This paper considers the dynamic economic dispatch problem for a group of generators with storage that communicate over a weight-balanced strongly connected digraph. The objective of the generators is to collectively meet a certain load profile, specified over a finite time horizon, while minimizing the aggregate cost. At each time slot, each generator decides on the amount of generated power and the amount sent to/drawn from the storage unit. The amount injected into the grid by each generator to satisfy the load is equal to the difference between the generated and stored powers. Additional constraints on the generators include bounds on the amount of generated power, ramp constraints on the difference in generation across successive time slots, and bounds on the amount of power in storage. We synthesize a provably-correct distributed algorithm that solves the resulting finite-horizon optimization problem starting from any initial condition. Our design consists of two interconnected dynamical systems, one estimating the mismatch in the injection and the total load at each time slot, and another using this estimate as a feedback to reduce the mismatch and optimize the total cost of generation, while meeting the constraints.