Aggregated battery energy storage systems (ABESSs) play an important role in smart grids. This study considers distributed ABESSs containing electric vehicle (EV) aggregators and battery energy storage systems (BESSs). Due to randomness and uncertainty caused by EV aggregators, the distributed ABESSs face heterogeneous random communication failures between the EV aggregators and the BESSs. Aiming to solve this problem, this study proposes a distributed state-of-charge (SoC) and power balance estimation strategy for ABESSs with an event-triggered mechanism. Different from the traditional SoC balance problem definition in the BESSs, this study introduces an innovative balance convergence condition to solve the problem of power imbalance between the EV aggregators and the BESS units. In addition, an adaptively robust estimation algorithm is designed to estimate the precise value of SoC and power under heterogeneous random failures caused by EV aggregators. Further, a distributed periodic dynamic event-triggered mechanism is developed considering redundant messages in the ABESS communication channels. This mechanism can both handle the Zeno phenomenon and obtain a larger minimum time interval with fewer sent measurements. The simulation results demonstrate the effectiveness of the proposed distributed approach in both the discharging and charging modes of ABESSs.