  Evolving power systems with increasing levels of stochasticity call for a
need to solve optimal power flow problems with large quantities of random
variables. Weather forecasts, electricity prices, and shifting load patterns
introduce higher levels of uncertainty and can yield optimization problems that
are difficult to solve in an efficient manner. Efficient solution methods for
single chance constraints in optimal power flow problems have been considered
in the literature; however, joint chance constraints have predominantly been
solved via scenario-based approaches or by utilizing the overly conservative
Boole's inequality as an upper bound. In this paper, joint chance constraints
are used to solve an AC optimal power flow problem which maintain desired
levels of voltage magnitude in distribution grids under high penetrations of
photovoltaic systems. A tighter version of Boole's inequality is derived and
used to provide a new upper bound on the joint chance constraint, and
simulation results are shown demonstrating the benefit of the proposed upper
bound.
