## 2 Mathematical Modelling  
A coupled electro-thermal-aging model is the base of the analysis for LiFePO\({}_{4}\) batteries [34]. The model consists of a two-state thermal model, a two RC pair electrical model, and a semi-empirical aging model. The electrical model of LiFePO\({}_{4}\) batteries contains an open-circuit voltage (OCV, \(V_{OC}\)), an ohmic resistor (\(R_{0}\)), and two resistor-capacitor (RC) pairs (\(R_{1}\), \(C_{1}\), \(R_{2}\), \(C_{2}\)). The state-space model is given by:  
\[\frac{\text{d}SOC}{\text{d}t}(t)=\frac{I(t)}{C_{bat}} \tag{1}\]  
\[\frac{\text{d}V_{2}}{\text{d}t}(t)=-\frac{V_{2}(t)}{R_{2}C_{2}}+\frac{I(t)}{C _{2}} \tag{2}\]  
\[V_{t}(t)=V_{OC}(SOC)+V_{1}(t)+V_{2}(t)+R_{0}I(t) \tag{3}\]  
where \(I(t)\) is the current (positive for charging), \(C_{bat}\) is the battery nominal capacity, and \(V_{1}(t)\) is the terminal voltage. Three state variables are \(SOC\) and voltages across the two RC pairs \(V_{1}\), \(V_{2}\). The identification of the electrical parameters is in [35]. For this model, the listed equations in Appendix A would be followed for the derivation of these parameters based on the SOC (\(I<0\)) or discharge (\(I\geq 0\)). As the core temperature could be greater than the surface one under high current rates [35], a two-state thermal system was hereby introduced to capture both core and surface temperature dynamics. The model of the radial heat transfer dynamics of a cylindrical battery could be as follows:  
\[\frac{\text{d}T_{c}(t)}{\text{d}t}=\frac{T_{s}(t)-T_{c}(t)}{R_{c}C_{c}}+\frac{Q (t)}{C_{c}} \tag{4}\]  
\[\frac{\text{d}T_{s}(t)}{\text{d}t}=\frac{T_{f}(t)-T_{s}(t)}{R_{u}C_{s}}+\frac{ T_{s}(t)-T_{c}(t)}{R_{c}C_{s}} \tag{5}\]  
\(C_{c}\), \(C_{s}\), \(R_{c}\), and \(R_{u}\) represent the core heat capacity, surface heat capacity, heat conduction resistance, and convection resistance, respectively. Table 1 shows their values. The ambient temperature \(T_{f}\) is treated as uncontrollable input; the two-state variables are surface temperature \(T_{s}\) and core temperature \(T_{c}\). \(Q(t)=|I(V_{\text{OC}}-V_{t})|\) is heat generation including energy dissipated by electrode overpotentials and joule heating. Based on Equations (3) and (4), this could be rewritten as:  
\[\frac{\text{d}T_{c}(t)}{\text{d}t}=\frac{T_{s}(t)-T_{c}(t)}{R_{c}C_{c}}+\frac {I(t)(V_{1}(t)+V_{2}(t)+R_{0}I(t)}{C_{c}} \tag{6}\]  
The aging model is based upon a matrix of cycling tests from [34]. Results of the experiment suggest that capacity fade depends strongly on temperature and C-rate in the cell at low charge/discharge rates while neglecting the sensitivity to depth-of-discharge. The semi-empirical life model adopted the next equation, which describes the correlation between the capacity loss (\(\Delta Q_{b}\), in%) and the discharged ampere-hour (Ah) throughput (\(A\), the discharge current, depends on the rated capacity (C-rate)):  
\[\Delta Q_{b}=M(c)\exp\left(\frac{-E_{a}(c)}{RT_{c}}\right)A(c)^{z} \tag{7}\]  
where \(M(c)\) is the pre-exponential factor as a function of the rated capacity, which is denoted by \(c\), and \(E_{a}\) is the activation energy. \(R\) is the gas constant, which equals 8.3145 J mol\({}^{-1}\) K\({}^{-1}\). Table 2 shows the C-rate and \(M(c)\) relation. The power-law factor \(z\) and the activation energy \(E_{a}\) are given by:  
\[E_{a}(c)=31,700-370.3c\ \&\ z=0.55 \tag{8}\]  
This model considers a capacity loss of 20% for an automotive battery as the end-of-life (EOL). The corresponding Ah throughput \(A_{tol}\) and the number of cycles \(N\) are, therefore, computed as:  
\[A_{tol}(c,T_{c})=\left[\frac{20}{M(c)\exp\left(\frac{-E_{a}(c)}{RT_{c}}\right) }\right]^{\frac{1}{z}} \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \(R_{\epsilon}\) (KW\({}^{-1}\)) & \(R_{u}\) (KW\({}^{-1}\)) & \(C_{c}\) (JK\({}^{-1}\)) & \(C_{s}\) (JK\({}^{-1}\)) \\ \hline
1.94 & 3.08 & 62.7 & 4.5 \\ \hline \end{tabular}
\end{table} 
Table 1: The thermal parameters.