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 different- tial-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.
Highlights
In an effort to better understand dynamical properties of power systems, their stability features and the impact of various parameters, the differenttial-algebraic equations (DAEs) approach seems to be very important as was shown in a number of recent papers.Some of these papers deal with the existence of Hopf bifurcations in the singularly perturbed systems of nonlinear ODEs
A second order matrix pencil would normally be preferred for power systems as it may directly lead to double singularity induced bifurcations (SIBs) points [1,21]
It turns out that under the assumption of the SIB theorem (with an algebraically simple zero of gy(x∗, y∗, λ0) [9]), the SIB phenomenon for semi-explicit parameter dependent DAEs may be equivalent to Hopf bifurcation of the singularly perturbed ODEs
Summary
In an effort to better understand dynamical properties of power systems, their stability features and the impact of various parameters, the DAEs approach seems to be very important as was shown in a number of recent papers (see for example [1,2,3,4,5,6,7,8,9,10]) Some of these papers deal with the existence of Hopf bifurcations in the singularly perturbed systems of nonlinear ODEs. DAEs are closely related to singularly perturbed ODEs, it is natural to expect similar types of behavior.
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