Este trabalho propõe estudos de casos em sistemas de energia elétrica por meio do Fluxo de Potência Ótimo (FPO) e da Análise de Sensibilidade em diferentes cenários de operação. Para isso, foram obtidos dados teóricos, a partir de levantamento bibliográfico, que explicitaram os conceitos de otimização aplicados ao sistema estático de energia elétrica. A pesquisa fundamentou-se metodologicamente no método primal-dual barreira logarítmica e nas condições necessárias de primeira-ordem de Karush-Kuhn-Tucker (KKT) para o problema de FPO, e no teorema proposto por Fiacco (1976) para a Análise de Sensibilidade. Os sistemas de equações resultantes das condições de estacionaridade, da função Lagrangiana, foram resolvidos pelo método de Newton. Na implementação computacional foram usadas técnicas de esparsidade. Estudos de casos foram realizados nos sistemas 3, IEEE 14, 30, 118, 300 barras e no equivalente CESP 440 kV com 53 barras, em que foi verificada a eficiência das técnicas apresentadas. / This work proposes a study of cases in power systems by Optimal Power Flow (OPF) and Sensitivity Analysis in different operation scenarios. For this purpose, theoretical data were obtained, starting from a bibliographical review, which enlightened the optimization concepts applied to the static system of electrical energy. The research was methodologically based on the primal-dual logarithmic barrier method and in the first-order necessary Karush-Kuhn-Tucker conditions to the OPF problem and in the theorem proposed by Fiacco (1976) to the Sensitivity Analysis. The equation sets generated by the first-order necessary conditions of the Lagrangian function, were solved by Newton\'s method. In the computational implementation, sparsity techniques were used. Studies of cases were carried out in the 3, IEEE 14, 30, 118, 300 buses and in the equivalent CESP 440 kV 53 bus, where the efficiency of the presented techniques was verified.

Este trabalho propõe estudos de casos em sistemas de energia elétrica por meio do Fluxo de Potência Ótimo (FPO) e da Análise de Sensibilidade em diferentes cenários de operação. Para isso, foram obtidos dados teóricos, a partir de levantamento bibliográfico, que explicitaram os conceitos de otimização aplicados ao sistema estático de energia elétrica. A pesquisa fundamentou-se metodologicamente no método primal-dual barreira logarítmica e nas condições necessárias de primeira-ordem de Karush-Kuhn-Tucker (KKT) para o problema de FPO, e no teorema proposto por Fiacco (1976) para a Análise de Sensibilidade. Os sistemas de equações resultantes das condições de estacionaridade, da função Lagrangiana, foram resolvidos pelo método de Newton. Na implementação computacional foram usadas técnicas de esparsidade. Estudos de casos foram realizados nos sistemas 3, IEEE 14, 30, 118, 300 barras e no equivalente CESP 440 kV com 53 barras, em que foi verificada a eficiência das técnicas apresentadas. / This work proposes a study of cases in power systems by Optimal Power Flow (OPF) and Sensitivity Analysis in different operation scenarios. For this purpose, theoretical data were obtained, starting from a bibliographical review, which enlightened the optimization concepts applied to the static system of electrical energy. The research was methodologically based on the primal-dual logarithmic barrier method and in the first-order necessary Karush-Kuhn-Tucker conditions to the OPF problem and in the theorem proposed by Fiacco (1976) to the Sensitivity Analysis. The equation sets generated by the first-order necessary conditions of the Lagrangian function, were solved by Newton\'s method. In the computational implementation, sparsity techniques were used. Studies of cases were carried out in the 3, IEEE 14, 30, 118, 300 buses and in the equivalent CESP 440 kV 53 bus, where the efficiency of the presented techniques was verified.