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.

The electric power system is designed to economically and reliably transmit electricity to homes, industry, and businesses. The economic impact of the electric grid was demonstrated by the 2003 blackout's visible impact in the graph of the yearly gross domestic product of the Unites States. However, because the number of customers is so large and economies of scale are leveraged to keep electricity prices low, utilities are strongly interconnected. Performing comprehensive engineering analyses to ensure reliable operation is still impossible. Instead, heuristics and safety factors are incorporated into planning processes to continually meet demand in a way that complies with Federal regulations. As evidenced by the infrequency of blackouts in the United States, the sophisticated planning processes have up to date been relatively successful. However, the power system is constantly changing. Electrical generators based on renewable energies are a beneficial addition to the grid, but these and other technological changes like high-voltage power electronic converters also come with their own challenges. These systems as currently employed tend to have a different impact on the reliability of operation than traditional fossil fuel based generators. As the system changes, so do the engineering analyses required to ensure reliable operation. In particular, the wind energy conversion systems (WECS) negatively impact the response of the grid to disturbances in certain ways due to inherent challenges harnessing the wind as an energy sources. These negative impacts (and the advent of powerful personal computing) require an increase in the sophistication of power system models. Thus, there are competing challenges: the scale of the power system necessitates computationally efficient modeling, but the complexity of analysis required to maintain reliable operation is also increasing. The primary aim of this study is to develop models and methods for more detailed yet computationally manageable simulation. To this aim, higher order linearizations and the properties of linear systems (graph theory and linear algebra) are exploited. More specifically, this document contains three studies. In the short term planning and situational awareness context, a method is proposed to quickly check credible outages of important grid equipment. This methodology enables the inspection of a wider breadth of system conditions to ameliorate the negative impacts of the unpredictability of the wind. A linear model in the traditional sense is also developed to model any arbitrary number of wind turbines in a wind farm. This enables industry players to study the impacts wind turbine interaction on the dynamic stability of the grid in response to small disturbances. Finally, a wind farm is modeled as a large matrix to model even nonlinear behavior of wind farms. This helps industry players analyze the impact of large disturbances on the grid. / Doctor of Philosophy / The electric power system is designed to economically and reliably transmit electricity to homes, industry, and businesses. The economic impact of the electric grid was demonstrated by the 2003 blackout’s visible impact in the graph of the yearly gross domestic product of the United States. However, because the number of customers is so large and economies of scale are leveraged to keep electricity prices low, utilities are strongly interconnected. Performing comprehensive engineering analyses to ensure reliable operation is still impossible. Instead, heuristics and safety factors are incorporated into planning processes to continually meet demand in a way that complies with Federal regulations. As evidenced by the infrequency of blackouts in the United States, the sophisticated planning processes have up to date been relatively successful. However, the power system is constantly changing. Electrical generators based on renewable energies are a beneficial addition to the grid, but these and other technological changes like high-voltage power electronic converters also come with their own challenges. These systems as currently employed tend to have a different impact on the reliability of operation than traditional fossil fuel based generators. As the system changes, so do the engineering analyses required to ensure reliable operation. In particular, the wind energy conversion systems (WECS) negatively impact the response of the grid to disturbances in certain ways due to inherent challenges harnessing the wind as an energy sources. These negative impacts (and the advent of powerful personal computing) require an increase in the sophistication of power system models. Thus, there are competing challenges: the scale of the power system necessitates computationally efficient modeling, but the complexity of analysis required to maintain reliable operation is also increasing. The primary aim of this study is to develop models and methods for more detailed yet computationally manageable simulation. To this aim, higher order linearizations and the properties of linear systems (graph theory and linear algebra) are exploited. More specifically, this document contains three studies. In the short term planning and situational awareness context, a method is proposed to quickly check credible outages of important grid equipment. This methodology enables the inspection of a wider breadth of system conditions to ameliorate the negative impacts of the unpredictability of the wind. A linear model in the traditional sense is also developed to model any arbitrary number of wind turbines in a wind farm. This enables industry players to study the impacts wind turbine interaction on the dynamic stability of the grid in response to small disturbances. Finally, a wind farm is modeled as a large matrix to model even nonlinear behavior of wind farms. This helps industry players analyze the impact of large disturbances on the grid.