  We study the distributions of the resilience of power flow models against
transmission line failures via a so-called backup capacity. We consider three
ensembles of random networks and in addition, the topology of the British
transmission power grid. The three ensembles are Erd\H{o}s-R\'enyi random
graphs, Erd\H{o}s-R\'enyi random graphs with a fixed number of links, and
spatial networks where the nodes are embedded in a two dimensional plane. We
investigate numerically the probability density functions (pdfs) down to the
tails to gain insight in very resilient and very vulnerable networks. This is
achieved via large-deviation techniques which allow us to study very rare
values which occur with probability densities below $10^{-160}$. We find that
the right tail of the pdfs towards larger backup capacities follows an
exponential with a strong curvature. This is confirmed by the rate function
which approaches a limiting curve for increasing network sizes. Very resilient
networks are basically characterized by a small diameter and a large power sign
ratio. In addition, networks can be made typically more resilient by adding
more links.
