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An important monitoring task for power networks is to estimate accurately the underlying grid state, which is useful for security-constrained dispatch and power system control. For nonlinear AC power systems, the state estimation (SE) problem is inherently nonconvex giving rise to many local optima. As a result, existing estimators used extensively in practice rely on iterative optimization methods, which are destined to return only locally optimal solutions. A semidefinite programming (SDP) based approach is introduced in this paper, which relies on convex relaxation of the original SE problem and thereby renders it efficiently solvable. A sufficient condition also becomes available to guarantee that the dual SDP problem attains zero duality gap, and thus ensure that the globally optimal SE solution is achievable in polynomial time. The novel scheme's ability to markedly outperform existing iterative alternatives is corroborated through numerical tests on the standard IEEE 14-bus benchmark system.