Failure Warning at the End of Service-Life of Lead–Acid Batteries for Backup Applications

: The prediction of remaining useful life is an important function of battery management systems. Existing research typically focused on factors that determine the quantity of the remaining useful capacity, and are able to determine the remaining useful capacity several years before battery failure to counter hysteresis of variables of lead–acid batteries. These techniques are not suitable at the end of service-life for backup batteries. This paper proposes a linear-superposition–voltage-aging model with three improvements. First, the estimation of the deep-discharge of the proposed voltage model does not require the remaining useful capacity. Second, the internal resistance of the deep-discharge is predicted from the contacting resistance of electrochemical impedance spectroscopy. Third, a morphology correction factor of internal resistance is about to saturate at the end of battery service-life. The model accurately forecasts battery failure at the end of service-life in two groups of accelerated-aging experiments. The proposed method in this paper focuses on the factors that determine quality of remaining useful capacity to counter hysteresis of variables of lead–acid batteries and judge battery failure at the end of service-life.


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, 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. Figure 1a is a typical DC-equivalent circuit model of a lead-acid battery [32,33]. The open-circuit voltage (V OCP ) is the initial voltage of electrolyte bulk capacitor (c bulk ). The charge-transfer resistances (R ct,pd R ct,nd ) of discharge are functions of the electrode utilization coefficients (EUC) and morphology correction factors (ξ) [32,33]. In Equation (1), EUC is the remaining state of charge (SOC), subtracted from unit one. The Q i is capacity of discharge and AH o is rated design capacity (560 Ah). The ξ is power index of EUC in Equations (2) and (3). Both r ct,p and r ct,n are charge-transfer resistances of the full state of charge and are calculated according to Tafel law [34].

Morphology Correction Factor of Battery Internal Resistance
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 22 failure at the end of service-life. First, the model reduces circuit order by replacing the two chargetransfer 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. Figure 1a is a typical DC-equivalent circuit model of a lead-acid battery [32,33]. The open-circuit voltage (V OCP ) is the initial voltage of electrolyte bulk capacitor (c bulk ). The charge-transfer resistances ( R ct,pd , R ct,nd ) of discharge are functions of the electrode utilization coefficients (EUC) and morphology correction factors (ξ) [32,33]. In Equation (1), EUC is the remaining state of charge (SOC), subtracted from unit one. The Q i is capacity of discharge and Ah o is rated design capacity (560 Ah). The ξ is power index of EUC in Equations (2) and (3). Both r ct,p and r ct,n are charge-transfer resistances of the full state of charge and are calculated according to Tafel law [34]. This paper proposes that battery contacting resistance (R C ) and internal resistance (R Ω ) also appear as morphology correction phenomena. Equation (4) represents the discharging surface area (a e,d ) [35,36] of the electrode surface area (a e ). The a e,d is physical contact area between the chargetransfer and contacting resistances. The discharge current passes through the contacting resistance with the same area (a e,d ). Hence, the morphology-correction phenomenon appears on contacting resistance and is the physical basis upon which existing research [22] estimates battery state of charge by contacting resistance. In Equation (5), r C is contacting resistance of the full state of charge.

Morphology Correction Factor of Battery Internal Resistance
Internal resistance consists of contacting and charge-transfer resistances in series connection. Both contacting and charge-transfer resistances lead internal resistance with morphology correction phenomena as per Equation (6). The r Ω is internal resistance of the full state of charge and is defined as the sum of the contacting and charge-transfer resistances of the full state of charge. Reference electrode is necessary to check morphology correction factor of individual electrode, but reference electrode is not available in commercial lead-acid batteries. Equation (6) roughly defines a This paper proposes that battery contacting resistance (R C ) and internal resistance (R Ω ) also appear as morphology correction phenomena. Equation (4) represents the discharging surface area (a e,d ) [35,36] of the electrode surface area (a e ). The a e,d is physical contact area between the charge-transfer and contacting resistances. The discharge current passes through the contacting resistance with the same area (a e,d ). Hence, the morphology-correction phenomenon appears on contacting resistance and is the physical basis upon which existing research [22] estimates battery state of charge by contacting resistance. In Equation (5), r C is contacting resistance of the full state of charge.
Internal resistance consists of contacting and charge-transfer resistances in series connection. Both contacting and charge-transfer resistances lead internal resistance with morphology correction phenomena as per Equation (6). The r Ω is internal resistance of the full state of charge and is defined as the sum of the contacting and charge-transfer resistances of the full state of charge. Reference electrode is necessary to check morphology correction factor of individual electrode, but reference electrode is not available in commercial lead-acid batteries. Equation (6) roughly defines a morphology correction factor (ξ Ω ). According to the morphology-correction phenomenon, the proposed Thevenin model replaces two charge-transfer resistances and one contacting resistance with one internal resistance to reduce the circuit order. The proposed Thevenin model is presented in Equation (7) and Figure 1b. The V ti is battery terminal voltage of SOC i and I is the discharging current.

