Development and improvement of a transient temperature model of PV modules: Concept of trailing data

The development of a transient temperature model of photovoltaic (PV) modules is presented in this paper. Currently, there are a few steady‐state temperature models targeted at assessing and predicting the PV module temperature. One of the most commonly used models is the Faiman thermal model. This model is derived from the modified Hottel‐Whillier‐Bliss (HWB) model for flat‐plate solar‐thermal collector under steady‐state conditions and assumes low or no thermal mass in the modules (i.e., short time constants such that transients are neglected, and steady‐state conditions are assumed). The transient extension of the Faiman model we present in this paper introduces a thermal mass, which provides two advantages. First of all, it improves the temperature prediction under dynamic conditions. Second, our transient extension to the Faiman model allows the accurate parametrization of the Faiman model under dynamic conditions. We present our model and parametrization method. Furthermore, we applied the model and parametrization method to a 1‐year data set with 5‐min resolved outdoor module measurements. We demonstrate a significant improvement in temperature prediction for the transient model, especially under dynamic conditions.


| INTRODUCTION
The accurate determination of the operating temperature of PV modules, especially operated under outdoor conditions, is of key importance in the field of photovoltaics.0][11][12][13][14] Moreover, two important quantities in photovoltaics: the Climatic Energy Rating of PV modules (CSER) according to International Electrotechnical Commission (IEC) 61853-3 standard series 15 and the Nominal Module Operating Temperature (NMOT), 16,17 require an accurate module temperature prediction.These quantities depend heavily on the standard Faiman temperature model.The Faiman model is widely used as it is very simple, using linearized effective heat loss, and for many applications adequate. 4,7,15,18wever, the model neglects the thermal mass of the PV module and thus neglects transients.This makes the Faiman model, like every other steady-state model, inaccurate under rapidly changing conditions. 11,19,20Furthermore, the inability to accurately describe temperature transients hampers model parametrization as measured data must be filtered to only select those datapoints where the conditions were sufficiently stable to assume a steady-state temperature. 16ne way to overcome this limitation of the Faiman model is to extend it to a transient thermal model.1][22][23][24][25][26] However, these models are generally much more complex than the Faiman model, distinguishing between convective and radiative heat loss and applying nonlinear heat loss terms.

| Data description
The dataset used in this work was collected from a PV test facility in Widderstall, in the south of Germany.The data set includes the module temperature (T m ) and weather data.The weather data consists of the plane of array (POA) irradiance (G), wind speed (w), and rack ambient temperature (T a ).The data have a temporal resolution of 5 min, collected over the year 2015.Exploratory data analysis was conducted on the dataset to filter out missing sensor data.The resulting data set consists of 78,777 complete measurements.

| Parametrization of Faiman model
The Faiman model in its original form 27 is given by the interdependence of the module temperature T m on the ambient temperature T a given by where T m (K) is the module temperature, T a (K) is the ambient temperature, G (W/m 2 ) is the irradiance, w is the wind speed, and u 0 (W/[m 2 K]) is a convective heat loss term, and u 1 (Ws/[m 3 K]) describes the wind dependent forced convection.
Usually, the model is parameterized using linear least squares fit of According to the IEC 61853-2 standard, 28 prior to the linear fitting to determine the model parameters, a minimum of 10 days high frequency measurements taken on a suitably clear day is usually required.Furthermore, the input data are preprocessed by rejecting certain data points to achieve steady-state conditions as much as possible.Consequently, the following filtering procedure is followed 28 : • Data points with irradiance values below 400 W/m 2 are rejected.
• Irradiance values fluctuating more than ±10% from the maximum to the minimum value within 10 min interval are also rejected.
• Instantaneous windspeed data with a deviation below 0.25 m/s or wind gusts above +200% from a 5 min running average within 10 min interval are rejected.
• After rejecting gusts and low wind speeds, all data when the 5 min running average is less than 1 m/s or greater than 8 m/s are also rejected.
Unfortunately, this data filtering procedure poses a strict reduction in the number of usable data points for analysis.In fact, other researchers have clearly noted that only about 0.3% of data points are usable after filtering. 16Furthermore, wind speed ranges with higher point density exert more weights in the regression, leading to a lower regression accuracy. 16 we cannot strictly follow the Faiman parametrization procedure according to the IEC norm due to the low resolution of our dataset, we parameterize the Faiman model using an equivalent form given by This equivalent form is more robust to numerical errors in case (T m À T a ) is small.The parameters determined from this approach as well as the predicted temperature plots are presented in Figure 1.
Note that by applying linear least squares on Equations ( 2) and (3), the parameters u 0 and u 1 are determined such that the errors in the right-hand side of the equations are minimized.It should be noted that this is generally not identical to the solution where the error in the module temperature is minimized.

