Classical alternating direction (AD) methods for parabolic equations, based on some standard implicit time stepping procedure such as Crank-Nicolson, can have errors associated with the AD perturbations that are much larger than the errors associated with the underlying time stepping procedure . We plan to show that minor modifications in the AD procedures can virtually eliminate the perturbation errors at an minor additional computational cost. A mixed finite element method is applied in the spactial variables. Similar to the finite difference and finite element methods in spactial variables, we plan to have the same accuracy in time. A convergence analysis can also be shown .

Notre société étant plus dépendante que jamais au vecteur électrique, la moindre perturbation du transport ou de l’acheminement de l’électricité a un impact social et économique important. La fiabilité et la sécurité des réseaux électriques sont donc cruciales pour les gestionnaires de réseaux, en plus des aspects économiques. De plus, les réseaux de transport sont interconnectés pour réduire les coûts des opérations et pour améliorer la sécurité. Un des plus grand défis des gestionnaires des réseaux de transport est ainsi de se coordonner avec les réseaux voisins, ce qui soulève des problèmes liés à la taille du problème, à l’interopérabilité et à la confidentialité des données.Cette thèse se focalise principalement sur la sécurité des opérations sur les réseaux électriques, c’est pourquoi l’évolution des principales caractéristiques des blackouts, qui sont des échecs de la sécurité des réseaux, sont étudiés sur la période 2005-2016. L’approche de cette étude consiste à déterminer quelles sont les principales caractéristiques des incidents de ces 10 dernières années, afin d’identifier ce qui devrait être intégré pour réduire le risque que ces incidents se reproduisent. L’évolution a été étudiée et comparé avec les caractéristiques des blackouts qui se sont produit avant 2005. L’étude se focalise sur les préconditions qui ont mené à ces blackouts et sur les cascades, et particulièrement sur le rôle de la vitesse des cascades. Les caractéristiques importante sont extraites et intégrées dans la suite de notre travail.Un algorithme résolvant un problème préventif d’Optimal Power Flow avec contraintes de sécurité (SCOPF) de manière distribuée est ainsi développé. Ce problème consiste en l’ajout de contraintes qui assure qu’après la perte de n’importe quel appareil d’importance, le nouveau point d’équilibre, atteint suite au réglage primaire en fréquence, respecte les contraintes du système. L’algorithme développé utilise une décomposition fine du problème et est implémenté sous le paradigme multi-agent, basé sur deux catégories d’agents : les appareils et les bus. Les agents sont coordonnés grâce à l’ « Alternating Direction Method of Multipliers (ADMM)» et grâce à un problème de consensus. Cette décomposition procure l’autonomie et la confidentialité nécessaire aux différents acteurs du système, mais aussi, un bon passage à l’échelle par rapport à la taille du problème. Cet algorithme a aussi pour avantage d’être robuste à n’importe quelle perturbation, incluant la séparation du système en plusieurs régions.Puis, pour prendre en compte l’incertitude sur la production créée par les erreurs de prédiction des fermes éoliennes, une approche distribuée à deux étapes est développée pour résoudre un problème d’Optimal Power Flow avec contraintes probabilistes (CCOPF), d’une manière complétement distribuée. Les erreurs de prédiction des fermes éoliennes sont modélisées par des lois normales indépendantes et les écarts par rapport aux plannings de production sont considérés compensés par le réglage primaire en fréquence. La première étape de l’algorithme a pour but de déterminer des paramètres de sensibilités nécessaires pour formuler le problème. Les résultats de cette étape sont ensuite des paramètres d’entrée de la seconde étape qui, elle, résout le problème de CCOPF. Une extension de cette formulation permet d’ajouter de la flexibilité au problème en permettant la réduction de la production éolienne. Cet algorithme est basé sur la même décomposition fine que précédemment où les agents sont également coordonnés par l’ADMM et grâce à un problème de consensus. En conclusion, cet algorithme en deux étapes garantit la confidentialité et l’autonomie des différents acteurs, et est parallèle et adaptée aux plateformes hautes performances. / Our societies are more dependent on electricity than ever, thus any disturbance in the power transmission and delivery has major economic and social impact. The reliability and security of power systems are then crucial to keep, for power system operators, in addition to minimizing the system operating cost. Moreover, transmission systems are interconnected to decrease the cost of operation and improve the system security. One of the main challenges for transmission system operators is therefore to coordinate with interconnected power systems, which raises scalability, interoperability and privacy issues. Hence, this thesis is concerned with how TSOs can operate their networks in a decentralized way but coordinating their operation with other neighboring TSOs to find a cost-effective scheduling that is globally secure.The main focus of this thesis is the security of power systems, this is why the evolution of the main characteristics of the blackouts that are failures in power system security, of the period 2005-2016 is studied. The approach consists in determining what the major characteristics of the incidents of the past 10 years are, to identify what should be taken into account to mitigate the risk of incidents. The evolution have been studied and compared with the characteristics