## 4 Aging Model Establishment
### Superposition  
Because the aging behavior of batteries is nonlinear and is affected by more than one variable, the capacity decay caused by aging cannot be superposed directly. Such as the example shown in Figure 10, we can regard the aging event that \"maintains the SOC-level at 50% for three months\" as three identical continuous aging events that \"maintains the SOC-level at 50% for one month\". If we apply the direct superposition method on the latter, which means to superpose the capacity decay by directly adding them together, we will obtain a result that has a considerable disparity from the result of the former.  
Seeing the incompleteness of the direct superposition method, another method, named the mapping superposition method, is proposed in this study, as shown in Figure 11. In other words, before the superposition, the equivalent quantity of aging features should be calculated according to its former state of the amount capacity fade.  
Figure 8: A schematic diagram for aging feature extraction.  
Figure 9: A schematic diagram of the applied rainflow counting method of SOC. (**a**) Raw SOC curve. (**b**) Curve After filtering continuous values. (**c**) Only the relative peaks and valleys are left, and the values between the peaks and valleys are removed. (**d**) Start rainflow cycle counting method.  
In calendar aging, the former decay quantity must be \"mapped\" to the curve corresponding to the SOC level of the aging event to be superposed, so as to obtain the equivalent placing time. The equivalent placing time plus the placing time of this aging event is the equivalent total placing time, which can be used to calculate the aging quantity further.  
The relevant calculation formula can be derived from Equation (1):  
\[t_{eq}=9.00831\cdot C_{f,cal}^{t}{}^{1.25}\cdot e^{-0.009235\cdot SOC} \tag{7}\]  
\[C^{\prime}_{f,cal}=0.1723\cdot e^{0.007388\cdot SOC}\cdot(t_{eq}+\Delta t)^{0.8} \tag{8}\]  
In Equation (7), \(t_{eq}\) represents the equivalent placing time, \(C_{f,cal}\) represents the former calendar aging amount, and \(SOC\) represents the storage SOC level of this aging event. According to this formula, the equivalent placing time \(t_{eq}\) under the conditions of the aging event can be derived. Meanwhile, in Equation (8), \(C^{\prime}_{f,cal}\) and \(\Delta t\) represent the final calendar aging capacity fade after superposing and the placing time in the aging event, respectively.  
On the other hand, in the case of cycling aging, the former aging amount should be \"mapped\" to the curve corresponding to the conditions including cycle depth and the average SOC of the aging event to be superposed, so that the equivalent number of cycles can be derived. The equivalent number of cycles plus the number of cycles of this aging event is the equivalent total number of cycles, which can be used to calculate the aging amount further.  
\[nc_{eq}=2267.573696\cdot C_{f,cyc}{}^{2}\cdot e^{0.03886\cdot SOC}\cdot cd^{-1. 4324} \tag{9}\]  
\[C^{\prime}_{f,cyc}=0.021\cdot e^{-0.01943\cdot SOC}\cdot cd^{0.7162}\cdot(nc_{ eq}+\Delta nc)^{0.5} \tag{10}\]  
Similarly, in Equation (9), \(nc_{eq}\), \(C_{f,cyc}\), \(cd\), and \(SOC\) represent the equivalent number of cycles, the former cycling aging amount, the cycle depth, and the average SOC of this aging  
Figure 11: A schematic diagram for superposition.  
Figure 10: A schematic diagram for direct superposition.  
event, respectively. In Equation (10), \(C^{\prime}_{f,cyc}\) stands for the capacity fade after superposing, while \(\Delta nc\) refers to the number of cycles of the aging event.  
Applying the mapping superposition method, a result closer to the theoretical values than that when using the direct superposition method can be obtained, as shown in Figure 12.