## 3 Semi-Empirical Model of Retired Battery Capacity Degradation Based on Interval Number Similarity
### Interval Number Similarity Degree Computing Model  
The \(\Delta SOC\) is a concept of interval. To obtain the retired battery capacity degradation model of the required \(\Delta SOC\) based on the test data of the limited \(\Delta SOC\), it is necessary to know the interval relationship between the \(\Delta SOC\) and the known \(\Delta SOC\). For this reason, this paper introduces the concept of interval number similarity to describe the relationship between them [34]. Based on the concept of interval number similarity, the interval measurement model is established and applied to the empirical model of retired battery capacity degradation. Let \(m=[m^{-},m^{+}]\) and \(n=[n^{-},n^{+}]\), while \(p_{j}(j=1,2,3,4)\) is the \(j\)-th largest in \(m^{-}\), \(m^{+}\), \(n^{-}\), and \(n^{+}\), and then the similarity between interval number \(m\) and \(n\) is defined as [35]. \[S_{m,n}=\frac{(p_{2}-p_{3})[1-\text{sgn}(n^{-}-m^{+})][1-\text{sgn}(m^{-}-n^{+ })]/4}{(p_{1}-p_{4})-(p_{2}-p_{3})|\text{sgn}(n^{-}-m^{+})-\text{sgn}(m^{-}-n^{ +})|/2} \tag{10}\]  
\[\text{sgn}(x)=\left\{ \begin{array}{cl}1,&x>0\\ 0,&x=0\\ -1,&x<0\end{array} \right. \tag{11}\]If we decompose Equation (6), that is:  
\[S_{m,n}= \begin{cases}0,&n^{+}\leq m^{-}\text{ or }m^{+}\leq n^{-}\\ \frac{n^{+}-m^{-}}{m^{+}-m^{-}},&n^{-}<m^{-}\leq n^{+}<m^{+}\\ \frac{m^{+}-n^{-}}{n^{+}-m^{-}},&m^{-}<n^{-}\leq m^{+}<n^{+}\\ l_{m}l_{n}^{-1},&n^{-}\leq m^{-}<m^{+}\leq n^{+}\\ l_{n}l_{m}^{-1},&m^{-}\leq n^{-}<n^{+}\leq m^{+}\end{cases}  \tag{12}\]  
where \(l_{m}=m^{+}-m^{-}\), \(l_{n}=n^{+}-n^{-}\). Semi-Empirical Model of Capacity Degradation of Retired Battery Based on Model Interval Number Similarity  
The method of the capacity degradation model based on the interval number similarity is to normalize the interval similarity as the weight of the semi-empirical model of each known \(\Delta SOC\), calculate the sum of the weights of the semi-empirical model of each known \(\Delta SOC\), and obtain the model of the \(\Delta SOC\) to be determined, as shown in Equations (13) and (14):  
First of all, the known interval number similarity can be normalized by Equation (12):  
\[S^{\prime}_{m,n_{i}}=\frac{S_{m,n_{i}}}{S_{m,n_{1}}+S_{m,n_{2}}+\cdots+S_{m,n_ {k}}} \tag{13}\]  
where \(m\) represents the \(\Delta SOC\) to be simulated, \(n_{i}\) (\(i=1,2,\ldots,k\)) represents \(k\)\(\Delta SOC\)s with known capacity degradation, \(S_{m,n_{i}}\) represents the similarity between the \(\Delta SOC\) to be simulated and the \(i\)-th known \(\Delta SOC\), while \(S^{\prime}_{m,n_{i}}\) represents the weight of the \(i\)-th expression. The capacity degradation model based on the similarity method of model interval numbers is as follows:  
\[SOH_{model}=S^{\prime}_{m,n_{1}}\times SOH_{1}+S^{\prime}_{m,n_{2}}\times SOH _{2}+\cdots+S^{\prime}_{m,n_{k}}\times SOH_{k} \tag{14}\]  
where \(SOH_{i}(i=1,2,\ldots,k)\) denotes the retired batteries capacity degradation model obtained by Equation (9) fitting from the known \(\Delta SOC\) data, and \(SOH_{model}\) is the result of the similarity method based on the interval number of the model. The flow chart of the semi-empirical model of the capacity degradation of retired batteries based on model interval number similarity is shown in Figure 1:  
Semi-Empirical Model of Capacity Degradation of Retired Batteries Based on Parameter Interval Number Similarity  
The method of the capacity degradation model based on the parameter interval number similarity is to take the interval similarity as the weight of each fitting parameter in the semi-empirical model, and the fitting parameter of the model to be solved is determined by the weight of each known parameter. Finally, the semi-empirical model of the \(\Delta SOC\) based on the parameter interval number similarity method is obtained, as shown in Equations (12) and (13). First of all, the semi-empirical model with parameters in the known \(\Delta SOC\) can be obtained from Equation (9) as follows:  
\[\left\{