# State of Health Estimation for Lithium-Ion Batteries Based on TCN-RVM
## 2 Materials and Methods
### Extraction and Optimization of Health Indicators
#### 2.1.2 Extraction of Health Factors  
Health indicators of lithium-ion batteries are key parameters for evaluating the degree of battery degradation. These indicators are typically extracted from operational data or aging characteristics of the battery. In this study, feature extraction is performed based on charge-discharge curves. During the charge and discharge process of a lithium-ion battery, the voltage exhibits relatively stable stages, during which the migration rate of lithium ions and the reaction rate reach a dynamic equilibrium. This stage is referred to as the voltage plateau. Incremental capacity (IC) analysis is a method that reveals the battery aging state by analyzing the differential characteristics of capacity with respect to voltage during charging and discharging. It enables the transformation of voltage plateau regions into identifiable peaks on the IC curve. The basic principle is as follows:  
\[\frac{dQ}{dV}\approx\frac{\Delta Q}{\Delta V}=\frac{Q_{k}-Q_{k-1}}{V_{k}-V_{k- 1}} \tag{1}\]  
In the expression, \(Q_{k}\) and \(Q_{k-1}\) represent the charged capacities at time steps \(k\) and \(k-1\), respectively, while \(V_{k}\) and \(V_{k-1}\) denote the corresponding battery voltages at those time steps. Based on the above principle, this study extracts five health indicators from  
Figure 1: Parameters of CS2 lithium-ion batteries. the incremental capacity (IC) curve: the peak height, peak position, left peak area, right peak area, and total IC curve area. These health indicators are designated as HF1, HF2, HF3, HF4, and HF5, respectively, to analyze the battery degradation level. Since battery aging phenomena often exhibit strong dependencies, Mutual Information (MI) is computed to quantify the nonlinear correlations between the health indicators (HF1-HF5) and battery capacity degradation. Table 1 presents the MI values for cells CS2_36, CS2_37, and CS2_38. As shown in Table 1, for cells CS2_36, CS2_37, and CS2_38, the MI values of health indicators HF1, HF2, HF4, and HF5 all exceed 4.1 \(\times\) 10\({}^{-1}\), demonstrating strong nonlinear dependencies. The MI value of HF3 for CS2_36 is 3.9298 \(\times\) 10\({}^{-1}\), while for CS2_37 and CS2_38, the values are 4.2102 \(\times\) 10\({}^{-1}\) and 4.0741 \(\times\) 10\({}^{-1}\), respectively, indicating a consistently significant nonlinear dependency. Since the incremental capacity (IC) curve is essentially the derivative of capacity with respect to voltage, slight fluctuations in voltage will be significantly amplified in differential calculations, resulting in considerable noise interference and affecting the accuracy of the curve morphology and feature extraction. To effectively mitigate this issue, this paper applies a Gaussian filter to smooth the original voltage-capacity data before differential processing. This method can effectively remove high-frequency noise while preserving the overall trend of the signal. The Gaussian filter for smoothing the capacity-voltage data can be expressed in the form of one-dimensional discrete convolution:  
\[\widetilde{Q}(V_{i})=\sum_{j=-k}^{k}G(j)\cdot Q\big{(}V_{i+j}\big{)} \tag{2}\]  
where \(\widetilde{Q}(V_{i})\) denotes the smoothed capacity value at voltage point \(V_{i}\) after filtering; \(Q\big{(}V_{i+j}\big{)}\) represents the original capacity value at voltage point \(V_{i+j}\) before filtering; \(G(j)\) is the Gaussian kernel function (weight); \(k\) is the half-width of the window; and \(j\in[-k,k]\) is the index range of the smoothing window. The Gaussian weight function \(G(j)\) is defined as follows:  
\[G(j)=\frac{1}{\sqrt{2\pi\sigma}}\text{exp}\bigg{(}-\frac{j^{2}}{2\sigma^{2}} \bigg{)} \tag{3}\]  
In the formula, \(j\) is the discrete offset relative to the central point (an integer, \(-k\leq j\leq k\)) and \(\sigma\) is the standard deviation of the Gaussian kernel, which controls the smoothing degree of the filter. The larger \(\sigma\) is, the stronger the filtering effect and the more obvious the smoothing effect; \(\text{exp}(\cdot)\) is the exponential function.