Proposed Models to Improve Predicting the Operating Temperature of Different Photovoltaic Module Technologies under Various Climatic Conditions

: The operating temperature is an essential parameter determining the performance of a photovoltaic (PV) module. Moreover, the estimation of the temperature in the absence of measurements is very complex, especially for outdoor conditions. Fortunately, several models with and without wind speed have been proposed to predict the outdoor operating temperature of a PV module. However, a problem for these models is that their accuracy decreases when the sampling interval is smaller due to the thermal inertia of the PV modules. In this paper, two models, one with wind speed and the other without wind speed, are proposed to improve the precision of estimating the operating temperature of outdoor PV modules. The innovative aspect of this study is two novel thermal models that consider the variation of solar irradiation over time and the thermal inertia of the PV module. The calculation is applied to different types of PV modules, including crystalline silicon, thin ﬁlm as well as tandem technology at different locations. The models are compared to models that are described in the literature. The results obtained in different time steps show that our proposed models achieve better performance and can be applied to different PV technologies.


Introduction
Renewable energy, especially PV energy, has experienced strong growth in recent years. According to the International Energy Agency, solar PV generation increased by 131 TWh in 2019 and represented the second-largest generation growth of all renewable technologies, slightly behind wind energy [1]. With this increase, power generation from PV is approximately 720 TWh and shares almost 3% in global electricity generation. To optimize PV energy yield, besides inventing new technologies and reducing cost, predicting the performance of a PV module under on-site conditions plays an important role in selecting a PV module and installation method.
Various studies showed that the performance of a PV module depends not only on its own properties such as material, glazing-cover transmittance, and plate absorptance but also on the actual weather conditions such as ambient temperature, wind speed, and the spectral distribution of incident irradiance [2][3][4][5]. Even operating under conditions similar to Standard Test Conditions (STC) solar irradiance, the outdoor operating efficiency of multi-crystalline PV was found to be on average 18.1% lower than the given STC efficiency [6]. Typically, every 1 • C of increasing temperature results in a 0.5% reduction in efficiency of crystalline PV module, while this figure of amorphous silicon PV module is 0.27% on average [7].
One of the most important loss factors, which have a negative impact on the final energy produced by a PV system, is the operating temperature of the modules. This loss is represented by the temperature loss coefficient, which depends on the technology of the module as well as on the materials. Notwithstanding that those coefficient values are 1 Selected data from [20]. 2 Selected data from [22][23][24].
The data in Group 1 were used for both training and validating the proposed models, while the data in Group 2 were used for evaluating the accurate prediction of those models on different PV module technologies compared to other literature models. The PV module technologies and their characteristics used in this study are presented in Table 2. Due to the lack of wind measurements, which needs to be measured on-site at the same conditions as other parameters, measurements of the wind speed at the 10-m height [24] and the nearest measured station [23] were applied.

Previous Models
There are many temperature models in the literature, so we have implemented and compared seven existing models presented in Table 3 with our proposed models in this study. In these selected models, some of them are able to predict the module temperature while others calculate the cell temperature directly. For the models predicting cell temperature, which becomes significantly different from the back-surface module temperature in high solar radiation intensities [3], the following relationship was applied: Faiman: PVSyst1: Akhsassi1: Without wind PVSyst2: Akhsassi2: The temperature difference between the PV cell and module back surface was evaluated by King et al. [3] to be about ∆T m = 3 • C at an irradiance level of G g = G 0 for an open-rack installation PV system.
For the first Akhsassi model, the module efficiency and temperature coefficient of maximum power at STC were taken directly from the PVs' characteristics. While the solar irradiance coefficient γ P mp is between 0.03 and 1.12 for silicon and most often simplified to γ P mp = 0 [10,11,25], it was chosen to be 0.04 in this study.

