# Thermal and Performance Analysis of a Photovoltaic Module with an Integrated Energy Storage System
## 2 Thermal Model of a Single PV Module
### Models  
For a PV module, the steady-state thermal balance can be written as [9]:  
\[0=GA-P_{pv}-Q_{tot},\ \text{being}\ \left\{ \begin{array}{l}G=G_{n}\cdot A\ -G_{ rif}\cdot A\\ G_{rif}=\rho\cdot G_{n}\\ P_{pv}=\eta\cdot G_{n}\cdot A\\ Q_{tot}=Q_{conv,f}+Q_{rad,f}+Q_{conv,b}+Q_{rad,b}\end{array} \right. \tag{1}\]  
With reference to the thermal balance (Equation (1)), the heat exchanges of the commercial PV module are described in Figure 2a: \(G_{n}\) is the specific radiation incident on the module's surface (W/m\({}^{2}\)); \(G_{n}\) must be multiplied by the area of the PV module \(A\) (m\({}^{2}\)) in order to get the total incident power. \(P_{pv}\) (W) is the electrical power generation, proportional to the total incident power, and \(\eta\) is the photovoltaic conversion efficiency. \(G_{rif}\) is the total radiative power reflected from the surface of the PV module that is proportional to the reflection index \(\rho\). The \(Q\) terms [W] consider convective \(Q_{conv}\) and radiative \(Q_{rad}\) exchanges of the front (\(f\)) and back (\(b\)) sides of the PV module with the surrounding environment. As for the commercial PV module, the heat exchanges of the PV module with an energy storage system are shown in Figure 2b. By introducing flat batteries on the back side of the PV module it is evident that both convective and radiative heat exchanges were limited. In particular, radiative exchange was limited due to the reflection of infrared (IR) rays by the aluminum flat plates (having extremely low emissivity). The PV module backsheetemited IR rays, proportional to its temperature (T\({}^{\text{\text{\text{\text{\text{th}}}}}}\)). The aluminum plate acted as a thermal shield, reducing overheating of the batteries. However, the IR rays reflected from the backsheet increased the temperature of the PV module. In addition, the presence of batteries led to a restriction in the convective heat exchange, even though air circulation was still possible through the airgap (i.e., almost 2 cm). The decrease of the radiative and conductive exchanges of the backsheet due to the batteries led to higher backsheet and PV cell temperatures compared to commercial PV modules. In the thermal models under consideration, the following assumptions have been made:  
* One-dimensional (1D) thermal models. * Isothermal surface was approximated as a flux node, so edge effects were neglected. Figure 2: Heat exchanges of PV module: (**a**) without battery storage system; (**b**) with storage battery system. \(G_{n}\) is the specific radiation incident on the module’s surface (W/m\({}^{2}\)); \(P_{pv}\) (W) is the electrical power generation, proportional to the total incident power; \(\eta\) is the photovoltaic conversion efficiency; \(G_{rif}\) is the total radiative power reflected from the surface of the PV module that is proportional to the reflection index \(\rho\). Negligible thermal capacitances [10]. * Material properties of PV module layers were constant, as shown in Table 1.
* PV cell temperature was considered uniform due to the higher value of its thermal conductivity (\(k\)) compared to any of the other layers. * Apparent sky temperature \(T_{s}\) was calculated according to [11]: \[T_{s}=T_{a}-\delta T\] (2) where \(T_{a}\) is the air temperature and \(\delta T\) is the variation depending on atmospheric conditions. * Ground temperature (on the back side of the PV module) was equal to air temperature (\(T_{g}=T_{a}\)). * Convective heat exchange coefficients were evaluated with empirical formulations and it was assumed that wind flowed around the module, both front and back sides [12]. * Reflection (\(\rho\)), transmission (\(\tau\)), and absorption (\(\alpha\)) coefficients were independent from temperature. * Internal reflection phenomena between the layers of PV module were neglected. * Emissivity of surfaces were independent from temperature and wavelength, values are given in Table 2.
* According to the radiative heat transfer, the view factor was assumed to be unity.