	215	−14.20	229.20	430	−42.42	472.42	645	−99.69	744.69
	③	215	38.91	176.10	430	68.01	361.99	645	100.57	544.43
1/2	①	0	−45.39	45.39	0	−30.15	30.15	0	−46.98	46.99
	②	215	−420.76	635.76	430	−506.21	936.21	645	−1079.62	1724.62
	③	215	31.41	183.60	430	51.45	378.55	645	75.99	569.01
3/4	①	0	−69.35	69.35	0	−47.97	47.97	0	−81.52	81.52
	②	215	−549.75	764.75	430	−810.46	1240.46	645	−1583.86	2228.86
	③	215	19.23	195.77	430	31.52	398.47	645	42.07	602.93
1	①	0	−75.93	75.93	0	−83.38	83.38	0	−122.35	122.35
	③	215	7.28	207.72	430	7.70	422.30	645	9.06	635.95

Note: The positions ①, ②, and ③ in the table correspond to those shown in Figure 2.

. It can be seen that the heat-source power density at Area ② of the shaded cell is the highest, being 1 to 4 times greater than that at Area ③ and 10 to 200 times greater than at Area ①. This suggests that Area ② tends to exhibit a hot spot, while Area ① tends to show a cold spot, as will be confirmed later in the text. The heat-source power density at Area ② increases with both the shading ratio and solar irradiance, peaking when R = 3/4 and the irradiance is 750 W/m². Additionally, the heat-source power density of the unshaded cells (Area ③) is positively correlated with the shading ratio (R) across various irradiance levels. This implies that the temperature of the PV module (unshaded cells) will increase as R increases.

3.3. Temperature Distribution within a Locally Shaded PV Module

The temperature distribution within a locally shaded PV module was simulated based on the heat-source power densities. To validate the reliability and accuracy of these simulations, the results were compared with actual measured temperatures. Specifically, the lowest temperature at Area ①, the highest temperature at Area ②, and the average temperature of the unshaded cells (Area ③) were examined for R = 1/2 under various irradiance levels, as given in

Table 4. Simulated and measured results for the lowest temperature at the shaded Area ①, the highest temperature at the unshaded Area ②, and the average temperature of the unshaded cells ③ when R = 1/2.

Irradiance (W/m2)	The Lowest Temp. in ① (°C)		The Highest Temp. in ② (°C)		The Average Temp. in ③ (°C)
	Measured	Simulated	Measured	Simulated	Measured	Simulated
750	44.7	43.04	74.1	77.66	52.3	50.08
500	35.8	35.34	50.8	53.37	39.1	40.59
250	30.1	29.52	38.3	42.75	31.0	30.37

. The comparison indicates that the simulated temperatures align well with the experimental measurements, confirming the reliability and accuracy of both the heat-source calculations and temperature simulations.

It can be clearly observed from Table 4 that, at an irradiance of 750 W/m², the highest temperature in Area ② is approximately 27.58 °C higher than the average temperature in Area ③, demonstrating a significant hot spot. Conversely, the lowest temperature in Area ① is about 7.04 °C lower than the average temperature in Area ③, indicating a notable cold spot. The causes of these hot- and cold-spot phenomena will be discussed in detail later.

The simulation results of the average, highest, and lowest temperatures at different positions of the PV module under various shading ratios and irradiance levels are presented in

Table 5. The simulated average, highest, and lowest temperatures at different positions of the PV module, considering five shading ratios (R) and three irradiance levels. ① and ② denote the shaded and unshaded area of the shaded cell, respectively; and ③ denotes the unshaded cell.

