## 3 Electrothermal and Lifetime Model Explanation for LiCs
### Electrical and Thermal Model  
The electrical and thermal models as shown in Figure 4, are fully dependent on each other.  
The electrical part of the above-mentioned model is a first order electric circuit as shown in Figure 5. The parameters values (\(OCV,~{}R_{O},~{}C_{P},~{}\text{and}~{}R_{P}\)) are extracted during experiments at different conditions, including different temperatures, states of charge (SoC), and current rates. The terminal voltage in the first order electric circuit is calculated based on the Equation (6):  
\[V_{t}=OCV(SoC)-I_{L}R_{0}-V_{cp} \tag{6}\]  
where \(R_{o}\) is the ohmic resistance, \(R_{P}\) is concentration polarization resistance, \(OCV\) is the open circuit voltage, \(I_{L}\) is the LiC current, \(I_{CP}\) is the current in the \(C_{P}\) branch, \(I_{RP}\) is the current in the \(R_{P}\) branch, and \(V_{t}\) is the terminal voltage of the LiC.  
Figure 4: The block diagram of an electrothermal model [20].  
Figure 5: First-order electrical model.  
All parameters shown in Figure 5 are temperature-dependent. In order to have a more accurate estimation, the surface temperature of the cell is modeled with a first-order electric circuit. As shown in Figure 6, the heat source, initial temperature, and ambient temperature are essential. The heat source represents power loss in the cell and arises from two references:  
1. internal resistance (\(R_{int}=\ R_{o}+\ R_{p}\)), which is called \"irreversible heat source\"; and
2. the entropy changes, which is called \"reversible heat source\".  
The first one is always positive and is due to the ion and electron movement, while the second one can be positive or negative and is generated in the chemical reaction. The power loss is calculated by Equation (7) [20]:  
\[P_{l}=\ l_{cell}^{2}\ (R_{int}(I_{cell},SoC))-\ I_{cell}T_{cell}\frac{\partial V_{ OCV}(T_{cell,}\ SoC)}{\partial T_{cell}} \tag{7}\]  
where the \(I_{cell}\) is the cell's current, \(R_{int}\) represents the entire internal resistance, and \(T_{cell}\) is the cell's surface temperature. The circuit components shown in Figure 6 are described as follow: the \(P_{\bar{g}}\) is the power loss generated in the cell, \(C_{th}\) is the thermal capacity of the cell, \(R_{th}\) represents the thermal conductivity resistance from the cell's center to the surface, \(R_{con}\) is the conductivity resistance from the surface to the ambient, and \(T_{a}\) is the ambient temperature.  
Finally, in order to calculate the power loss for charge and discharge conditions, the power loss is calculated by considering the proportional coefficients (\(\alpha\) and \(\beta\)) as shown in Equation (8):  
\[P_{Itot}=\ \alpha P_{clh}+\ \beta P_{diss} \tag{8}\]  
which \(P_{clh}\) represents the power loss during the charge process and \(P_{diss}\) gives the power loss during the discharge phase. It has been observed in the experiments that LiCs absorb heat while charging and release heat while discharging. However, the absorbed heat and generated heat are not equal when a same value of current is applied. Moreover, at higher value of charge and discharge current, the exothermic part dominates the endothermic one. As shown in Equation (8), \(\alpha\) and \(\beta\) are used to consider this phenomenon. In this equation, \(\alpha\) represents the portion of the total heat source that is generated during the charge process and is mostly negative, and \(\beta\) donates the portion of the total heat source during the discharge times and is always positive. These parameters are current-dependent and are estimated by applying dynamic and steady state load profiles to the cell at different ambient temperatures. A non-linear least square technique is used for estimation. These values are given in Table 1 and are detailed in [20]. \(P_{tot}\) is the total heat source (\(P_{\bar{g}}\)) in the presented model in Figure 6.  
Figure 6: First-order electric circuit for the thermal part of the electrothermal model [20].