# Thermal and Performance Analysis of a Photovoltaic Module with an Integrated Energy Storage System
## 4 Temperature Extrapolation  
In order to determinate the over-temperature limit of PV cells due to the battery placement in the back side of the commercial PV module in different environmental conditions, the proposed thermal models were evaluated by setting different inputs (\(T_{a}\) and \(G_{n}\)) corresponding to the average temperature for each month of the year, at the solar midday. Specifically, the reference conditions are in accordance with an installation of Bologna, Italy (latitude 44\({}^{\circ}\)30\({}^{\prime}\), tilt angle of PV modules 30\({}^{\circ}\)). Concurrently, the temperature of the batteries was determined in order to verify the restrictions given by the manufacturer. Thermal powers \(G_{1}\), \(G_{2}\), \(G_{3}\) and \(G_{4}\) were calculated according to Equation (3). Wind speed was set to 1 m/s for both the front to back sides of PV module (i.e., mixed convection). Table 3 shows the resulting temperatures for each month, in the worst case scenario without electric power conversion (open circuit, \(P_{pv}\) = 0). In particular, for the commercial PV model (without batteries) \(T_{f}\), \(T_{c,pv}\), and \(T_{b}\) represent the front, PV cell, and back temperatures, according to Figure 3. The same temperatures are shown for the two relevant regions of the modified PV module with integrated energy storage: the area covered by the batteries, and the remaining area (not covered by batteries). In this case, the average PV cell temperature \(T_{c,pv+batt}\) was calculated as the weighted average of \(T_{c,pv}\) in the two regions, with respect to the size of the corresponding areas. The temperatures of the batteries were considered as well, both of the backsheet and the ground side (\(T_{batt-b}\) and \(T_{batt-g}\)), according to Figure 3. In the end, the over-temperature \(\Delta T\) was calculated to illuminate the difference in PV cell temperatures between the original commercial PV module and the modified one with the integrated energy storage. Regarding the modified PV module with integrated energy storage, the thermal exchanges related to the area not covered by batteries were treated similarly to the case of the commercial PV module without batteries. In this case, a reduced convective coefficient (decreased to 20%) was taken into account, and a unity view factor for the radiative exchange with the ground was still assumed, but a reduced radiative surface was assumed (decreased to 80%). Table 3 shows that the thermal radiation shield (aluminum plates) causes an important increase in the PV cell temperature in the area covered by batteries, in the order of 20 \({}^{\circ}\)C. However, because of this shield, the over-temperature of the batteries compared to the air was limited to 5-10 \({}^{\circ}\)C, and the temperature of the batteries didn't exceed 40 \({}^{\circ}\)C, far from the limit given by the manufacturer for this type of storage element (i.e., max 50-55 \({}^{\circ}\)C). Table 3 referred to the worst case scenario of open circuit operation. Table 4 shows the temperature in the case of active PV conversion (\(\eta\) = 12%) in order to understand the corresponding decrease, estimated at about 4-5 \({}^{\circ}\)C over the year. In general, with or without PV conversion, the average PV cell temperature over the module area in the case of energy storage exceeds the corresponding temperature of the commercial PV module of 10-15 \({}^{\circ}\)C.  
The last column of Table 4 reports the decrease (in percent) of the electric power produced by the PV module with integrated storage batteries, obtained in consideration of the over-temperature \(\Delta\)T and the temperature-power coefficient (\(-\)0.52%/\({}^{\circ}\)C) for the case study. Corresponding to Table 4, Figure 10 shows the power produced by the PV module with integrated storage batteries (in percent) compared to the power produced by the original PV module without the batteries.