# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 3 Results and Discussion
### Battery-Failure Warning Results of the Proposed Linear Aging Model
The parameters of measurements from deep-discharge and estimations from linear-aging model are plotted in Figures 7-10. The internal resistances predicted from contacting resistances of EIS did not show delays in Figure 7. The fitting errors of the internal resistances are shown in Table 2.  
In Figure 8, open-circuit voltages of E-20# and E-02# typically decreased along aging. the open-circuit voltage of E-52# fluctuated during tests. The open-circuit voltage of E-11# showed hysteresis among the sixth and seventh experiments. Figure 6: Battery capacities of parallel samples in 70 \({}^{\circ}\)C aging experiments. (**a**) E-52#; (**b**) E-11#; (**c**) E-20#; (**d**) E-02#. In Figure 9, electrolyte bulk capacitance of E-20# typically decreased with aging. The electrolyte bulk capacitance of E-52# showed typical hysteresis during the fifth to seventh tests. The capacitance first increased at sixth test, then sharply reduced at the seventh test. The electrolyte bulk capacitance of E-11# showed hysteresis from the fifth to seventh tests. The electrolyte bulk capacitance of E-02# also showed minor hysteresis at the last two experiments. Predictions of electrolyte bulk capacitors and open-circuit voltages clearly showed a delay of one aging cycle in Figures 8 and 9. It was difficult to accurately estimate the quantity of the remaining capacity of battery with the delay predictions of the parameters. The accuracy of the linear-aging model in determining battery failure depends on the inaccuracy of the prediction of the remaining capacity. The worst case aging rates were an active overshoot for the linear model to catch up saturation of the morphology correction factor. All predictions of the morphology correction factor were below 2.0 after saturation in Figure 10. The morphology correction factor was extracted by fitting deep-discharge curves. When a healthy battery terminal was approaching the end voltage (1.80 V), the internal resistance of 1.80 V was generally three times the resistance as that of the full state of charge. While the morphology correction factor was saturating, the internal resistance of 1.80 V \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cell** & **MBE** & **RMSE** \\ \hline E-52\# & \(0.28\times 10^{-3}\) & \(0.47\times 10^{-3}\) \\ E-11\# & \(0.12\times 10^{-3}\) & \(0.45\times 10^{-3}\) \\ E-20\# & \(0.16\times 10^{-3}\) & \(0.19\times 10^{-3}\) \\ E-02\# & \(0.29\times 10^{-3}\) & \(0.30\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitting errors of estimations of internal resistances of proposed aging model. Figure 8: Electrolyte bulk capacitors (\(c_{bulk}\)) of parallel samples in 70 \({}^{\circ}\)C aging experiments. (**a**) E-52\(\#\); (**b**) E-11\(\#\); (**c**) E-20\(\#\); (**d**) E-02\(\#\). Figure 9: _Cont._Figure 10: Morphology correction factors (\(\xi\)) of parallel samples in 70 \({}^{\circ}\)C aging experiments. (**a**) E-52\(\#\); (**b**) E-11\(\#\); (**c**) E-20\(\#\); (**d**) E-02\(\#\). Figure 9: Open-circuit voltages (\(V_{OCP}\)) of parallel samples in 70 \({}^{\circ}\)C aging experiments. (**a**) E-52\(\#\); (**b**) E-11\(\#\); (**c**) E-20\(\#\); (**d**) E-02\(\#\). Because of saturation phenomenon, delay of predictions of the morphology correction factor are not dominant in Figure 10. The worst-case algorithm of aging rates guarantees that missing alarm do not continue. If missing alarm of the remaining capacity happens at the beginning of saturation, then next prediction of the morphology correction factor is smaller than measurement. If prediction of the morphology correction factor is minus, algorithm replaces minus value with 0.1 in Equation (50) and Figure 10.  
Increasing frequency of sampling is a direct way to reduce parameter prediction delays. The aging effect of one month is less in 55 \({}^{\circ}\)C tests than in 70 \({}^{\circ}\)C tests.