# Lithium Battery SOH Estimation Based on Manifold Learning and LightGBM
## 5 Instance Validation
### Feature Extraction
#### 5.1.1 Capacity Degradation Characteristics  
The capacity of a Li-ion battery is related to its complex internal physicochemical and thermal effects. It is necessary to find the health factors that replace the degradation state of the battery, so it is necessary to extract the characteristic sequence of the degradation state of the lithium battery for the parameters that can be easily monitored and to construct the set of health factors. First is the voltage decay characteristic. The time used by a brand new fully charged battery to discharge the power will gradually decrease with the increase in the number of discharge cycles. As can be seen from the discharge voltage curves in Figure 4, within 500-1500 s, the value of the voltage drop in different cycle curves increases with the number of cycles, i.e., the distance between the nominal voltage and the real-time voltage increases, so the voltage decay can be used to quantify the capacity decay. Several discharge voltage curves in Figure 4 were randomly selected, and the discharge voltage decay schematic is shown in Figure 5.  
As shown in Figure 5, in order to describe the voltage decay more comprehensively, this paper chooses to use the mean voltage decay (MVF) as the health factor. Specify 500-1500 s as the time range for extracting the health factor, and define MVF by averaging 50 voltage points within the defined time range, with each sampling time point defined as \(j\), where the MVF during the \(i\)th discharge cycle can be expressed as follows:  
\[MVF_{i}=\frac{\sum\limits_{j=1}^{50}\left|V_{n}-V_{j}\right|}{50} \tag{17}\]  
where \(MVF_{i}\) is the discharge voltage drop of the \(i\)th charge/discharge cycle; \(V_{j}\) is the \(j\)th discharge voltage point in the defined time range; \(V_{n}\) is the nominal voltage of 4.2 V.  
In addition to the average voltage drop sequence, there are some physical quantities that can also reflect the battery capacity degradation state, such as the capacity increment curve [31]. The essence of the capacity increment curve is the differentiation of the capacity \(Q\) charged or discharged by the battery during charging and discharging against the voltage \(V\), which is the magnitude of the capacity change corresponding to the unit voltage change in the battery when it occurs. The voltage plateau appearing in the charging and discharging voltage change can be transformed into the capacity increment with significant characteristics (\(dQ/dV\)). The specific equation for the IC in the discharging mode is as follows:  
\[\frac{dQ}{dV}\approx\frac{\Delta Q}{\Delta V}=\frac{Q_{v2}-Q_{v1}}{V_{2}-V_{1}} \tag{18}\]  
Figure 5: Schematic diagram of discharge voltage decay. where \(Q_{v1}\) represents the battery capacity at \(V_{1}\) and \(Q_{v2}\) represents the battery capacity at \(V_{2}\). The IC curves in the discharge experiment are shown in Figure 6a. As can be seen from Figure 6a, with the decline in the battery, there is a more obvious change in the IC curve, and the change is more significant at the peak of the IC curve and its corresponding position. To quantitatively analyze the battery aging state by using the capacity increment curve, it is also necessary to filter and smooth the curve, so that it is easier to find the characteristics and change patterns of the curve. In this paper, the generalized moving average method is chosen to smooth and filter the curve. The filtered discharge IC curve is shown in Figure 6b. As can be seen from Figure 6b, the IC curve after filtering is obviously smooth and the peak height is clearly visible, and the peak height shows a better correlation with capacity decline as the battery aging degree deepens. Meanwhile, analyzing the IC curve characteristics of different aging degrees during the battery cycle aging process, such as the left and right positions, the area contained in the peak, and other parameters, the mode and mechanism of battery aging decline can be analyzed and judged, and then the health status of the battery can be diagnosed [31; 32].