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This paper presents a novel hierarchical framework for real-time, network-admissible coordination of responsive grid resources aggregated into virtual batteries (VBs). In this context, a VB represents a local aggregation of directly controlled loads, such as smart inverters, electric water heaters, and air-conditioners. The coordination is achieved by solving an optimization problem to disaggregate a feeder's desired reference trajectory into constraint-aware set-points for the VBs. Specifically, a novel, provably-tight, convex relaxation of the AC optimal power flow (OPF) problem is presented to optimally dispatch the VBs to track the feeder's desired power trajectory. In addition to the optimal VB dispatch scheme, a real-time, corrective control scheme is designed, which is based on optimal proportional-integral (PI) control with anti-windup, to reject intra-feeder and inter-feeder disturbances that arise during operation of the power system. Simulation results conducted on a modified IEEE test system demonstrate the effectiveness of the proposed multi-layer VB coordination framework.