  In this paper, we consider the problem of state estimation through
observations possibly corrupted with both bad data and additive observation
noises. A mixed $\ell_1$ and $\ell_2$ convex programming is used to separate
both sparse bad data and additive noises from the observations. Through using
the almost Euclidean property for a linear subspace, we derive a new
performance bound for the state estimation error under sparse bad data and
additive observation noises. Our main contribution is to provide sharp bounds
on the almost Euclidean property of a linear subspace, using the
"escape-through-a-mesh" theorem from geometric functional analysis. We also
propose and numerically evaluate an iterative convex programming approach to
performing bad data detections in nonlinear electrical power networks problems.
