# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 2 Linear Superposition-Voltage Aging Model  
### Morphology Correction Factor of Battery Internal Resistance  
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 (\(\xi\)) [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 \(\xi\) 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 (\(R_{\Omega}\)) also appear as morphology correction phenomena. Equation (4) represents the discharging surface area (\(a_{c,d}\)) [35; 36] of the electrode surface area (\(a_{c}\)). The \(a_{c,d}\) is physical contact area between the charge-transfer and contacting resistances. The discharge current passes through the contacting resistance with the same area (\(a_{c,d}\)). 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 \(r_{\Omega}\) 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  
Figure 1: Equivalent discharge circuit models. (**a**) Typical DC-equivalent circuit model; (**b**) proposed Thevenin model.  
not available in commercial lead-acid batteries. Equation (6) roughly defines a morphology correction factor (\(\xi_{\Omega}\)). 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.  
\[EUC_{i}=\ 1-SOC_{i}=\frac{Q_{i}}{Ah_{o}},\ i=\ 0,1,2\cdots,j. \tag{1}\]  
\[R_{ct,pd}=\frac{r_{ct,p}}{1-EUC_{i}^{\xi_{p}}} \tag{2}\]  
\[R_{ct,nd}=\frac{r_{ct,n}}{1-EUC_{i}^{\xi_{n}}} \tag{3}\]  
\[a_{e,d}=\ a_{e}\Big{(}1-EUC_{i}^{\xi}\Big{)} \tag{4}\]  
\[R_{C}=\frac{r_{C}}{1-EUC_{i}^{\xi_{C}}} \tag{5}\]  
\[R_{\Omega}=\frac{r_{\Omega}}{1-EUC_{i}^{\xi_{\Omega}}},\ r_{\Omega}=\ r_{C}+r_{ ct,p}+r_{ct,n} \tag{6}\]  
\[V_{ti}\!\!=V_{OCP}\ -\frac{Ah_{o}EUC_{i}}{c_{bulk}}\!\!-\!\!\left(\frac{r_{ \Omega}}{1-EUC_{i}^{\xi_{\Omega}}}\right) \tag{7}\]