of the blackouts before 2005. The study focuses on the pre-conditions that led to those blackouts and on the cascades, and especially the role of the cascade speed. Some important features are extracted and later integrated in our work.An algorithm that solve the preventive Security Constrained Optimal Power Flow (SCOPF) problem in a fully distributed manner, is thus developed. The preventive SCOPF problem consists in adding constraints that ensure that, after the loss of any major device of the system, the new steady-state reached, as a result of the primary frequency control, does not violate any constraint. The developed algorithm uses a fine-grained decomposition and is implemented under the multi-agent system paradigm based on two categories of agents: devices and buses. The agents are coordinated with the Alternating Direction method of multipliers in conjunction with a consensus problem. This decomposition provides the autonomy and privacy to the different actors of the system and the fine-grained decomposition allows to take the most of the decomposition and provides a good scalability regarding the size of the problem. This algorithm also have the advantage of being robust to any disturbance of the system, including the separation of the system into regions.Then, to account for the uncertainty of production brought by wind farms forecast error, a two-step distributed approach is developed to solve the Chance-Constrained Optimal Power Flow problem, in a fully distributed manner. The wind farms forecast errors are modeled by independent Gaussian distributions and the mismatches with the initials are assumed to be compensated by the primary frequency response of generators. The first step of this algorithm aims at determining the sensitivity factors of the system, needed to formulate the problem. The results of this first step are inputs of the second step that is the CCOPF. An extension of this formulation provides more flexibility to the problem and consists in including the possibility to curtail the wind farms. This algorithm relies on the same fine-grained decomposition where the agents are again coordinated by the ADMM and a consensus problem. In conclusion, this two-step algorithm ensures the privacy and autonomy of the different system actors and it is de facto parallel and adapted to high performance platforms.

L’objectif principal de cette thèse est l’exploration du recalage d’images à l’aide de champs aléatoires de Markov d’ordres supérieurs, et plus spécifiquement d’intégrer la connaissance de transformations globales comme une transformation rigide, dans la structure du graphe. Notre cadre principal s’applique au recalage 2D-2D ou 3D-3D et utilise une approche hiérarchique d’un modèle de champ de Markov dont le graphe est une grille régulière. Les variables cachées sont les vecteurs de déplacements des points de contrôle de la grille.Tout d’abord nous expliciterons la construction du graphe qui permet de recaler des images en cherchant entre elles une transformation affine, rigide, ou une similarité, tout en ne changeant qu’un potentiel sur l’ensemble du graphe, ce qui assure une flexibilité lors du recalage. Le choix de la métrique est également laissée à l’utilisateur et ne modifie pas le fonctionnement de notre algorithme. Nous utilisons l’algorithme d’optimisation de décomposition duale qui permet de gérer les hyper-arêtes du graphe et qui garantit l’obtention du minimum exact de la fonction pourvu que l’on ait un accord entre les esclaves. Un graphe similaire est utilisé pour réaliser du recalage 2D-3D.Ensuite, nous fusionnons le graphe précédent avec un autre graphe construit pour réaliser le recalage déformable. Le graphe résultant de cette fusion est plus complexe et, afin d’obtenir un résultat en un temps raisonnable, nous utilisons une méthode d’optimisation appelée ADMM (Alternating Direction Method of Multipliers) qui a pour but d’accélérer la convergence de la décomposition duale. Nous pouvons alors résoudre simultanément recalage affine et déformable, ce qui nous débarrasse du biais potentiel issu de l’approche classique qui consiste à recaler affinement puis de manière déformable. / The main objective of this thesis is the exploration of higher order Markov Random Fields for image registration, specifically to encode the knowledge of global transformations, like rigid transformations, into the graph structure. Our main framework applies to 2D-2D or 3D-3D registration and use a hierarchical grid-based Markov Random Field model where the hidden variables are the displacements vectors of the control points of the grid.We first present the construction of a graph that allows to perform linear registration, which means here that we can perform affine registration, rigid registration, or similarity registration with the same graph while changing only one potential. Our framework is thus modular regarding the sought transformation and the metric used. Inference is performed with Dual Decomposition, which allows to handle the higher order hyperedges and which ensures the global optimum of the function is reached if we have an agreement among the slaves. A similar structure is also used to perform 2D-3D registration.Second, we fuse our former graph with another structure able to perform deformable registration. The resulting graph is more complex and another optimisation algorithm, called Alternating Direction Method of Multipliers is needed to obtain a better solution within reasonable time. It is an improvement of Dual Decomposition which speeds up the convergence. This framework is able to solve simultaneously both linear and deformable registration which allows to remove a potential bias created by the standard approach of consecutive registrations.

