## 3 Model Development
### Hybrid RVM-GM Prediction Method  
Our purpose is to get a more accurate battery RUL long-term prediction. Hence, taking into account the effect of the regenerative phenomena, we extract the SOH values of the regeneration cycle, the regeneration cycle number, and normal degradation from the SOH sequence and predict them, respectively. Figure 2 shows the RVM and GM prediction curves of the #5 battery starting prediction at cycle 80. Starting prediction at cycle 80 means the length of the training data is 80, which is a long-term prediction for the battery's RUL. Although the RVM and GM can describe future trends of the degradation, the long-term prediction performance is still unsatisfactory. The SOH prediction results of the RVM are accurate over the first few cycles, but the accuracy will lose or even diverge after a few cycles. This problem occurs because the relevance vectors are easily affected by data noise and the coefficients matrix of the trained RVM cannot be updated during the prediction process. The GM identifies the different degree of the development trend among the system factors to predict the trend of the system, but disregards the uncertainty among the factors in the system. To predict the normal degradation trends more accurately, we propose a hybrid model for fusing the RVM and GM. Figure 3 illustrates  
Figure 1: Degradation curve of the state of health (SOH) with cycle number and calendar time for #5 battery. the framework of the hybrid RVM-GM prediction method. This framework includes the modeling phase and the prediction phase. The execution steps are as follows:  
**Step 1** decoupling the training data into two parts of normal degradation and regeneration. **Step 2** modeling of the capacity regeneration based on rest time. **Step 3** modeling of the normal degradation by the RVM and the GM. **Step 4** predicting the normal degradation and regeneration region, respectively. In particular, the normal degradation is predicted by the alternating iterative method of the RVM-GM. **Step 5** adding the predicted SOH results of each regeneration region into the normal degradation prediction sequence and getting the final SOH prediction results. We assume that the battery is fully charged and discharged in each cycle. Thus, we can get the time sequence of the SOH \(\{H_{k}\}_{k=1}^{a}\) and the start time sequence of two adjacent discharge cycles from cycle 1 to cycle \(\alpha\), where \(\alpha\) is the length of training data. In the modelling phase, we decouple normal degradation and regeneration from the training data, using \(s_{k}\) to distinguish whether there is a visible SOH regeneration at the \(k\)th cycle. \[s_{k}=\left\{ \begin{array}{c}1,DH_{k}<DH_{th}\\ -1,DH_{k}\geq DH_{th}\end{array} \right., \tag{23}\]  
where \(DH_{k}=H_{k+1}-H_{k}\) is the forward difference of the SOH between the \(k\)th and the \((k+1)\)th cycle, \(DH_{th}\) is the threshold for capacity regeneration. \(s_{k}=1\) means normal degradation, and \(s_{k}=-1\) means that there is a visible SOH regeneration from cycle \(k\)th to \((k+1)\)th. The set of cycle indices before a regeneration phenomenon can be represented as \(A=\{k|s_{k}=-1,k\in\{1,2,\cdots,\alpha-1\}|\}\). The number of elements in the set \(A\) is \(N_{a}\), which indicates the  
Figure 3: The framework of the hybrid RVM and GM model (RVM-GM) prediction method. Figure 2: Relevance vector machine (RVM) and gray model (GM) prediction curve starting prediction at the cycle 80 of the #5 battery. number of regeneration phenomena in the training data. The cycle index of the regeneration region can be extracted according to \(A\), which can be represented as \(B=\left\{k+j\big{|}H_{k+j}>H_{k},k\in A,j\in N^{+}\right\}\). Continuous cycles in \(B\) means a regeneration region. According to the division, the cycles of each regeneration region can be determined. As a result, the cycle index of the normal degenerate can be extracted, which can be represented as \(E=\{k\}k\in\{1,2,\cdots,\alpha\},k\
otin B\}\).