Selected Metrics for Evaluating the Methods
In order to evaluate the differences between the measured (T measured ) and predicted (T predicted ) operating temperature of the PV modules, three statistical metrics have been applied:

•
The correlation coefficient R 2 defines the relationship between the estimated and measured data as the following expression: • The root mean square error, used to evaluate the fluctuations around the model and defined by the expression: As mentioned above, several studies evidently showed that there is no best option, as we could not apply a unique temperature model for all PV module technologies. However, it was clear that the Sandia (King) model is frequently on the top list of matching temperature models. Besides, the Sandia model, unlike other temperature models, only takes on-site parameters such as plane-of-array (POA) irradiance, wind speed, and installation method into consideration. This reduction of input parameters leads to a simple implementation since it does not require many input parameters and a deep understanding of module technologies and materials.
As PV thermal models usually describe the module temperature based on the assumption of thermal equilibrium, the thermal mass of the module has a high impact on the responding time of the changes in the temperature [26]. Moreover, Segado et al. proved that the temperature prediction of a PV module is strongly affected by its own thermal inertia [9]. However, literature studies showed that Sandia is one of the models that is least affected by meteorological conditions. In other words, the changes in its empirical efficiencies are insignificant under different weather conditions and PV technologies [9,18,26]. Nevertheless, it was recognized that this model usually performed well under the low intensity of POA irradiance and less fluctuation of weather conditions; otherwise, it could suffer performance loss in practical implementation.
In this research, therefore, in order to improve the calculation of the module backsurface temperature, a modification of the Sandia model (Equation (2)) by adding a correction term was put forward, based on the idea of the relevance between module thermal inertia and interval sampling, presented by the following Equation: where ∆G g (t) = G g (t) − G g (t − 1) is a dimensional constant describing the difference of POA irradiance between two consecutive sampling points (∆G g = 0 W/m 2 at t = 0); ∆t is the time recording interval in minutes; and ∆T * re f and W re f are empirical coefficients depending on module installation configuration and are equivalent to the coefficients a an b in the Sandia model from Ref. [3] and Equation (2). With ∆T * re f = G 0 e a and W re f = − 1 b , this correction term becomes exactly the wind correction of Equation (2).

Model without Wind
As wind data are not always available for every PV system, there were many models without wind speed were presented to estimate the operating temperature of a PV cell/module. Some of them took into account the manufacturing and installing characteristics [15,27], while others neglected those parameters [11,14]. In this paper, we proposed a model based on the proportion of the POA irradiance to the solar irradiance at STC to calculate the operating temperature of a PV module without using wind velocity. At this suggestion, we assumed that the changing temperature of a PV module is proportional to the varying POA irradiance at operating conditions compared to the laboratory conditions.
To improve the accuracy, the thermal mass of the PV module is once again taken into account, and our proposed model is expressed as the following Equation:

Parametric Identification
The measurements in 2018 from Group 1 in Table 1, which include POA irradiance, air temperature, wind speed, and module temperature, were selected to determine the coefficients of the suggested models. In these locations, the multi-crystalline silicon modules were installed with fixed open cracks; therefore, the coefficients ∆T * re f and W re f in Equation (11) were taken as 28.4 • C and 13.3 m/s, respectively. These values correspond with a = −3.56 and b = −0.075 s/m in the original Sandia model [3].
In the next stage, the best fit for the values y = ∆T re f G 0 f (∆t), describing the relevance of the thermal inertia of the module and the interval sampling, were obtained as a function of the time step ∆t for both models, based on measured data in three cities in 2018. The measurement data are available for every minute, and to determine y based on the time step ∆t, the measurement data of T m and ∆G g are averaged over the duration of the time step ∆t. The time step is limited to 10 min because otherwise, the time dependency is not described in sufficient detail. The resulting values of y and the corresponding correlation (R 2 ) and error (RMSE) between the simulated values of the temperature T m and the measurement data for the proposed model with wind speed are presented in Table 4. The mean y, averaged over the three locations, and the related standard deviation (SD) were also calculated.
The obtained values of y as a function of ∆t for both proposed models are illustrated in Figure 1. The relation between y and ∆t is approximately exponential and a least-squares fit yields R 2 equal to 0.97 and 0.99 for the model with wind speed and the model without wind speed, respectively: y = 0.0175 exp(−0.061∆t) and y = 0.0162 exp(−0.056∆t). These relations can be used to eliminate the function f (∆t) that we have introduced previously. Table 4. Calculated values of y = ∆T re f G 0 f (∆t), correlation (R 2 ), and error (RMSE) between measured and simulated temperature T m for three cities and for different values of the time step ∆t.

