Nonlinear dynamics in power systems

We use a perturbation analysis to predict some of the instabilities in a single-machine quasi-infinite busbar system. The system’s behavior is described by the so-called swing equation, which is a nonlinear second-order ordinary-differential equation with additive and multiplicative harmonic terms having the frequency Ω. When Ω≈ω₀, and Ω≈2ω₀, where ω₀ is the linear natural frequency of the machine, we use digital-computer simulations to exhibit some of the complicated responses of the machine, including period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism). To predict the onset of these complicated behaviors, we use the method of multiple scales to develop approximate closed-form expressions for the periodic responses of the machine. Then, we use various techniques to determine the stability of the analytical solutions. The analytically predicted periodic solutions and conditions for their instability are in good agreement with the digital-computer results.