Stability of Nonlinear Systems with Parameter Uncertainty

Abstract

A novel approach is introduced to assess stability of nonlinear systems in the presence of parameter uncertainty. The idea is to consider the deterministic dynamics of the system as a function of parameter values, where the parameter-dependent initial condition may for example be the output of a particular finite-time disturbance. The goal is to numerically determine the boundary in parameter space, referred to as the recovery boundary, between parameter values which lead to recovery and those which lead to a failure to recover to an initial stable equilibrium point. Critical parameter values, which are defined to be those parameter values whose corresponding initial conditions lie on the boundary of the region of attraction of their corresponding stable equilibrium points, have the potential to provide an explicit connection between the recovery boundary in parameter space and the region of attraction boundary in state space that can be exploited for algorithm design. However, examples are provided to illustrate that the recovery boundary may not contain critical parameter values when the boundary of the region of attraction of the stable equilibrium point varies discontinuously with parameter. Fortunately, it is shown that, for a large class of vector fields possessing stable equilibrium points, the boundaries of the regions of attraction of these equilibrium points vary continuously with respect to small variations in parameter values. This region of attraction boundary continuity ensures that the recovery boundary consists entirely of critical parameter values and that the nearest critical parameter value to any non-critical parameter value lies on the recovery boundary. Two classes of theoretically motivated algorithms are developed to compute critical parameter values by exploiting the structure and behavior of the region of attraction boundary under parameter variation. The system trajectory corresponding to a critical parameter value converges to an invariant set and, therefore, spends an infinite amount of time in any neighborhood of that invariant set. A first class of algorithms proceed by varying parameter values so as to maximize the time in a neighborhood of the invariant set. Under reasonable assumptions, the system trajectory corresponding to a critical parameter value becomes infinitely sensitive to small changes in parameter value. A second class of algorithms proceed by varying parameter values so as to maximize the trajectory sensitivities to parameters. Theoretical motivation is provided for both classes of algorithms, and shows that under reasonable assumptions they will drive parameter values to their critical values. Both of these approaches transform the abstract problem of finding critical parameter values into concrete numerical optimization problems. Based on these approaches, algorithms are developed to find the closest parameter value on the recovery boundary in the case of one-dimensional parameter space, trace the recovery boundary in two dimensional parameter space, and find the nearest point on the recovery boundary in parameter space of arbitrary dimension. The algorithms are applied to assess fault vulnerability in power systems. Results from the test cases of a modified IEEE 37-bus feeder and a modified IEEE 39-bus system illustrate the algorithm performance. The emphases in these test cases are, respectively, to explore the onset of induction motor stalling during Fault Induced Delayed Voltage Recovery, and to analyse stability of a system of synchronous machines under high levels of load uncertainty.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155156/1/fishermw_1.pd

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