abstract: In the recent past, due to regulatory hurdles and the inability to expand transmission systems, the bulk power system is increasingly being operated close to its limits. Among the various phenomenon encountered, static voltage stability has received increased attention among electric utilities. One approach to investigate static voltage stability is to run a set of power flow simulations and derive the voltage stability limit based on the analysis of power flow results. Power flow problems are formulated as a set of nonlinear algebraic equations usually solved by iterative methods. The most commonly used method is the Newton-Raphson method. However, at the static voltage stability limit, the Jacobian becomes singular. Hence, the power flow solution may fail to converge close to the true limit. To carefully examine the limitations of conventional power flow software packages in determining voltage stability limits, two lines of research are pursued in this study. The first line of the research is to investigate the capability of different power flow solution techniques, such as conventional power flow and non-iterative power flow techniques to obtain the voltage collapse point. The software packages used in this study include Newton-based methods contained in PSSE, PSLF, PSAT, PowerWorld, VSAT and a non-iterative technique known as the holomorphic embedding method (HEM). The second line is to investigate the impact of the available control options and solution parameter settings that can be utilized to obtain solutions closer to the voltage collapse point. Such as the starting point, generator reactive power limits, shunt device control modes, area interchange control, and other such parameters. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2017

abstract: Recently, a novel non-iterative power flow (PF) method known as the Holomorphic Embedding Method (HEM) was applied to the power-flow problem. Its superiority over other traditional iterative methods such as Gauss-Seidel (GS), Newton-Raphson (NR), Fast Decoupled Load Flow (FDLF) and their variants is that it is theoretically guaranteed to find the operable solution, if one exists, and will unequivocally signal if no solution exists. However, while theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not. Numerically, the HEM may require extended precision to converge, especially for heavily-loaded and ill-conditioned power system models. In light of the advantages and disadvantages of the HEM, this report focuses on three topics: 1. Exploring the effect of double and extended precision on the performance of HEM, 2. Investigating the performance of different embedding formulations of HEM, and 3. Estimating the saddle-node bifurcation point (SNBP) from HEM-based Thévenin-like networks using pseudo-measurements. The HEM algorithm consists of three distinct procedures that might accumulate roundoff error and cause precision loss during the calculations: the matrix equation solution calculation, the power series inversion calculation and the Padé approximant calculation. Numerical experiments have been performed to investigate which aspect of the HEM algorithm causes the most precision loss and needs extended precision. It is shown that extended precision must be used for the entire algorithm to improve numerical performance. A comparison of two common embedding formulations, a scalable formulation and a non-scalable formulation, is conducted and it is shown that these two formulations could have extremely different numerical properties on some power systems. The application of HEM to the SNBP estimation using local-measurements is explored. The maximum power transfer theorem (MPTT) obtained for nonlinear Thévenin-like networks is validated with high precision. Different numerical methods based on MPTT are investigated. Numerical results show that the MPTT method works reasonably well for weak buses in the system. The roots method, as an alternative, is also studied. It is shown to be less effective than the MPTT method but the roots of the Padé approximant can be used as a research tool for determining the effects of noisy measurements on the accuracy of SNBP prediction. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2018

Orientador: Vanderlei Minori Horita / Banca: Ali Tahzibi / Banca: Paulo Ricardo Silva / Resumo: Estudamos intersecções homoclínicas de variedades estável e instável de pontos peródicos. Toda intersecção homoclínica produz um comportamento curioso na dinâmiôa. Nosso modelo de tal fenômeno é a famosa ferradura de Smale, a qual é um conjunto hiperbólico para um difeomorfismo. Além disso, estudamos dinâmica não hiperbólica cuja perda de hiperbolicidade é divido à tangências homoclínicas. Elas tem um papel central na teoria de sistemas dinâmicos. O desdobramento de uma tangência homoclínica produz dinâmicas muito interessantes. Neste trabalho estudamos a criação de cascatas de bifurcações de duplicação de período e um esquema de renormalização para uma tangência homoclínica. / Abstract: We study homoclinic intersection of stable and unstable manifolds of periodic points. Every homoclinic intersection produce a intricate behavior of the dynamics. Our model of such phenomena is the so called Smalesþs horseshoe, which is a hyperbolic set for a di eomorphism. We also study non hyperbolic dynamics whose lack of hyperbolicity is due to homoclinic tangencies. They play a central role in the theory of dynamical systems. The unfolding of a homoclinic tangency produce many interesting dynamics. In this work we study creation of cascade of period doubling bifurcations and a renormalization scheme for a homoclinic tangency. / Mestre

