Invariant Sets in Quasiperiodically Forced Dynamical Systems

This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set. We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction. We provide a characterization of invariant sets in the quasiperiodically forced systems. A theoretical result on uniform boundedness of the invariant sets is presented. The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages. Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability. We show that our theoretical results can be used to understand stability regions in such complex systems.Comment: 23 pages, 4 figure

Estimation of Power System Inertia Using Nonlinear Koopman Modes

We report a new approach to estimating power system inertia directly from time-series data on power system dynamics. The approach is based on the so-called Koopman Mode Decomposition (KMD) of such dynamic data, which is a nonlinear generalization of linear modal decomposition through spectral analysis of the Koopman operator for nonlinear dynamical systems. The KMD-based approach is thus applicable to dynamic data that evolve in nonlinear regime of power system characteristics. Its effectiveness is numerically evaluated with transient stability simulations of the IEEE New England test system.Comment: 10 pages, 4 figures, conferenc

Applied Koopman Operator Theory for Power Systems Technology

Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.Comment: 31 pages, 11 figure

Dynamic Modeling and Renewable Integration Studies on the U.S. Power Grids

Wind and solar generation have gained a significant momentum in the last five years in the United States. According to the American Wind Energy Association, the installed wind power capacity has tripled from 25,410 MW in early 2009 to 74,472 MW as of the end of 2015. Meanwhile, solar photovoltaic (PV) is reported that its capacity has skyrocketed from 298 MW in 2009 to 7,260 MW in 2015 by the Solar Energy Industries Association. Despite the fact that wind and solar only make up 4.4% and 0.4% , respectively, of total electricity generation in 2014, the nation is right on its track to the Department of Energy (DOE)’s goal of 20% wind and 14% solar by year 2030. The future of renewable energy is aspiring. The rapid growth in renewable generation results in an urge to studying the reliability implication of renewable integration. For this purpose, two DOE projects were funded to the University of Tennessee, Knoxville, and the Oak Ridge National Laboratory. The first project, Grid Operational Issues and Analyses of the Eastern Interconnection (EI), is aimed at studying the dynamic stability impact of high wind penetration on the U.S. EI system in year 2030. The second project, Frequency Response Assessment and Improvement of Three Major North American Interconnections due to High Penetrations of Photovoltaic Generation, concentrates on the influence of high solar penetration on primary frequency response. This thesis documents the efforts of the above-mentioned two projects. Chapter 1 gives an introduction on power system dynamic modeling. Chapter 2 describes the process of dynamic models development. Chapter 3 discusses the adoption of synchro-phasor measurement for system-level dynamic model validation and the impact of turbine governor deadband on system dynamic response. Chapter 4 presents a stability impact study of high wind penetration on the U.S. Eastern Grid. Chapter 5 documents the modeling and simulation of the EI system under high solar penetration. Chapter 6 summaries two dynamic model reduction studies on the EI system. Conclusions, a summary of the major contribution of the Ph.D. work, and a discussion of possible future work are given in Chapter 7