  Recent studies have considered thwarting false data injection (FDI) attacks
against state estimation in power grids by proactively perturbing branch
susceptances. This approach is known as moving target defense (MTD). However,
despite of the deployment of MTD, it is still possible for the attacker to
launch stealthy FDI attacks generated with former branch susceptances. In this
paper, we prove that, an MTD has the capability to thwart all FDI attacks
constructed with former branch susceptances only if (i) the number of branches
$l$ in the power system is not less than twice that of the system states $n$
(i.e., $l \geq 2n$, where $n + 1$ is the number of buses); (ii) the
susceptances of more than $n$ branches, which cover all buses, are perturbed.
Moreover, we prove that the state variable of a bus that is only connected by a
single branch (no matter it is perturbed or not) can always be modified by the
attacker. Nevertheless, in order to reduce the attack opportunities of
potential attackers, we first exploit the impact of the susceptance
perturbation magnitude on the dimension of the \emph{stealthy attack space}, in
which the attack vector is constructed with former branch susceptances. Then,
we propose that, by perturbing an appropriate set of branches, we can minimize
the dimension of the \emph{stealthy attack space} and maximize the number of
covered buses. Besides, we consider the increasing operation cost caused by the
activation of MTD. Finally, we conduct extensive simulations to illustrate our
findings with IEEE standard test power systems.