Analysis of Deep-Discharge Curve by Linear Superposition
The principle of linear superposition was used to sequentially calculate parameters of the Thevenin circuit model. First, the open-circuit voltage was read from the terminal ports before discharge as per Equation (8). Second, r Ω was calculated at the first discharge point as per Equation (9). Third, c bulk was averaged among the electrode utilization coefficient interval of [0.1, 0.3] as per Equation (10). In Equation (10), internal resistance was approximated with r Ω and influence of morphology correction was ignored. Fourth, while terminal voltage (V tj ) was approaching the end of the voltage (1.80 V), ξ Ω was calculated by Equation (11). If there were many samples around 1.80 V, ξ Ω could be calculated by least square method. Figure 2a marks points cited by above algorithm. In the figure of the deep-discharge curve, the horizontal axis is the rated state of charge and the vertical axis is the terminal voltage. Figure 2b plots the fitting curve of the proposed Thevenin model.

Interactive Analysis of Internal Resistance of Discontinuous Current
In the proposed Thevenin model, r Ω depends on rated load discharge current (19 A). However, discharge current abruptly changes from 19 A to 50 A in some experiments, such as Figure 3a. The charge-transfer resistance decreases due to Tafel law [34]. Contacting resistance increases with current density increasement. At the beginning of deep-discharge, r Ω, 19 of 19 A discharge current is extracted. At the end of discharging curve, r Ω,50 of 50 A discharge current is hard to be extracted from Thevenin model.
The CIEMAT model can extract r Ω,50 of 50-A discharge current. The parameters of the original CIEMAT model are fixed [8,9] as per Equation (12). This study proposes to loosen four parameters (V OCP , a1, a2 and a3) of CIEMAT model as per Equation (13). Deep-discharge experiments were carried out at room temperature, so that ∆T is ignored. Equations (14) to (15) were identical to original CIEMAT model [8,9]. The C is remaining useful capacity. The I 10 discharges rated battery capacity (C 10 = 560 Ah) for 10 h.
Equations (16) and (17) are terminal voltages before and after discontinuous current (I 11.2 = 50 A, I 29.4 = 19 A). These two equations and Equation (18) are combined as a set of linear equations to calculate parameters (a1, a2 and a3). With a1, a2 and a3, CIEMAT model extracts r Ω,50 as per Equation (19). We bring r Ω,50 back into Equation (11) to calculate morphology correction factor. Figure 3a marks the points cited by above algorithm. Figure 3b plots fitting curve of proposed Thevenin and 3-point CIEMAT models.

Interactive Analysis of Internal Resistance of Discontinuous Current
In the proposed Thevenin model, r Ω depends on rated load discharge current (19 A). However, discharge current abruptly changes from 19 A to 50 A in some experiments, such as Figure 3a. The charge-transfer resistance decreases due to Tafel law [34]. Contacting resistance increases with current density increasement. At the beginning of deep-discharge, r Ω,19 of 19 A discharge current is extracted. At the end of discharging curve, r Ω,50 of 50 A discharge current is hard to be extracted from Thevenin model.