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-03-03Bitstream added on 2014-06-13T20:27:28Z : No. of bitstreams: 1 bronzi_ma_me_sjrp.pdf: 904425 bytes, checksum: 2344eb35a112034c2f1741b2e229f1ec (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Estudamos intersecções homoclínicas de variedades estável e instável de pontos peródicos. Toda intersecção homoclínica produz um comportamento curioso na dinâmiôa. Nosso modelo de tal fenômeno é a famosa ferradura de Smale, a qual é um conjunto hiperbólico para um difeomorfismo. Além disso, estudamos dinâmica não hiperbólica cuja perda de hiperbolicidade é divido à tangências homoclínicas. Elas tem um papel central na teoria de sistemas dinâmicos. O desdobramento de uma tangência homoclínica produz dinâmicas muito interessantes. Neste trabalho estudamos a criação de cascatas de bifurcações de duplicação de período e um esquema de renormalização para uma tangência homoclínica. / We study homoclinic intersection of stable and unstable manifolds of periodic points. Every homoclinic intersection produce a intricate behavior of the dynamics. Our model of such phenomena is the so called Smalesþs horseshoe, which is a hyperbolic set for a di eomorphism. We also study non hyperbolic dynamics whose lack of hyperbolicity is due to homoclinic tangencies. They play a central role in the theory of dynamical systems. The unfolding of a homoclinic tangency produce many interesting dynamics. In this work we study creation of cascade of period doubling bifurcations and a renormalization scheme for a homoclinic tangency.

abstract: The holomorphic embedding method (HEM) applied to the power-flow problem (HEPF) has been used in the past to obtain the voltages and flows for power systems. The incentives for using this method over the traditional Newton-Raphson based nu-merical methods lie in the claim that the method is theoretically guaranteed to converge to the operable solution, if one exists. In this report, HEPF will be used for two power system analysis purposes: a. Estimating the saddle-node bifurcation point (SNBP) of a system b. Developing reduced-order network equivalents for distribution systems. Typically, the continuation power flow (CPF) is used to estimate the SNBP of a system, which involves solving multiple power-flow problems. One of the advantages of HEPF is that the solution is obtained as an analytical expression of the embedding parameter, and using this property, three of the proposed HEPF-based methods can es-timate the SNBP of a given power system without solving multiple power-flow prob-lems (if generator VAr limits are ignored). If VAr limits are considered, the mathemat-ical representation of the power-flow problem changes and thus an iterative process would have to be performed in order to estimate the SNBP of the system. This would typically still require fewer power-flow problems to be solved than CPF in order to estimate the SNBP. Another proposed application is to develop reduced order network equivalents for radial distribution networks that retain the nonlinearities of the eliminated portion of the network and hence remain more accurate than traditional Ward-type reductions (which linearize about the given operating point) when the operating condition changes. Different ways of accelerating the convergence of the power series obtained as a part of HEPF, are explored and it is shown that the eta method is the most efficient of all methods tested. The local-measurement-based methods of estimating the SNBP are studied. Non-linear Thévenin-like networks as well as multi-bus networks are built using model data to estimate the SNBP and it is shown that the structure of these networks can be made arbitrary by appropriately modifying the nonlinear current injections, which can sim-plify the process of building such networks from measurements. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2017

