## 3 System Modeling and Mathematical Formulation
In this study, we set the sampling rate to 1 h, a commonly used interval for data collection in battery monitoring systems. The SoC is a critical parameter in battery management systems, indicating the remaining capacity of the battery relative to its fully charged state. SoC is defined as the ratio of the remaining capacity to the battery capacity, typically expressed as a percentage ranging from 0 to 1 or 0 to 100%. Here, 0 represents a fully discharged battery, while 1 represents a fully charged battery. During each iteration of the data collection process, SoC is calculated using the current capacity estimated by the EKF and the battery's initial capacity. SoH is commonly associated with the capacity or ability of the battery to store and deliver charge effectively over time. We select a degradation model that best represents the aging process of the battery. Let \(X\) represent the state vector estimated by the EKF, with one of its components denoting the battery capacity, labeled as \(C\). Denoting the initial capacity by \(C_{i}\), we express a capacity fade degradation model as Equation (13), where \(C\) is the estimated current capacity of the battery, \(C_{i}\) is its initial capacity when new, and SoH represents the state of health of the battery. In this model, as the battery capacity \(C\) diminishes over time due to aging, the SoH value will decrease accordingly. When \(C=C_{i}\), SoH will be 1, indicating the battery is in perfect health. As \(C\) decreases over time, SoH will also decrease accordingly. The SoH model update rate refers to how frequently the SoH estimation is conducted. In our model, SoH is estimated at regular intervals corresponding to the data collection frequency. The SoH model can be evaluated at different SoC levels to assess its performance under various operating conditions. This study introduces a novel approach to EV battery SoH prediction through the integration of EKF and DL models. Our research stands out for developing a comprehensive system that harmonizes the strengths of EKF's deterministic modeling with the data-driven capabilities of deep learning techniques. This integration facilitates more precise and dependable predictions of EV battery SoH by capitalizing on the prior knowledge of battery behavior embedded in the EKF framework and the deep learning models' ability to discern complex patterns from data. We devised an algorithm utilizing the EKF to estimate the SoH of the EV battery, named the EKF-SoH algorithm. The symbols and definitionsin the algorithm are presented in Table 1. The pseudocode of the EKF-SoH algorithm is shown in Algorithm 1. By leveraging EKF's Bayesian estimation principles, our system adeptly integrates prior knowledge of battery dynamics and nonlinear behaviors into the prediction process. This ensures resilience and adaptability to diverse operational conditions and degradation patterns inherent in real-world EV applications. Prior knowledge of battery dynamics utilized in the prediction process involves understanding the relationship between battery internal states (such as voltage, current, temperature, and state of charge) and the observed measurements. This knowledge is integrated into the dynamic and measurement models employed in the EKF. The battery dynamics were modeled at a specific time scale, which may vary depending on the application. Specifically, the battery dynamics were modeled at the sampling rate scale, representing the time interval between consecutive measurements of the battery's characteristics during the estimation process. Additionally, the incorporation of deep learning models augments the system's capacity to capture intricate relationships and subtleties in battery health data, thereby enhancing prediction accuracy and generalization performance. The fusion of EKF and deep learning models in our system represents a significant advancement in EV battery SoH prediction methodologies.