Linear Aging filter of Internal Resistance based on Contacting Resistance of EIS
The AC current of EIS was 5 A (rms) and the DC current of deep-discharge was 19 A. With the current density increasing from the EIS to deep-discharge, the charge-transfer resistance decreased according to Tafel law [34]. Tafel law was not active for contacting resistance, so that contacting resistance increased with current. Under the full state of charge, the increasing contacting resistance (r C ) was more significant than the variation of charge-transfer resistance (r ct,p , r ct,n ). The contacting resistance usually nonlinearly increased with battery corrosion [37]. The internal resistance also nonlinearly increased with battery corrosion [18]. The ratio (α k ) between contacting (r C,k ) and internal resistances (r Ω,k ) is the linearized by least square method in Equation (20). The CIEMAT model can extract r Ω,50 of 50-A discharge current. The parameters of the original CIEMAT model are fixed [8,9] as per Equation (12). This study proposes to loosen four parameters (V OCP , a 1 , a 2 and a 3 ) of CIEMAT model as per Equation (13). Deep-discharge experiments were carried out at room temperature, so that ∆Tis ignored. Equations (14) to (15) were identical to original CIEMAT model [8,9]. The C is remaining useful capacity. The I 10 discharges rated battery capacity (C 10 = 560 Ah) for 10 h.
Equations (16) and (17) are terminal voltages before and after discontinuous current (I 11.2 = 50 A, I 29.4 = 19 A). These two equations and Equation (18) are combined as a set of linear equations to calculate parameters (a 1 , a 2 and a 3 ). With a 1 , a 2 and a 3 , CIEMAT model extracts r Ω,50 as per Equation (19). We bring r Ω,50 back into Equation (11) to calculate morphology correction factor. Figure 3a marks the points cited by above algorithm. Figure 3b plots fitting curve of proposed Thevenin and 3-point CIEMAT models. 1+0.67

Linear Aging Filter of Internal Resistance Based on Contacting Resistance of EIS
The AC current of EIS was 5 A (rms) and the DC current of deep-discharge was 19 A. With the current density increasing from the EIS to deep-discharge, the charge-transfer resistance decreased according to Tafel law [34]. Tafel law was not active for contacting resistance, so that contacting resistance increased with current. Under the full state of charge, the increasing contacting resistance (r c ) was more significant than the variation of charge-transfer resistance (r ct,p , r ct,n ). The contacting resistance usually nonlinearly increased with battery corrosion [37]. The internal resistance also nonlinearly increased with battery corrosion [18]. The ratio (α k ) between contacting (r C,k ) and internal resistances (r Ω,k ) is the linearized by least square method in Equation (20).
At least two sets of initial measurements were required to start the algorithm. The linear Kalman filter algorithm [16] was executed after each accelerated aging cycle. Implementation involves following steps: Step 1. Initialization: In Equation (21), α2 is the linear ratio between contacting and internal resistances of the beginning two measurements. In Equation (22), α1 is the linear ratio between contacting and internal resistances of the first measurement. In Equation (23),r Ω,1 andr Ω,2 are estimated internal resistances. In Equation (24), P − 2 is the error covariance of estimation. In Equation (25), M Ω is the covariance matrix of measurement errors. In Equation (26), H Ω is the transformation matrix between measurement and state vector. The r Ω,k α k is state vector and r Ω,2 α 2 is initializing as r Ω,2 α 2 . In Equation (27), K 2 is the Kalman gain. In Equation (28), P 2 is the error covariance for updated estimation: r Ω,1 r Ω,2 = r C,1 r C,2 α 2 (23) H Step 2. EIS Sampling and state prediction: After EIS sampling, a new contacting resistance r C,k is obtained. In Equation (29), ∆r C,k is the incremental contacting resistance. In Equation (30), Φ Ω,k is the state transition matrix. In Equation (31), r − Ω,k is prediction of aging internal resistance at the moment of EIS sampling,â − k is the same asâ (k−1) . In Equation (32), P − k is the error covariance for following estimation. In Equation (33), K k is updated before jumping to Step 3.
Φ Ω,k = 1 0 Step 3. Updating the estimation with measurement of the deep-discharge: After measurement of the deep-discharge, r Ω,k and α k is calculated in Equation (20). In Equation (34), state vector estimation (r Ω,k andα k ) is updated with measurement (r Ω,k and α k ). In Equation (35), P k is updated before jumping back to Step 2.
Until battery failure, the algorithm returns to Step 2. In this paper, the sampling interval between k and (k + 1) is limited by period of accelerated aging. However, in future applications, the sampling of EIS can be shorter than the deep-discharge counterpart.

