Sufficient conditions for power flow insolvability considering reactive power limited generators with applications to voltage stability margins

For the non-linear power flow problem with PQ and reactive power limited slack and PV buses, we present two sufficient conditions under which the specified set of nonlinear algebraic equations has no solution. The first condition uses a semidefinite programming relaxation of the power flow equations along with binary variables to model the generators' reactive power capabilities. As a byproduct, this condition yields a voltage stability margin to the power flow solvability boundary. The second condition formulates the power flow equations, including generator reactive power limits, as a system of polynomials and uses real algebraic geometry and sum of squares programming to create infeasibility certificates which prove power flow insolvability.