# Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging
## 2 Cell Modelling
### Description of the Electrochemical Part of the Model  
The equations given in [18; 19] were used to define the electrochemical part of the model. These equations result in [PERSON]'s modelling approach [20], which leads to a pseudo 2D model and assumes that each electrode of a cell is replaced by a single spherical particle in which the Li+ ions are inserted. In the resulting particle, and through several simplifications that are given in [3], the lithium ions' concentration gradient is described by the following diffusion equation:  
\[\frac{\partial c_{s}}{\partial t}=\frac{D_{s}}{r^{2}}\frac{\partial}{\partial r }\bigg{(}r^{2}\frac{\partial c_{s}}{\partial r}\bigg{)}\left\{\begin{array}{c }\frac{\partial c_{s}}{\partial t}\bigg{|}_{r=0}=0\\ D_{s}\frac{\partial c_{s}}{\partial r}\bigg{|}_{r=R_{s}}=-\frac{\frac{\mu_{s} }{\mu_{s}}}{\mu_{s}F}\end{array}\right. \tag{1}\]In Relation (1), \(J_{mean}^{Li}\) is the average of the electrode current density, \(r\) is the sphere radius and \(D_{s}\) is the diffusion coefficient. Resolution of the differential Equation (1) leads to the following transfer function linking the lithium concentration at the surface of the particle denoted \(C_{s}(r=R_{s})\) to \(J_{mean}^{Li}\):  
\[G(s)=\frac{C_{s}(r=R_{s})}{J_{mean}^{Li}}=-\frac{R_{S}}{FD_{s}a_{s}\bigg{(} \sqrt{\frac{s}{D_{s}}}\text{coth}\bigg{(}\sqrt{\frac{s}{D_{s}}}R_{S}\bigg{)}-1 \bigg{)}} \tag{2}\]  
The transfer function (2) cannot be easily simulated due to the \(\coth\) function. As demonstrated in [3], it can be approximated by the transfer  
\[H(s)=\frac{K_{1}}{s}\bigg{(}1+\frac{s}{\omega_{cs}}\bigg{)}^{0.5} \tag{3}\]  
where the parameters \(K_{1}\) and \(\omega_{cs}\) are functions of the coefficients of the diffusion Equation (1) (see [3]).  
For a consistent initialization of the model, and also to make the variable SOC appear, the transfer function \(H(s)\) can be split into two transfer functions, such that  
\[H(s)=H_{avg}(s)+H_{part}(s) \tag{4}\]  
with  
\[H_{avg}(s)=\frac{C_{s,avg}}{j_{mean}^{Li}}=\frac{K_{1}}{s} \tag{5}\]  
and  
\[H_{part}(s)=\frac{\Delta C_{s}}{j_{mean}^{Li}}=H(s)-H_{avg}(s)=\frac{K_{1}}{s} \bigg{(}\bigg{(}1+\frac{s}{\omega_{cs}}\bigg{)}^{0.5}-1\bigg{)} \tag{6}\]  
In Relation (4), \(H_{avg}(s)\) represents the average concentration (from the centre to the surface of the particle) and \(H_{part}(s)\) is the partial gradient of ion concentration on the particle surface.  
For the implementation of the battery model in a vehicle control unit, a discrete form of the transfer functions \(H_{avg}(s)\) and \(H_{part}(s)\) can be computed, as shown in [6]. Also, a polynomial can be used to link the lithium concentration at the surface of the particles, denoted SOC\({}_{\text{surf}}\), to the cell Open Circuit Potential (OCP), denoted \(U_{\text{n}}\) and \(U_{\text{p}}\) respectively for the anode and the cathode. If the charge transfer resistance is denoted R\({}_{\text{ct}}\) (R\({}_{\text{ct}}\) can be deduced from the Butler-Volmer equation linearization), the resulting electrochemical sub-model is represented by Figure 1.  
Figure 1: Coupled electro-aging thermal model designed.