Singularity analysis of differential-algebraic power system models

Differential-algebraic power system models exhibit singularity induced (SI) bifurcations as the parameters vary. The SI bifurcation point, which occurs when the system equilibria meet the singularity of the algebraic part of the model, belongs to a large set of other singular points called a singular set. At any singular point, the DAE model breaks down; meaning a vector field on a set satisfying algebraic part of the DAE model is not defined. Thus, DAE's cannot be reduced to a set of ordinary differential equations (ODEs) at the singular points. In terms of power system dynamics, at the singular points, relationship between generator angles and load bus variables breaks down and the DAE model cannot predict system behavior. Thus, singular points impose a dynamic constraint to the system model and computational tools are essential to locate them. We propose a novel method to compute singular points of the DAE model for the multi-machine power systems. The identification of the singular points is formulated as a bifurcation problem of a set of algebraic equations whose parameters are the generator angles. The method implements Newton-Raphson (NR)/Newton-Raphson-Seydel (NRS) methods to compute singular points. The proposed method is implemented into Voltage Stability Toolbox (VST) and simulations results are presented for several power system examples. Singular points are depicted together with the system equilibria to visualize static and dynamic limits.