An important monitoring task for power systems is accurate estimation of the system operation state. Under the nonlinear AC power flow model, the state estimation (SE) problem is inherently nonconvex giving rise to many local optima. In addition to nonconvexity, SE is challenged by data integrity and cyber-security issues. Unfortunately, existing robust (R-) SE schemes employed routinely in practice rely on iterative solvers, which are sensitive to initialization and cannot ensure global optimality. A novel R-SE approach is formulated here by capitalizing on the sparsity of an overcomplete outlier vector model. Observability and identifiability issues of this model are investigated, and neat links are established between R-SE and error control coding. The convex semidefinite relaxation (SDR) technique is further pursued to render the nonconvex R-SE problem efficiently solvable. The resultant algorithm markedly out-performs existing iterative alternatives, as corroborated through numerical tests on the standard IEEE 30-bus system.