# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 1 Introduction  
Lead-acid batteries are widely accepted across industries, and commonly used as clean-energy storage, in automotive vehicles, and as standby and emergency power supplies [1, 2]. The estimation of remaining service-life is an important output of battery management systems (BMS) [3, 4] and failure warning of BMS is direct instruction for battery replacement. Nonlinear electrical passive load samplings and intensive deep-discharge recycles, in vehicle and photovoltaic systems, are not applied in backup application. Only periodical deep-discharges and active samplings of electrochemical impedance spectroscopy (EIS) are allowed in backup applications. The remaining useful capacity can be directly calculated in deep-discharge curves using the voltage model [5, 6]. However, in typical voltage models, such as Shepherd [7], CIEMAT [8, 9] and Monegon [10], the remaining useful capacity itself is one of the parameters. The interpretation of polynomials and nonlinear functions of deep-discharge curves needs brute-force numeric efforts,including generic [11; 12; 13] and artificial neural-network [14] intelligent algorithms. The remaining useful capacity can also be directly calculated with floating service time by the linear Kalman filter, but the turning point from stability to deterioration is required [15; 16]. Moreover, intensive deep-discharge recycles are carried out to accumulate empirical data and provide typical coefficients for aging model [6]. The remaining and loss capacity are modeled to calculate the remaining useful capacity of lead-acid batteries. Remaining capacity is modeled by physical-chemical governing differential equations of lead-acid battery [4; 17]. Loss capacity is modeled by various Ah throughput during specific operations deviating from the standard conditions [18; 19]. The loss-capacity model basically consists of the voltage and aging model. The voltage model is decomposed into static and transient electrical equivalent circuits [19]. The aging model includes corrosion, gassing, acid stratification phenomena [18; 19] and degradation of active material [20; 21] equations. The initial value of the parameters, sampling of terminal electrical variables and filtering of aging rates are core items of the (remaining and loss) capacity model. The initial value of the parameters are generally determined according to manufacturers' datasheet [19] or calculated from an intelligent algorithm [4]. The terminal electrical variables of multiple states of charge are passively sampled from regular operation, such as in automobile and photovoltaic systems [3; 4]. However, the sampling frequency should be high enough to achieve a nonlinear filtering effect. Active sampling of electrochemical impedance spectroscopy is alternative for long-term floating service backup batteries [22; 23]. Recently, switch-mode power supply has been integrated with electrochemical impedance spectroscopy functions [24], and contacting resistance is a basic inspection item [25]. The parameters of the aging model are generally nonlinear and monotonic. However, the aging rates of these parameters fluctuate during service life. In order to suppress fluctuations, researchers have proposed nonlinear filtering techniques such as particle filter [3], Kalman filter [26] and Bayesian approach [27]. In ideal theory, the physical and electrochemical variables of lead-acid batteries continue to increase (decrease) in the direction of deterioration during service life operation. However, battery variables fluctuate during aging tests and field operations. Some fluctuation is so much in the counter deterioration direction that a hysteresis phenomenon appears. Hysteresis includes two parts: the beginning part is that variables significantly change or hold up in the opposite direction of degradation; the following part is that variables rapidly change in the positive direction of degradation. The main reason for hysteresis is the nature of complicated nonlinear system dynamics of lead-acid batteries [28].