# Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging
## 2 Cell Modelling
### Description of the Aging Part of the Model  
Depending on the intended application, fast charging or battery cycling until End of Life (EOL), a short-term or a long-term model is required. In this work, the developed model is devoted to designing fast charging algorithms, and aging is considered to be mainly caused by the growth of a Solid Electrolyte Interphase (SEI) layer on the anode. This degradation mostly happens during short to middle-term battery usage during fast charging cycles. The aging model used is based on the equations proposed in [22] and, following the observations from post-mortem analysis of the investigated cells, aging is only considered on the anode electrode. For a constant temperature \(T\), the current produced by the side reaction leading to the undesired reduction of Li ions is defined by the Tafel equation  
\[j_{sr}=-a_{s}^{n}j_{0,sel}exp\left(-\frac{nFa_{n}}{RT}\eta_{sei}\right) \tag{10}\]  
If \(\varphi_{en}\), \(R_{sei}\) and \(U_{sei}^{ref}\) denote respectively the anode potential, the inner resistance of the SEI layer and the reference potential of SEI formation (assumed to be 0.4 V), the SEI overpotential is given by the relation  
\[\eta_{sei}=\varphi_{en}-R_{sei}I-U_{sei}^{ref} \tag{11}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Unit** \\ \hline \(m\) & Mass of the cell & kg \\ \(C_{p}\) & Specific heat capacity & J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\) \\ \(U_{pol}\) & Polarization voltage & V \\ \(R_{f}\) & High frequency & \(\Omega\) \\ \(I\) &
\begin{tabular}{c} Input current \\ Heat transfer coefficient \\ \end{tabular} & W\(\cdot\)m\({}^{-2}\cdot\)K \\ \(A\) & Cell surface area & m\({}^{2}\) \\ \(T_{amb}\) & Ambient temperature & K \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the thermal model. in which the anode potential \(\varphi_{en}\) is defined by  
\[\varphi_{en}=\eta_{n}+U_{n}+R_{sei}I \tag{12}\]  
According to [23], \(\eta_{n}\) is deduced from the Butler-Volmer equation which defines the kinetics of Li ion intercalation, and can be approximated by  
\[\eta_{n}=\frac{2 RT}{F}\log\left(\left(\frac{j_{l}}{2a_{s}^{n}j_{0}^{n}}\right) +\sqrt{\left(\frac{j_{l}}{2a_{s}^{n}j_{0}^{n}}\right)^{2}+1}\right) \tag{13}\]  
for an easier implementation. The electrochemical parameters associated with Relations (10) to (13) are defined in Table 2. Some parameters are taken from [22]. The others were estimated using measurements on the cell under investigation. The increase in temperature resulting from fast charging operation has a large impact on the side reaction rate. Thus, Relation (10) must be modified to take into account the Arrhenius dependency of the side reaction current on the temperature variation. The modified Equation (10) is thus given by  
\[j_{sr}=-a_{s}^{n}j_{0,sei}exp\left(\frac{E_{a}}{R}\left(\frac{1}{T_{ref}}- \frac{1}{T}\right)\right)exp\left(-\frac{nF\alpha_{n}}{RT}\eta_{sei}\right) \tag{14}\]  
The SEI thickness (\(\delta_{sei}\)) and resistance (\(R_{sei}\)) can be computed by the following Relations [22]:  
\[\frac{\partial\delta_{sei}}{\partial t}=\frac{M_{sei}}{2F\rho}\;j_{sr} \tag{15}\]  
\[R_{p}(t)=\frac{\delta_{sei}}{\kappa_{sei}} \tag{16}\]  
\[R_{sei}(t)=R_{sei,init}+R_{p}(t) \tag{17}\]  
Experimental observations [24] have shown that, after a primary formation and after some cycles, the SEI layer thickness growth at the surface of the electrode decreases. To model such a behavior, an exponential function was associated to Relation (18), which becomes  
\[j_{sr}=-a_{s}^{n}j_{0,sei}exp(-\lambda\delta_{sei})exp\left(\frac{E_{a}}{R} \left(\frac{1}{T_{ref}}-\frac{1}{T}\right)\right)exp\left(-\frac{nF\alpha_{n} }{RT}\eta_{sei}\right) \tag{18}\]