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-03-22T20:23:25Z No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-23T11:30:21Z (GMT) No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-23T11:30:21Z (GMT). No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We consider Newton’s method for solving a generalized equation of the form f(x) + F(x) 3 0, where f : Ω → Y is continuously differentiable, X and Y are Banach spaces, Ω ⊆ X is open and F : X ⇒ Y has nonempty closed graph. Assuming strong regularity of the equation and that the starting point satisfies Kantorovich’s conditions, we show that the method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. In addition, a local convergence analysis of this method is presented. Moreover, using convex optimization techniques introduced by S. M. Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), we prove a robust convergence theorem for inexact Newton’s method for solving nonlinear inclusion problems in Banach space, i.e., when F(x) = −C and C is a closed convex set. Our analysis, which is based on Kantorovich’s majorant technique, enables us to obtain convergence results under Lipschitz, Smale’s and Nesterov-Nemirovskii’s self-concordant conditions. / N´os consideraremos o m´etodo de Newton para resolver uma equa¸c˜ao generalizada da forma f(x) + F(x) 3 0, onde f : Ω → Y ´e continuamente diferenci´avel, X e Y s˜ao espa¸cos de Banach, Ω ⊆ X ´e aberto e F : X ⇒ Y tem gr´afico fechado n˜ao-vazio. Supondo regularidade forte da equa¸c˜ao e que o ponto inicial satisfaz as hip´oteses de Kantorovich, mostraremos que o m´etodo ´e quadraticamente convergente para uma solu¸c˜ao, a qual ´e ´unica em uma vizinhan¸ca do ponto inicial. Uma an´alise de convergˆencia local deste m´etodo tamb´em ´e apresentada. Al´em disso, usando t´ecnicas de otimiza¸c˜ao convexa introduzida por S. M. Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), provaremos um robusto teorema de convergˆencia para o m´etodo de Newton inexato para resolver problemas de inclus˜ao n˜ao–linear em espa¸cos de Banach, i.e., quando F(x) = −C e C ´e um conjunto convexo fechado. Nossa an´alise, a qual ´e baseada na t´ecnica majorante de Kantorovich, nos permite obter resultados de convergˆencia sob as condi¸c˜oes Lipschitz, Smale e Nesterov-Nemirovskii auto-concordante.

Cette thèse porte sur certains aspects symboliques-numériques du problème des paires invariantes pour les polynômes de matrices. Les paires invariantes généralisent la définition de valeur propre / vecteur propre et correspondent à la notion de sous-espaces invariants pour le cas nonlinéaire. Elles trouvent leurs applications dans le calcul numérique de plusieurs valeurs propres d’un polynôme de matrices; elles présentent aussi un intérêt dans le contexte des systèmes différentiels. En utilisant une approche basée sur les intégrales de contour, nous déterminons des expressions du nombre de conditionnement et de l’erreur rétrograde pour le problème du calcul des paires invariantes. Ensuite, nous adaptons la méthode des moments de Sakurai-Sugiura au calcul des paires invariantes et nous étudions le comportement de la version scalaire et par blocs de la méthode en présence de valeurs propres multiples. Le résultats obtenus à l’aide des approches directes peuvent éventuellement être améliorés numériquement grâce à une méthode itérative: nous proposons ici une comparaison de deux variantes de la méthode de Newton appliquée aux paires invariantes. Le problème des solvants de matrices est très proche de celui des paires invariants. Le résultats présentés ci-dessus sont donc appliqués au cas des solvants pour obtenir des expressions du nombre de conditionnement et de l’erreur, et un algorithme de calcul basé sur la méthode des moments. De plus, nous étudions le lien entre le problème des solvants et la transformation des polynômes de matrices en forme triangulaire. / In this thesis, we study some symbolic-numeric aspects of the invariant pair problem for matrix polynomials. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. They have applications in the numeric computation of several eigenvalues of a matrix polynomial; they also present an interest in the context of differential systems. Here, a contour integral formulation is applied to compute condition numbers and backward errors for invariant pairs. We then adapt the Sakurai-Sugiura moment method to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues, and we analyze the behavior of the scalar and block versions of the method in presence of different multiplicity patterns. Results obtained via direct approaches may need to be refined numerically using an iterative method: here we study and compare two variants of Newton’s method applied to the invariant pair problem. The matrix solvent problem is closely related to invariant pairs. Therefore, we specialize our results on invariant pairs to the case of matrix solvents, thus obtaining formulations for the condition number and backward errors, and a moment-based computational approach. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

