Modernized and advanced electricity transmission and distribution infrastructure ensures reliable, efficient, and affordable delivery of electric power. The complexity of fault location problem increases with the proliferation of unusual topologies and with the advent of renewable energy-based power generation in the smart grid environment. The proliferation of new Intelligent Electronic Devices (IEDs) provides a venue for the implementation of more accurate and intelligent fault location methods. This dissertation focuses on intelligent fault location methods for smart power grids and it aims at improving fault location accuracies and decreasing the cost and the mean time to repair damaged equipment in major power outages subsequently increasing the reliability of the grid. The developed methods utilize wavelet transformation to extract the traveling wave information in the very fast voltage and current transients which are initiated immediately after a fault occurs, support vector machines to classify the fault type and identify the faulted branches and finally Bewley diagrams to precisely locate the fault. The approach utilizes discrete wavelet transformation (DWT) for analysis of transient voltage and current measurements. The transient wavelet energies are calculated and utilized as the input for support vector machine (SVM) classifiers. SVM learns the mapping between inputs (i.e. transient voltages and/or currents wavelet energies) and desired outputs (i.e. faulty phase and/or faulty section) through processing a set of training cases. This dissertation presents the proposed methodologies applied to three complex power transmission systems. The first transmission system is a three-terminal (teed) three-phase AC transmission network, a common topology in high- and extra high-voltage networks. It is used to connect three substations that are wide apart from each other through long transmission lines with a tee-point, which is not supported by a substation nor equipped with a measuring device. The developed method overcomes the difficulties introduced by the discontinuity: the tee point. The second topology is a hybrid high voltage alternative current (HVAC) transmission line composed of an overhead line combined with an underground cable. The proposed fault location method is utilized to overcome the difficulties introduced by the discontinuity at the transition point from the overhead line to the underground cable and the different traveling wave velocities along the line and the cable. The third topology is a segmented high voltage direct current (HVDC) transmission line including an overhead line combined with an underground cable. This topology is widely utilized to transmit renewable energy-based electrical power from remote locations to the load centers such as from off-shore wind farms to on-shore grids. This dissertation introduces several enhancements to the existing fault type and fault location algorithms: improvement in the concept of fault type classification and faulty section identification by using SVMs with smaller inputs and improvements in the fault location in the complex configurations by utilizing less measurements from the terminals. / Ph. D.

Recent studies have utilized spanwise traveling waves to alter the turbulent boundary layer with the aim of reducing skin friction drag. Spanwise traveling waves are a promising active drag reduction technique; however, the wave generation methods used in previous studies are bulky and could not be practically implemented. This research has developed an implementable traveling wave generation method and then fundamentally demonstrated how it changes the turbulent boundary layer, which is in a manner consistent with skin friction/shear stress reduction. Traveling waves were generated on a two-dimensional surface using low-profile piezoelectric actuators, in an open-loop fashion, and with minimal frequency limitations. The wave generation method was developed to generate tailored traveling wave patterns; thus, yielding control over the propagation direction, number of wave-fronts, and regions of the surface containing traveling waves. These tailored traveling waves have the capacity not just for affecting the boundary layer, but also for other applications such as propulsion. The implementable traveling wave generation method was then tested in a low-speed wind tunnel and shown to alter the structure of the turbulent boundary layer. The boundary layer is pushed off the wall, and the viscous sublayer is thickened, indicating a reduction in shear stress. Analysis of the boundary layer at positions phase-locked to the wave oscillation suggests that the traveling waves induce a phase-lag effect in the flow. This phase-lag produces a stretching of the viscous sublayer and may contribute to the skin friction reduction. The effects of standing waves on the turbulent boundary layer were also investigated and compared with traveling waves. The results indicate that both wave types alter the boundary layer in the same manner. Standing waves are simpler to generate than traveling waves, suggesting that standing waves may be an effective skin friction reduction method. Before traveling or standing waves can be implemented, further research is necessary to investigate the interaction between the wave pattern and the turbulent phenomena and also to quantify the skin friction reduction and overall net energy usage. / Ph. D. / Recent studies have utilized spanwise traveling waves to alter the turbulent boundary layer with the aim of reducing skin friction drag. Spanwise traveling waves are a promising active drag reduction technique; however, the wave generation methods used in previous studies are bulky and could not be practically implemented. This research has developed an implementable traveling wave generation method and then fundamentally demonstrated how it changes the turbulent boundary layer, which is in a manner consistent with skin friction/shear stress reduction. Traveling waves were generated on a two-dimensional surface using low-profile piezoelectric actuators, in an open-loop fashion, and with minimal frequency limitations. The wave generation method was developed to generate tailored traveling wave patterns; thus, yielding control over the propagation direction, number of wave-fronts, and regions of the surface containing traveling waves. These tailored traveling waves have the capacity not just for affecting the boundary layer, but also for other applications such as propulsion. The implementable traveling wave generation method was then tested in a low-speed wind tunnel and shown to alter the structure of the turbulent boundary layer. The boundary layer is pushed off the wall, and the viscous sublayer is thickened, indicating a reduction in shear stress. Analysis of the boundary layer at positions phase-locked to the wave oscillation suggests that the traveling waves induce a phase-lag effect in the flow. This phase-lag produces a stretching of the viscous sublayer and may contribute to the skin friction reduction. The effects of standing waves on the turbulent boundary layer were also investigated and compared with traveling waves. The results indicate that both wave types alter the boundary layer in the same manner. Standing waves are simpler to generate than traveling waves, suggesting that standing waves may be an effective skin friction reduction method. Before traveling or standing waves can be implemented, further research is necessary to investigate the interaction between the wave pattern and the turbulent phenomena and also to quantify the skin friction reduction and overall net energy usage.

