This article proposes a BLS-STCKF estimation and diagnosis scheme of state of charge (SOC) and RUL prediction for power batteries in electric vehicles (EVs). First, the process of SOC estimation is considered as a time series problem. The walk-forward validation method has been applied in BLS (Broad Learning System) estimator training and validation. Then, a sliding window with variable width is used to help the time series SOC estimation improve. Meanwhile, STCKF (Strong Tracking Cubature Kalman Filter) is combined with BLS to help search for an optimal hyperparameter for BLS. Then, this article takes the SOC diagnosis as a problem of diagnosing abnormal time segments and uses HOT SAX(Heuristically Ordered Time series using Symbolic Aggregate ApproXimation) to detect SOC anomalies. Moreover, a prediction method of RUL (Remain Useful Life) has been designed by BLS-STCK. In light of result of RUL, the capacity degradation is considered to improve SOC estimation, and capacity retention coefficient is posed to adjust BLS-STCKF when battery aged. Finally, the proposed methods has also been evaluated by real driving data of EVs. Based on those critical experiments, rational analysis and conclusion have been illustrated that the proposed method has competitive accuracy.