[pt] O objetivo deste trabalho é propor uma nova metodologia de solução para o problema de planejamento da expansão de sistemas hidrotérmicos, levando em consideração as incertezas nas afluências que chegam aos reservatórios do sistema. O problema consiste em determinar o conjunto de investimentos em usinas hidroelétricas e termoelétricas que minimize a soma dos custos de construção mais o valor esperado do custo de operação ao longo de um determinado período. O problema de expansão pode ser resolvido através de técnicas de decomposição, sendo dividido em um problema mestre que contém as variáveis de investimento e várias subproblemas que contém as variações de operação. O problema mestre pode ser resolvido por técnicas de programação linear ou inteira mista. Os subproblemas de operação, por sua vez, podem ser resolvidos pro Programação Dinâmica Estocástica. Como neste trabalho os reservatórios são representados de forma individualizada, será aplicada a técnica de Programação Dinâmica Dual Estocástica, de forma a evitar o problema da maldição da dimensionalidade da Programação Dinâmica Estocástica. Portanto, o objetivo principal do estudo é investigar mecanismos de coordenação eficientes entre o problema mestre e os subproblemas de operação que levem à otimização global do problema de planejamento da expansão de sistemas elétricos. / [en] The objective of this paper is to present a new methodology for the solution of hydrothermal systems expansion planning problems that takes into account uncertainties of future water inflows to the system reservoirs. The problem consists on the determination of a set of hydroeletric and thermoelectric plants that will be constructed in order to minimize the sum of investiment costs and expected operation costs during the planning period. The expansion problem can be solved by decomposition techniques in which the problem in divided into a master problem that contains the investiment variables and many subproblems that contain the operation variables. The master problem can be solved by linear or integer/linear programming recursion. In this paper, since the system reservoirs are individually represented, Stochastic Dual Dynamic Programming techniques will be applied in order to avoid the "curse of dimensionality" of dynamic programming. The fundamental objective of the study is to investigate efficient coordination mechanisms between master problem and operation subproblems that lead to a global optimization of power systems expansion planning problem.

A successful power system in military applications (warship, aircraft, armored vehicle etc.) must operate acceptably under a wide range of conditions involving different loading configurations; it must maintain war fighting ability and recover quickly and stably after being damaged. The introduction of energy storage for the power system of an electric warship integrated engineering plant (IEP) may increase the availability and survivability of the electrical power under these conditions. Herein, the problem of energy storage control is addressed in terms of maximizing the average performance. A notional medium-voltage dc system is used as the system model in the study. A linear programming model is used to simulate the power system, and two sets of states, mission states and damage states, are formulated to simulate the stochastic scenarios with which the IEP may be confronted. A genetic algorithm is applied to the design of IEP to find optimized energy storage control parameters. By using this algorithm, the maximum average performance of power system is found.

Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as $10^{10}$ discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.

Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as $10^{10}$ discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.