In this thesis we develop tools for carrying out computer assisted proof of the stability of traveling wave solutions of the spatially one-dimensional compressible Navier-Stokes equations with an ideal gas equation of state. In particular, we obtain rigorous, tight error bounds on a high-accuracy numerical approximation of the traveling wave profile for parameters corresponding to air, and we obtain rigorous representations in a neighborhood of positive and negative infinity of the solution to the first order ODE associated with linearizing the PDE equations about the traveling wave solution. We also develop supporting tools for rigorous verification of wave stability.

The Fermi-Pasta-Ulam-Tsingou (FPUT) lattice became of great mathematical interest when it was observed that it exhibited a near-recurrence of its initial condition, despite it being a nonlinear system. This behavior was explained by showing that the Korteweg-de Vries (KdV) equation serves as a continuum limit for the FPUT and has soliton solutions. Much work has been done into analyzing the solitary wave solutions of the FPUT and the relationship between the lattice and its continuum limit. For certain potentials the modified KdV (mKdV) instead serves as the continuum limit for the FPUT. However, there has been little research done to examine how the defocusing mKdV can be used a modulation equation for the FPUT or how the kink solutions of the mKdV relate to solutions of the FPUT. This thesis first addresses the existence of kink-like solutions of the FPUT and shows that their profiles can be approximated by the profiles of the kink solutions of the mKdV. Next, it is shown that the defocusing mKdV can be used more widely as a modulation equation for small-amplitude, long-wavelength solutions of the FPUT lattice. Finally, the issue of stability of the kink-like solutions is discussed, and some results toward linear stability are given.

