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

Structural Analysis and Control of a Model of Two-site Electricity and Heat Supply

This paper introduces a control problem of regulation of energy flows in a two-site electricity and heat supply system, where two Combined Heat and Power (CHP) plants are interconnected via electricity and heat flows. The control problem is motivated by recent development of fast operation of CHP plants to provide ancillary services of power system on the order of tens of seconds to minutes. Due to the physical constraint that the responses of the heat subsystem are not necessary as fast as those of the electric subsystem, the target controlled state is not represented by any isolated equilibrium point, implying that stability of the system is lost in the long-term sense on the order of hours. In this paper, we first prove in the context of nonlinear control theory that the state-space model of the two-site system is non-minimum phase due to nonexistence of isolated equilibrium points of the associated zero dynamics.Instead, we locate a one-dimensional invariant manifold that represents the target controlled flows completely. Then, by utilizing a virtual output under which the state-space model becomes minimum phase, we synthesize a controller that achieves not only the regulation of energy flows in the short-term regime but also stabilization of an equilibrium point in the long-term regime. Effectiveness of the synthesized controller is established with numerical simulations with a practical set of model parameters

Analyzing feasibility of waste to energy technologies using life cycle assessment method based on waste stream composition in Surakarta City

Willie Susuki. A131408022. 2016. Analyzing feasibility of waste to energy technologies using life cycle assessment method based on waste stream composition in Surakarta City. Thesis. Supervisor I: Prof. Ir.Ari Handono Ramelan, MSc. (Hons), Ph.D. Supervisor II: Prof. Dr. Dwi Aries Himawanto, S.T., M.T ABSTRACT Increase in volume of waste generated by municipal residents, change in the quality of waste composition and the treatment and disposal method of waste collected are of major concern in Surakarta City which needs urgent workable solutions, hence, this research study has been necessitated. The objectives of this study is to (1) determine the existing condition of waste stream composition in Surakarta City, and (2) evaluate, based on the result of waste stream composition study, under what circumstances waste to energy technologies will be feasible in Surakarta City using life cycle assessment. In carrying out this research simple random sampling was used. At the final landfill site, measurement on weight, separation of waste into different categories were made and recorded. Proximate analysis was undertaken at the laboratory to determine the moisture content, volatile matter content, ash content, fixed carbon content and calorific value of selected samples of MSW. Finally, life cycle assessment method was used as an analytical tool to assess the feasibility of suitable waste to energy technologies based on the results of waste quantity and composition study undertaken in Surakarta City. The results showed that out of the samples of MSW collected, 11 types of wastes were identified including food waste, plastic, wood waste, paper, leaf, vegetable, metal, yard waste, textile, rubber and others. Out of these wastes, leave waste was found to be highest with 21.96%, followed by food waste 18.35%, vegetable waste, 16.08% paper with 11.07%, and plastic with 10.92%, wood waste with (5.8%), yard waste (4.63%), textile (4.13%), rubber (2.93%) and other smaller fractions which make up of 4%. In general, MSW in Surakarta City is largely made up of 67% of organic components of waste while inorganic components of wastes make up another 33%. The results of proximate analysis indicated that organic wastes had high moisture content of 81.67% compared to inorganic wastes with 25.8%. On average, the samples of MSW contributed high volatile solids of 35%, ash content of 6.3% and fixed carbon content of 5.5% respectively. The overall result of calorific value was high and found to be 15,770 MJ/kg as dry basis. In this case, inorganic fractions of wastes have high calorific value in general as compared to organic wastes. Based upon the condition of waste composition identified, when performing life cycle assessment, the result showed that out of all the waste to energy technologies discussed, incineration technology was found to be the most viable form of waste to energy technology option. Incinerating MSW decrease the space they take up and reduce the stress on landfills experienced at Putri Cempo landfill site. Key word: waste composition, waste to energy, life cycle assessmen