  This paper investigates distributed control and incentive mechanisms to
coordinate distributed energy resources (DERs) with both continuous and
discrete decision variables as well as device dynamics in distribution grids.
We formulate a multi-period social welfare maximization problem, and based on
its convex relaxation propose a distributed stochastic dual gradient algorithm
for managing DERs. We further extend it to an online realtime setting with
time-varying operating conditions, asynchronous updates by devices, and
feedback being leveraged to account for nonlinear power flows as well as reduce
communication overhead. The resulting algorithm provides a general online
stochastic optimization algorithm for coordinating networked DERs with discrete
power setpoints and dynamics to meet operational and economic objectives and
constraints. We characterize the convergence of the algorithm analytically and
evaluate its performance numerically.
