A Closed-Loop Hybrid Framework for SOC Estimation of Lithium Batteries: Integrating EKF with Improved ELM

Accurate state of charge (SOC) estimation of lithium-ion batteries is essential for enhancing energy utilization and safety performance. Traditional single SOC estimation methods often lack reliability. To address these limitations, this paper proposes a novel closed-loop hybrid framework that integrates an extended Kalman filter (EKF) with an improved extreme learning machine (ELM) optimized by the Aquila optimizer (AO), termed AO-ELM-EKF. The key innovation lies in the closed-loop structure that dynamically couples the model-based EKF and the data-driven AO-ELM. The EKF provides preliminary SOC estimates and filtering parameters, which are then used as inputs to the AO-ELM to predict and compensate for the estimation error in real time. This feedback mechanism effectively suppresses error accumulation and enhances estimation reliability. The AO is employed to optimize the initial weights and thresholds of the ELM, significantly improving its nonlinear mapping capability and generalization performance. Experimental results under multiple driving cycles (FUDS, BJDST, US06) demonstrate that the proposed method achieves root-mean-square errors (RMSE) as low as 0.13 and 0.18% with 80% and 60% training data, respectively, outperforming conventional EKF, ELM-EKF, and several state-of-the-art hybrid algorithms. The proposed framework offers a promising solution for high-precision SOC estimation in real-world BMS applications.