Magíster en Ciencias de la Ingeniería, Mención Matemáticas Aplicadas. Ingeniero Civil Matemático / Desde los inicios de la la teoría ergódica la teoría espectral ha representado una herramienta potente para entender distintos aspectos de la dinámica de un sistema. La relación entre estas teorías se establece a través del operador de Koopman definido a partir de un sistema diná- mico en distintos espacios funcionales. Entre los resultados notables que se han demostrado utilizando ésta idea se pueden mencionar dos atribuídos a John von Neumann: un teorema ergódico y una caracterización de los sistemas de espectro discreto (ver [29]). Los operadores de Koopman se pueden estudiar a partir de las medidas espectrales, debido al teorema de representación espectral. Si bien ésta es una manera útil de caracterizar esos operadores, calcular las medidas espectrales es un problema difícil en el contexto general. Por esta razón es que se busca obtener información sobre ellas de forma indirecta, por ejemplo, a través de sus decaimientos asintóticos. Los sistemas dinámicos en donde ha sido posible describir las medidas espectrales son muy pocos y una categoría muy explorada es la de aquellos provenientes de substituciones y en particular de substituciones de largo constante [29]. Más recientemente, inspirados en [17], Bufetov y Solomyak prueban en [7] módulos de continuidad para las medidas espectrales asociadas a sistemas de tilings substitutivos uni- dimensionales. En [11] se generaliza uno de los resultados de [7] al contexto de sistemas de tilings susbtitutivos del espacio euclideano R d . Más precisamente se dan cotas del decaimiento de las medidas espectrales en torno al orígen. En el presente trabajo de tesis se generalizan las ideas de [7] encontrando un módulo de continuidad de tipo log-Hölder para las medidas espectrales en sistemas substitutivos de tilings de R d , pero para puntos alejados del orígen. Entre las técnicas esenciales que se usan para la demostración están la de representar las medidas espectrales como productos de Riesz matriciales, estimaciones del crecimiento de sumas torcidas de Birkhoff y su relación con los decaimientos de las medidas espectrales, la descomposición en sistemas de torres del sistema substitutivo de tilings y argumentos de teoría de números de Pisot. El resultado principal permite entender la parte continua del espectro de un sistema di- námico de tiling substitutivo. Los decaimientos de las medidas espectrales entregan tasas de débil mezcla como se hace en [23], las que son invariantes de conjugación topológica. Ésto podría ser una herramienta útil para distinguir sistemas dinámicos de espectro continuo. / Este trabajo ha sido parcialmente financiado por el Centro de Modelamiento Matemático

Traditional generators have been decommissioned or replaced by renewable energy generation due to utility long-standing goals. However, instead of flattening the entire plant, the rotating mass of generator can be utilized as a storage unit (inertia resource) to mitigate the frequency swings during transient caused by the renewables. The goal of this work is to design a control strategy utilizing the decommissioned generator interfaced with power grid via a back-to-back converter to provide inertia support. This is referred to as decoupled synchronous machine system (DSMS). On top of that, the grid-side converter is capable of providing reactive power as an auxiliary voltage controller. However, in a practical setting, for power utilities, the detailed state equations of such device as well as the complicated nonlinear power system are usually unobtainable making the controller design a challenging problem. Therefore, a model free, purely data-driven strategy for the nonlinear controller design using Koopman operator-based framework is proposed. Besides, the time delay embedding technique is adopted together with Koopman operator theory for the nonlinear system identification. Koopman operator provides a linear representation of the system and thereby the classical linear control algorithms can be applied. In this work, model predictive control is adopted to cope with the constraints of the control signals. The effectiveness and robustness of the proposed system are demonstrated in Kundur two-area system and IEEE 39-bus system. / Doctor of Philosophy / Power system is facing an energy transformation from the traditional fuel to sustainable renewable such as solar, wind and so on. Unlike the traditional fuel energized generators, the renewable has very little inertia to maintain frequency stability. Therefore, this work proposes a new system referred to as decoupled synchronous machine system (DSMS) to support the grid frequency. DSMS consists of the rotating mass of generator and a back-to-back converter which can be utilized as an inertia resource to mitigate the frequency oscillations. In addition, the grid-side converter can provide reactive power to improve voltage performance during faults. This work aims to design a control strategy utilizing DSMS to support grid frequency and voltage. However, an explicit mathematical model of such device is unobtainable in a practical setting making data-driven control the only option. A data-driven technique which is Koopman operator-based framework together with time delay embedding algorithm is proposed to obtain a linear representation of the system. The effectiveness and robustness of the proposed system are demonstrated in Kundur two-area system and IEEE 39-bus system.