Models Comparison
At this stage, the recorded data in 2019 from three locations in Vietnam (Group 1 in Table 1) were taken out for evaluating the proposed models. After removing the samples which were recorded in the nighttime and all incorrect intervals, the total number of sampling points and corresponding time steps are presented in Table 5. Besides comparing experimental measurements and model predictions, a comparison between proposed and literature models was also implemented and presented in Table A1 (see Appendix A). As was to be expected, our proposed model with wind presented the best correlation between calculations and measurements with R 2 greater than 95% compared to other models, while the without-wind proposed model was the best in the group of models without wind speed. Moreover, the proposed model with wind illustrates that it has the most stable performance compared to all other models from the literature, as it is the only one that provided RMSE values less than 3 ℃ for all regions. In The Equation (11), therefore, becomes: where ∆T re f = 18 • C and τ = 16.7 min. The same method was applied for the second model and then the Equation (12) becomes: with ∆T * re f = T 0 = 25 • C, ∆T re f = 16 • C and τ = 16.7 min.

Models Comparison
At this stage, the recorded data in 2019 from three locations in Vietnam (Group 1 in Table 1) were taken out for evaluating the proposed models. After removing the samples which were recorded in the nighttime and all incorrect intervals, the total number of sampling points and corresponding time steps are presented in Table 5. Besides comparing experimental measurements and model predictions, a comparison between proposed and literature models was also implemented and presented in Table A1 (see Appendix A). As was to be expected, our proposed model with wind presented the best correlation between calculations and measurements with R 2 greater than 95% compared to other models, while the without-wind proposed model was the best in the group of models without wind speed. Moreover, the proposed model with wind illustrates that it has the most stable performance compared to all other models from the literature, as it is the only one that provided RMSE values less than 3 • C for all regions. In contrast, the RMSE values for other models changed significantly between the three locations. Moreover, this method also obtained better results for both R 2 and RMSE coefficients at all time steps compared to the original Sandia model.
It was clear that the accuracy of all models was proportional to the interval time. As mentioned above, these models describe the temperature of PV modules under the assumption of their thermal equilibrium. Therefore, if this condition is not satisfied, then large differences can result. Moreover, it is known that the time response to changes in the temperature depends on the thermal mass of the module and the prevailing conditions [26], so that if the time resolution of the data is shorter than the typical thermal response times, the performance of the models will not be as good.
The difference between the predicted and measured values of the PV module backsurface temperature, ∆T, is illustrated in Figure 2. As we can see in the picture, the calculation for Sandia is proportional to the magnitude of fluctuating POA between two consecutive sampling points. The larger the variation, the greater the difference is. While the corresponding values for our proposed models tend to approach zero and are evenly distributed around zero across the range of ∆G g . contrast, the RMSE values for other models changed significantly between the three locations. Moreover, this method also obtained better results for both 2 and coefficients at all time steps compared to the original Sandia model.
It was clear that the accuracy of all models was proportional to the interval time. As mentioned above, these models describe the temperature of PV modules under the assumption of their thermal equilibrium. Therefore, if this condition is not satisfied, then large differences can result. Moreover, it is known that the time response to changes in the temperature depends on the thermal mass of the module and the prevailing conditions [26], so that if the time resolution of the data is shorter than the typical thermal response times, the performance of the models will not be as good.
The difference between the predicted and measured values of the PV module backsurface temperature, ∆ , is illustrated in Figure 2. As we can see in the picture, the calculation for Sandia is proportional to the magnitude of fluctuating POA between two consecutive sampling points. The larger the variation, the greater the difference is. While the corresponding values for our proposed models tend to approach zero and are evenly distributed around zero across the range of ∆ . We also took into account the implementation of temperature models at different POA levels, which strongly affected the accuracy of the predicting process. The results displayed in Figure 3 demonstrate that the accuracy of the temperature models will decrease when the radiation intensity reaches the module surface increase. This circumstance occurred because the fluctuations among high POA values happen more frequently than those of low POA levels. Moreover, the greater POA the