Sparse Error Correction From Nonlinear Measurements With Applications in Bad Data Detection for Power Networks

In this paper, we consider the problem of sparse error correction from general nonlinear measurements, which has applications in state estimation of electrical power networks, when bad data (outliers) are present. An iterative mixed $\ell _1$ and $\ell _2$ convex program is used to estimate the true state by locally linearizing the nonlinear measurements. In the special case when the measurements are linear, 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 noise. As a byproduct, in this paper we provide sharp bounds on the almost Euclidean property of a linear subspace, using the “escape-through-the-mesh” theorem from geometric functional analysis. When the measurements are nonlinear, we give conditions under which the solution of the iterative algorithm converges to the true state even though the locally linearized measurements may not be the actual nonlinear measurements. We are able to use a semidefinite program to verify the conditions for convergence of the proposed iterative sparse recovery algorithms from nonlinear measurements. We then numerically evaluate our iterative convex programming approach of performing bad data detections in nonlinear electrical power networks problems.