Mechanics of structure genome (MSG) is a unified homogenization theory that provides constitutive modeling of three-dimensional (3D) continua, beams and plates. In present work, the author extends the MSG to study the buckling of structures such as stiffened and sandwich panels. Such structures are usually slender or flat and easily buckle under compressive loads or bending moments which may result in catastrophic failure.<div><br><div>Buckling studies of stiffened and sandwich panels are found to be scattered. Most of the existed theories employ unnecessary assumptions or only apply to certain types of structures. There are few unified approaches that are capable of studying the buckling of different kinds of structures altogether. The main improvements of current approach compared with other methods in the literature are avoiding unnecessary assumptions, the capability of predicting all possible buckling modes including the global and local buckling modes, and the potential in studying the buckling of various types of structures.<br></div><div><br></div><div>For global buckling that features small local rotations, MSG mathematically decouples the 3D geometrical nonlinear problem into a linear constitutive modeling using structure genome (SG) and a geometrical nonlinear problem defined in a macroscopic structure. As a result, the original structures are simplified as macroscopic structures such as beams, plates or continua with effective properties, and the global buckling modes are predicted on macroscopic structures. For local buckling that features finite local rotations, Green strain is introduced into the MSG theory to achieve geometrically nonlinear constitutive modeling. Newton’s method is used to solve the nonlinear equilibrium equations for fluctuating functions. To find the bifurcated fluctuating functions, the fluctuating functions are then perturbed under the Bloch-periodic boundary conditions. The bifurcation is found when the tangent stiffness associated with the perturbed fluctuating functions becomes singular. Moreover, the arc-length method is introduced to solve the nonlinear equilibrium equations for post-local-buckling predictions because of its robustness. The imperfection is included in the form of geometrical imperfection by superimposing the scaled buckling modes in linear perturbation analysis on mesh.<br></div><div><br></div><div>Extensive validation case studies are carried out to assess the accuracy of the MSG theory in global buckling analysis and post-global-buckling analysis, and assess the accuracy of the extended MSG theory in local buckling and post-local-buckling analysis. Results using MSG theory and extended MSG theory in buckling analysis are compared with direct numerical solutions such as 3D FEA results and results in literature. Parametric studies are performed to reveal the relative influence of selective geometric parameters on buckling behaviors. The extended MSG theory is also compared with representative volume element (RVE) analysis with Bloch-periodic boundary conditions using commercial finite element packages such as Abaqus to assess the efficiency and accuracy of the present approach.<br></div></div>

In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

