# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 1 Introduction
Nonlinear filters accurately predict remaining useful life about 4-5 years before end of service-life [3; 29]. This counters a battery's hysteresis, but these results are too far in the future to be useful for battery replacement in practice. Prediction generally assumes that aging rates are continuous during the remaining years. Nonlinear hysteresis of the remaining capacity brings fluctuations and uncertainty of battery-replacement decision when approaching the end of service-life. If some aging parameters will soon saturate near the end of service-life, aging models can alert of impending battery failure according to the saturation to counter terminal hysteresis characteristics. In this paper, the morphology correction factor has similar saturation characteristic. As early as 1970s, researchers have [30; 31] proposed that a basic characteristic of lead-acid batteries is that the main reaction surface area of porous electrodes clearly reduces with a decrease of charge state. This feature is parameterized by a morphology correction factor that has been gradually developed by recent literatures [32; 33]. Variation of main reaction surface area of porous electrode is greater under the deep lower state of charge than under middle or the full state of charge. The morphology correction factor of porous electrodes clearly significantly increases charge-transfer resistance under deep lower state of charge. According to the Tafel law, under middle- or full-state of charge, charge-transfer resistance reduces with increased discharge current. Based on morphologic correction phenomena and the Tafel law of porous electrodes, this paper improves the linear-aging model. This paper proposes a linear superposition-voltage aging model to analyze and predict deep-discharging curves for lead-acid batteries. The linear-aging model is focused on backup battery failure at the end of service-life. First, the model reduces circuit order by replacing the two charge-transfer and one contacting resistances with only one internal resistance, according to the morphology-correction phenomenon. Second, the model obtains initial value of the morphology correction factor through interactive analysis between an equivalent (Thevenin) circuit and a mathematical (CIEMAT) model. Third, the model predicts the internal resistance of deep-discharge from the contacting resistance of EIS. Fourth, the model estimates battery failure mainly according to saturation of the morphology correction factor. Accelerated aging experiments show that the proposed model is effective in deep-discharging curve analysis and accurate for backup battery failure warning-failure warnings. This paper is organized as follows: in Section 2, a linear-superposition-voltage-aging model based on morphology-correction phenomenon of internal resistance are demonstrated; in Section 3, fitting errors of deep-discharge curves and battery-failure warnings of accelerated aging tests are presented and discussed. Physical and chemical meanings of various Parameters in following sections are summarized in Table A1 of Appendix A.