  Time-synchronization attacks on phasor measurement units (PMU) pose a real
threat to smart grids; it was shown that they are feasible in practice and that
they can have a non-negligible negative impact on the state estimation, without
triggering the bad-data detection mechanisms. Previous works identified
vulnerability conditions when targeted PMUs measure a single phasor. Yet, PMUs
are capable of measuring several quantities. We present novel vulnerability
conditions in the general case where PMUs measure any number of phasors and can
share the same time reference. One is a sufficient condition that does not
depend on the measurement values. We propose a security requirement that
prevents it and provide a greedy offline algorithm that enforces it. If this
security requirement is satisfied, there is still a possibility that the grid
can be attacked, although we conjecture that it is very unlikely. We identify
two sufficient and necessary vulnerability conditions which depend on the
measurement values. For each, we provide a metric that shows the distance
between the observed and vulnerability conditions. We recommend their
monitoring for security. Numerical results, on the IEEE-39 bus benchmark with
real load profiles, show that the measurements of a grid satisfying our
security requirement are far from vulnerable.
