# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 2 Linear Superposition-Voltage Aging 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 (\(r_{c,p}\), \(r_{c,n}\)). The contacting resistance usually nonlinearly increased with battery corrosion [37]. The internal resistance also nonlinearly increased with battery corrosion [18]. The ratio (\(\alpha_{k}\)) between contacting (\(r_{C,k}\)) and internal resistances (\(r_{\Omega,k}\)) is the linearized by least square method in Equation (20). \[\alpha_{k}=\left[ \begin{array}{c}r_{C,1}\\ \cdots\\ r_{C,k}\end{array} \right]\left[ \begin{array}{c}r_{\Omega,1}\\ \cdots\\ r_{\Omega,k}\end{array} \right] \tag{20}\]  
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), \(\alpha 2\) is the linear ratio between contacting and internal resistances of the beginning two measurements. In Equation (22), \(\alpha 1\) is the linear ratio between contacting and internal resistances of the first measurement. In Equation (23), \(\hat{r}_{\Omega,1}\) and \(\hat{r}_{\Omega,2}\) are estimated internal resistances. In Equation (24), \(P_{2}^{-}\) is the error covariance of estimation. In Equation (25), \(M_{\Omega}\) is the covariance matrix of measurement errors. In Equation (26), \(H_{\Omega}\) is the transformation matrix between measurement and state vector. The \(\left[ \begin{array}{c}r_{\Omega,k}\\ \alpha_{k}\end{array} \right]\) is state vector and \(\left[ \begin{array}{c}\hat{r}_{\Omega,2}\\ \hat{\alpha}_{2}\end{array} \right]\) is initializing as \(\left[ \begin{array}{c}r_{\Omega,2}\\ \alpha_{2}\end{array} \right]\). In Equation (27), \(K_{2}\) is the Kalman gain. In Equation (28), \(P_{2}\) is the error covariance for updated estimation:  
\[\alpha_{2}=\left[ \begin{array}{c}r_{C,1}\\ r_{C,2}\end{array} \right]\left[ \begin{array}{c}r_{\Omega,1}\\ r_{\Omega,2}\end{array} \right] \tag{21}\]  
\[\alpha_{1}=\ r_{C,1}\
u_{\Omega,1} \tag{22}\]  
\[\left[ \begin{array}{c}\hat{r}_{\Omega,1}\\ \hat{r}_{\Omega,2}\end{array} \right]=\left[ \begin{array}{c}r_{C,1}\\ r_{C,2}\end{array} \right]\alpha_{2} \tag{23}\]  
\[P_{2}^{-}=\left[ \begin{array}{cc}E\big{(}(r_{\Omega,1}-\hat{r}_{\Omega,1})^ {2}+(r_{\Omega,2}-\hat{r}_{\Omega,2})^{2}\big{)}&0\\ 0&E\Big{(}\big{(}\alpha_{2}-\frac{\alpha_{2}+\alpha_{1}}{2}\big{)}^{2}\Big{)} \end{array} \right] \tag{24}\]  
\[M_{\Omega}=\left[\begin{array}{cc} \begin{array}{cc}\left(10^{-5}\right)^ {2}&0\\ 0&\left(10^{-1}\right)^{2}\end{array} \right] \tag{25}\]  
\[H_{\Omega}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right] \tag{26}\]  
\[K_{2}=\ P_{2}^{-}H_{\Omega}^{T}\big{[}H_{\Omega}P_{2}^{-}H_{\Omega}^{T}+M_{ \Omega}\big{]}^{-1} \tag{27}\]  
\[P_{2}=[I-K_{2}H_{\Omega}]P_{2}^{-} \tag{28}\]  
Step 2. EIS Sampling and state prediction:  
After EIS sampling, a new contacting resistance \(r_{C,k}\) is obtained. In Equation (29), \(\Delta r_{C,k}\) is the incremental contacting resistance. In Equation (30), \(\Phi_{\Omega,k}\) is the state transition matrix. In Equation (31), \(\hat{r}_{\Omega,k}^{-}\) is prediction of aging internal resistance at the moment of EIS sampling, \(\hat{a}_{k}^{-}\) is the same as \(\hat{a}_{(k-1)}\). In Equation (32), \(P_{k}^{-}\) is the error covariance for following estimation. In Equation (33), \(K_{k}\) is updated before jumping to Step 3. \[\Delta r_{C,k}=r_{C,k}-r_{C,(k-1)} \tag{29}\]  
\[\Phi_{\Omega,k}=\left[ \begin{array}{cc}1&\Delta r_{C,k}\\ 0&1\end{array} \right] \tag{30}\]  
\[\left[