In this paper, a fully distributed hierarchical control strategy is proposed for operating networked gridsupporting inverters (GSIs) in islanded ac microgrids (MGs). The primary control level implements frequency and voltage control of an ac MG through a cascaded structure, consisting of a droop control loop, a virtual impedance control loop, a mixed H 2 /H ∞-based voltage control loop, and a sliding-mode-control-based current loop. Compared to conventional proportional-plus-integral-based cascaded control, the proposed cascaded control does not require a precise model for the GSI system. The proposed secondary control level implements distributed-consensus-based economic automatic generation control and distributed automatic voltage control, which integrates the conventional secondary control and tertiary control into a single control level by bridging a gap between traditional secondary control and tertiary control. Simulation results demonstrate the effectiveness of the proposed hierarchical control strategy. Index Terms-Automatic voltage control (AVC), distributed consensus, distributed hierarchical control, droop control, economic automatic generation control (EAGC), grid-supporting inverter (GSI).