A fast-transient current control strategy for three-phase four-wire cascaded H-bridge inverter is presented for versatile operation of grid-connected battery energy storage systems. The proposed control method enables both symmetric and asymmetric inverter operation requirements of a stationary on-grid battery energy storage system by employing three independent pairs of synchronous reference frame PI controllers in each phase of the three-phase four-wire inverter. The independent single-phase grid synchronizations in the control structure enable reliable operation of fault-free phase(s) during grid fault conditions. The proposed control system characterizes fast-transient current tracking in asymmetric control mode. It can exchange active and/or reactive power with independent single-phase loads and/or generations connected to the three-phase system, by following given current references. MATLAB and Simulink based simulations are used to evaluate the system performance and verified by hardware experiments.