## 3 Results and Discussions
### Model Performance Comparison  
In this section, we evaluate the efficacy of the proposed MHA-BiLSTM model against four other deep learning architectures: GRU, LSTM, Attention-LSTM (A-LSTM), and Attention-BiLSTM (A-BiLSTM). The evaluation is divided into two parts. First, the performances of all five models are assessed on the training dataset to observe their learning capabilities under identical hyperparameter settings. Second, the models are evaluated on an untrained simulated dataset to examine their accuracy and generalization in realistic prediction tasks. Figure 11 presents the R\({}^{2}\) and RMSE results for voltage and SOC predictions during both the training phase and on the unseen validation dataset for all five models. In the battery voltage prediction task, the five models were evaluated on both the training and validation datasets, using identical hyperparameter settings. On the training set, the MHA-BiLSTM model demonstrated the best performance, with its R\({}^{2}\) surpassing the baseline LSTM and achieving a lower RMSE. On the validation dataset, MHA-BiLSTM also outperforms, with an R\({}^{2}\) approximately 3% higher and an RMSE reduction of 10.43% compared to the baseline LSTM. These results highlight the model's ability to obtain complex relationships between the battery parameters and its superior generalization performance. A-BiLSTM performed comparably to MHA-BiLSTM with a slightly lower R\({}^{2}\) and RMSE values. The integration of MHA in the MHA-BiLSTM model further enhanced its ability to capture intricate dependencies. Unlike A-BiLSTM, which leverages bidirectional dependencies and a basic attention mechanism, MHA-BiLSTM utilizes a multi-head attention mechanism to effectively learn the complex interrelations among the voltage, SOC, and other battery parameters. This capability allows MHA-BiLSTM to achieve more accurate predictions, demonstrating its superiority over models with only single-layer attention mechanisms. The GRU model showed a weaker performance than the LSTM-based models, likely due to its simpler architecture, which struggles with capturing the more complex temporal dependencies in the data. A-LSTM exhibited the lowest performance, likely due to suboptimal training under the current hyperparameter settings, resulting in reduced accuracy and higher prediction errors. In SOC prediction, a similar trend was observed. The MHA-BiLSTM model consistently achieved the highest R\({}^{2}\) and lowest RMSE in both the training and validation datasets,  
Figure 11: Accuracy and error statistics of five prediction models in the train and validate set: (**a**) Train R\({}^{2}\), (**b**) Train RMSE, (**c**) Validate R\({}^{2}\), and (**d**) Validate RMSE. further demonstrating its ability to find the intricate dependencies among the battery parameters. A-BiLSTM again showed a competitive performance. These findings emphasize the effectiveness of the MHA-BiLSTM model in learning complex interdependencies between battery parameters, such as the voltage and SOC, resulting in a consistently superior prediction performance across different datasets. The superior performance of the MHA-BiLSTM model can be attributed to its ability to effectively learn the complex relationships among multiple battery parameters, such as the voltage and SOC. By incorporating MHA, the model captures interdependencies between these parameters, enabling more accurate and robust predictions. Across both the voltage and SOC prediction tasks on the training dataset, the MHA-BiLSTM consistently outperformed other models by reducing the prediction error and enhancing the accuracy. This highlights its comprehensive effectiveness in addressing the challenges of the battery parameters' prediction and its capability to capture complex dependencies in battery data more effectively. To further evaluate the model's performance, a segment of untrained simulated battery charge-discharge cycle data was selected to test the model's ability to predict the SOC and voltage over a five-minute interval.