Energy functions for power systems with load dynamics

This thesis develops and applies new results in power system energy function analysis. It has been motivated by the recognition that load dynamics play an important role in dynamic voltage collapse. Strict Lyapunov (energy) functions are derived for power system structure-preserving models. Generators are assumed to obey the classical model, and reactive power loads are of the exponential recovery type. The inclusion of the latter represents the major contribution towards energy function theory. Two methods are used to obtain energy functions: a 'first integral' type of method, and the multivariable Popov criterion for linear systems with nonlinear feedback. The power system equations need to be singularly perturbed before the multivariable Popov criterion becomes applicable. Energy functions are then obtained by allowing the perturbation terms to vanish. Equivalence amongst the results of the two methods is demonstrated. Validity of the energy functions is shown to be verified by the sign of the eigenvalues of a power flow Jacobian, and by the form of the time-derivative of energy along trajectories. Numerical examples are constructed to illustrate the new energy function results. By calculating a critical energy, it is possible to determine critical fault-clearing times, or capacitor switching times. Also, the behaviour of dynamic recovery loads is examined within the energy function setting. An important feature is the damping which is associated with such loads. The effects of changing the various parameters are investigated. For power systems which normally result in path-dependent energy integrals, it is more difficult to obtain strict Lyapunov functions. This problem is first examined using a passivity form of input/output approach. It leads to a non-passivity result but it provides some insights into system energy exchanges. Then, optimal control theory is used to motivate energy functions. Known solutions for linearised systems point the way towards solutions for the associated nonlinear systems.