This thesis consists of four chapters including two main contributions on the application of machine learning and artificial intelligence on process modeling and controller design. Chapter 2 will talk about applying AI to controller design. This chapter proposes and implements a novel reinforcement learning (RL)--based controller design on chemical engineering examples. To address the issue of costly and unsafe training of model-free RL-based controllers, we propose an implementable RL-based controller design that leverages offline MPC calculations, that have already developed based on a step-response model. In this method, a RL agent is trained to imitate the MPC performance. Then, the trained agent is utilized in a model-free RL framework to interact with the actual process so as to continuously learn and optimize its performance under a safe operating range of processes. This contribution is marked as the first implementable RL-based controller for practical industrial application. Chapter 3 will focus on AI applications in process modeling. As nonlinear dynamics are widely encountered and challenging to simulate, nonlinear MPC (NMPC) is recognized as a promising tool to tackle this challenge. However, the lack of a reliable nonlinear model remains a roadblock for this technique. To address this issue, we develop a novel data-driven modeling method that utilizes the nonlinear autoencoder, to result in a modeling technique where the nonlinearity in the model stems from the analysis of the measured variables. Moreover, a quadratic program (QP) based MPC is developed based on this model, by utilizing an autoencoder as a transformer between the controller and process. This work contributes as an extension of the classic Koopman operator modeling method and a remarkable linear MPC design that can outperform other NMPCs such as neural network-based MPC. / Thesis / Master of Applied Science (MASc)

Koopman theory is widely used for data-driven modeling of nonlinear dynamical systems. One of the well-known algorithms that stem from this approach is the Extended Dynamic Mode Decomposition (EDMD), a data-driven algorithm for uncontrolled systems. In this thesis, we will start by discussing the EDMD algorithm. We will discuss how this algorithm encompasses Dynamic Mode Decomposition (DMD), a widely used data-driven algorithm. Then we will extend our discussion to input-output systems and identify ways to extend the Koopman Operator Theory to input-output systems. We will also discuss how various algorithms can be identified as instances of this framework. Special care is given to Wavelet-based Dynamic Mode Decomposition (WDMD). WDMD is a variant of DMD that uses only the input and output data. WDMD does that by generating auxiliary states acquired from the Wavelet transform. We will show how the action of the Koopman operator can be simplified by using the Wavelet transform and how the WDMD algorithm can be motivated by this representation. We will also introduce a slight modification to WDMD that makes it more robust to noise. / Master of Science / To analyze a real-world phenomenon we first build a mathematical model to capture its behavior. Traditionally, to build a mathematical model, we isolate its principles and encode it into a function. However, when the phenomenon is not well-known, isolating these principles is not possible. Hence, rather than understanding its principles, we sample data from that phenomenon and build our mathematical model directly from this data by using approximation techniques. In this thesis, we will start by focusing on cases where we can fully observe the phenomena, when no external stimuli are present. We will discuss how some algorithms originating from these approximation techniques can be identified as instances of the Extended Dynamic Mode Decomposition (EDMD) algorithm. For that, we will review an alternative approach to mathematical modeling, called the Koopman approach, and explain how the Extended DMD algorithm stems from this approach. Then we will focus on the case where there is external stimuli and we can only partially observe the phenomena. We will discuss generalizations of the Koopman approach for this case, and how various algorithms that model such systems can be identified as instances of the EDMD algorithm adapted for this case. Special attention is given to the Wavelet-based Dynamic Mode Decomposition (WDMD) algorithm. WDMD builds a mathematical model from the data by borrowing ideas from Wavelet theory, which is used in signal processing. In this way, WDMD does not require the sampling of the fully observed system. This gives WDMD the flexibility to be used for cases where we can only partially observe the phenomena. While showing that WDMD is an instance of EDMD, we will also show how Wavelet theory can simplify the Koopman approach and thus how it can pave the way for an easier analysis.

