# An Integrated Thermal-Electrical Model for Simulations of Battery Behavior in CubeSats
## 2 Methodology
### Electrical Model
#### 2.6.1 Solar Panel  
The solar panel is composed of several photovoltaic cells, which ideally can be modeled by a diode DI (PN junction) in parallel with a current source \(I_{\text{ph}_{w}}\) (photovoltaic effect) [12]. A practical model includes a bypass resistor \(R_{p_{w}}\) with a series resistor \(R_{s_{w}}\), the first one due to current leakage, spikes diffusion, or other effects. The second is the series resistance, mainly due to metal contacts with the semiconductor. Figure 6 shows the solar cell equivalent circuit, where \(I_{\text{ph}_{w}}\) is the photogenerated current, \(I_{\text{D}_{w}}\) is the inverse saturation current of the diode, \(I_{R_{p_{w}}}\) is the parallel resistor current, \(I_{\text{PV}_{w}}\) is the solar cell current, and \(V_{\text{PV}_{w}}\) is solar cell voltage. As mentioned previously, in this EPS model \(V_{\text{PV}_{w}}\) is the battery voltage \(V_{B}\).  
Figure 5: Simplified electrical power system (EPS) circuit diagram.  
The current \(I_{\text{PV}_{w}}\) for a panel \(w\) is defined with the following equation:  
\[I_{\text{PV}_{w}}(t)=I_{\text{ph}_{w}}(t)-I_{\text{DI}_{w}}(t)-I_{R_{p_{w}}}(t) \tag{43}\]  
\(I_{\text{ph}_{w}}\) can be obtained from the solar panel short circuit current \(I_{\text{sc}_{w}}\), for a given solar irradiance \(Q^{\prime\prime}_{s_{w}}\) (given by Equation (19)), and solar cell temperature \(T_{w}\) obtained with the proposed thermal model. \(I_{\text{ph}_{w}}\) for a panel \(w\) is described as:  
\[I_{\text{ph}_{w}}(t)=I_{\text{sc}_{w}}\frac{Q^{\prime\prime}_{s_{w}}(t)}{Q^{ \prime\prime}_{s_{w}}}[1+\Delta I_{\text{sc}_{w}}(T_{w}(t)-T_{w_{0}})], \tag{44}\]  
where \(Q^{\prime\prime}_{s_{w0}}\) is the solar irradiance and \(T_{w_{0}}\) the temperature for which \(I_{\text{sc}_{w}}\) has been obtained, \(\Delta I_{\text{sc}_{w}}\) is the coefficient that expresses the \(I_{\text{sc}_{w}}\) variation with temperature. These parameters are usually obtained from the solar cell datasheet.  
The diode current is expressed by Equation (45) [24], where \(I_{S_{w}}\) is the saturation current, \(\eta_{w}\) is the ideality factor of the diode, \(k\) being the Boltzmann constant and \(q\) the charge of the electron.  
\[I_{\text{DI}_{w}}(t)=I_{S_{w}}\exp\left(\frac{-q(V_{B}(t)+I_{\text{PV}_{w}}(t) R_{s_{w}})}{\eta_{w}kT_{w}(t)}-1\right) \tag{45}\]  
The saturation current of the diodes \(I_{S_{w}}\) in calculated by Equation (46), where \(V_{q}\) is the voltage of the Crystaline Silicon.  
\[I_{S_{w}}(t)=I_{S_{0w}}\frac{T_{w}(t)^{\frac{3}{\text{ps}}}}{T_{w_{0}}}\exp \frac{qV_{q}}{\eta_{w}k(\frac{1}{T_{w}}(t)-\frac{1}{T_{w_{0}}})}, \tag{46}\]  
where \(I_{S_{0w}}\) is calculated using the open-circuit voltage \(V_{OC_{w}}\) and short circuit current \(I_{SC_{w}}\) of the solar panel at the temperature \(T_{w_{0}}\) (47).  
\[I_{S_{0w}}=\frac{I_{S_{OC_{w}}}(T_{w_{0}})}{\exp\frac{qV_{OC_{w}}(T_{w_{0}})} {\eta_{w}kT_{w_{0}}}-1} \tag{47}\]  
Combining Equations (44), and (45) with Equation (43), the solar cell current is described in terms of circuit parameters, as shown in Equation (48).  
Figure 6: Solar cell equivalent circuit.  
\[I_{\text{PV}_{w}}(t)=I_{\text{Scw}}\frac{Q_{sw}^{\prime\prime}(t)}{Q_{ sw_{w0}}^{\prime\prime}}[1+\Delta I_{\text{Scw}}(T_{w}(t)-T_{w_{0}})]\] \[-I_{\text{Scw}}\exp\left(\frac{-q(V_{\text{B}}(t)+I_{\text{PV}_{w}} (t)R_{\text{Scw}})}{\eta_{wk}T_{w}(t)}-1\right) \tag{48}\] \[-\frac{V_{\text{B}}(t)+I_{\text{PV}_{w}}(t)R_{\text{Scw}}}{R_{p_{w }}}\]  
Notice that in order to solve (48) for \(I_{\text{PV}_{w}}(t)\), one can rely on iterative methods like Newthon-Raphson [33]. The data presented in Table 3 are used in this paper for the solar panel model.