A collection of optimization problems central to power system operation requires distributed solution architectures to avoid the need for aggregation of all information at a central location. In this paper, we study distributed dual subgradient methods to solve three such optimization problems. Namely, these are tie-line scheduling in multi-area power systems, coordination of distributed energy resources in radial distribution networks, and joint dispatch of transmission and distribution assets. With suitable relaxations or approximations of the nonconvex power flow equations, all three problems can be reduced to a multi-agent constrained convex optimization problem. We utilize a constant step-size dual subgradient method with averaging on these problems. For this algorithm, we provide a convergence guarantee that is shown to be order-optimal. We illustrate its application on the grid optimization problems.