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We develop a stochastic energy scheduling model for a local-area smart-grid system with a single energy source and multiple energy consumers. The tasks of the energy consumers are classified into two categories, namely, the stochastic background tasks and the deterministic dynamic tasks. The objective is to schedule the energy consumptions of the dynamic tasks to maximize the expected system utility under the given energy consumption and energy generation constraints. To make this problem tractable, using rolling horizon optimization and Gaussian approximation we transform the original stochastic optimization problem into a convex optimization problem with linear constraints. We then derive a distributed Newton's method to solve this problem, and design a message-passing mechanism for a distributed implementation of the algorithm with limited information exchange between the energy consumers and the energy source. In simulations, the proposed distributed Newton's method converges for the system under consideration, while the traditional dual decomposition method does not converge to a primary feasible solution; and thus it is a powerful practical tool for real-time control of smart-grid systems.