Analysis of local bifurcation mechanisms in large differential-algebraic systems such as the power system

The dynamics of a large class of physical systems such as the general power system can be represented by parameter dependent differential-algebraic models of the form x˙=f and 0=g. Typically such constrained models have singularities or noncausal points. When the system parameters change, physical system operation which is generally around a stable equilibrium point, loses dynamic stability at local bifurcation points. This paper analyzes the generic local bifurcations for the large system, including those which are directly related to the singularity. It is shown that generically loss of local stability at the equilibrium results from one of three different local bifurcations namely the well-known saddle node and Hopf bifurcations, and the singularity induced bifurcation. The latter results when an equilibrium point is at the singular surface. Under certain transversality conditions, the change in the eigenstructure of the system Jacobian at the equilibrium is established and the local dynamical structure of the trajectories near this bifurcation point is analyzed. Physical phenomena connected with the bifurcations, called the bifurcation mechanisms, are analyzed with an eye on the limitations of the constrained model. It is shown by genericity arguments that the singularity induced bifurcation in the constrained DAE model arises as the limiting case of Hopf bifurcations in the singularly perturbed models