Linear Aging Filter of Parameters to Counter Hysteresis
The accelerating aging is periodic and consistent in experiments. If the measured parameters (V OCP , c bulk , ξ Ω ) appear hysteresis, then the aging rates are adjusted according to the higher value of the two beginning points in Equation (36). This guarantees the prediction to decrease at average rates. If the measured parameters (V OCP , c bulk , ξ Ω ) do not appear hysteresis, the aging rates are adjusted according to the last value as a starting point in Equation (37). This guarantees the prediction to decline at the latest rates. Two aging rates are compared in Equation (38); the lower one is to be used in prediction: Step 1. Initialization: The initial value of the parameters (V OCP , c bulk , ξ Ω ) requires three measurements because of bias points in Equations (36) and (37).
In Equation (39), β 3 is the linear aging rate between continuous three accelerated experiments, x o,3 is bias point according to least square method. In Equation (40), β 2 is the linear aging rate between the beginning two accelerated experiments, x o,2 is bias point according to least square method. In Equation (41),x 1 ,x 2 andx 3 are estimations of the parameters. In Equation (42), P − 3 is the error covariance of estimation. In Equation (43), M x is the covariance matrix of measurement error. In Equation (44), H x is the transformation matrix between measurement and state vector. The x k β k is state vector and x 3 β 3 is initializing as x 3 β 3 . In Equation (45), K 3 is the Kalman gain. In Equation (46), P 3 is the error covariance for updated estimation.
Step 2. State prediction: In Equation (47), Φ x,k is the state transition matrix, to pass incremental aging round. In Equation (48),x − k is prediction of aging parameter at the moments of the next deep-discharge measurement,β − k is the same asβ (k−1) . In Equation (49), P − k is the error covariance for following estimation. In Equation (50), K k is updated before jumping to Step 3.
In Equation (51), prediction parameters (V OCP,k ,ĉ bulk,k ,ξ Ω,k andr Ω,k ) are applied to approximate EÛC j,k by table look-up method. EÛC j,k corresponds to the end of terminal voltage (1.80 V). If EÛC j,k is below 80%, filter loop is halted, and the battery is removed out of stack. The battery's state of health can be further verified by deep-discharge and EIS: Step 3. Updating estimation with measurement of deep-discharge: After measurement of deep-discharge, x k and β k is calculated in Equation (52). In Equation (53), state vector estimation (x k andβ k ) is updated with measurement (x k and β k ). In Equation (54), P k is updated before jumping back to Step 2.
Step 4. Increment (k + 1) and go back to Step 2. Until battery failure, algorithm returns to Step 2. In filtering loop, sampling interval between k and (k + 1) is limited by accelerated aging period.

Linear Aging Model of Quantity of Remaining Useful Capacity
As a comparison, the prediction of quantity of remaining useful capacity was modeled the same as variables in Section 2.5. In Equation (55), M capacity is the covariance matrix of measurement error. In Equation (56), H capacity is the transformation matrix between measurement and state vector:

Accelerated Aging Experiments
Four parallel samples were float-charged at 70 • C for nearly seven months. Moreover, another four parallel samples were float-charged at 55 • C for nearly six months. After float charging in a high-temperature chamber for one month, all samples were deep-discharged for the remaining capacity and recharged back. More than 24 h later after recharging, all cells were individually sampled by EIS for contacting resistances of the full state of charge. Contacting resistances of parallel samples are presented in Figure 4.

Accelerated Aging Experiments
Four parallel samples were float-charged at 70 °C for nearly seven months. Moreover, another four parallel samples were float-charged at 55 °C for nearly six months. After float charging in a hightemperature chamber for one month, all samples were deep-discharged for the remaining capacity and recharged back. More than 24 h later after recharging, all cells were individually sampled by EIS for contacting resistances of the full state of charge. Contacting resistances of parallel samples are presented in Figure 4.