Group sparsity reveals underlying sparsity patterns and contains rich structural information in data. Hence, exploiting group sparsity will facilitate more efficient techniques for recovering large and complicated data in applications such as compressive sensing, statistics, signal and image processing, machine learning and computer vision. This thesis develops efficient algorithms for solving a class of optimization problems with group sparse solutions, where arbitrary group configurations are allowed and the mixed L21-regularization is used to promote group sparsity. Such optimization problems can be quite challenging to solve due to the mixed-norm structure and possible grouping irregularities. We derive algorithms based on a variable splitting strategy and the alternating direction methodology. Extensive numerical results are presented to demonstrate the efficiency, stability and robustness of these algorithms, in comparison with the previously known state-of-the-art algorithms. We also extend the existing global convergence theory to allow more generality.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22862号 / 情博第741号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 山下 信雄, 教授 太田 快人, 教授 鹿島 久嗣 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Due to the increase in the advances in wireless communication, there has been an increase in the use of multi-agents systems to complete any given task. In various applications, multi-agent systems are required to solve an underlying optimization problem to obtain the best possible solution within a feasible region. Solving such multi-agent optimization problems in a distributed framework preferable over centralized frameworks as the former ensures scalability, robustness, and security. Further distributed optimization problem becomes challenging when the decision variables of the individual agents are coupled. In this thesis, a distributed optimization problem with coupled constraints is considered, where a network of agents aims to cooperatively minimize the sum of their local objective functions, subject to individual constraints. This problem setup is relevant to many practical applications like formation flying, sensor fusion, smart grids, etc. For practical scenarios, where agents can solve their local optimal solution efficiently and require fewer assumptions on objective functions, the Alternating Direction Method of Multipliers(ADMM)-based approaches are preferred over gradient-based approaches. For such a constraint coupled problem, several distributed ADMM algorithms are present that guarantee convergence to optimality but they do not discuss the complete analysis for the rate of convergence. Thus, the primary goal of this work is to improve upon the convergence rate of the existing state-of-the-art Tracking-ADMM (TADMM) algorithm to solve the above-distributed optimization problem. Moreover, the current analysis in literature does not discuss the convergence in the case of a time-varying communication network. The first part of the thesis focuses on improving the convergence rate of the Tracking-ADMM algorithm to solve the above-distributed optimization problem more efficiently. To this end, an upper bound on the convergence rate of the TADMM algorithm is derived in terms of the weight matrix of the network. To achieve faster convergence, the optimal weight matrix is computed using a semi-definite programming (SDP) formulation. The improved convergence rate of this Fast-TADMM (F-TADMM) is demonstrated with a simple yet illustrative, coupled constraint optimization problem. Then, the applicability of F-TADMM is demonstrated to the problem of distributed optimal control for trajectory generation of aircraft in formation flight. In the second part of the thesis, the convergence analysis for TADMM is extended while considering a time-varying communication network. The modified algorithm is named as Time-Varying Tracking (TV-TADMM). The formal guarantees on asymptotic convergence are provided with the help of control system analysis of a dynamical system that uses Lyapunov-like theory. The convergence of this TV-TADMM is demonstrated on a simple yet illustrative, coupled constraint optimization problem with switching topology and is compared with the fixed topology setting.