A power system will experience voltage collapse when the loads increase up to a certain critical limit, where the system physically cannot support the amount of connected load. This point identified as a Saddle- Node Bifurcation (SNB), corresponds to a generic instability of parameterized differential equation models and represents the intersection point where different branches of equilibria meet. At this point the jacobian matrix of the system is singular and the system loses stability bringing the typical scenario of voltage collapse. To prevent voltage instability and collapse, the computation of the closest distance from a present operating point to the saddle-node bifurcation set can be used as a loadability index useful in power system operation and planning. The power margin is determined by applying the iterative or direct method described in [16]. Numerical examples of both methods applied to IEEE 9-bus system and IEEE 39-bus system shows that the iterative method is more reliable although it requires a longer computation time. The stability of the system is negatively affected in two ways when generators reach their reactive power limits: the voltage stability margin is deteriorated, or immediate voltage instability and collapse is produced.

Estimar a região de estabilidade de um ponto de equilíbrio assintoticamente estável é importante em aplicações tais como sistemas de potência, economia e ecologia. A compreensão da estrutura qualitativa da fronteira da região de estabilidade é fundamental para estimar com eficiência a região de estabilidade. Caracterizações topológicas e dinâmicas da fronteira da região de estabilidade foram desenvolvidas ao longo das últimas décadas. Estas caracterizações foram desenvolvidas sob hipóteses de hiperbolicidade dos pontos de equilíbrio na fronteira e transversalidade. Para sistemas que dependem de parâmetros, a condição de hiperbolicidade pode ser violada em pontos de bifurcações. Estaremos interessados em estimar a região de estabilidade, para sistemas sujeitos a variações de parâmetros, onde ocorre a violação da condição de hiperbolicidade dos pontos de equilíbrio na fronteira da região de estabilidade devido ao aparecimento de uma bifurcação sela-nó do tipo zero nesta fronteira. Apresentaremos neste trabalho uma caracterização completa da fronteira da região de estabilidade na presença de um ponto de equilíbrio não hiperbólico sela-nó do tipo zero. Motivados também em oferecer um algoritmo conceitual para obter estimativas da região de estabilidade perturbada via conjunto de nível de uma dada função energia na vizinhança de um parâmetro de bifurcação sela-nó do tipo zero, buscaremos exibir resultados que permitam compreender o comportamento da região de estabilidade e de sua fronteira sob a influência das variações do parâmetro, incluindo variações do parâmetro próximo a um parâmetro de bifurcação sela-nó do tipo zero. / Estimating the stability region of an asymptotically stable equilibrium point is fundamental in applications such as power systems, economy and ecology. The knowledge of the qualitative structure of the stability boundary is essential to estimate with efficiency the stability region. Topological and dynamical characterizations of the stability boundary have been developed over the past decades. These characterizations were developed under assumptions of hyperbolicity of equilibrium points on the stability boundary and transversality. For systems that depend on parameters, the condition of hyperbolicity can be violated at points of bifurcations. We will be primarily interested in estimating the stability region, for systems subjected to parameter variations, when the condition of hyperbolicity of equilibrium points on the stability boundary is violated due to the appearance of a type-zero saddle-node bifurcation on the stability boundary. We will develop in this work, a complete characterization of the stability boundary in the presence of a type-zero saddle-node non-hyperbolic equilibrium point. Also, motivated to providing a conceptual algorithm to obtain estimates of the perturbed stability region via level sets of a given energy function in the neighborhood of a type-zero saddle-node bifurcation parameter, we offer results that explain the behavior of the stability region and its boundary under the influence of parameter variations, including variations of the parameter close to a type-zero saddle-node bifurcation parameter.