The main focus of this present work is to study how mechanical steady state traveling waves can be generated and propagated through one dimensional media by applying forces. By steady state traveling waves we refer to propagating mechanical waves in a finite medium that never exhibit reflections at the boundaries and continuously move from one end of the structure to the other. Mechanical waves can be classified as traveling, standing and hybrid waves that are the results of the interplay of excitation forces, applied force locations, and the boundary conditions of the structure. Traveling waves carry energy through a defined medium while standing waves keep energy at certain areas that are associated with the modes of excitation. To understand the interaction of systems that exhibit traveling waves with their surrounding media (i.e., swimming flagella, manta ray locomotion), it is crucial to first understand the wave propagation and what is desired in these structural systems. The parameters that affect the generation and propagation of waves should be welldefined to control and manipulate the desired system’s response. One-dimensional string and rod equations are studied with various boundary conditions to generate steady-state traveling waves in a string and longitudinal traveling waves in a rod. Two excitation forces are applied to a string and a rod near the boundaries to understand the generation and propagation of traveling and standing waves at various frequencies. The work examines the quality of the wave propagation in a string, and in a rod. A cost function approach is applied to identify the quality of such waves. Furthermore, steady-state square traveling waves are generated in a string and in-plane in a rod, both theoretically and experimentally. To the authors’ knowledge this is the first time this has been attempted in the literature. Determining the quality of traveling waves and understanding the parameters on the wave propagation of a string and rod can lead to further understand and leverage various engineering disciplines such as mechanical actuation mechanisms, propulsion of flagella, and the basilar membrane in the ear’s cochlea. / Master of Science / This work presents how mechanical steady state traveling waves can be generated and propagated through structures by applying forces. By steady state traveling waves we refer to propagation of mechanical waves in a finite medium that never exhibits reflections at the boundaries and continuously moves from one end of the structure to the other. Mechanical waves can be classified as traveling, standing and hybrid waves that are the results of applied forces, their locations, and the boundary conditions of the structure. Traveling waves carry energy through a defined medium while standing waves keep energy at certain areas. To understand the interaction of systems that exhibit traveling waves with their surrounding media, it is crucial to first understand the wave propagation and what is desired in these structural systems. The parameters that affect the generation and propagation of waves should be well-defined to control and manipulate the desired system’s behavior. In this study, two excitation forces are applied to a string and a rod near the boundaries to understand the generation and propagation of traveling and standing waves at various frequencies. The work examines the quality of the wave propagation in a string, and in a rod. Steady-state square traveling waves are generated in a string and in-plane in a rod, both theoretically and experimentally. To the authors’ knowledge this is the first time this has been attempted in the literature. Determining the quality of traveling waves and understanding the parameters on the wave propagation of a string and rod can lead to further understand and leverage various engineering disciplines such as mechanical actuation mechanisms, propulsion of flagella, and the basilar membrane in the ear’s cochlea.

Les systèmes de réaction-diffusion interviennent pour décrire les transitions de phase dans de nombreux champs d'application. Cette thèse porte sur l'analyse mathématique de modèles de propagation dans des milieux diffusifs, non bornés et hétérogènes, et s'inscrit ainsi dans la lignée d'une recherche particulièrement active. La première partie concerne l'équation simple: on s'y intéressera à la structure interne des fronts, mais on exhibera aussi de nouvelles dynamiques où la vitesse d'un profil de propagation n'est pas unique. Dans la seconde partie, on s'intéresse aux systèmes à deux équations, pour lesquels l'absence de principe du maximum pose de nombreuses difficultés. Ces travaux, en portant sur un vaste éventail de situations, offrent une meilleure compréhension des phénomènes de propagation, et mettent en avant de nouvelles propriétés des problèmes de réaction-diffusion, aidant ainsi à améliorer l'analyse théorique comme alternative à l'approche empirique. / Reaction-diffusion systems arise in the description of phase transitions in various fields of natural sciences. This thesis is concerned with the mathematical analysis of propagation models in some diffusive, unbounded and heterogeneous media, which comes within the scope of an active research subject. The first part deals with the single equation, by looking at the inside structure of fronts, or by exhibiting new dynamics where the profile of propagation may not have a unique speed. In a second part, we take interest in some systems of two equations, where the lack of maximum principles raises many theoretical issues. Those works aim to provide a better understanding of the underlying processes of propagation phenomena. They highlight new features for reaction-diffusion problems, some of them not known before, and hence help to improve the theoretical approach as an alternative to empirical analysis.

