## 2 System Dynamics Modeling
### Battery Model
#### 2.3.2 Battery Capacity Decay Model  
At present, many scholars have found through experiments that a battery's charge-discharge rate, temperature, and discharge depth are the main reasons for the battery's capacity decline [39]. Therefore, this study uses a semi-empirical model based on experimental data to estimate the battery's capacity decay. The evaluation index is expressed as Formula (5):  
\[\left\{\begin{array}{l}Q_{loss}=A\cdot\exp(-\frac{E_{a}+B\cdot C_{tall}}{R \cdot T})\cdot\text{Ah}^{z}\\ Ah=2\times N_{cyc}\times DOD\times Q_{cell}\end{array}\right. \tag{5}\]  
where \(Q_{loss}\) is the battery capacity loss; \(A\), \(B\) and \(z\) are the parameters to be fitted; \(E_{a}\) is the activation energy(J mol\({}^{-1}\)), take \(E_{a}\) = 31,700; \(R\) is the gas constant, take \(R\) = 8.314; \(T\) is the  
Figure 4: OCV-SOC curve.  
Figure 3: Battery test system.  
working ambient temperature of the battery in K; Ah is the accumulated current flow of the battery; \(DOD\) is the depth of discharge; and \(Q_{cell}\) is the capacity of the battery.  
In this paper, we defined the SOH of the battery based on its capacity and, in accordance with the capacity decay of the battery, we obtained the current SOH and actual available capacity of the battery through Formula (6):  
\[\left\{\begin{array}{l}\text{SOH}=\frac{Q_{\text{now}}-Q_{\text{EOL}}}{Q_{ new}-Q_{\text{EOL}}}\\ Q_{now}=(1-Q_{loss})\times Q_{ini}\end{array}\right. \tag{6}\]  
[PERSON] et al. [40] pointed out that there is a linear relationship between battery capacity decline and battery internal resistance, which is expressed as:  
\[Q_{loss}=k\cdot R_{0,now}+b \tag{7}\]  
where \(R_{0,now}\) is the ohmic resistance value of the current battery. Substituting Formula (7) into Formula (6), we can obtain:  
\[\frac{Q_{\text{now}}-Q_{\text{EOL}}}{Q_{\text{new}}-Q_{\text{EOL}}}=\frac{Q_{ \text{loss,EOL}}-Q_{\text{loss,now}}}{Q_{\text{loss,EOL}}-Q_{\text{loss,new}}} \tag{8}\]  
where \(Q_{EOL}\) is the capacity of the battery to reach the EOL state; \(Q_{new}\) is the capacity of the new battery; and \(Q_{now}\) is the current capacity of the battery. Substituting Formula (7) into Formula (8), we can obtain:  
\[\frac{Q_{now}-Q_{EOL}}{Q_{new}-Q_{EOL}}=\frac{R_{0,EOL}-R_{0,now}}{R_{0,EOL}-R _{0,new}} \tag{9}\]  
where \(R_{0,EOL}\) is the resistance value of the battery at the end of life (EOL) and \(R_{0,new}\) is the internal resistance of the new battery. By transforming Formula (9), the internal resistance in the battery model is expressed as:  
\[R_{0,now}=R_{0,EOL}-\frac{Q_{now}-Q_{EOL}}{Q_{new}-Q_{EOL}}(R_{0,EOL}-R_{0,new}) \tag{10}\]  
Based on the derivation analysis of the above formula, we obtained the relationship between battery capacity and internal resistance in the equivalent circuit model. Next, the lithium-ion battery was subjected to charging and discharging experiments under different test conditions to obtain the battery capacity degradation curve. As the number of charge and discharge cycles increased, the Ah passing through the battery also gradually increased. The relationship between the Ah passing through the battery and the battery capacity degradation rate is shown in Figure 5. The battery was tested at different DODs and ambient temperatures. The capacity measured at different temperatures was compensated by a compensation coefficient, and the battery capacity equivalent to 25 \({}^{\circ}\)C was obtained. Then, the experimental results were curve-fitted, and the fitting parameters were A = 53.86, B = \(-\)9.868, and z = 0.6749.