R	Position	The Average Temp. (°C)			The Highest Temp. (°C)			The Lowest Temp. (°C)
		250 W/m2	500 W/m2	750 W/m2	250 W/m2	500 W/m2	750 W/m2	250 W/m2	500 W/m2	750 W/m2
0	-	26.43	33.15	40.05	27.33	34.94	42.76	24.22	28.62	33.15
1/4	①	25.91	31.83	37.93	27.20	34.48	41.86	24.89	29.58	34.66
	②	27.29	34.86	42.79	28.12	36.62	45.47	25.38	30.74	36.50
	③	26.79	33.75	40.37	27.79	35.73	43.26	24.39	28.80	33.11
1/2	①	32.87	40.23	51.58	38.21	46.99	65.52	29.52	35.34	43.04
	②	40.17	49.83	71.35	42.75	53.37	77.66	35.10	42.07	56.67
	③	30.37	40.59	50.08	32.12	44.01	54.70	26.90	33.93	39.45
3/4	①	31.29	37.57	47.24	36.83	45.99	63.44	28.21	32.63	38.85
	②	37.27	46.97	65.68	38.88	49.54	70.02	33.40	40.17	54.00
	③	31.02	41.59	51.69	32.87	45.15	56.54	27.34	34.59	40.52
1	①	26.98	31.18	35.72	31.07	40.43	48.19	23.67	24.95	27.11
	③	31.66	42.77	53.23	33.63	46.54	58.29	27.82	35.43	41.52

. For clearer detail, Figure 3a shows the relationship between the average temperatures of different positions and solar irradiance when R = 1/2. It can be seen that the average temperature of the unshaded cells (Area ③) is approximately proportional to the solar irradiance. This is because the heat-source power density on the unshaded cells, P, can be estimated by the formula [86% − (1−R) × η] × P_In, where η represents the photovoltaic conversion efficiency of the cell, which is approximately proportional to the solar irradiance under a fixed shading ratio, as illustrated in Figure 3b. However, the average temperature at Area ② of the shaded cell shows a super-linear growth trend as solar irradiance increases. This is due to the increasing reverse-bias voltage and reverse-bias current at both ends of the shaded cell with increasing solar irradiance, bringing additional heat-source power density.

Figure 4a illustrates the variation of the highest temperatures at different positions within the locally shaded PV module with the shading ratio under an irradiance of 750 W/m². It shows that the temperature of the shaded cell (Areas ① and ②) initially increases and then decreases as R increases, with the highest temperature occurring at Area ② reaching 77.66 °C when R = 1/2. This trend appears to contrast with the variation of the heat-source power density depicted in Figure 4b, where the heat-source power density at Area ② peaks when R = 3/4. The reason for this inconsistency is as follows: at R = 3/4, Area ② dissipates more heat to the external environment compared to R = 1/2. Heat exchange between Area ② and the external environment were calculated, and the net heat-source power density, P_net, responsible for the temperature rise by excluding this exchange is derived. At R = 1/2 and 3/4, the calculated values of P_net at Area ② are 1390.13 W/m² and 1371.74 W/m², respectively.

From Figure 4a, it can also be observed that as R approaches 1/2, the highest temperature in the PV module varies sharply with R. This is because as R increases, the reverse-bias voltage at both ends of the shaded cell rises rapidly (Table 2); meanwhile, the current density at Area ② increases rapidly. The rapid increase in voltage and current density leads to a sharp increase in the heat-source power density: it increases by 1484.17 W/m² as R increases from 1/4 to 3/4, as shown in Figure 4b.

Additionally, from Figure 4a, it can also be observed that, under the same shading ratio, there is a significant temperature difference at different positions within the PV module. This can be attributed to the variation in the heat-source power densities at different positions (Figure 4b). However, when R = 1/2, although the heat-source power density at Area ① is much lower than that at Area ③, its temperature is higher. This can be attributed to the excellent thermal conductivity of crystalline silicon, which allows Area ① to receive a substantial amount of conductive heat from Area ②.

Table 5 also shows that the temperatures of the unshaded cells (Area ③) in the locally shaded PV module are significantly higher than those of the cells in an unshaded PV module, and these temperatures further increase with R. For instance, at 750 W/m², when R = 1/4, 1/2, 3/4, and 1, the average temperatures at Area ③ are 40.37 °C, 50.08 °C, 51.69 °C, and 53.23 °C, respectively; while that of the cells in the unshaded PV module is 40.05 °C. This result is consistent with Ref. [18]. This occurs because the increase in the shading ratio more severely restricts the overall current of the module, leading to a decrease in the electrical output power (P_E) of Area ③, and more solar energy is converted into thermal energy. This is also evident from the data in Table 3: when R = 1, the P_E of Area ③ is 9.06 W/m², much lower than that in an unshaded PV module, which is 111.96 W/m².