Ce travail de thèse est constitué de l'étude de deux problèmes inverses associés à des équations paraboliques à coefficients périodiques. Dans la première partie, on a considéré une équation parabolique à coefficients et condition initiale périodiques. Notre travail a consisté à aborder le cas de coefficient à régularité faible et à minimiser les contraintes d'observations requises pour établir notre résultat de reconstruction du potentiel. On a commencé par établir un résultat d'existence et d'unicité de la solution dans un espace d'énergie adéquat. Ensuite, on a énoncé un principe du maximum adapté aux hypothèses du problème étudié et on a travaillé avec des coefficients mesurables et bornés. Enfin, on a reconstruit le potentiel en établissant une inégalité de Carleman. Le résultat d'identification a été obtenu via une inégalité de stabilité de type Lipschitz. Dans le second travail, on s'est intéressé à la détermination d'un coefficient périodique en espace du terme de réaction dans une équation de réaction-diffusion définie dans l'espace entier $mathbb{R}$. On établit un résultat d'unicité en utilisant un nouveau type d'observations. La nature du problème étudié, posé dans l'espace $mathbb{R}$, nous a permis d'utiliser la notion de vitesse asymptotique de propagation. On a prouvé l'existence de cette vitesse et on l'a caractérisé. On a surdéterminé le problème inverse en choisissant une famille de conditions initiales à décroi-ssance exponentielle. Notre principal résultat est que ce coefficient est déterminé de façon unique, à une symétrie près, par l'observation d'un continuum de vitesses asymptotiques de propagation. / This thesis consists in the study of two problems associated to inverse para-bolic equations with periodic coefficients. We are interested in identifying one coefficient by using two different methods. In the first part, we consider a parabolic equation with periodic coefficients and periodic initial condition. Our work consists to consider the case of coefficient with weak regularity and to minimize the constraints of observations which are required to establish our reconstruction result. We establish a result of existence and uniqueness of the solution in adequate energy space. Then we prove a maximum principle adapted to the hypothesis of the problem studied and we work with measurable and bounded coefficients. Finally, we reconstruct the potential by establishing a Carleman estimate. The identification result was achieved via an inequality of stability. In the second work, we are interested to determine a periodic coefficient of the reaction term defined in the whole space $mathbb{R}$. We establish a uniqueness result by using a new type of observations. The nature of the studied problem allowed us to use the notion of asymptotic speed of propagation. We prove the existence of this speed and we give its characterization. We overdetermin the inverse problem by choosing a family of initial conditions exponentially decaying. Our main result is that the coefficient is uniquely determined up to a symmetry, by the observation of a continuum of asymptotic speed of propagation.

Neste trabalho, será abordada a proteção digital fundamentada na teoria de ondas viajantes e da ferramenta Transformada Wavelet. Um algoritmo completo para proteção digital foi implementado, consistindo basicamente nos módulos de detecção, classificação e localização da falta, com a conseqüente determinação das zonas de proteção do relé. Para este esquema completo de releamento, empregou-se como ferramenta de análise a Transformada Wavelet. Sinais transitórios de alta freqüência gerados por uma situação de falta podem ser analisados pela Transformada Wavelet, detectando-se com precisão o instante de chegada das ondas em um ou nos dois terminais da linha. Conhecendo o instante de chegada dessas ondas e sua velocidade de propagação, a zona de proteção na qual a falta está alocada pode ser estimada de maneira fácil e rápida. A capacidade de acusar com precisão os instantes de chegada das ondas torna a Transformada Wavelet adequada ao problema de determinação das zonas de proteção. Nessa implementação, o usuário pode escolher entre as técnicas de aquisição de dados provenientes de um ou de dois terminais. O algoritmo foi implementado através do software Matlabâ e posteriormente testado, quando se utilizaram dados de faltas obtidos através de simulações no software ATP (Alternative Transients Program). Nas simulações aplicadas a este software, considerou-se um sistema de transmissão de 440 kV. As simulações produziram uma base de dados, variando-se os seguintes parâmetros: tipos de faltas, localizações ao longo da linha, ângulos de incidência, resistências de faltas, entre outros. Pelos resultados alcançados, pode-se afirmar que a aplicação é bastante adequada para uso em relés digitais de distância. / The present work deals with digital protection based on the traveling wave theory and Wavelet transform. An complete algorithm for digital protection was implemented. It is composed of modules of fault detection, classification, location, with protection zone determination. For this complete relaying scheme the Wavelet transform was employed as analysis tool. Transient signals of high frequency generated by the fault occurrence was analyzed by the Wavelet transform, and also the instant of the waves arrival was accurately detected in one or two terminals of the line. By knowing the instant of the arrival of the waves and their propagation velocity, the protection zone, in which the fault is located, can be estimated easily and quickly. The Wavelet transform property of accurate detection the instant of waves arrival make it appropriate to the problem of protection zone determination. In this implementation, the user can choose between the location techniques using data from one or two terminals. The algorithm was implemented by Matlab® software and then tested utilizing data of faults obtained by means of simulations of ATP software (Alternative Transients Program). In these simulations, a transmission line of 440 kV was considered. These simulations produced a database composed of various fault types considering different fault locations, fault inception angles and fault resistances. According to the analysis of the test results, it is possible to verify that the application is very adequate for use in digital distance relays.

