# Low-Order Electrochemical State Estimation for Li-Ion Batteries
## 3 Evaluation Study
### Numerical Evaluation  
As commented in Section 2.3 the DMDc approach is repeated six times, one for each of the main variables of the electrochemical model using a time discretization step of \(\Delta t=0.5\)s. More detail on the choice of training data can be found in [39]. Note that for the training scenarios with different charging and discharging scenarios and different temperatures were combined. The dimensions of the original system and the identified ROM are summarized in Table 1. The obtained dynamics matrix \(\boldsymbol{\bar{A}}\) has all eigenvalues within the unit circle, so that the identified system dynamics is input-output stable. To test the accuracy and efficiency of the ROM with the improved ECMwith continuous parameter adaptation (CPE), and the combination with the EKF, as well as comparing these approaches against a simple constant parameter ECM with CC, a simulation was carried out for a 70 min discharge and charge process with the following C current rates: \(\frac{1}{2}\)C, \(\frac{1}{2}\)C, \(1\)C, \(1.5\)C, \(-1.5\)C, \(-1\)C, \(-\frac{1}{2}\)C, \(-\frac{1}{3}\)C with the external temperature being set to 30 \({}^{\circ}\)C. An initial error of 5% was considered for the values of the SOC of both electrodes at \(t=0\). The values of the parameters for the numerical model simulation are provided in Appendix B. The effectiveness of the ROM based on the considered DMDc identification procedure outlined above is evaluated first. Figure 3 shows the approximated profiles according to (28) for the concentrations of the electrodes (left column, top and center) and the electrolyte (left column, bottom), with the associated approximation errors (right column) in comparison to the high-fidelity numerical model. It can be clearly seen that the approximations are quite convincing inspite of the considerable order reduction according to Table 1. This illustrates clearly the strong approximation potential of the DMDc approach and its usefullness for the prediction of cell internal electrochemical states. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Numerical Model** & **ROM** \\ \hline \(C_{s}^{-}\) & 60 & 9 \\ \hline \(C_{s}^{+}\) & 60 & 6 \\ \hline \(\Phi_{s}^{-}\) & 60 & 2 \\ \hline \(\Phi_{s}^{+}\) & 60 & 2 \\ \hline \(C_{e}\) & 130 & 4 \\ \hline \(\Phi_{e}\) & 130 & 9 \\ \hline total & 500 & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DMDc approximation orders used for the model with parameters given in Appendix B. Figure 4 shows the error between the ROM, Coulomb counting (CC) and EKF and the numerical model for the surface SOC for both electrodes, which is derived from the Li-ion concentration. It can be seen that in comparison to an ECM with constant optimized parameters and simple CC, the ROM achieves a better prediction, which is further improved by the EKF, achieving a convergence over the simulation time, compensating the initial error. Subsequently, Figure 5 shows the error of the output voltage \(V\) for the simulated discharge/charge process for the ECM with CC, ECM with continous parameter adaptation (CPE) based on the ROM and the ROM with the EKF, showing that the inclusion of the ECM by using the SOC predicted by the ROM according to (36) already achieves an improvement in comparison to the constant parameter ECM with CC, and that this is further improved significantly by coupling the ROM with the EKF. Figure 6 shows the associated output voltage. In both figures the superiority of the ROM-based approaches can be clearly seen in comparison to the ECM with CC using constant parameters and no correction terms, while a considerable additional further improvement is achieved with the ROM coupled to the EKF. Figure 3: ROM approximation of concentration profiles (**left** column) for negative electrode (**top**), positive electrode (**center**) and electroylte (**bottom**), with the associated approximation error evolutions over time (**right** column). Figure 4: Error for the Surface SOC of electrodes.