## 2 Empirical Model of Retired Battery Capacity Degradation  
The capacity loss of a battery is usually quantified by changes in the state of health (SOH) of the battery. The \(SOH\) of the retired battery can be used to quantify its capacity decay, that is, the ratio of the current maximum discharge capacity to the initial capacity of the retired battery [28], as shown in Equation (1):  
\[SOH=\frac{C_{i}}{C_{0}}\times 100\% \tag{1}\]  
where \(C_{i}\) represents the maximum discharge capacity of the retired battery and \(C_{0}\) denotes the initial capacity of the new battery, with their unit's ampere-hours (Ah). \(SOH\) reflects the current aging degree of the battery. According to IEEE Standard 1188-1996 [29], when the initial capacity of the battery declines to 80%, the battery should be replaced. At the same time, the percentage of battery capacity loss can be defined as follows:  
\[Q_{loss}=\left(1-\frac{C_{i}}{C_{0}}\right)\times 100\% \tag{2}\]  
where \(Q_{loss}\) denotes the percentage of capacity loss. The relationship between the \(Q_{loss}\) and \(SOH\) is:  
\[SOH+Q_{loss}=1 \tag{3}\]  
According to the survey report released by Trend Force, with the large-scale decomposition of power and energy storage batteries in the future, it is expected that the global power and energy storage battery recycling scale will exceed 1 TWh by 2030, of which the lithium iron phosphate battery recycling scale will account for more than 58% [30]. The research on the echelon utilization of retired power batteries should focus on lithium iron phosphate batteries, and this paper will carry out a follow-up analysis of retired lithium iron phosphate power batteries. Reference [31] researched the law of capacity degradation of LiFePO4 batteries caused by cycles through experiments and established an empirical model of battery cycle life under the influence of temperature, DOD, and charge/discharge ratio. The model uses a general equation to describe the battery capacity degradation under all conditions, and its specific function form is available as follows:  
\[Q_{loss}=\delta\cdot\exp\left[\frac{-E_{a}}{R\cdot T}\right]\left(Ah\right)^{z} \tag{4}\]  
\[Ah=N\cdot DOD\cdot Q_{b} \tag{5}\]  
where \(\delta\) is the pre-exponential factor; \(E_{a}\) is the activation energy of the battery at the current temperature, and it defaults to a fixed value at the standard operating temperature; \(R\) is the molar gas coefficient and its value is 8.314 J/K\(\cdot\)mol; \(T\) is the absolute temperature; and \(z\) is the power law factor. \(Ah\) is the Ah throughput, which is expressed as Equation (5), in which \(N\) is the cycle number of the retired batteries, and is the full cell capacity. The exponential term \(z\) indicates that the temperature obeys the Arrhenius Law. However, this model is based on the capacity degradation stage of the new battery from the initial capacity to the battery's retirement, and it may not be suitable for decommissioning batteries. Reference [32] conducted several sets of cyclic aging experiments on retired 18650 lithium iron phosphate power batteries and established a cycle life model of retired batteries related to battery charge/discharge rate, DOD, and ambient temperature based on the measured data. The form of the model is the same as Equation (4), which just verifies the feasibility and accuracy of Equation (4) for fitting the capacity degradation rule of a retired 18650 lithium iron phosphate power battery. Reference [33] improved the empirical model to overcome the problem of poor fitting accuracy of the above-mentioned retired battery capacity degradation model because it did not involve the separate effects of charge/discharge rate and DOD. The inverse power law model is used to describe the separate effects of charge/discharge rate \(C_{Rate}\) and \(DOD\) on the cycle life of lithium-ion batteries.