Fitting Errors of Deep-discharge Curves of 70 °C Aging Experiments
Due to space limitations, only discharge curves of No. E-52# cell is shown in Figure 5.

Fitting Errors of Deep-Discharge Curves of 70 • C Aging Experiments
Due to space limitations, only discharge curves of No. E-52# cell is shown in Figure 5.

Accelerated Aging Experiments
Four parallel samples were float-charged at 70 °C for nearly seven months. Moreover, another four parallel samples were float-charged at 55 °C for nearly six months. After float charging in a hightemperature chamber for one month, all samples were deep-discharged for the remaining capacity and recharged back. More than 24 h later after recharging, all cells were individually sampled by EIS for contacting resistances of the full state of charge. Contacting resistances of parallel samples are presented in Figure 4.

Fitting Errors of Deep-discharge Curves of 70 °C Aging Experiments
Due to space limitations, only discharge curves of No. E-52# cell is shown in Figure 5. The mean biased error (MBE), root mean square error (RMSE) and correlation coefficient (R 2 ) are shown in Table 1. Fitting effect proves that internal resistance has morphology-correction The mean biased error (MBE), root mean square error (RMSE) and correlation coefficient (R 2 ) are shown in Table 1. Fitting effect proves that internal resistance has morphology-correction phenomenon. Capacity of seventh experiment of cell E-02# is only 38 Ah, and only three measurement points are above 1.80 V. Hence, data of this experiment is not shown in Table 1.

Battery-Failure Warning Results of the Proposed Linear Aging Model
Measurements and estimations of (quality and quantity) the remaining useful capacity of parallel samples are shown in Figure 6. Approximately 450 Ah as 80% of the rated design capacity (560 Ah) was set up as a red line of battery failure for the linear-aging model.
Hysteresis of measurement of the remaining capacity near 400 Ah appeared in the fifth and sixth tests of E-52#. The fifth and sixth tests of E-02# near 210 Ah also showed hysteresis. The value of the remaining capacity of E-20# continuously reduced during aging experiments.
During fifth to seventh tests, measurement of remaining useful capacity of E-11# around 450 Ah showed hysteresis. Prediction of quantity of the remaining capacity was according to model in Section 2.6. Prediction of the fifth test was very close to measured value which was above red line. The model carried out the sixth prediction and determined battery failure; however, the capacity verification confirmed that the measured remaining capacity was still above red line. The model adjusted its prediction of the seventh test with a result of 458 Ah. However, at this round, the measured remaining capacity (418 Ah) was below the red line. The model of Section 2.6 continuously showed a false alarm at the sixth test and missing the alarm at the seventh alarm. The missing alarm was a serious error of class-1E nuclear backup batteries. The hysteresis of the remaining useful capacity-especially at the end of battery service life-was an obvious difficult problem for the typical aging model.
The proposed linear-aging model accurately predicted the battery failures of E-52#, E-20# and E-02#. There were no false alarms at the fourth test, and no missing alarms at the fifth test. The proposed linear-aging model continued to present failure warnings for E-11# from the fifth to the seventh tests. E-11# presented saturation of the morphology correction factor from the fifth test. The proposed linear-aging model successfully predicted this saturation and the failure warning of E-11# at the fifth test was an effective alarm of the quality of the remaining capacity. This alarm should not be judged as a false alarm, although measurement of the remaining capacity was slightly more than the red line. The red line of measurement of the remaining capacity was average and was individually different.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 22 The proposed linear-aging model accurately predicted the battery failures of E-52#, E-20# and E-02#. There were no false alarms at the fourth test, and no missing alarms at the fifth test. The proposed linear-aging model continued to present failure warnings for E-11# from the fifth to the seventh tests. E-11# presented saturation of the morphology correction factor from the fifth test. The proposed linear-aging model successfully predicted this saturation and the failure warning of E-11# at the fifth test was an effective alarm of the quality of the remaining capacity. This alarm should not be judged as a false alarm, although measurement of the remaining capacity was slightly more than the red line. The red line of measurement of the remaining capacity was average and was individually different. 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 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 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 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 was generally as twice the resistance of the full state of charge. This illustrates that the electrical characteristic of battery terminal was soft at the end of service-life.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 22 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 Figure 8 and Figure 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 was generally as twice the resistance of the full state of charge. This illustrates that the electrical characteristic of battery terminal was soft at the end of service-life.         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 • C tests than in 70 • C tests. Predictions of the remaining capacity are shown in Figure 11; the fitting errors are list in Table 3. The errors between predictions and measurements gradually decrease in Figure 11. The remaining useful capacity of the parallel samples also presents bathtub curve of hysteresis in Figure 11.
In practice, it was difficult to determine threshold of saturation of the morphology correction factor. In Figure 10, saturation values of the morphology correction factors are individually different. Therefore, comparison with the red line (450 Ah) of the remaining capacity was still necessary. The advantage of this treatment could counter hysteresis of the morphology correction factor instead of saturation, as shown in Figure 12. The proposed linear-aging model do not have false alarm in 55 • C tests in Figure 11.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 22 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 °C tests than in 70 °C tests. Predictions of the remaining capacity are shown in Figure 11; the fitting errors are list in Table 3. The errors between predictions and measurements gradually decrease in Figure 11. The remaining useful capacity of the parallel samples also presents bathtub curve of hysteresis in Figure 11.
In practice, it was difficult to determine threshold of saturation of the morphology correction factor. In Figure 10, saturation values of the morphology correction factors are individually different. Therefore, comparison with the red line (450 Ah) of the remaining capacity was still necessary. The advantage of this treatment could counter hysteresis of the morphology correction factor instead of saturation, as shown in Figure 12. The proposed linear-aging model do not have false alarm in 55 °C tests in Figure 11.

