  Physical security of power networks under power injection attacks that alter
generation and loads is studied. The system operator employs Phasor Measurement
Units (PMUs) for detecting such attacks, while attackers devise attacks that
are unobservable by such PMU networks. It is shown that, given the PMU
locations, the solution to finding the sparsest unobservable attacks has a
simple form with probability one, namely, $\kappa(G^M) + 1$, where
$\kappa(G^M)$ is defined as the vulnerable vertex connectivity of an augmented
graph. The constructive proof allows one to find the entire set of the sparsest
unobservable attacks in polynomial time. Furthermore, a notion of the potential
impact of unobservable attacks is introduced. With optimized PMU deployment,
the sparsest unobservable attacks and their potential impact as functions of
the number of PMUs are evaluated numerically for the IEEE 30, 57, 118 and
300-bus systems and the Polish 2383, 2737 and 3012-bus systems. It is observed
that, as more PMUs are added, the maximum potential impact among all the
sparsest unobservable attacks drops quickly until it reaches the minimum
sparsity.