The mountain pine beetle (MPB, Dendroctonus ponderosae), a tree-killing bark beetle, has historically been part of the normal disturbance regime in lodgepole pine (Pinus contorta) forests. In recent years, warm winters and summers have allowed MPB populations to achieve synchronous emergence and successful attacks, resulting in widespread population outbreaks and resultant tree mortality across western North America. We develop an age-structured forest demographic model that incorporates temperature-dependent MPB infestations: the Susceptible-Infested-Juvenile (SIJ) model. Stability of fixed points is analyzed as a function of population growth rates, and indicates the existence of periodic outbreaks that intensify as growth rates increase. We devise analytical methods to predict outbreak severity and duration as well as outbreak return time. To assess the vulnerability of natural resources to climate change, we develop a thermally-driven mechanistic model to predict MPB population growth rates using a distributional model of beetle phenology in conjunction with criteria for successful tree colonization. The model uses projected daily minimum and maximum temperatures for the years 2025 to 2085 generated by three separate global climate models. Growth rates are calculated each year for an area defined by latitude range 42° N to 49° N and longitude range 108° W to 117° W on a Cartesian grid of approximately 4km resolution. Using these growth rates, we analyze how the optimal thermal window for beetle development is changing with respect to elevation as a result of climate change induced warming. We also use our combined model to evaluate if thermal regimes exist that would promote life cycle bivoltinism and discuss how yearly growth rates would change as a result. Outbreaks of MPB are largely driven by host tree stand demographics and spatial effects of beetle dispersal. We augment the SIJ model to account for the spatial effects of MPB dispersal throughout a forest landscape by coupling it with a Gaussian redistribution kernel. The new model generates a train of sustained solitary waves of infestation that move through a forest with constant speed. We convert the resulting integrodifference equation into a partial differential equation and search for travelling wave solutions. The resulting differential equation provides predictions of the shape of an outbreak wave profile and of peak infestation as functions of wave speed, which can be calculated analytically. These results culminate in the derivation of an explicit formula for predicting the severity of an outbreak based on the net reproductive rate of MPB and host searching efficiency.

Neste trabalho, será abordada a proteção digital fundamentada na teoria de ondas viajantes e da ferramenta Transformada Wavelet. Um algoritmo completo para proteção digital foi implementado, consistindo basicamente nos módulos de detecção, classificação e localização da falta, com a conseqüente determinação das zonas de proteção do relé. Para este esquema completo de releamento, empregou-se como ferramenta de análise a Transformada Wavelet. Sinais transitórios de alta freqüência gerados por uma situação de falta podem ser analisados pela Transformada Wavelet, detectando-se com precisão o instante de chegada das ondas em um ou nos dois terminais da linha. Conhecendo o instante de chegada dessas ondas e sua velocidade de propagação, a zona de proteção na qual a falta está alocada pode ser estimada de maneira fácil e rápida. A capacidade de acusar com precisão os instantes de chegada das ondas torna a Transformada Wavelet adequada ao problema de determinação das zonas de proteção. Nessa implementação, o usuário pode escolher entre as técnicas de aquisição de dados provenientes de um ou de dois terminais. O algoritmo foi implementado através do software Matlabâ e posteriormente testado, quando se utilizaram dados de faltas obtidos através de simulações no software ATP (Alternative Transients Program). Nas simulações aplicadas a este software, considerou-se um sistema de transmissão de 440 kV. As simulações produziram uma base de dados, variando-se os seguintes parâmetros: tipos de faltas, localizações ao longo da linha, ângulos de incidência, resistências de faltas, entre outros. Pelos resultados alcançados, pode-se afirmar que a aplicação é bastante adequada para uso em relés digitais de distância. / The present work deals with digital protection based on the traveling wave theory and Wavelet transform. An complete algorithm for digital protection was implemented. It is composed of modules of fault detection, classification, location, with protection zone determination. For this complete relaying scheme the Wavelet transform was employed as analysis tool. Transient signals of high frequency generated by the fault occurrence was analyzed by the Wavelet transform, and also the instant of the waves arrival was accurately detected in one or two terminals of the line. By knowing the instant of the arrival of the waves and their propagation velocity, the protection zone, in which the fault is located, can be estimated easily and quickly. The Wavelet transform property of accurate detection the instant of waves arrival make it appropriate to the problem of protection zone determination. In this implementation, the user can choose between the location techniques using data from one or two terminals. The algorithm was implemented by Matlab® software and then tested utilizing data of faults obtained by means of simulations of ATP software (Alternative Transients Program). In these simulations, a transmission line of 440 kV was considered. These simulations produced a database composed of various fault types considering different fault locations, fault inception angles and fault resistances. According to the analysis of the test results, it is possible to verify that the application is very adequate for use in digital distance relays.