Conclusions
The red line of the quantity of the remaining capacity is often different among batteries; the same red line ignores differentiation. The red line of the quality of the remaining capacity can be unified, such as by the saturation characteristic of the morphology correction factor. The saturation of the morphology correction factor at the end of service-life proposed in this paper is more suitable for battery failure decision, although the remaining useful capacity of an individual cell (E-11#) may above average alarm threshold. State-of-the-art approaches focus on factors that determine quantity of remaining useful capacity and determine remaining useful capacity before several years far away from battery failure to counter hysteresis of lead-acid battery. The proposed model in this paper

Conclusions
The red line of the quantity of the remaining capacity is often different among batteries; the same red line ignores differentiation. The red line of the quality of the remaining capacity can be unified, such as by the saturation characteristic of the morphology correction factor. The saturation of the morphology correction factor at the end of service-life proposed in this paper is more suitable for battery failure decision, although the remaining useful capacity of an individual cell (E-11#) may above average alarm threshold. State-of-the-art approaches focus on factors that determine quantity of remaining useful capacity and determine remaining useful capacity before several years far away from battery failure to counter hysteresis of lead-acid battery. The proposed model in this paper focuses on factors which determine quality of remaining useful capacity and model battery failure at the end of service-life to counter hysteresis of lead-acid battery.
The linear-superposition-voltage-aging model proposed in this paper directly predicts internal resistances by contacting resistances from EIS. This method is particularly suitable for long-term floating service backup applications and for continuous prediction of remaining useful capacity between deep-discharging tests. While frequency of deep-discharging tests is increased, error of prediction is reduced. The linear-superposition-voltage-aging model can be extended to other batteries for prediction of remaining useful capacity.
Interactive analysis of nonlinear element is an alternative to avoid limitations brought by numeric calculations. Advantages of equivalent circuit and mathematical models are combined by interactively analysis. Generally, equivalent circuit model is more intuitive to extract parameters by local linear superposition; and mathematical model is more flexible to extract parameters by global linear equations. Acknowledgments: Thanks to Cheng Cheng, Hanchuan Huang and XueXue Li, from Zhejiang Narada Power Source Co., Ltd., for helpful discussions and experimental configuration.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript or in the decision to publish the results.