Electric power systems are undergoing significant changes with the deployment of large-scale wind and solar plants connected to the transmission system and small-scale Distributed Energy Resources (DERs) and microgrids connected to the distribution system, making the latter an active system. A microgrid is a small-scale power system that interconnects renewable and non-renewable generating units such as solar photo-voltaic panels and micro-turbines, storage devices such as batteries and fly wheels, and loads. Typically, it is connected to the distribution feeders via power electronic converters with fast control responses within the micro-seconds. These new developments have prompted growing research activities in stability analysis and control of the transmission and the distribution systems. Unfortunately, these systems are treated as separated entities, limiting the scope of the applicability of the proposed methods to real systems. It is worth stressing that the transmission and distribution systems are interconnected via HV/MV transformers and therefore, are interacting dynamically in a complex way. In this research work, we overcome this problem by investigating the dynamics of the transmission and distribution systems with parallel microgrids as an integrated system . Specifically, we develop a generic model of a microgrid that consists of a DC voltage source connected to an inverter with real and reactive power control and voltage control. We analyze the small-signal stability of the two-area four-machine system with four parallel microgrids connected to the distribution feeders though different impedances. We show that the conventional PQ control of the inverters is insufficient to stabilize the voltage at the point-of-common coupling when the feeder impedances have highly unequal values. To ensure the existence of a stable equilibrium point associated with a sufficient stability margin of the system, we propose a new voltage control implemented as an additional feedback control loop of the conventional inner and outer current control schemes of the inverter. Furthermore, we carry out a modal analysis of the four-machine system with microgrids using Koopman mode analysis. We reveal the existence of local modes of oscillation of a microgrid against the rest of the system and between parallel microgrids at frequencies that range between 0.1 and 3 Hz. When the control of the microgrid becomes unstable, the frequencies of the oscillation are about 20 Hz. Recall that the Koopman mode analysis is a new technique developed in fluid dynamics and recently introduced in power systems by Suzuki and Mezic. It allows us to carry out small signal and transient stability analysis by processing only measurements, without resorting to any model and without assuming any linearization. / Ph. D. / Electric power systems are undergoing significant changes with the deployment of large-scale wind and solar plants connected to the transmission system and small-scale Distributed Energy Resources (DERs) and microgrids connected to the distribution system, making the latter an active system. A microgrid is a small-scale power system that interconnects renewable and non-renewable generating units such as solar photo-voltaic panels and micro-turbines, storage devices such as batteries and fly wheels, and loads. Typically, it is connected to the distribution feeders via power electronic converters with fast control responses within the micro-seconds. These new developments have prompted growing research activities in stability analysis and control of the transmission and the distribution systems. Unfortunately, these systems are treated as separated entities, limiting the scope of the applicability of the proposed methods to real systems. It is worth stressing that the transmission and distribution systems are interconnected via HV/MV transformers and therefore, are interacting dynamically in a complex way. In this research work, we overcome this problem by investigating the dynamics of the transmission and distribution systems with parallel microgrids as an integrated system . Specifically, we develop a generic model of a microgrid that consists of a DC voltage source connected to an inverter with real and reactive power control and voltage control. We show that the conventional PQ control of the inverters is insufficient to stabilize the voltage at the point-of-common coupling when the feeder impedances have highly unequal values. Furthermore, we carry out a modal analysis of the four-machine system with microgrids using Koopman mode analysis. Koopman mode analysis is a new technique developed in fluid dynamics and recently introduced in power systems by Suzuki and Mezic. It allows us to carry out small signal and transient stability analysis by processing only measurements, without resorting to any model and without assuming any linearization.