Figure 5 presents line scans of the temperature distribution within the PV module when R = 1/2 and P_In = 750 W/m². On the right-side line scan, one can observe an obvious “hot spot” and “cold spot” as described in Table 4: the hot spot occurs in Area ② of the shaded cell, whereas the cold spot occurs in Area ①. The temperature of the hot spot is about 22.5 °C higher and the cold spot is about 7.5 °C lower than that of the surrounding cells, respectively. The temperature in the PV module quickly drops from the highest point outward, indicating that the monocrystalline silicon material has a good thermal conductivity, which allows the heat generated by the cells to be rapidly dissipated. The excellent thermal conductivity of monocrystalline silicon also results in almost identical temperatures on the front and back faces of the PV module (a temperature difference less than 0.1 °C), as evident from the three-dimensional cross-sectional temperature distribution shown in Figure 5. Furthermore, because the cells at the edges of the PV module can dissipate heat more effectively, their temperature is about 6 °C lower than that at the center of the PV module.

In the prediction of hot and cold spots under local shading in a PV module, the key point is to extract the micro-electrical characteristics of shaded and unshaded portions based on actual measurements. This enables us to accurately predict the temperatures and locations of hot and cold spots and explain the mechanisms behind them. Recently, Oulefki et al. employed Multi-view VR imaging [22] and its integration with algorithms [23] to achieve intuitive detection and precise segmentation of deteriorated areas in PV modules, demonstrating excellent performance in identifying and locating hot spots under operational conditions. These two technologies, one focusing on the underlying mechanisms and the other on practical application, are complementary. Combining our method with the VR imaging technology holds promise for providing a powerful tool and comprehensive explanation for the state diagnosis of PV modules.

It should be noted that the results obtained above are not only applicable to a single PV module but also to PV strings with multiple modules connected in series. As mentioned earlier, the bypass diode in parallel with the locally shaded PV module will conduct, causing the module to be in an almost short-circuit condition. The voltage across the shaded cell will be much higher, and the current will also increase. According to the Joule heat equation, the shaded cell in a real PV string will generate significantly more heat compared to when it is in a single PV module operating at MPPT. Therefore, the function of the bypass diode is to prevent the “energy” generated by the other PV modules from being dissipated in the shaded cell. However, it cannot prevent the “energy” generated by the module itself from being lost in the shaded cell.

4. Conclusions

This study aims to reveal the intrinsic mechanisms of the uneven temperature distribution, particularly the hot and cold spots, by investigating the relationships between temperature distribution, micro-electrical characteristics, and shading ratio within a locally shaded PV module.

It is found that the voltage across the shaded cell is negative, indicating that it will consume electrical energy, thereby generating heat output. The key technology in this work is extracting the current densities of the shaded and unshaded portions of the shaded cell. The current density of the shaded area is replaced by that of a single cell under the same voltage in a dark environment; while that of the unshaded portion is calculated by subtracting the current of the shaded portion from the total current of the cell, then dividing by the area of the unshaded portion. The heat-source power densities at different positions were calculated by subtracting optical losses and electrical output from the incident power density. Taking the heat-source power density as an input parameter, the temperature distribution at different positions in the PV module was obtained using a thermal simulation method. This procedure enables us to accurately predict the temperatures and locations of hot and cold spots and explain the mechanisms behind them.

The simulation results present an evident “hot-spot” and “cold-spot” phenomenon in the unshaded and shaded areas of the shaded cell. When the shading ratio is 1/2 and the irradiance is 750 W/m², the hot-spot temperature reaches 77.66 °C, which is about 22.5 °C higher than the surrounding cells, while the cold-spot temperature is about 7.5 °C lower than surrounding cells. This is because the unshaded portion not only absorbs additional heat from solar irradiance (645 W) but also generates significantly more Joule heat than the shaded portion (1079.62 vs. 46.98 W/m²). In addition, the temperature of the hot spot initially rises and then decreases with the shading ratio, with the highest temperature occurring around a shading ratio of 1/2. This is because, although the Joule heat generated by the unshaded portion increases with the shading ratio, its heat dissipation to the environment also increases. The results obtained in this work are not only applicable to a single PV module but also to PV strings with multiple modules (with bypass diodes) connected in series, where the hot-spot temperature will be much higher.

The insights from this research are valuable for establishing the relationships between the shading conditions, temperature distribution, and power output of the PV module. This is useful in accurately predicting the power-output capability and monitoring the operational status of PV modules.