# A Data-Driven Battery Degradation Estimation Method for Low-Earth-Orbit (LEO) Satellites
## 3 Proposed Method
### Proposed Network Training  
The overall process of network training for the voltage model proposed in this study consists of three stages: data preprocessing, network selection, and model training and prediction. First, in the data preprocessing stage, various sensor data related to the battery are collected; in this study, the timespan of the data was approximately six years. The input and output features are then selected, and the data are preprocessed into a format suitable for training. The initial conditions and data, such as position, quaternion, solar angle, battery SOC, current, voltage, and mission mode, are labeled accordingly. Next, the dataset is divided to optimize training efficiency; in this study, 70% of the data were used as training data, 10% were used as validation data for designing the network structure, and the remaining 20% were used as test data for evaluating the model's performance. This structured approach ensured the model's reliability and effectiveness in predicting battery voltage. Next, in the network selection stage, the optimal structure of the deep learning model is explored. Parameters such as the initial learning rate, the number of transformer layers, the number of multi-head attention mechanisms, model dimensions, and dropout rate are configured. The model is then trained using the validation dataset designated specifically for network selection. During this process, the loss is computed using the validation dataset, and parameter adjustments are iteratively performed until the loss converges or the predefined number of iterations is reached. Ultimately, the network with the best performance is selected and saved. In this study, to ensure that the selected network structure was optimized for the satellite's unique characteristics, Bayesian optimization was employed in this process. This approach enables efficient and precise determination of the most suitable network configuration. Bayesian optimization is an efficient method for exploring the optimal parameters with minimal iterations [23]. It involves the use of a probabilistic model and an acquisition function to search for the parameter combination that maximizes model performance. In this study, the number of network layers (\(L\)), embedding dimensions (\(D\)), self-attention heads (\(H\)), and transformer stages (\(T\)) were optimized. The process begins with the initialization stage, in which an initial set of parameter combinations is randomly selected within the search space. Based on this initial dataset, a Gaussian process (GP) regression model is built to approximate the objective function, which represents model performance as a function of the parameters. Next, an acquisition function is defined to determine which parameter combinations to evaluate next. In this study, we used the expected improvement (EI) acquisition function, designed to prioritize regions with a high potential for performance improvement. The acquisition function identifies the combination of parameters that maximizes its value, and the transformer model is trained and evaluated using this combination. The resulting performance metric is added to the dataset, and the GP model is updated to refine future searches. This process is iteratively repeated until the maximum number of iterations is reached or the performance metric meets a predefined threshold. When the termination condition is satisfied, the parameter combination with the highest recorded performance for the dataset is returned as the final result. The cost function used in this optimization process is defined as follows:  
\[cost\ function=W1.\frac{1}{n}\sum\lvert Y_{ture}-Y_{prediction}\rvert+W2\cdot maximum\ error\, \tag{12}\]  
where \(n\) is the number of verification data points, \(Y_{ture}\) is the actual verification data value, \(Y_{prediction}\) is the model-predicted data value, and \(W1\) and \(W2\) are weight factors. Optimizing all data, network structures, and parameters is challenging due to the development of time constraints.