A direct method for the computation of the oscillatory voltage stability boundary

This paper proposes a two-stage approach for directly computing Hopf bifurcation (HB) points and the associated oscillatory voltage stability boundary in electric power systems. The mathematical formulation of the proposed approach is developed in a unified framework based on both a set of nonlinear differential algebraic equations representing the power system and a set of algebraic equations related to the conditions that must be satisfied for the occurrence of HBs. Both sets of equations are simultaneously solved in a two-stage approach for overcoming the difficulty faced by traditional Newton-based techniques regarding the definition of proper initial conditions for improving convergence towards the problem's solution. In this context, the original set of equations is first reformulated and solved by using a homotopy method, such that proper initial conditions for the set of state variables to be solved are computed. The original set of equations is then solved in a second stage by the Newton method using the computed initial conditions. Because the oscillatory voltage stability boundary is only composed of HBs points, a practical procedure for directly computing this boundary based on the proposed approach is also presented. Lastly, the effectiveness of the proposed methodology is presented through numerical examples using the IEEE 3-generator 9-bus test system.