# Low-Order Electrochemical State Estimation for Li-Ion Batteries
## 2 Methodology and Models
### Improved Equivalent Circuit Model  
From the numerical model, optimized parameters can be computed to create an ECM. Figure 2 shows the considered circuit. On the other hand, from the ROM, a continuous parameter estimation (CPE) or adaptation can be performed in order to improve the accuracy of the ECM.  
In the simplest way, ECM uses Coulomb Counting CC  
\[\frac{\text{dSOC}}{\text{d}t}(t)=\frac{I(t)}{Q_{\text{max}}},\quad t>0,\quad \text{SOC}(0)\text{=SOC}_{0}, \tag{32}\]  
with \(Q_{\text{max}}\) denoting the maximum charge possible, and \(\text{SOC}_{0}\) being an estimate of the initial SOC, to keep track of the SOC of the battery, or as in our model, of each electrode. Leaving aside the separated nature of two electrodes of different material and all related electrochemical processes and taking into account the inability of directly measuring the concentration distribution on the electrode surface, on the long run this approach requires a correction term as typically obtained from some state estimation technique.  
Using the concentration of Li-ions obtained from the ROM, the ECM can be improved. As not all the data of \(\bar{\mathbf{x}}_{k}\) is needed and just some part of it, the variable \(\mathbf{z}_{k}\) is defined in the following for the reduced model extraction. In addition, the voltage \(U_{p}\) of the capacitor \(C_{p}\) is converted to a discrete form.  
\[\mathbf{z}_{[k]}=\begin{bmatrix}\mathbf{z}_{1,[k]}\\ \mathbf{z}_{2,[k]}\end{bmatrix}=\begin{bmatrix}\mathbf{C}_{s}^{-}\\ \mathbf{\hat{C}}_{s}^{+}\end{bmatrix}. \tag{33}\]  
The variable \(\bar{\mathbf{\zeta}}\) represents the states of the improved ECM model, i.e.,  
\[\bar{\mathbf{\zeta}}_{[k]}=\begin{bmatrix}\bar{\mathbf{\zeta}}_{1,[k]}\\ \bar{\mathbf{\zeta}}_{2,[k]}\\ \bar{\mathbf{\zeta}}_{3,[k]}\end{bmatrix}=\begin{bmatrix}U_{p}\\ \mathbf{z}_{[k]}\end{bmatrix}, \tag{34}\]  
Figure 2: First Order RC Equivalent Circuit Model.  
where in discrete time, e.g., using a simple Euler discretization with step \(\Delta t\) the voltage \(U_{p}\) is approximated by:  
\[U_{p[k]}=\frac{R_{p}C_{p}-\Delta t}{R_{p}C_{p}}\cdot U_{p[k-1]}+ \frac{\Delta t}{C_{p}}\cdot I_{[k-1]}. \tag{35}\]  
The tracking of the internal potential will use the improved ECM approach creating the following discrete-time system:  
\[\xi_{[k]} =\mathbf{A}_{d}\mathbf{\xi}_{[k-1]}+\mathbf{B}_{d}\mathbf{u}_{[k-1]} \tag{36a}\] \[V =h(\mathbf{\xi},\mathbf{u})=\xi_{1}+U_{\text{OCP}}^{+}(SOC^{+})-U_{\text {OCP}}^{-}(SOC^{-})+R_{o}u_{1} \tag{36b}\]  
with  
\[\mathbf{A}_{d} =\begin{bmatrix}\frac{R_{p}C_{p}-\Delta t}{R_{p}C_{p}}&0&0\\ 0&\mathbf{\bar{A}}_{C_{s}^{-}}&0\\ 0&0&\mathbf{\bar{A}}_{C_{s}^{+}}\end{bmatrix},\quad\mathbf{B}_{d}=\begin{bmatrix}\frac{ \Delta t}{C_{p}}&0\\ \mathbf{\bar{B}}_{C_{s}^{-}}\\ \mathbf{\bar{B}}_{C_{s}^{+}}\end{bmatrix} \tag{36c}\] \[SOC^{+}_{[k]} =\mathbf{\bar{U}}_{C_{s}^{+}}(n,:)\cdot\mathbf{\bar{A}}_{C_{s}^{+}}\cdot \mathbf{\bar{5}}_{3,[k]}\cdot(C_{s,max}^{+})^{-1}\] (36d) \[SOC^{-}_{[k]} =\mathbf{\bar{U}}_{C_{s}^{-}}(n,:)\cdot\mathbf{\bar{A}}_{C_{s}^{-}}\cdot \mathbf{\bar{5}}_{2,[k]}\cdot(C_{s,max}^{-})^{-1}. \tag{36e}\]  
In the system performed by (36b) until (36e), \(\xi_{1,[k]}\) is the RC sub-network voltage in discrete time, \(\mathbf{\xi}_{2,[k]}\) is the reduced state of the negative electrode concentration with the dimension \(\hat{r}_{1}\), and \(\mathbf{\bar{5}}_{3,[k]}\) is the reduced state of the concentration of Lithium in the positive electrode with dimension \(\hat{r}_{2}\). The input \(\mathbf{u}_{[k]}\) is composed by the current and temperature. In (36d)-(36e) the original surface concentration of each electrode is recovered by using the state transformation \(\hat{U}_{i}\) from (31). Note that the concentration profile over the complete electrode can be recovered, but as for the output voltage only the surface concentration is required, only the \(n^{th}\) columns of the state transformation matrices are used.