## 3 Model Development
### Hybrid RVM-GM Prediction Method
Based on the decoupled data, the RVM and the GM for normal degradation and the rest time-based model for regeneration phenomenon are trained. Moreover, the training data set for the RVM is \(R_{RVM}=\{(H_{k},H_{k+1})\big{|}k,k+1\in E\}\), and the original sequences training data set for the GM is \(X^{(0)}_{GRAY}=\{H_{k}|k\in E\}\). To predict the regeneration phenomenon, we need to model the regeneration amplitude and the regeneration cycle number, respectively. Figure 4 shows the data points and fitting curves of the rest time and regeneration amplitude of the experimental data of four groups of batteries, indicating a nonlinear relationship. Furthermore, it can be seen that after a visible regeneration, the SOH degradation rate at the next several cycles will be much faster than before. [PERSON] et al. [30] assumed that the growth rate of the regeneration amplitude decreased with an increase in rest time and that the degradation of the capacity regeneration region was a uniform linear process, and therefore the regeneration amplitude and the regeneration cycle number are accurately predicted. However, the modeling of the degradation process in the regeneration region was not accurate. By observing the data of [33], we find that the degradation process of the regeneration zone was decreasing exponentially. Therefore, by obtaining the regeneration amplitude and the regeneration quantity prediction model, the regeneration area is modelled using the exponential model. The rest time is defined as the forward difference of the beginning time of each cycle, which can be represented as:  
\[DT_{k}=T_{k+1}-T_{k}, \tag{24}\]  
The rest times dataset for estimating regeneration phenomena is  
\[DH^{e}_{i}=s\cdot\phi\big{(}DT^{e}_{i}\big{)}, \tag{25}\]  
\[NUM^{e}_{i}=\left[\phi\big{(}DT^{e}_{i}\big{)}\cdot\Bigg{(}\frac{1}{N_{a}}\sum _{j=1}^{N_{a}}\frac{NUM_{j}}{\phi\big{(}DT^{e}_{j}\big{)}}\Bigg{)}+\frac{1}{2} \right], \tag{26}\]  
Figure 4: The relationship between rest time and the amplitude of the SOH regeneration. where \(s\) is a scale parameter, \(\phi(\bullet)\) is a hyperbolic tangent function, \(NUM_{i}^{e}\) is the regeneration cycle number to be estimated, \(N_{a}\) is the number of times that regeneration occurs in the training data, and \(NUM_{j}\) is the cycle number of the \(j\)th regeneration in the training data. When using the exponential degradation model to model the regeneration region, it is assumed that \(j\) is the cycle number before the \(i\)th regeneration and the health state sequence of the regeneration region can be expressed as follows:  
\[RH_{k}^{e}=H_{j}+DH_{i}^{e}-a\cdot\exp(b\cdot(k-j)), \tag{27}\]  
where \(a\) and \(b\) are undetermined parameters. In the prediction phase, we use the RVM and the GM alternately to carry out a multi-step prediction. We suppose that the plan for future use of the battery is known, which means \(\left\{T_{k}\right\}_{k=a+1}^{n}\) is known. The rest time sequence \(\left[DT_{k}^{p}\right]_{k=a+1}^{n-1}\) used for prediction is obtained by Equation (26). Because the SOH of the prediction process is unknown, we use the rest time threshold \(DT_{th}\) to indicate the occurrence of regeneration. The regeneration amplitude set \(\left\{DH_{k}^{p}\right\}_{k=1}^{M}\), and the regeneration quantity set \(\left\{NUM_{k}^{p}\right\}_{k=1}^{M}\) is predicted by Equations (25) and (26), where \(M\) is the number of times that regeneration occurs in the prediction process. Our proposed hybrid prediction algorithm is summarized as Algorithm 1.