Chaos in the swing dynamics

The swing equations model the interactions of generators (or subsystems of generators) in an electric power system. They are used to study "transient stability", or stability of the system right after a large disturbance (e.g. a fault). For a system of n generators, the two swing equations are given. The equations take into account the mechanical rotor angle of each generator, and its rotor angular velocity, as well as the inertia constant, the damping constant and the (constant) mechanical power input at each machine. The equations are coupled only through the terms that represent the electrical power output at each machine demanded by the network. Integral equations are developed assuming that there is no damping and no resistence in the transmission lines, and that resistive loads are located only at generator busses. These equations can be used to reduce the system by one degree of freedom (two dimensions). The equations are broadened to cover the case of 3-generator system (2 degrees of freedom).