Estimar a região de estabilidade de um ponto de equilíbrio assintoticamente estável é importante em aplicações tais como sistemas de potência, economia e ecologia. A compreensão da estrutura qualitativa da fronteira da região de estabilidade é fundamental para estimar com eficiência a região de estabilidade. Caracterizações topológicas e dinâmicas da fronteira da região de estabilidade foram desenvolvidas ao longo das últimas décadas. Estas caracterizações foram desenvolvidas sob hipóteses de hiperbolicidade dos pontos de equilíbrio na fronteira e transversalidade. Para sistemas que dependem de parâmetros, a condição de hiperbolicidade pode ser violada em pontos de bifurcações. Estaremos interessados em estimar a região de estabilidade, para sistemas sujeitos a variações de parâmetros, onde ocorre a violação da condição de hiperbolicidade dos pontos de equilíbrio na fronteira da região de estabilidade devido ao aparecimento de uma bifurcação sela-nó do tipo zero nesta fronteira. Apresentaremos neste trabalho uma caracterização completa da fronteira da região de estabilidade na presença de um ponto de equilíbrio não hiperbólico sela-nó do tipo zero. Motivados também em oferecer um algoritmo conceitual para obter estimativas da região de estabilidade perturbada via conjunto de nível de uma dada função energia na vizinhança de um parâmetro de bifurcação sela-nó do tipo zero, buscaremos exibir resultados que permitam compreender o comportamento da região de estabilidade e de sua fronteira sob a influência das variações do parâmetro, incluindo variações do parâmetro próximo a um parâmetro de bifurcação sela-nó do tipo zero. / Estimating the stability region of an asymptotically stable equilibrium point is fundamental in applications such as power systems, economy and ecology. The knowledge of the qualitative structure of the stability boundary is essential to estimate with efficiency the stability region. Topological and dynamical characterizations of the stability boundary have been developed over the past decades. These characterizations were developed under assumptions of hyperbolicity of equilibrium points on the stability boundary and transversality. For systems that depend on parameters, the condition of hyperbolicity can be violated at points of bifurcations. We will be primarily interested in estimating the stability region, for systems subjected to parameter variations, when the condition of hyperbolicity of equilibrium points on the stability boundary is violated due to the appearance of a type-zero saddle-node bifurcation on the stability boundary. We will develop in this work, a complete characterization of the stability boundary in the presence of a type-zero saddle-node non-hyperbolic equilibrium point. Also, motivated to providing a conceptual algorithm to obtain estimates of the perturbed stability region via level sets of a given energy function in the neighborhood of a type-zero saddle-node bifurcation parameter, we offer results that explain the behavior of the stability region and its boundary under the influence of parameter variations, including variations of the parameter close to a type-zero saddle-node bifurcation parameter.

Signalverarbeitung im Nervensystem hängt sowohl von der Netzwerkstruktur, als auch den zellulären Eigenschaften der Nervenzellen ab. In dieser Abhandlung werden zwei zelluläre Eigenschaften im Hinblick auf ihre funktionellen Anpassungsmöglichkeiten untersucht: Es wird gezeigt, dass neuronale Morphologie die Signalweiterleitung unter Berücksichtigung energetischer Beschränkungen verstärken kann, und dass selbst kleine Änderungen in biophysikalischen Parametern die Aktivierungsbifurkation in Nervenzellen, und damit deren Informationskodierung, wechseln können. Im ersten Teil dieser Abhandlung wird, unter Verwendung von mathematischen Modellen und Daten, die Hypothese aufgestellt, dass Energie-effiziente Signalweiterleitung als starker Evolutionsdruck für unterschiedliche Zellkörperlagen bei Nervenzellen wirkt. Um Energie zu sparen, kann die Signalweiterleitung vom Dendrit zum Axon verstärkt werden, indem relativ kleine Zellkörper zwischen Dendrit und Axon eingebaut werden, während relativ große Zellkörper besser ausgelagert werden. Im zweiten Teil wird gezeigt, dass biophysikalische Parameter, wie Temperatur, Membranwiderstand oder Kapazität, den Feuermechanismus des Neurons ändern, und damit gleichfalls Aktionspotential-basierte Informationsverarbeitung. Diese Arbeit identifiziert die sogenannte "saddle-node-loop" (Sattel-Knoten-Schlaufe) Bifurkation als den Übergang, der besonders drastische funktionale Auswirkungen hat. Neben der Änderung neuronaler Filtereigenschaften sowie der Ankopplung an Stimuli, führt die "saddle-node-loop" Bifurkation zu einer Erhöhung der Netzwerk-Synchronisation, was möglicherweise für das Auslösen von Anfällen durch Temperatur, wie bei Fieberkrämpfen, interessant sein könnte. / Signal processing in nervous systems is shaped by the connectome as well as the cellular properties of nerve cells. In this thesis, two cellular properties are investigated with respect to the functional adaptations they provide: It is shown that neuronal morphology can improve signal transmission under energetic constraints, and that even small changes in biophysical parameters can switch spike generation, and thus information encoding. In the first project of the thesis, mathematical modeling and data are deployed to suggest energy-efficient signaling as a major evolutionary pressure behind morphological adaptations of cell body location: In order to save energy, the electrical signal transmission from dendrite to axon can be enhanced if a relatively small cell body is located between dendrite and axon, while a relatively large cell body should be externalized. In the second project, it is shown that biophysical parameters, such as temperature, membrane leak or capacitance, can transform neuronal excitability (i.e., the spike onset bifurcation) and, with that, spike-based information processing. This thesis identifies the so-called saddle-node-loop bifurcation as the transition with particularly drastic functional implications. Besides altering neuronal filters and stimulus locking, the saddle-node-loop bifurcation leads to an increase in network synchronization, which may potentially be relevant for the initiation of seizures in response to increased temperature, such as during fever cramps.

