# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 2 Linear Superposition-Voltage Aging Model
### Interactive Analysis of Internal Resistance of Discontinuous Current  
In the proposed Thevenin model, \(r_{\Omega}\) 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, \(r_{\Omega,19}\) of 19 A discharge current is extracted. At the end of discharging curve, \(r_{\Omega,50}\) of 50 A discharge current is hard to be extracted from Thevenin model.  
The CIEMAT model can extract \(r_{\Omega,50}\) 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 (\(V_{OCP}\), \(\text{a}_{1}\), \(\text{a}_{2}\) and \(\text{a}_{3}\)) of CIEMAT model as per Equation (13). Deep-discharge experiments were carried out at room temperature, so that \(\Delta T\)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  
Figure 3: Deep-discharge curves of cell E-52# at the third experiment. (**a**) Measured points cited by algorithm; (**b**) fitting curves of proposed Thevenin and CIEMAT models.  
Figure 2: Deep-discharge curves of cell E-52# at the fourth experiment. (**a**) Measured points cited by algorithm; (**b**) fitting curve of proposed Thevenin model.  
calculate parameters (a\({}_{1}\), a\({}_{2}\) and a\({}_{3}\)). With a\({}_{1}\), a\({}_{2}\) and a\({}_{3}\), CIEMAT model extracts \(r_{\Omega,50}\) as per Equation (19). We bring \(r_{\Omega,50}\) 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.  
\[V_{ti}=\left(2.085\text{--}0.12\frac{Q_{i}}{C}\right)-\frac{I}{C_{10}}\Bigg{(} \frac{4}{1+l^{1/3}}+\frac{0.27}{\left(1-Q_{i}/C_{T}\right)^{1.5}}+0.02\Bigg{)} \left(1\text{--}0.007\Delta T\right) \tag{12}\]  
\[V_{ti}=\left(V_{OCP}\text{--}a_{1}\frac{Q_{i}}{C}\right)-\frac{I}{C_{10}}\Bigg{(} \frac{a_{2}}{1+l^{1/3}}+\frac{a_{3}}{\left(1-Q_{i}/C_{T}\right)^{1.5}}+0.02 \Bigg{)} \tag{13}\]  
\[C=\frac{C_{T}}{1\text{--}0.67\Big{(}\frac{l_{29.4}}{l_{10}}\Big{)}^{0.9}} \tag{14}\]  
\[C_{T}=1.67C_{10} \tag{15}\]  
\[V_{tm}=\left(V_{OCP}\text{--}a_{1}\frac{Q_{n}}{C}\right)-\frac{I_{29.4}}{C_{10 }}\Bigg{(}\frac{a_{2}}{1+I_{29.4}}^{1/3}+\frac{a_{3}}{\left(1-Q_{n}/C_{T} \right)^{1.5}}+0.02\Bigg{)} \tag{16}\]  
\[V_{t(m+1)}=\left(V_{OCP}\text{--}a_{1}\frac{Q_{(n+1)}}{C}\right)-\frac{I_{11.2 }}{C_{10}}\Bigg{(}\frac{a_{2}}{1+I_{11.2}}^{1/3}+\frac{a_{3}}{\left(1-Q_{(n+1) }/C_{T}\right)^{1.5}}+0.02\Bigg{)} \tag{17}\]  
\[r_{\Omega,19}\approx\left.\frac{V_{OCP}-V_{II}}{I_{29.4}}\right|_{Q\approx 0}\approx\left.\frac{1}{C_{10}}\Bigg{(}\frac{a_{2}}{1+I_{29.4}}^{\frac{1}{3} }\text{--}a_{3}+0.02\Bigg{)} \tag{18}\]  
\[r_{\Omega,50}\approx\left.\frac{V_{OCP}-V_{II}}{I_{11.2}}\right|_{Q\approx 0}\approx\left.\frac{1}{C_{10}}\Bigg{(}\frac{a_{2}}{1+I_{11.2}}^{\frac{1}{3} }\text{--}a_{3}+0.02\Bigg{)} \tag{19}\]