In many situations, fluid flows can exhibit a wide range of temporal and spatial phenomena. It has become common to extract physically important features, called modes, as a first step in the analysis of flows with high complexity. One of the most prominent modal analysis techniques in the context of fluid dynamics is Proper Orthogonal Decomposition (POD), which enables extraction of energetically coherent structures present in the flow field. This method does, however, suffer from the lack of connection with the mathematical theory of dynamical systems and its utility in the analysis of arbitrarily complex flows might therefore be limited. In the present work, we instead consider application of the Koopman Mode Decomposition (KMD), which is an approach based on spectral decomposition of the Koopman operator. This technique is employed for modal analysis of the incompressible, two-dimensional ow past two side-by-side cylinders at Re = 60 and with a non-dimensional cylinder gap spacing g* = 1. This particular configuration yields a wake ow which exhibits in-phase vortex shedding during finite time, while later transforming into the so-called flip-flopping phenomena, which is characterised by a slow, periodic switching of the gap ow direction during O(10) vortex shedding cycles. The KMD approach yields modal structures which, in contrary to POD, are associated with specific oscillation frequencies. Specifically, these structures are here vorticity modes. By studying these modes, we are able to extract the ow components which are responsible for the flip-flop phenomenon. In particular, it is found that the flip-flop instability is mainly driven by three different modal structures, oscillating with Strouhal frequencies St1 = 0:023, St2 = 0:121 and St3 = 0:144, where it is noted that St3 = St1 + St2. In addition, we study the in-phase vortex shedding regime, as well as the transient regime connecting the two states of the flow. The study of the in-phase vortex shedding reveals| - not surprisingly - the presence of a single fundamental frequency, while the study of the transient reveals a Koopman spectrum which might indicate the existence of a bifurcation in the phase space of the flow field; this idea has been proposed before in Carini et al. (2015b). We conclude that the KMD offers a powerful framework for analysis of this ow case, and its range of applications might soon include even more complex flows.

The study of nonlinear dynamical systems via the spectrum of the Koopman operator has emerged as a paradigm shift, from the Poincaré's geometric picture that centers the attention on the evolution of states, to the Koopman operator's picture that focuses on the evolution of observables. The Koopman operator-theoretic framework rests on the idea of lifting the states of a nonlinear dynamical system to a higher dimensional space; these lifted states are referred to as the Koopman eigenfunctions. To determine the Koopman eigenfunctions, one performs a nonlinear transformation of the states by relying on the so-called observables, that is, scalar-valued functions of the states. In other words, one executes a change of coordinates from the state space to another set of coordinates, which are denominated Koopman canonical coordinates. The variables defined on these intrinsic coordinates will evolve linearly in time, despite the underlying system being nonlinear. Since the Koopman operator is linear, it is natural to exploit its spectral properties. In fact, the theory surrounding the spectral properties of linear operators has well-known implications in electric power systems. Examples include small-signal stability analysis and direct methods for transient stability analysis based on the Lyapunov function. From the applications' standpoint, this framework based on the Koopman operator is attractive because it is capable of revealing linear and nonlinear modes by only applying well-established tools that have been developed for linear systems. With the challenges associated with the high-dimensionality and increasing uncertainties in the power systems models, researchers and practitioners are seeking alternative modeling approaches capable of incorporating information from measurements. This is fueled by an increasing amount of data made available by the wide-scale deployment of measuring devices such as phasor measurement units and smart meters. Along these lines, the Koopman operator theory is a promising framework for the integration of data analysis into our mathematical knowledge and is bringing an exciting perspective to the community. The present dissertation reports on the application of the Koopman operator for identification, estimation, and control of electric power systems. A dynamic state estimator based on the Koopman operator has been developed and compares favorably against model-based approaches, in particular for centralized dynamic state estimation. Also, a data-driven method to compute participation factors for nonlinear systems based on Koopman mode decomposition has been developed; it generalizes the original definition of participation factors under certain conditions. / PHD / Electric power systems are complex, large-scale, and given the bidirectional causality between economic growth and electricity consumption, they are constantly being expanded. In the U.S., some of the electric power grid facilities date back to the 1880s, and this aging system is operating at its capacity limits. In addition, the international pressure for sustainability is driving an unprecedented deployment of renewable energy sources into the grid. Unlike the case of other primary sources of electric energy such as coal and nuclear, the electricity generated from renewable energy sources is strongly influenced by the weather conditions, which are very challenging to forecast even for short periods of time. Within this context, the mathematical models that have aided engineers to design and operate electric power grids over the past decades are falling short when uncertainties are incorporated to the models of such high-dimensional systems. Consequently, researchers are investigating alternative data-driven approaches. This is not only motivated by the need to overcome the above challenges, but it is also fueled by the increasing amount of data produced by today’s powerful computational resources and experimental apparatus. In power systems, a massive amount of data will be available thanks to the deployment of measuring devices called phasor measurement units. Along these lines, the Koopman operator theory is a promising framework for the integration of data analysis into our mathematical knowledge, and is bringing an exciting perspective on the treatment of high-dimensional systems that lie in the forefront of science and technology. In the research work reported in this dissertation, the Koopman operator theory has been exploited to seek for solutions to some of the challenges that are threatening the safe, reliable, and efficient operation of electric power systems.