  The advent of renewable energy has huge implications for the design and
control of power grids. Due to increasing supply-side uncertainty, traditional
reliability constraints such as strict bounds on current, voltage and
temperature in a transmission line have to be replaced by computationally
demanding chance constraints. In this paper we use large deviations techniques
to study the probability of current and temperature overloads in power grids
with stochastic power injections, and develop corresponding safe capacity
regions. In particular, we characterize the set of admissible power injections
such that the probability of overloading of any line over a given time interval
stays below a fixed target. We show how enforcing (stochastic) constraints on
temperature, rather than on current, results in a less conservative approach
and can thus lead to capacity gains.