Decreasing gas prices and the pressing need for fast-responding electric power generators are currently transforming natural gas networks. The intermittent operation of gas-fired plants to balance wind generation introduces spatiotemporal fluctuations of increasing gas demand. At the heart of modeling, monitoring, and control of gas networks is a set of nonlinear equations relating nodal gas injections and pressures to flows over pipelines. Given gas demands at all points of the network, the gas flow task aims at finding the rest of the physical quantities. For a tree network, the problem enjoys a closed-form solution; yet solving the equations for practical meshed networks is non-trivial. This problem is posed here as a feasibility problem involving quadratic equalities and inequalities, and is further relaxed to a convex semidefinite program (SDP) minimization. Drawing parallels to the power flow problem, the relaxation is shown to be exact if the cost function is judiciously designed using a representative set of network states. Numerical tests on a Belgian gas network corroborate the superiority of the novel method in recovering the actual gas network state over a Newton-Raphson solver. This thesis also considers the coupled infrastructures of natural gas and electric power systems. The gas and electric networks are coupled through gas-fired generators, which serve as shoulder and peaking plants for the electric power system. The optimal dispatch of coupled natural gas and electric power systems is posed as a relaxed convex minimization problem, which is solved using the feasible point pursuit (FPP) algorithm. For a decentralized solution, the alternating direction method of multipliers (ADMM) is used in collaboration with the FPP. Numerical experiments conducted on a Belgian gas network connected to the IEEE 14 bus benchmark system corroborate significant enhancements on computational efficiency compared with the centralized FPP-based approach. / Master of Science / The increase in penetration of renewable energy in the electric power grid has led to increased fluctuations in the power. The conventional coal based generators are inept to handle these fluctuations and thus, natural gas generators, which have fast response times are used to handle the intermittency caused by renewable energy sources. This manuscript solves the problem of finding the optimal dispatch of coupled natural gas and electric power systems. First, the optimal dispatch problem is framed as a optimization problem and then mathematical solvers are developed. Using the mathematical tools of Feasible point pursuit and Alternating direction method of multipliers, a distributed solver is developed, which can solve the optimal dispatch for large power and natural gas networks. The proposed algorithm is tested on a part of a Belgian gas network and the IEEE 14 bus power system. The algorithm is shown to converge to a feasible point.

Distributed computing is an approach where computationally heavy problems are broken down into more manageable sub-tasks, which can then be distributed across a number of different computers or servers, allowing for increased efficiency through parallelization. This thesis explores an established distributed computing setting, in which the computationally heavy task involves a number of users requesting a linearly separable function to be computed across several servers. This setting results in a condition for feasible computation and communication that can be described by a matrix factorization problem. Moreover, the associated costs with computation and communication are directly related to the number of nonzero elements of the matrix factors, making sparse factors desirable for minimal costs. The Alternating Direction Method of Multipliers (ADMM) is explored as a possible method of solving the sparse matrix factorization problem. To obtain convergence results, extensive convex analysis is conducted on the ADMM iterates, resulting in a theorem that characterizes the limiting points of the iterates as KKT points for the sparse matrix factorization problem. Using the results of the analysis, an algorithm is devised from the ADMM iterates, which can be applied to the sparse matrix factorization problem. Furthermore, an additional implementation is considered for a noisy scenario, in which existing theoretical results are used to justify convergence. Finally, numerical implementations of the devised algorithms are used to perform sparse matrix factorization. / Distribuerad beräkning är en metod där beräkningstunga problem bryts ner i hanterbara deluppgifter, som sedan kan distribueras över ett antal olika beräkningsenheter eller servrar, vilket möjliggör ökad effektivitet genom parallelisering. Denna avhandling undersöker en etablerad distribuerad beräkningssmiljö, där den beräkningstunga uppgiften involverar ett antal användare som begär en linjärt separabel funktion som beräknas över flera servrar. Denna miljö resulterar i ett villkor för tillåten beräkning och kommunikation som kan beskrivas genom ett matrisfaktoriseringsproblem. Dessutom är det möjligt att relatera kostanderna associerade med beräkning och kommunikation till antalet nollskilda element i matrisfaktorerna, vilket gör glesa matrisfaktorer önskvärda. Alternating Direction Method of Multipliers (ADMM) undersöks som en möjlig metod för att lösa det glesa matrisfaktoriseringsproblemet. För att erhålla konvergensresultat genomförs omfattande konvex analys på ADMM-iterationerna, vilket resulterar i ett teorem som karakteriserar de begränsande punkterna för iterationerna som KKT-punkter för det glesa matrisfaktoriseringsproblemet. Med hjälp av resultaten från analysen utformas en algoritm från ADMM-iterationerna, vilken kan appliceras på det glesa matrisfaktoriseringsproblemet. Dessutom övervägs en ytterligare implementering för ett brusigt scenario, där befintliga teoretiska resultat används för att motivera konvergens. Slutligen används numeriska implementeringar av de framtagna algoritmerna för att utföra gles matrisfaktorisering.

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