En aquest treball s'ha estudiat la dinàmica dels làsers de classe A i de classe B en termes del potencial de Lyapunov. En el cas que s'injecti un senyal al làser o es modulin alguns dels paràmetres, apareix un comportament moltmés complex i s'estudia el conjunt de bifurcacions.1) Als làsers de classe A, la dinàmica determinista s'ha interpretat com el moviment damunt el potencial de Lyapunov. En la dinàmica estocàstica s'obté un flux sostingut per renou per a la fase del camp elèctric.2) Per als làsers de classe A amb senyal injectat, s'ha descrit el conjunt de bifurcacions complet i s'ha determinat el conjunt d'amplituds i freqüències en el quals el làser responajustant la seva freqüència a la del camp extern. 3) S'ha obtingut un potencial de Lyapunov pels làsers de classe B, només vàlid en el cas determinista, que inclou els termes de saturació de guany i d'emissió espontània.4) S'ha realitzat un estudi del conjunt de bifurcacions parcial al voltant del règim tipus II de la singularitat Hopf--sella--node en un làser de classe B amb senyal injectat.5) S'han identificat les respostes òptimes pels làsers de semiconductor sotmesos a modulació periòdica externa. S'han obtingut les corbes que donen la resposta màxima per cada tipus de resonància en el pla definit per l'amplitud relativa de modulació i la freqüència de modulació. / In this work we have studied the dynamics of both class A and class B lasers in terms of Lyapunov potentials. In the case of an injected signal or when some laser parameters are modulated, and more complex behaviour is expected, the bifurcation set is studied. The main results are the following:1) For class A lasers, the deterministic dynamics has been interpreted as a movement on the potential landscape. In the stochastic dynamics we have found a noise sustained flow for the phase of the electric field. 2) For class A lasers with an injected signal, we have been able to describe the whole bifurcation set of this system and to determine the set of amplitudes frequencies for which the laser responds adjusting its frequency to that of the external field. 3) In the case of class B lasers, we have obtained a Lyapunov potential only valid in the deterministic case, including spontaneous emission and gain saturation terms. The fixed point corresponding to the laser in the on state has been interpreted as a minimum in this potential. Relaxation to this minimum is reached through damped oscillations. 4) We have performed a study of the partial bifurcation set around the type II regime of the Hopf-saddle-node singularity in a class B laser with injected signal. 5) We have identified the optimal responses of a semiconductor laser subjected to an external periodic modulation. The lines that give a maximum response for each type of resonance are obtained in the plane defined by the relative amplitude modulation and frequency modulation.