| Transient model development
We propose the following differential equation for a transient temperature model: where C a (J/[m 2 K]) is the area-specific thermal capacitance; that is, it is the heat capacity per unit area, and all other parameters are as earlier defined.
Under the assumption that the irradiance, wind, and ambient temperature do not vary, this differential equation has an analytical solution of the form where Also, under steady-state conditions, Equation (5) reduces to Equation ( 6), with T m = T ss , where T ss is the module temperature at steady-state.Note that Equation ( 6) is the Faiman equation as in Equation ( 1), with T m = T ss .Thus, a parametrization of the Faiman model is directly obtained from our model parameters.

| Parametrization approach
We parameterize our transient model from a time series of data points (G i , T a,i , w i , and T m,i ) to determine the parameters, u 0 , u 1 , and C a , by considering time intervals between consecutive data points.Consequently, we employ the finite difference approach while assuming constant conditions within an interval [t i , t i + 1 ].A good choice for the constant value would be the first order approximation of the value in the center of each interval: , where X is substituted with one of the quantities G, T a , w, T m , and i is the time series measurement index (i = 1 … N), and t i < t i + 1 for all i, and N is the number of measurements in the time series.Hence, we linearize Equation ( 4) to obtain Equation (8) as To compute the temperature at a specific time, we consider the thermal history of the module by trailing back in time.Here, we simply assume that we have a steady-state condition at t-Δt t minutes, where Δt t is the trailing time.We numerically integrate Equation (8) up to the desired time to get the module temperature, T m (t).The simplified process flow is as follows: I. ) At t-Δt t minutes, we assume steady-state conditions, that is, II. ) We compute the temperature for the next time step (T m,i + 1 ), using ).The final module temperature at time t is obtained by integrating N time steps using Equation ( 9), where 3 | RESULTS AND DISCUSSIONS

| Faiman model results
The ordinary least squares fitting was done using the equivalent equation of the Faiman model in Equation ( 3), for all the dataset, which yielded the Faiman parameters, u 0 and u 1 as 29.66 Wm À2 K À1 and 5.64 Wm À3 sK À1 , respectively.These values are comparable to those obtained by other authors in the literature. 16,27The Faiman parameters are subsequently used in Equation ( 1) to predict the module temperature, and the plots are presented in Figure 1.
Figure 1a shows a scatter density plot for the measured versus the Faiman predicted module temperature (T m ).The color scale depicts the density of points in the scatter plot (note the logarithmic scale).Large density of points especially at low temperatures is reasonably well predicted by the model.However, for these points, the irradiance is generally low, and thus, the module temperature is close to the ambient temperature.At higher temperatures, the scatter fans out a bit.Nevertheless, the plot clearly depicts that even without data filtering and other extraneous data preprocessing, the Faiman model works fairly well.The coefficient of determination, R 2 , and the rootmean-square error, RMSE, are 0.9610 and 2.19 K, respectively.We expect the Faiman model to be less accurate under dynamic conditions.To show this, we filter our dataset by selecting datapoints where the magnitude of ΔT m for the preceding interval is greater than 2 C (jΔT m j > 2 C); that is, we select those points in the dataset where the temperature is rapidly changing.The predicted versus the measured module temperatures for the thus filtered dataset is shown in Figure 1b.It is clearly seen that during these dynamic moments, the model is much less accurate with an R 2 value of 0.5200 and an RMSE of 6.72 K, which is more than a factor 3 larger as compared to the RMSE for the complete dataset.This is seen in the large scatter of points in Figure 2b with only very few points lying on the regression line.

