# Low-Order Electrochemical State Estimation for Li-Ion Batteries
## 2 Methodology and Models
### Numerical Model
#### Electrolyte current  
First of all, notice that equation (3c) can be solved analytically. Before and after the separator the current is the same, no additional boundaries conditions are needed:  
\[i_{e}(x,t)=\begin{cases}\frac{I(t)}{\delta_{-}}x&0\leq x\leq\delta_{-}\\ I(t)&\delta_{s}<x<\delta_{+}\\ I(t)-\frac{I(t)}{\delta_{+}}(x-\delta_{-}-\delta_{s})&\delta_{+}\leq x\leq L. \end{cases} \tag{9}\]Electrolyte concentrationIntroducing the short-hand notation \(C_{e,i}(t)=C_{e}(x_{i},t)\), and the vector  
\[\mathbf{C}_{e}(t)=\big{[}C_{e,1}(t)\quad\cdots\quad C_{e,n}(t)\big{]}^{T},\]  
Equation (3a) can be approximated as:  
\[\dot{\mathbf{C}}_{e}(t)=A_{C_{e}}\mathbf{C}_{e}(t)+\mathbf{b}_{C_{e}}I(t) \tag{10}\]  
where the matrix \(A_{C_{e}}\) and the vector \(\mathbf{b}_{C_{e}}\) are defined in Appendix A.  
Electrolyte potentialIn the same fashion, introducing \(\Phi_{e,i}(t)=\Phi_{e}(x_{i},t)\) and  
\[\mathbf{\Phi}_{e}(t)=\big{[}\Phi_{e,1}(t)\quad\cdots\quad\Phi_{e,n}(t)\big{]}^{T},\]  
Equation (4a) can be approximated by using central differences as:  
\[A_{\Phi_{e}}\mathbf{\Phi}_{e}(t)=\mathbf{b}_{\Phi_{e}}I(t)+C_{\Phi_{e}}\text{ln}(C_{e} (t)) \tag{11}\]  
where the matrices \(A_{\Phi_{e}},C_{\Phi_{e}}\) and the vector \(\mathbf{b}_{\Phi_{e}}\) defined in Appendix A.  
Electrode concentrationFor the case of the solid phase, the concentration is approximated by considering a spatial homogeneity with respect to the \(x\)-direction. With the creation of the vector \(\mathbf{C}_{s}^{\mp}(x,t)=\big{[}C_{s}^{\mp}(r_{1},x,t)\quad\cdots\quad C_{s}^{\mp }(r_{m},x,t)\big{]}\), Equation (1a) can be expressed as:  
\[\dot{\mathbf{C}}_{s}^{\mp}(x,t)=A_{\overline{C}_{s}}^{\mp}C_{s}^{\mp}(x,t)+\mathbf{b}_ {\overline{C}_{s}}^{\mp}I(t) \tag{12}\]  
where the matrix \(A_{\overline{C}_{s}}^{\mp}\) and the vector \(\mathbf{b}_{\overline{C}_{s}}^{\mp}\) are given in Appendix A.  
Output voltageThe open circuit potential of each material is assumed to be known. Mathematical approximations for different materials can be found, e.g., in [6]. Considering (6) and the boundary condition (4b), the only unknown in (5a) is the surface potential which can be solved to obtain:  
\[\Phi_{ssurf}^{\mp}(t)=\frac{RT}{\alpha F}\text{asin}\left[\frac{F_{J\overline{ u}}^{\mp}(t)}{2\overline{\dot{\sigma}}_{0}^{\mp}(t)}\right]+U_{OCP}^{\mp}(C_{s} ^{\mp}(t))+FR_{f}^{\mp}j_{n}^{\mp}(t)+\Phi_{e}^{+}(t) \tag{13}\]  
so that the terminal output in (7) can be computed.  
Electrode potentialsFinally, (2a) can be analytically integrated to obtain the potentials across the electrodes by making use of the boundary conditions (2b) and (2c):  
\[\Phi_{s}^{\mp}(x,t)=\Phi_{s,surf}^{\mp}-\frac{I(t)}{\sigma^{\mp}}x_{\mp}\pm \frac{I(t)}{2\sigma^{\mp}\delta_{\mp}}x_{\mp}^{2},\qquad 0\leq x_{\mp}\leq\delta_{\mp}. \tag{14}\]  
Here, the electrodes are considered as separated items, i.e., the equation starts at \(x_{-}=0\), or \(x_{+}=\delta_{-}+\delta_{s}\) artificially set to zero, respectively.  
State of chargeBy considering the saturation point in the concentration of Li-ions in each electrode and the concentration's profile over time, it is possible to estimate the SOC of each electrode independently.  
Regarding the SOC, two types can be determined from the model:
* Surface SOC, regarding only the surface of the particle: \[SOC^{\mp}_{surf}(t)=\frac{C^{\mp}_{s,surf}}{C^{\mp}_{s,max}}\] (15)
* Bulk SOC, which contemplates the spatial profiles of the whole spheres: \[SOC^{\mp}_{bulk}(t)=\frac{3}{\delta(R_{s})^{3}}\int_{0}^{\delta}\int_{0}^{R_{ s}}r^{2}\frac{C^{\mp}_{s}(x,r,t)}{C^{\mp}_{s,max}}\,\mathrm{d}r\mathrm{d}x\] (16)