Abstract: To address the frequency deviation issue in distributed generation systems caused by power mismatches, this paper proposes a distributed finite-time control algorithm based on low-bandwidth communication networks for multi-agent AC microgrids with battery energy storage systems (BESS). This control algorithm, founded on the Lyapunov method and homogeneous approximation theory, ensures that system nodes converge rapidly within a certain time without dependence on initial conditions. Moreover, to significantly alleviate communication burdens, an event-triggered communication mechanism is designed to avoid Zeno behavior, and the sufficient conditions for the triggered boundary behavior are derived. Simulation analyses are conducted on multiple islanded AC microgrid cases. The research results show that, compared with traditional controllers, the proposed control algorithm can coordinate BESS to eliminate deviations from the standard frequency while ensuring that the convergence time does not vary with initial conditions; simultaneously, the problem of the state of charge (SoC) balance can be solved, improving synchronization speed, reducing communication burdens, and ensuring the stability and reliability of the entire system.