A robust Hammerstein-based method for state-of-charge estimation of lithium-ion batteries using a multi-strategy whale optimization algorithm

Accurate and robust state-of-charge (SOC) estimation remains challenging under nonlinear battery behavior, stochastic disturbances, and varying operating conditions. This study develops a Hammerstein-based SOC estimation framework that combines an extreme learning machine (ELM) for static nonlinear mapping with an autoregressive moving-average model with exogenous input (ARMAX) for dynamic and stochastic behavior. A multi-strategy whale optimization algorithm (MSWOA), integrating Sobol initialization, Cauchy mutation, a nonlinear convergence factor, and adaptive weighting, is used exclusively to identify the five ARMAX coefficients, while the ELM parameters are determined sequentially. Experiments under ICCDT, DST, and FUDS profiles show that the proposed ELM–ARMAX model outperforms BP, LSTM, CNN-GRU, alternative block-oriented structures, and a conventional EKF baseline. In the main SOC comparison, RMSE remains below 0.17% across the investigated temperatures, with a 32.4% reduction relative to EKF under DST. Repeated-run ablation tests show that Full MSWOA reduces mean training RMSE by approximately 45.2% compared with standard WOA, and all paired comparisons remain significant after Holm correction. Using the identified model with fixed parameters in a closed-loop recursive implementation, robustness tests further demonstrate stable recovery from initial-SOC uncertainty and a mean RMSE below 1% under 20 dB measurement noise. The estimator exhibits good physical consistency, while a P95 online processing time of 0.0141 ms per sample provides substantial computational margin for real-time SOC estimation.