Regeneration-aware online SOH estimation framework for liquid metal batteries based on fragmented data

Liquid metal batteries (LMBs) are regarded as a promising candidate for grid energy storage, owing to their long lifespan and high safety. However, their accurate state of health (SOH) estimation is significantly hindered by the prevalence of fragmented data and the pronounced capacity regeneration phenomena observed during their high-temperature operation. To address these dual challenges, a hybrid modeling method that fuses physical prior knowledge with data-driven techniques is proposed. The framework is structured into two stages. In the first stage, to tackle data fragmentation, a curve reconstruction model based on a weak physics-informed neural network (W-PINN) augmented CNN-BiLSTM architecture is constructed. This model is capable of reconstructing complete LMB profiles from arbitrary sparse segments covering only 7% of the discharge range and remains robust under continuous and aging-biased data loss, effectively mitigating the physical distortion issues inherent in purely data-driven reconstruction models. In the second stage, to specifically address the capacity regeneration observed over the ultra-long lifecycle, a robust online SOH estimation strategy is developed. Based on the reconstructed curves, eight base health features are enhanced via canonical correlation analysis (CCA) and then processed by an online sequential extreme learning machine (OSELM) with prediction-based capacity-memory temporal features, enabling accurate SOH tracking under a strictly leak-free sequential inference protocol. Experimental results demonstrate that the proposed method achieves a reconstruction root mean square error (RMSE) of 0.0898 Ah and an SOH estimation RMSE of 0.1443% under fragmented input conditions with 20% reconstructed curves, exhibiting superior performance compared to existing baselines. Further evaluation on a 50 Ah LMB dataset validates the scalability of the complete two-stage framework, while the Oxford lithium-ion battery dataset provides supplementary evidence for the cross-chemistry adaptability of the Stage-1 reconstruction methodology. Ultimately, this work offers a practical and reliable solution for health monitoring of LMB-based energy storage systems (ESSs).