Double singularity-induced bifurcation points and singular Hopf bifurcations

The singularity-induced bifurcation and singular Hopf bifurcation theorems and the degeneracies that arise when Newtonʹs laws are coupled to Kirchhoffʹs laws are explored. Such models are used in the electrical engineering literature to describe electrical power systems and they can take the form of either an index-1 differential-algebraic equation (DAE) or a singularly perturbed ordinary differential equation (ODE). As a consequence of the debate in the engineering literature as to which class of system is the ʹtrueʹ representation of power systems, a discussion is included of the consequences of the power engineerʹs load-flow singularityʹ for both ODE and DAE.