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An analytic study of various approximations of the power flow equations for electric power systems is presented. The approximate models examined are the decoupled power flow, the linearized decoupled power flow (including the dc load flow) and the adjoint network sensitivity model, all of which are commonly used in steady-state security assessment. Error bounds on the difference between the solution of each of the approximate models and the solution of the full power flow are derived. The results are applied to the steady-state contingency analysis problem, resulting in a proposed new approach to the problem.