# A Data-Driven Battery Degradation Estimation Method for Low-Earth-Orbit (LEO) Satellites
## 3 Proposed Method
### Battery Modeling  
To estimate the state of the battery, a state equation is required. Figure 5 illustrates an equivalent circuit model representing the dynamic characteristics of the battery using electrical components. The open-circuit voltage (OCV) represents the intrinsic voltage of the battery, whereas the differential resistance (R\({}_{\text{Diff}}\)) and differential capacitance (C\({}_{\text{Diff}}\)), connected in parallel, model the internal voltage dynamics of the battery. These two components are key to explaining the battery's dynamic behavior. A differential voltage (V\({}_{\text{Diff}}\)) is generated within this parallel circuit and manifests as the terminal voltage (V\({}_{\text{k}}\)) through the internal resistance (R\({}_{\text{i}}\)). The terminal current (I\({}_{\text{k}}\)) flows according to the charge and discharge processes of the battery. Through analysis of these parameters, the state of the battery, including its dynamic response and internal conditions, can be effectively estimated. The state variables are selected to represent the battery's state of charge, differential voltage, and internal resistance, respectively. These state variables vary over time and are expressed using state equations, shown herein as Equations (4)-(6). \[x=\left[SOC\ V_{diff}\ R_{i}\right]\,, \tag{4}\]  
where \(SOC\) represents the state of charge, \(V_{diff}\) is the differential voltage, and \(R_{i}\) is the internal resistance. \[x_{k+1}=A_{k}x_{k}+B_{k}i_{k}+w_{k} \tag{5}\]  
\[V_{k}=C_{k}x_{k}+v_{k} \tag{6}\]  
Here, \(x_{k}\) represents the state variables; \(A_{k}\), \(B_{k}\), and \(C_{k}\) are state equation matrices; wk denotes process noise; and vk represents sensing noise. The matrices \(A_{k}\) and \(B_{k}\) in the state equations are defined as shown in Equation (7). Matrix \(A_{k}\) is formulated in terms of the sampling time, differential resistance, and capacitance, reflecting the dynamic characteristics of the battery. Matrix \(B_{k}\) represents voltage changes driven by the input current. \[A_{k}= \begin{bmatrix}1&0&0\\ 0&e^{\frac{-M}{R_{diff}}\cdot\frac{-M}{0}}&0\\ 0&0&1\end{bmatrix} ,B_{k}= \begin{bmatrix}0&\\ R_{diff}\cdot\left(1-e^{\frac{-M}{R_{diff}}\cdot\frac{-M}{R_{diff}}}\right)\\ 0\end{bmatrix} , \tag{7}\]  
where \(x_{k}\) represents the sampling time. The measurement value incorporates the derivative of the function f(SOC), which describes the nonlinear relationship between the SOC and voltage, along with the input current. This function is a higher-order polynomial that explains the nonlinear relationship between the SOC and voltage, ensuring accurate modeling of the battery's SOC. Figure 6 illustrates the nonlinear relationship between the battery's OCV and SOC. In general, battery voltage and the SOC exhibit variations depending on temperature. Therefore, temperature can be included in the SOC and OCV models. However, medium-to-large low-Earth-orbit (LEO) satellites typically maintain strict temperature control to preserve battery performance, allowing the influence of temperature to be neglected. In the case of KOMPSAT-5, which was used for this study, battery temperature is set to remain around 20 \({}^{\circ}\)C on average. Therefore, the influence of temperature was neglected, and the OCV was expressed solely as a function of the SOC. To describe the relationship between voltage and the SOC, a curve-fitting method was employed [21]. The order of the fitting function was determined by minimizing the residual value through iterative adjustments. Finally, the coefficients for the selected order were determined to establish the  
Figure 5: Battery electrical circuit modeling diagram. model. In this study, \(f(SOC)\) is modeled as a sixth-order polynomial to effectively capture the intricacies of the SOC-voltage dynamics, as shown below. \[OCV=f(SOC)=a_{6}SOC^{6}+a_{5}SOC^{5}+a_{4}SOC^{4}+\dots+a_{1}SOC^{1}+a_{0}, \tag{8}\]  
where \(a_{6}\), \(a_{6}\), \(\dots\)\(a_{o}\) represent coefficients of each order.