Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces

In this paper a model reduction technique is introduced for piecewise-smooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a non-dense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is non-dense. It is shown that under our assumptions the trajectories of such PWS systems have unique forward-time continuation when the parameter of the system switches. A finite-dimensional reduced order model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.Comment: 11 figures, 55 pages. Accepted for publication in Journal of Nonlinear Scienc

A Geometric Approach to Stationary Defect Solutions in One Space Dimension

Analysis and Stochastic

A geometric approach to stationary defect solutions in one space dimension

Analysis and Stochastic

Self-healing in power systems: an approach using islanding and rate of frequency decline based load shedding

This dissertation provides a self-healing strategy to deal with catastrophic events such as simultaneous loss of several generating units or major transmission lines when power system vulnerability analysis indicates that the system is approaching an extreme emergency state. In our approach, the system is adaptively divided into smaller islands at a slightly reduced capacity with consideration of quick restoration. The basis for forming the islands is to minimize the load-generation imbalance in each island, thereby facilitating the restoration process. Then a carefully designed load shedding scheme based on the rate of frequency decline is applied to limit the extent of the disruption and expedite the restoration process. A slow coherency based islanding theory is provided. Issues regarding the linear and nonlinear applicability of the theory are discussed in detail. R-Rdot out of step relay is deployed to form the islands. The function of the relay can be enhanced with the help of phasor measurement technology and decision tree knowledge. An overall scheme including a new two-level load shedding scheme is proposed. The proposed scheme is tested on a 179-bus, 29-generator sample system and shows very good performance