## 5 Conclusions  
In this paper, a data-driven model incorporating mechanical force for SOE estimation is proposed. A data acquisition platform is established to simultaneously collect current, voltage, temperature, and mechanical force under four operating conditions consisting of DST, BJDST, FUDS, and US06. To reduce the complexity of manual parameter adjustment, the PSO is introduced to optimize the measurement noise covariance and process noise  
Figure 11: SOE estimation results under different preload forces: (**a**) SOE curves under 1000 N; (**b**) SOE errors under 1000 N; (**c**) SOE curves under 1500 N; (**d**) SOE errors under 1500 N.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Training Data** & **Test Data** & **RMSE\%** & **MAXE\%** & **MAE\%** \\ \hline LSTM & DST, BJDST & & 1.86 & 7.78 & 1.44 \\ BiLSTM & (1000 N) & US06 & 1.16 & 3.51 & 0.93 \\ Proposed & & & 0.59 & 1.09 & 0.46 \\ \hline LSTM & & & 2.38 & 8.34 & 1.85 \\ BiLSTM & (1500 N) & US06 & 1.65 & 6.70 & 1.28 \\ Proposed & & & 0.95 & 2.48 & 0.70 \\ \hline \hline \end{tabular}
\end{table}
Table 5: SOE estimation results under different preload forces.  
covariance of the KF. BiLSTM is then integrated with the KF, which further increases the accuracy of SOE estimation. Under different operating conditions, the RMSE is less than 0.65%, the MAXE is less than 2.23%, and the MAE is less than 0.54%. Under different temperatures, the RMSE, MAXE, and MAE are less than 0.9%, 2.59%, and 0.72%, respectively. In both of these operating conditions and variable temperatures, the introduction of the mechanical force results in an average improvement of over 66% in the RMSE, MAXE, and MAE compared to the conventional electrothermal signal model, which strongly proves the potential of force signals in SOE estimation. Additionally, the BiLSTM-PSO-KF model is compared with the LSTM and BiLSTM models under conditions of 35 \({}^{\circ}\)C, 500 N, and FUDS. The proposed model demonstrates a superior fit to the actual SOE, with RMSE, MAXE, and MAE reduced to 0.62%, 2.66%, and 0.48%, respectively. To further validate the generalization and robustness of the proposed model, SOE estimation is conducted under US06 at 25 \({}^{\circ}\)C with different preload forces. The results demonstrate that the RMSE is less than 0.96%, the MAXE is less than 2.49%, and the MAE is less than 0.71%. In summary, the offline training time of the model is less than one hour. The proposed novel framework introducing mechanical force signals into SOE estimation for lithium-ion batteries--a direction rarely explored in existing literature--offers a promising technical pathway for broader application in battery state estimation.  
In the future, the existing model will be extended to other types of lithium-ion battery SOE estimation. Moreover, a joint estimation model for SOE, SOC, and SOH will be developed to achieve a more comprehensive characterization of the state of LFP batteries and enhance the engineering application capability of system-level BMS.  
Not applicable.  
The authors have not been granted authorization to publicly disclose the relevant data. The authors are grateful for the guidance of the reviewing experts. At the same time, the authors would also like to thank the Hubei Key Laboratory for High-Efficiency Utilization of Solar Energy and Operation Control of Energy Storage System for its support.  
Author [PERSON] is employed by the Powerchina Equipment Research Institute Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.