Singular Hopf Bifurcations in DAE Models of Power Systems

Abstract

We investigate an important relationship that exists between the Hopf bifurcation in the singularly perturbed nonlinear power systems and the singularity induced bifurcations (SIBs) in the corresponding differenttial-algebraic equations (DAEs). In a generic case, the SIB phenomenon in a system of DAEs signals Hopf bifurcation in the singularly perturbed systems of ODEs. The analysis is based on the linear matrix pencil theory and polynomials with parameter dependent coefficients. A few numerical examples are included.

Generalization to differential–algebraic equations of Lyapunov–Schmidt type reduction at Hopf bifurcations

“…A similar system (with n g 2 and n l 1) has been considered in [22]. It will become clear later why in (5.4) damping coecients γ i have been introduced.…”

“…It is not pretended here to provide a meaningful parameter set in the context of electric power engineering, but rather to illustrate for a hopefully meaningful example the reduction procedure. Note that for instance in [22] similar parameter value magnitudes for a power system model with n g 2 and n l 1 have been considered (see also the examples in [6], chapter 4). The mechanical power injection parameters for the generators have been chosen as T 1 1, T 2 2, T 3 0.5…”

Singularity induced bifurcation and fold points in inviscid transonic flow

“…Matrix Pencils and Parameter Dependent Polynomials [22] For a pair of constant square n × n matrices, say A and L, with detA = 0, if det(sA−L) = 0, then there exist nonsingular matrices U and V , such that…”

Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles

“…The trajectory of a sliding orbit remaining partially inside the discontinuity boundary may be calculated by the Filippov convex method as in [4]. Systems with multiple regions and DBs are treated in [13], where an extended equation for Filippov systems is described in order to deal with the intersection of several discontinuity surfaces.…”