## 3 Semi-Empirical Model of Retired Battery Capacity Degradation Based on Interval Number Similarity
### An Improved Semi-Empirical Model Fitting Method Based on the Data-Driven Method  
To fit the capacity degradation model of retired batteries as accurately as possible based on the limited \(\Delta\)SOC data, and at the same time accurately simulate the cycle aging effect of \(\Delta\)SOC on batteries, this paper proposes a data-driven improved semi-empirical model fitting method. The flow chart of method is shown in Figure 2:The empirical model of the battery usually obeys the nonlinear relationship, such as exponential functions, quadratic functions, and so on. Moreover, the data-driven method has high requirements for the accuracy of function fitting. Thus, this paper adopts the nonlinear least squares algorithm for data fitting of the above problems [40]. It is known that the sample number of a set of measured data (\(x\), \(y\)) is \(n_{t}\), and it approximately obeys the nonlinear function \(f\) (\(s\), \(x\), \(t\)) with undetermined parameter vector \(s\); accordingly, the sum of squares of the fitting error, that is, the sum of squares of residuals, can be calculated as:  
\[Z=\sum_{t=1}^{n_{t}}w_{t}(y_{t}-\hat{y}_{t})^{2}=\sum_{t=1}^{n_{t}}w_{t}(y_{t}- f(\mathbf{s},x_{t},t))^{2} \tag{20}\]  
where \(w_{t}\) is the weight of the \(t\)-th sample; if the importance of each sample is identical, then \(w_{t}=1/n\); and \(\hat{y}_{t}\) is the estimated value of the \(t\)-th sample. To ensure the best effect of fitting the curve, it is necessary to minimize \(Z\) as below:  
\[\text{min}Z=\text{min}\sum_{t=1}^{n_{t}}w_{t}(y_{t}-f(\mathbf{s},x_{t},t))^{2} \tag{21}\]  
Considering that \(f\) (\(s\), \(x\), \(t\)) is a nonlinear function, a set of nonlinear equations will be obtained when solving the partial derivative of the undetermined parameter vector based on Equation (21), so the iterative method is used for the solution [40]. First, set the initial value of the undetermined parameter vector to \(\mathbf{s}^{0}\), and the vector \(\hat{\mathbf{Y}}(\mathbf{s})\) composed of the sample estimate \(\hat{y}_{t}\) performs Taylor series expansion around \(\mathbf{s}^{0}\), as shown in the following formula:  
\[\hat{\mathbf{Y}}(\mathbf{s})\approx\hat{\mathbf{Y}}\Big{(}\mathbf{s}^{0}\Big{)}+\left[\frac{ \partial\hat{\mathbf{Y}}(\mathbf{s})}{\partial\mathbf{s}^{T}}\right]_{\mathbf{s}=\mathbf{s}^{0}} \times\Big{(}\mathbf{s}-\mathbf{s}^{0}\Big{)} \tag{22}\]  
Next, update \(\mathbf{s}\) with \(\partial Z/\partial\mathbf{s}^{T}=0\) as a known condition, that is:  
\[\frac{\partial Z}{\partial\mathbf{s}^{T}}=-2\left[\frac{\partial\hat{\mathbf{Y}}(\bm {s})}{\partial\mathbf{s}^{T}}\right]^{T}\times\big{[}\mathbf{Y}-\hat{\mathbf{Y}}\big{]} \approx-2\mathbf{J}_{0}\Big{[}\mathbf{Y}-\hat{\mathbf{Y}}\Big{(}\mathbf{s}^{0}\Big{)}-\mathbf{J}_ {0}\times\Big{(}\mathbf{s}^{1}-\mathbf{s}^{0}\Big{)}\Big{]}=0 \tag{23}\]  
Figure 2: Flow chart of improved semi-empirical model fitting method based on data-driven method. where \(\mathbf{Y}\)is the vector formed by the sample value \(y_{t}\). \(\mathbf{J}_{0}\) is the Jacobian matrix corresponding to the initial value, which can be calculated according to:  
\[\mathbf{J}_{0}=\left[\frac{\partial\hat{\mathbf{Y}}(\mathbf{s})}{\partial s^{T} }\right]_{\mathbf{s}=\theta^{0}} \tag{24}\]  
Combine Equation (23) with Equation (24), that is:  
\[\mathbf{s}^{1}=\mathbf{s}^{0}+\left(\hat{J}_{0}^{T}\mathbf{J}_{0}\right)^{-1} \hat{J}_{0}^{T}\left[\mathbf{Y}-\hat{\mathbf{Y}}\left(\mathbf{s}^{0}\right)\right] \tag{25}\]  
Similarly, the Jacobian matrix \(\mathbf{J}_{p}\) under the \(p\)-th iteration can be obtained by updating the above Equations (22)-(25), that is:  
\[\mathbf{J}_{p}=\left[\frac{\partial\hat{\mathbf{Y}}(\mathbf{s})}{\partial s^{ T}}\right]_{\mathbf{s}=\mathbf{s}^{p}} \tag{26}\]  
In addition, the corresponding correction is:  
\[\left\{ \begin{array}{l}\mathbf{s}^{p+1}=\mathbf{s}^{p}+\Delta s^{p}\\ \Delta s^{p}=\left(\hat{J}_{p}^{T}\mathbf{J}_{p}\right)^{-1}\hat{J}_{p}^{T} \left[\mathbf{Y}-\hat{\mathbf{Y}}(\mathbf{s}^{p})\right]\end{array} \right.