Σε αυτή την εργασία, ασχολούμαστε με το πρόβλημα της επίλυσης συστημάτων μη γραμμικών αλγεβρικών ή/και υπερβατικών εξισώσεων και συγκεκριμένα αναφερόμαστε σε βελτιωμένες αλγοριθμικές τεχνικές επίλυσης τέτοιων συστημάτων. Μη γραμμικά συστήματα υπάρχουν σε πολλούς τομείς της επιστήμης, όπως στη Μηχανική, την Ιατρική, τη Χημεία, τη Ρομποτική, τα Οικονομικά, κ.τ.λ. Υπάρχουν πολλές μέθοδοι για την επίλυση συστημάτων μη γραμμικών εξισώσεων. Ανάμεσά τους η μέθοδος Newton είναι η πιο γνωστή μέθοδος, λόγω της τετραγωνικής της σύγκλισης όταν υπάρχει μια καλή αρχική εκτίμηση και ο Ιακωβιανός πίνακας είναι nonsingular. Η μέθοδος Newton έχει μερικά μειονεκτήματα, όπως τοπική σύγκλιση, αναγκαιότητα υπολογισμού του Ιακωβιανού πίνακα και ακριβής επίλυση του γραμμικού συστήματος σε κάθε επανάληψη. Σε αυτή τη μεταπτυχιακή διπλωματική εργασία αναλύουμε τη μέθοδο Newton και κατηγοριοποιούμε μεθόδους που συμβάλλουν στην αντιμετώπιση των μειονεκτημάτων της μεθόδου Newton, π.χ. Quasi-Newton και Inexact-Newton μεθόδους. Μερικές πιο πρόσφατες μέθοδοι που περιγράφονται σε αυτή την εργασία είναι η μέθοδος MRV και δύο νέες μέθοδοι Newton χωρίς άμεσες συναρτησιακές τιμές, κατάλληλες για προβλήματα με μη ακριβείς συναρτησιακές τιμές ή με μεγάλο υπολογιστικό κόστος. Στο τέλος αυτής της μεταπτυχιακής εργασίας, παρουσιάζουμε τις βασικές αρχές της Ανάλυσης Διαστημάτων και τη Διαστηματική μέθοδο Newton. / In this contribution, we deal with the problem of solving systems of nonlinear algebraic or/and transcendental equations and in particular we are referred to improved algorithmic techniques of such kind of systems. Nonlinear systems arise in many domains of science, such as Mechanics, Medicine, Chemistry, Robotics, Economics, etc. There are several methods for solving systems of nonlinear equations. Among them Newton's method is the most famous, because of its quadratic convergence when a good initial guess exists and the Jacobian matrix is nonsingular. Newton's method has some disadvantages, such as local convergence, necessity of computation of Jacobian matrix and the exact solution of linear system at each iteration. In this master thesis we analyze Newton's method and we categorize methods that contribute to the treatment of drawbacks of Newton's method, e.g. Quasi-Newton and Inexact-Newton methods. Some more recent methods which are described in this thesis are the MRV method and two new Newton's methods without direct function evaluations, ideal for problems with inaccurate function values or high computational cost. At the end of this master thesis, we present the basic principles of Interval Analysis and Interval Newton's method.

La modélisation du transport réactif du contaminant en milieu poreux est un problème complexe cumulant les difficultés de la modélisation du transport avec celles de la modélisation de la chimie et surtout du couplage entre les deux. Cette modélisation conduit à un système d'équations aux dérivées partielles et algébriques dont les inconnues sont les quantités d'espèces chimiques. Une approche possible, déjà utilisée par ailleurs, est de choisir la méthode globale DAE : l'utilisation d'une méthode de lignes, correspondant à la discrétisation en espace seulement, conduit à un système différentiel algébrique (DAE) qui doit être résolu par un solveur adapté. Dans notre cas, on utilise le solveur IDA de Sundials qui s'appuie sur une méthode implicite, à ordre et pas variables, et qui requiert à chaque pas de temps la résolution d'un grand système non linéaire associé à une matrice jacobienne. Cette méthode est implémentée dans un logiciel qui s'appelle GRT3D (Transport Réactif Global en 3D). Le présent travail présente une amélioration de la méthode GDAE, du point de vue de la performance, de la stabilité et de la robustesse. Nous avons ainsi enrichi les possibilités de GRT3D, par la prise en compte complète des équations de précipitation-dissolution permettant l'apparition ou la disparition d'une espèce précipitée. En complément de l'étude de la méthode GDAE, nous présentons aussi une méthode séquentielle non itérative (SNIA), qui est une méthode basée sur le schéma d'Euler explicite : à chaque pas de temps, on résout explicitement l'équation de transport et on utilise ces calculs comme données pour le système chimique, résolu dans chaque maille de façon indépendante. Nous présentons aussi une comparaison entre cette méthode et l'approche GDAE. Des résultats numériques pour deux cas tests, celui proposé par l'ANDRA (cas-test 2D) d'une part, celui proposé par le groupe MoMas (Benchmark "easy case") d'autre part, sont enfin présentés, commentés et analysés. / Modeling reactive transport of contaminants in porous media is a complex problem