  This paper develops a power management scheme that jointly optimizes the real
power consumption of programmable loads and reactive power outputs of
photovoltaic (PV) inverters in distribution networks. The premise is to
determine the optimal demand response schedule that accounts for the stochastic
availability of solar power, as well as to control the reactive power
generation or consumption of PV inverters adaptively to the real power
injections of all PV units. These uncertain real power injections by PV units
are modeled as random variables taking values from a finite number of possible
scenarios. Through the use of second order cone relaxation of the power flow
equations, a convex stochastic program is formulated. The objectives are to
minimize the negative user utility, cost of power provision, and thermal
losses, while constraining voltages to remain within specified levels. To find
the global optimum point, a decentralized algorithm is developed via the
alternating direction method of multipliers that results in closed-form updates
per node and per scenario, rendering it suitable to implement in distribution
networks with large number of scenarios. Numerical tests and comparisons with
an alternative deterministic approach are provided for typical residential
distribution networks that confirm the efficiency of the algorithm.
