# A Novel Real-Time Battery State Estimation Using Data-Driven Prognostics and Health Management
## 6 Results and Analysis
### Battery SOC Estimation  
Out of the nine drive cycles tests dataset used during the data-driven PHM comparison with the state-of-the-art Kalman filters, the HWFTa (Highway Fuel Economy Test Cycle, dataset A) has its results presented graphically. For each of the tested drive cycles, with the second-order battery ECM with optimized parameters, the ECM voltage derived from the battery model was compared with the respective drive cycle dataset, and the SOC was derived and computed according to Section 5.2. The results for the battery parameter optimization is depicted in Figure 6. The current drawn from the battery is depicted in the first plot. The x-axis is shown in seconds, the current is given in Amperes, and the voltage in Volts. The SOC is an adimensional variable. Figure 5: UDDS optimal input attribution of the most impactful features. The second graphic of Figure 6 shows the battery terminal voltage measured from the experiment in the dataset in black and the one resulting from the battery ECM with optimized parameters in blue. It is seen that the battery ECM voltage is not able to perfectly match the real voltage values. Some reasons are related to the limitation of the optimization process to find suitable \(R^{\prime}s\) and \(C^{\prime}s\) to better represent the battery cell nonlinearity and its intrinsic behavior. In addition, other aspects such as the battery's temperature and aging effects are difficult to model. Finally, the third plot shows the different values and computations for the SOC: Coulomb-counting derived from the dataset, the ones from Kalman filters (both EKF and UKF), and the data-driven PHM one. The graphical result shows an overall good accuracy for all models with regards to the Coulomb-counting method, which shows low data variance for each model. The only noticeable characteristic across the results is the overshoot seen at the SOC estimation with UKF. This behavior during a transient period was previously reported, such as in the case of inaccurate initial SOC [3,10]. The SOC estimation error for two cases, HWFTa and Cycle 3, is presented in Table 2. The results show the maximum absolute error and the mean absolute error (MAE) for each of the three estimators. The values outperform the ones reported in the literature [9,10,16], with mean absolute errors (MAE) below 2% for all methods. The data-driven PHM reached 0.40% and 0.47% for the drive cycles with the best MAE across the methods. Figure 6: HWFTa drive cycle estimation. The results for the ten different drive cycles are consolidated in Table 3. In general, the data-driven PHM method outperforms the Kalman filters by having a higher correlation with the real values, and by displaying a higher \(R^{2}\) (near the unit value) and lower MSE. Additionally, the confidence interval (CI) width is presented to compare the precision of the different methods. A confidence level (CL) of 95% is chosen, which gives an equivalent significance level of 5% (or 0.05). The confidence intervals (upper and lower) are then calculated and the CI width is then presented to demonstrate the method's precision. It \begin{table}
\begin{tabular}{l c c c c} \hline
**Drive Cycles** & **Method** & \
\begin{tabular}{c} **SOC Max** \\ **Absolute Error** \\ \end{tabular} &
\begin{tabular}{c} **SOC Mean** \\ **Absolute Error** \\ \end{tabular} \\ \hline HWFTa & EKF & 10.10\% & 0.66\% \\ UKF & 13.14\% & 1.90\% \\ PHM & 7.66\% & 0.40\% \\ \hline Cycle 3 & EKF & 11.26\% & 1.58\% \\ UKF & 10.22\% & 1.36\% \\ PHM & 2.66\% & 0.47\% \\ \hline \end{tabular}
\end{table} 
Table 2: SOC estimation error comparison. \begin{table}
\begin{tabular}{l c c c c c c} \hline
**Drive Cycles** & **Method** & \begin{tabular}{c} **Correlation** \\ **(corr)** \\ \end{tabular}  & \ \begin{tabular}{c} **Variance** \\ **(var)** \\ \end{tabular}  & **Bias** & \(R^{2}\) & **MSE** & \begin{tabular}{c} **CI** \\ **Width** \\ \end{tabular}