78 research outputs found
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
Characterizing scale dependence of effective diffusion driven by fluid flows
We study the scale dependence of effective diffusion of fluid tracers,
specifically, its dependence on the P\'{e}clet number, a dimensionless
parameter of the ratio between advection and molecular diffusion. Here, we
address the case that length and time scales on which the effective diffusion
can be described are not separated from those of advection and molecular
diffusion. For this, we propose a new method for characterizing the effective
diffusivity without relying on the scale separation. For a given spatial domain
inside which the effective diffusion can emerge, a time constant related to the
diffusion is identified by considering the spatio-temporal evolution of a test
advection-diffusion equation, where its initial condition is set at a pulse
function. Then, the value of effective diffusivity is identified by minimizing
the distance between solutions of the above test equation and the
diffusion one with mean drift. With this method, for time-independent gyre and
time-periodic shear flows, we numerically show that the scale dependence of the
effective diffusivity changes beyond the conventional theoretical regime. Their
kinematic origins are revealed as the development of the molecular diffusion
across flow cells of the gyre and as the suppression of the drift motion due to
a temporal oscillation in the shear.Comment: 10 pages, 4 figure
- …