Sparse Recovery from Nonlinear Measurements with Applications in Bad Data Detection for Power Networks

TL;DR

This paper introduces an iterative convex programming method for sparse recovery from nonlinear measurements, with applications in power network state estimation and bad data detection, providing theoretical bounds and convergence guarantees.

Contribution

It develops a new iterative convex approach for nonlinear sparse recovery, with performance bounds and convergence analysis, specifically tailored for power network applications.

Findings

Effective in detecting bad data in power networks

Provides theoretical bounds for linear and nonlinear measurement cases

Numerical validation confirms practical utility

Abstract

In this paper, we consider the problem of sparse recovery from nonlinear measurements, which has applications in state estimation and bad data detection for power networks. An iterative mixed $\ell_1$ and $\ell_2$ convex program is used to estimate the true state by locally linearizing the nonlinear measurements. 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 numerically evaluate our iterative convex programming approach to perform bad data detections in nonlinear electrical power networks problems. We are able to use semidefinite programming to verify the conditions for convergence of the proposed iterative sparse recovery algorithms from nonlinear measurements.