| Transient model results
In the next step, we parameterize the transient model.In principle, we can use an ordinary least squares (OLS) approach to find the coefficients C a , u 0 , and u 1 from Equation ( 8). Figure 2a shows the measured (ΔT meas ) and predicted (ΔT pred ) temperature difference for subsequent measurements in the time series (ΔT m = T m,i + 1 -T m,i ), where the parameters are obtained with OLS.However, we observe that the variance of the residuals is not constant across the ΔT meas .
When non-constant variance of the residuals is observed, it indicates that the variability of the errors is not consistent throughout the data range (in ΔT meas ).Moreover, this condition represents a violation of the assumption of homoscedasticity, a critical assumption in linear regression.[31] This problem arises from a rather unbalanced dataset where the majority of ΔT m values are close to 0. To overcome this, we apply a weighted least square (WLS) regression.The weights are obtained by binning the ΔT m /Δt values into 100 equally spaced bins and set weights inversely proportional to the number of data points per bin.
The results of the WLS fit on all the dataset are shown in Figure 2b.
From the parametrization procedure in Equation ( 8 1.As compared to the previous direct parametrization of the Faiman model, we observe the new u 0 is slightly smaller (was 29.66 Wm À2 K À1 ), whereas the new u 1 is slightly larger (was 5.64 Wm À3 sK À1 ).In any case, the exact values of these parameters are no more important than their impact on the annual energy yield for a specific location. 28rst, we compare the Faiman model parameterized from the transient model coefficients.In Figure 3a, we show the predicted versus measured module temperature using this alternative parametrization of the Faiman model.The results are nearly identical to Figure 1a.The coefficient of determination and root mean square error are also nearly the same (R 2 = 0.9598 and an RMSE = 2.23 K).
The results for the transient model are shown in Figure 3b, where we use a trailing time of 15 min.Here, we observe a better temperature prediction using the transient model, as compared to the Faiman model in Figure 3a.The point cloud gets narrower with more density of data points, which lie about the regression line.The model accuracy improves to having an R 2 value of 0.9786 and RMSE value of 1.63 K. Quantitatively, this RMSE value has been noted in the literature to be comparable to cell-to-cell temperature within any given module. 4,27Moreover, in terms of power rating of commonly available PV modules, such as HIT and c-Si modules with typical temperature coefficients of about À0.3%/K and À0.5%/K, respectively, the RMS error would translate to only 0.48% or 0.82% uncertainty in power.
Next, we evaluate the transient model performance under dynamic conditions.Figure 4 shows plots of filtered datasets for large transients, that is, for moments where the absolute module temperature difference is larger than 2 C (jΔT m j > 2 C). Figure 5a shows the results for the complete dataset and Figure 5b for the dynamic conditions only.In both figures, we observe that the accuracy improves rapidly up to a trailing time of about 15 min.Trailing further in the past, the less the module temperature is correlated with the current temperature.So if we integrate from a point far in the past, then we get a better temperature estimate.
From Equation ( 5 In this work, we propose a simple transient extension of the Faiman model with only one extra parameter for the thermal mass of the module.The model is compatible with the Faiman model such that the transient extension may be used to obtain a parametrization for the original steady-state model.This allows us to parameterize both the Faiman model as well as our extension thereof, without the usual data filtering to reject datapoints where the conditions are too dynamic.We demonstrate that our model provides a significant improvement in temperature prediction over the Faiman model, in particular under dynamic conditions.In Section 2, we briefly present the source and composition of the dataset used in the model development and the parametrization approach of the Faiman model.In Section 3, our proposed model and its parametrization are introduced.We compare our transient model with the Faiman model for all the datasets and further show the significant improvement realized by our model when we focus only on dynamical conditions.Finally, we summarize the key findings of our work in Section 4.

1
Scatter density plots for the predicted temperature (T pred ) versus the measured temperature (T meas ).The color of the scatter points shows the estimated density of scatter points in logarithmic scale.(a) the Faiman model prediction for the whole dataset.(b) Model prediction for a filtered dataset where jΔT m j > 2 C, that is, select only those points where two subsequent temperature measurements show a large difference.
), we obtain the three parameters C a , u 0 , and u 1 , from the weighted least squares fit as 19.57kJ m À2 K À1 , 26.31 Wm À2 K À1 , and 7.93 Wm À3 sK À1 , respectively.The results of the Faiman parameters obtained from our parametrization approach, as well as from the transient model and (B) (A) F I G U R E 2 Scatter density plots showing the predicted (ΔT pred ) versus the measured (ΔT meas ) module temperature difference from the transient model parametrization fits using (a) OLS and (b) WLS regression.The solid line is the line of best fits or the identity line for which the modeled and measured ΔT m are equal, and the color of the scatter points indicates the estimated probability density of the scatter points on a logarithmic scale.T A B L E 1 Values of Faiman parameters derived from equivalent equation, transient model, and published literature.

F
I G U R E 3 Scatter density plots of the predicted temperature (T pred ) versus the measured temperature (T meas ) for the transient model on all dataset (a) for trail = 0 min (Faiman equivalent) (b) trail = 15 min.how they compare with literature published values, are shown in Table

Figure
Figure 4a-c shows the improvement of our transient model from its steady-state counterpart in Figure 1b.In Figure 4a, we show the results without trailing data, which produces more or less identical results as before in Figure 1b, while Figure 4b,c shows results for 5-, 10-, and 15-min trailing data.It is observed that with an increasing trailing time, the model gets more accurate.Under these dynamic conditions, we can reduce the RMSE from 6.69 to 3.33 K. Similarly, in Figure 5, we show the development of the coefficient of determination and the RMSE, as a function of the trailing time.

4
), we see the time constant τ, for the transients equals Ca u0þu1w .Using this expression, we compute the average time constant over the entire dataset and obtain 9.74 min.Thus, a 15-min trailing time corresponds to about 1.7 time constants.Note that the time constant for a PV module may vary depending on the module type, mounting construction, and typical wind speeds.Our dataset is for a rack mounted glass module with a back sheet.Scatter plots showing the results of the transient model for (a) trail = 0 min (Faiman equivalent for filtered data set), (b) trail = 5 min, (c) trail = 10 min, and (d) trail = 15 min.