combining the difficulties of modeling the trasport with those of modeling the chemistry and especially the coupling between the two .This model leads to a system of partial differential equations and algebraic equations whose unknowns are the quantities of chemical species. One approach , already used elsewhere , is choosing the global DAE method : using the method of lines, discretization in space only, leads to a differential algebraic system (DAE ) to be solved by a suitable solver . In our case , the solver IDA Sundials relies on an implicit method, order is used but not variables, and requires at each time solving a large nonlinear system associated with a Jacobian matrix . This method is implemented in a software called GRT3D (Global Reactive Transport in 3D). This paper presents an improved GDAE method , from the standpoint of performance, the stability and robustness. We have enriched the possibilities of GRT3D , by taking full account of the equations of dissolution – precipitation for the appearance or disappearance of precipitated species. In addition to the study of the GDAE method, we also present a non-iterative sequential method ( SNIA ) which is a method based on the explicit Euler scheme : at each time step, we explicitly solve the transport equation and we use these calculations as data for the chemical system which is resolved in each cell independently. We also present a comparison between this method and GDAE approach . Numerical results for two test cases , one proposed by ANDRA ( 2D test case ) on one hand and one proposed by the group MOMAS ( Benchmark "easy case" ) on the other hand, are finally presented , discussed and analyzed.

The aim of this study was to investigate the effect of an active learning approach (ALA) on the achievement of 11 th grade learners in Newton’s laws of motion, The case study was done in one school by comparing the achievement of learners in Newton’s laws of motion through an Active Learning Approach (ALA) with that of learners taught through Traditional Direct I nstruction (TDI) and to determine whether learners taught with ALA retained the material better than those taught with TDI. Two grade 11 classes in one school in the Queenstown District of the Eastern Cape were selected on their performance in two differen t tests and used as case study . The Force Concept Inventory was used to determine the achievements and retention of knowledge of the two groups. The effect of the active lear ning approach on the treatment group was also measured by ask ing each learner in t he group to complete a learner assessment of instruction form. The treatment group was also subjected to a classroom group interview. T he following information emerged from the study: i) Learners taught using the ALA achieved significantly better in FCI pos t - test than those taught with traditional direct instruction. ii) In the retention of knowledge test, the mean score of the learners taught using the ALA was 4.8% higher than those taught with TDI although insignificant. iii) Learners subjected to the ALA liked the instructional approach and as such put in extra time to learn. The findings suggest that the active learning approach had positive effect on the achievement of the Grade 11 learners and thus, can be adapted to enhance learning in the classroom. / Physics

Wir betrachten die lineare inverse Probleme mit gestörter rechter Seite und gestörtem Operator in Hilberträumen, die inkorrekt sind. Um die Auswirkung der Inkorrektheit zu verringen, müssen spezielle Lösungsmethode angewendet werden, hier nutzen wir die sogenannte Tikhonov Regularisierungsmethode. Die Regularisierungsparameter wählen wir aus das verallgemeinerte Defektprinzip. Eine typische numerische Methode zur Lösen der nichtlinearen äquivalenten Defektgleichung ist Newtonverfahren. Wir schreiben einen Algorithmus, die global und monoton konvergent für beliebige Startwerte garantiert. Um die Stabilität zu garantieren, benutzen wir die Glattheit der Lösung, dann erhalten wir eine sogenannte bedingte Stabilität. Wir demonstrieren die sogenannte Interpolationsmethode zur Herleitung von bedingten Stabilitätsabschätzungen bei inversen Problemen für partielle Differentialgleichungen.