module gets, the higher the temperature of the module will be. Subsequently, the thermal inertia of the PV modules will have a stronger effect when POA increases, which explains why our proposed models perform better than literature models at high POA levels but remain equivalent when POA levels are low. We also took into account the implementation of temperature models at different POA levels, which strongly affected the accuracy of the predicting process. The results displayed in Figure 3 demonstrate that the accuracy of the temperature models will decrease when the radiation intensity reaches the module surface increase. This circumstance occurred because the fluctuations among high POA values happen more frequently than those of low POA levels. Moreover, the greater POA the module gets, the higher the temperature of the module will be. Subsequently, the thermal inertia of the PV modules will have a stronger effect when POA increases, which explains why our proposed models perform better than literature models at high POA levels but remain equivalent when POA levels are low. Figure 4 showed that the on-site wind speed of all regions remained stable during the observed period, while the average POA irradiance occurred in different ways. The POA irradiance in Tri An is distributed across the whole range of values, while the values of that parameter in Ha Noi are mainly concentrated below 500 W/m 2 . Moreover, Figure 4c illustrates that the distribution of the difference between two consecutive sampling points of wind is almost the same for all regions with variations between 0 and 3 m/s. While the largest fluctuation of POA irradiance occurred in Tri An, and the smallest fluctuation of that happened in Ha Noi. This means that the effect of thermal inertia on the thermal prediction in Tri An is greater than for the other regions due to its high intensity and frequent fluctuation of POA irradiance. Consequently, our proposed model with wind obtained the best result in Tri An where the on-site conditions fluctuated most significantly, while the proposed model without wind gained better performances in Tri An and Ha Noi, where the wind was less important than in Da Nang. Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 15 (a) (b)  Figure 4 showed that the on-site wind speed of all regions remained stable during the observed period, while the average POA irradiance occurred in different ways. The POA irradiance in Tri An is distributed across the whole range of values, while the values of that parameter in Ha Noi are mainly concentrated below 500 W/m 2 . Moreover, Figure  4c illustrates that the distribution of the difference between two consecutive sampling points of wind is almost the same for all regions with variations between 0 and 3 m/s. While the largest fluctuation of POA irradiance occurred in Tri An, and the smallest fluctuation of that happened in Ha Noi. This means that the effect of thermal inertia on the thermal prediction in Tri An is greater than for the other regions due to its high intensity and frequent fluctuation of POA irradiance. Consequently, our proposed model with wind obtained the best result in Tri An where the on-site conditions fluctuated most significantly, while the proposed model without wind gained better performances in Tri An and Ha Noi, where the wind was less important than in Da Nang.   Figure 4 showed that the on-site wind speed of all regions remained stable during the observed period, while the average POA irradiance occurred in different ways. The POA irradiance in Tri An is distributed across the whole range of values, while the values of that parameter in Ha Noi are mainly concentrated below 500 W/m 2 . Moreover, Figure  4c illustrates that the distribution of the difference between two consecutive sampling points of wind is almost the same for all regions with variations between 0 and 3 m/s. While the largest fluctuation of POA irradiance occurred in Tri An, and the smallest fluctuation of that happened in Ha Noi. This means that the effect of thermal inertia on the thermal prediction in Tri An is greater than for the other regions due to its high intensity and frequent fluctuation of POA irradiance. Consequently, our proposed model with wind obtained the best result in Tri An where the on-site conditions fluctuated most significantly, while the proposed model without wind gained better performances in Tri An and Ha Noi, where the wind was less important than in Da Nang.  The back-surface temperature of the PV module that is calculated for three regions by our proposed model with wind effect (Equation (13)) using optimized values of ∆ * and coefficients is presented in Table 6. By applying these best-fitted coefficients, the obtained values were decreased significantly to below 2 ℃ for all regions. The back-surface temperature of the PV module that is calculated for three regions by our proposed model with wind effect (Equation (13)) using optimized values of ∆T * re f and W re f coefficients is presented in Table 6. By applying these best-fitted coefficients, the obtained RMSE values were decreased significantly to below 2 • C for all regions. Table 6. Correlation coefficients (R 2 and RMSE) for measured and calculated module temperatures for three regions in Vietnam using optimal coefficients (5-min time step).