Au niveau atomique, le contrôle de spins est un objectif primordial en physique quantique. Malheureusement la présence de bruits gêne ce dernier. Le but est de trouver les conditions à imposer à l’environnement pour que le contrôle ne soit pas perturbé par le bruit. L’étude d’une chaîne de spins caractérisée par trois couplages : interactions d’Heisenberg, d’Ising-Z et d’Ising-X, évoluant librement est prise comme référence. Nous observons que l’interaction d’Heisenberg correspond à un couplage isotrope. Celle d’Ising-Z conserve l’ordre dans la chaîne tandis que celle d’Ising-X est très désordonnée. Nous rendons le système plus complexe en ajoutant du contrôle et en analysant le comportement adiabatique d’un système quantique. Ce dernier est composé d’un système et d’un environnement, dont le couplage est perturbatif. Trois régimes adiabatiques ont été mis en évidence. Des formules permettant d’obtenir la fonction d'onde au cours du temps ont alors été établies pour ces trois régimes. Cependant, dans la pratique, les systèmes quantiques ne sont en aucun cas isolés. L’interaction avec leur environnement peut entraîner des comportements plus complexes, rendant le contrôle très difficile. Nous avons alors étudié des systèmes de spins, couplés ou non, frappés par des trains d’impulsions magnétiques ultracourtes. Ces trains traversent un environnement classique (stationnaire, de dérive linéaire, Markovien, microcanonique) modifiant la force et le retard de chaque impulsion. La modification des trains par l’environnement classique est une des sources du désordre dans le système de spins. Ce désordre est transmis entre les spins par le couplage. Dans cette étude nous n’arrivons pas à contrôler le système lorsque les trains sont en présence des environnements précédents. Pour palier à ce problème, nous imposons aux impulsions magnétiques de traverser un environnement chaotique. Avant un temps t, appelé horizon de cohérence, le système couplé par une interaction d’Heisenberg et soumis à un environnement chaotique reste cohérent alors qu’après, la population et la cohérence d'un spin et du spin moyen du système tendent à se rapprocher de la distribution microcanonique. Pendant cet horizon, il est possible de réaliser du contrôle quantique soit par contrôle total (contrôle du système à chaque instant), soit par transmission d’information. Cette étude nous a permis de déterminer une formule empirique de l’horizon de cohérence. Finalement, nous nous sommes attachés à trouver une formule plus formelle de cet horizon. / At an atomic level, the spin control is an essential aim in quantum physics. Unfortunately, the presence of noises disturbs this last. The goal is to find the conditions which we have to impose to the environment in order that the control is not disturbed by the noise. The study of a spin chain characterized by three couplings (Heisenberg, Ising-Z and Ising-X interactions) freely evolving is taken as reference. We observe that the Heisenberg interaction corresponds to an isotropic coupling. The Ising-Z one conserves the order into the chain whereas the Ising-X one is really disordered. We consider a more complex quantum system by adding some control and analyzing its adiabatic behavior. This last is composed by a system and an environment, for which the coupling is perturbative. Three adiabatic regimes have been highlighted. Some formulas allowing to obtain the wave function across the time have been established for these three regimes. However, in practice, quantum systems are not isolated. The interaction with their environment can lead to more complex behaviors, driving the control more difficult. We have studied spin systems, coupled or not, kicked by some ultrashort magnetic pulse trains. These trains cross a classical environment (stationary, drift, Markovian, microcanonical) modifying the strength and the delay of each pulse. The modification of the trains by the environment is one of the sources of the disorder into the spin system. This disorder is transmitted between the spins by the coupling. In this study we do not succeed in controlling the system when the trains are in the presence of the previous environments. To remedy this situation, we force the magnetic pulses to cross a chaotic environment. Before a time t, called horizon of coherence, the system coupled by an Heisenberg interaction and submitted to a chaotic environment remains coherent whereas after, the population and the coherence of one spin and of the average spin of the system tend to go near the microcanonical distribution. During this horizon, it is possible to realize some quantum control either by total control (control of the system at every instants) or by information transmission. This study allows us to determine an empirical formula of the horizon of coherence. Finally, we have tried to find a more formal approach for this horizon.

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.

Flow control of transitional shear flows is investigated by means of numerical simulations. The attenuation of three-dimensional wavepackets of Tollmien-Schlichting (TS) and streaks in the boundary layer is obtained using active control in combination with localised sensors and actuators distributed near the rigid wall. Due to the dimensions of the discretized Navier-Stokes operator, reduced-order models are identified, preserving the dynamics between the inputs and the outputs of the system. Balanced realizations of the system are computed using balanced truncation and system identification. We demonstrate that the energy growth of the perturbations is substantially and efficiently mitigated, using relatively few sensors and actuators. The robustness of the controller is analysed by varying the number of actuators and sensors, the Reynolds number, the pressure gradient and by investigating the nonlinear, transitional case. We show that delay of the transition from laminar to turbulent flow can be achieved despite the fully linear approach. This configuration can be reproduced in experiments, due to the localisation of sensing and actuation devices. The closed-loop system has been investigated for the corresponding twodimensional case by using full-dimensional optimal controllers computed by solving an iterative optimisation based on the Lagrangian approach. This strategy allows to compare the results achieved using open-loop model reduction with model-free controllers. Finally, a parametric analysis of the actuators/ sensors placement is carried-out to deepen the understanding of the inherent dynamics of the closed-loop. The distinction among two different classes of controllers – feedforward and feedback controllers - is highlighted. A second shear flow, a confined turbulent jet, is investigated using particle image velocimetry (PIV) measurements. Proper orthogonal decomposition (POD) modes and Koopman modes via dynamic mode decomposition (DMD) are computed and analysed for understanding the main features of the flow. The frequencies related to the dominating mechanisms are identified; the most energetic structures show temporal periodicity. / <p>QC 20130207</p>

The Frobenius-Perron operator acting on integrable functions and the Koopman operator acting on essentially bounded functions for a given nonsingular transformation on the unit interval can be shown to have cyclic spectrum by referring to the theory of lattice homomorphisms on a Banach lattice. In this paper, it is verified directly that the peripheral point spectrum of the Frobenius-Perron operator and the point spectrum of the Koopman operator are fully cyclic. Under some restrictions on the underlying transformation, the Frobenius-Perron operator is known to be a well defined linear operator on the Banach space of functions of bounded variation. It is also shown that the peripheral point spectrum of the Frobenius-Perron operator on the functions of bounded variation is fully cyclic.