Coefficients
Implementing Results

Tri An Da Nang Ha Noi
∆T

Comparison between Different PV Technologies
Our proposed models were examined afterwards with different module technologies, which were deployed in three locations in the USA. The obtained results for all regions are shown in Table A2 (see Appendix A), while a visual presentation for Cocoa is depicted in Figure 5. As we can see, our proposed model with wind obtained the best compatibility of predicting module temperature since it showed the highest correlation coefficient R 2 and the lowest RMSE, whilst the other proposed model has the best performance among the models that ignore the influence of the wind, with highest R 2 and lowest RMSE values calculated for all PV technologies and locations.
The back-surface temperature of the PV module that is calculated for three regions by our proposed model with wind effect (Equation (13)) using optimized values of ∆ * and coefficients is presented in Table 6. By applying these best-fitted coefficients, the obtained values were decreased significantly to below 2 ℃ for all regions.

Comparison between Different PV Technologies
Our proposed models were examined afterwards with different module technologies, which were deployed in three locations in the USA. The obtained results for all regions are shown in Table A2 (see Appendix A), while a visual presentation for Cocoa is depicted in Figure 5. As we can see, our proposed model with wind obtained the best compatibility of predicting module temperature since it showed the highest correlation coefficient R 2 and the lowest RMSE, whilst the other proposed model has the best performance among the models that ignore the influence of the wind, with highest R 2 and lowest RMSE values calculated for all PV technologies and locations.
Previous studies have reported that it is difficult to apply a single model or a unique formula to precisely calculate the PV module/cell temperature [9,11,18,19]. Moreover, the thermal characteristics of PV modules are slightly different even if they are manufactured with the same technology and materials [12,13]. However, our proposed models have demonstrated that they can work well with different PV technologies and weather conditions without adjusting the empirical coefficients.  Table 7 shows the optimal coefficients, ∆ * and in Equation (13), and corresponding implementing results which were calculated for various PV technologies in three different locations. As we can see, the operating temperature of CdTe and CIGS Previous studies have reported that it is difficult to apply a single model or a unique formula to precisely calculate the PV module/cell temperature [9,11,18,19]. Moreover, the thermal characteristics of PV modules are slightly different even if they are manufactured with the same technology and materials [12,13]. However, our proposed models have demonstrated that they can work well with different PV technologies and weather conditions without adjusting the empirical coefficients. Table 7 shows the optimal coefficients, ∆T * re f and W re f in Equation (13), and corresponding implementing results which were calculated for various PV technologies in three different locations. As we can see, the operating temperature of CdTe and CIGS modules were least affected by on-site POA, with ∆T * re f is 31.25 ± 0.45 • C and 30.65 ± 0.75 • C, respectively. Silicon PV module operating temperatures, on the other hand, were influenced significantly by local POA irradiance conditions, with ∆T * re f equal to 28.6 ± 1.3 • C and 27.65 ± 1.95 • C, respectively, for xSi and mSi modules. Table 7. Correlation coefficient R 2 and RMSE between measured and calculated module temperatures for different PV technologies using optimal coefficients.

Conclusions
As the temperature has a significant effect on the electrical efficiency of solar cells, it is necessary to accurately estimate the photovoltaic module/cell temperature when predicting the energy yield. In this framework, we proposed two new models in order to calculate the PV module temperature. One model with the wind was developed based on the Sandia model by considering the effect of the thermal inertia of the module, whilst the other was determined without taking into account the wind speed. These proposed models have been compared to other models in the literature by using the statistical coefficients R 2 and RMSE. The results show that the inclusion of the thermal inertia of the PV module and the wind speed with the given models allow making a more accurate estimation of the module temperature, with R 2 correlation above 95% and the RMSE below 3 • C for most PV technologies.
As they do not need many input parameters and are simple to implement, these proposed models have demonstrated that they also work well under different environmental and operating conditions and PV technologies. The proposed model with wind, Equation (13), can be used to calculate PV module temperature with high precision for any location where wind data are available; otherwise, Equation (14) can be applied. These thermal models are useful for evaluating the thermal performance of a PV module under different environmental and operating conditions since they have proven their reliability when being examined in different locations and climates.
When climatological information such as wind speed and solar illumination are available in a time series for a certain location, our models make it possible to estimate the module temperature for different types of solar cells. The temperature information is essential in estimating the PV module efficiency and in deciding which kind of PV panel will be most economical for the given location.