A new MILP based approach for unit commitment in power production planning

This paper presents a complete, quadratic programming formulation of the standard thermal unit commitment problem in power generation planning, together with a novel iterative optimisation algorithm for its solution. The algorithm, based on a mixed-integer formulation of the problem, considers piecewise linear approximations of the quadratic fuel cost function that are dynamically updated in an iterative way, converging to the optimum; this avoids the requirement of resorting to quadratic programming, making the solution process much quicker. From extensive computational tests on a broad set of benchmark instances of this problem, the algorithm was found to be flexible and capable of easily incorporating different problem constraints. Indeed, it is able to tackle ramp constraints, which although very important in practice were rarely considered in previous publications. Most importantly, optimal solutions were obtained for several well-known benchmark instances, including instances of practical relevance, that are not yet known to have been solved to optimality. Computational experiments and their results showed that the method proposed is both simple and extremely effective.

Increase of the Delivered Power Probability in Distribution Networks using Pareto DC Programming

A methodology to increase the probability of delivering power to any load point through the identification of new investments in distribution network components is proposed in this paper. The method minimizes the investment cost as well as the cost of energy not supplied in the network. A DC optimization model based on mixed integer non-linear programming is developed considering the Pareto front technique in order to identify the adequate investments in distribution networks components which allow increasing the probability of delivering power for any customer in the distribution system at the minimum possible cost for the system operator, while minimizing the energy not supplied cost. Thus, a multi-objective problem is formulated. To illustrate the application of the proposed methodology, the paper includes a case study which considers a 180 bus distribution network

Aplicação de um modelo de programação linear inteira mista (PLIM) na expansão da rede de distribuição de energia elétrica em Angola

Em Angola, apenas cerca de 30% da população tem acesso à energia elétrica, nível que decresce para valores inferiores a 10% em zonas rurais mais remotas. Este problema é agravado pelo facto de, na maioria dos casos, as infraestruturas existentes se encontrarem danificadas ou não acompanharem o desenvolvimento da região. Em particular na capital angolana, Luanda que, sendo a menor província de Angola, é a que regista atualmente a maior densidade populacional. Com uma população de cerca de 5 milhões de habitantes, não só há frequentemente problemas relacionados com a falha do fornecimento de energia elétrica como há ainda uma percentagem considerável de municípios onde a rede elétrica ainda nem sequer chegou. O governo de Angola, no seu esforço de crescimento e aproveitamento das suas enormes potencialidades, definiu o setor energético como um dos fatores críticos para o desenvolvimento sustentável do país, tendo assumido que este é um dos eixos prioritários até 2016. Existem objetivos claros quanto à reabilitação e expansão das infraestruturas do setor elétrico, aumentando a capacidade instalada do país e criando uma rede nacional adequada, com o intuito não só de melhorar a qualidade e fiabilidade da rede já existente como de a aumentar. Este trabalho de dissertação consistiu no levantamento de dados reais relativamente à rede de distribuição de energia elétrica de Luanda, na análise e planeamento do que é mais premente fazer relativamente à sua expansão, na escolha dos locais onde é viável localizar novas subestações, na modelação adequada do problema real e na proposta de uma solução ótima para a expansão da rede existente. Depois de analisados diferentes modelos matemáticos aplicados ao problema de expansão de redes de distribuição de energia elétrica encontrados na literatura, optou-se por um modelo de programação linear inteira mista (PLIM) que se mostrou adequado. Desenvolvido o modelo do problema, o mesmo foi resolvido por recurso a software de otimização Analytic Solver e CPLEX. Como forma de validação dos resultados obtidos, foi implementada a solução de rede no simulador PowerWorld 8.0 OPF, software este que permite a simulação da operação do sistema de trânsito de potências.

A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.

Modeling of hybrid systems by piecewise decomposition

Piecewise linear models systems arise as mathematical models of systems in many practical applications, often from linearization for nonlinear systems. There are two main approaches of dealing with these systems according to their continuous or discrete-time aspects. We propose an approach which is based on the state transformation, more particularly the partition of the phase portrait in different regions where each subregion is modeled as a two-dimensional linear time invariant system. Then the Takagi-Sugeno model, which is a combination of local model is calculated. The simulation results show that the Alpha partition is well-suited for dealing with such a system

Importância da Programação Linear na Determinação de Informações de Gestão - Caso Moave, Sa

A Investigação Operacional vem demonstrando ser uma valiosa ferramenta de gestão nos dias de hoje em que se vive num mercado cada vez mais competitivo. Através da Programação Linear pode-se reproduzir matematicamente um problema de maximização dos resultados ou minimização dos custos de produção com o propósito de auxiliar os gestores na tomada de decisão. A Programação Linear é um método matemático em que a função objectivo e as restrições assumem características lineares, com diversas aplicações no controlo de gestão, envolvendo normalmente problemas de utilização dos recursos disponíveis sujeitos a limitações impostas pelo processo produtivo ou pelo mercado. O objectivo geral deste trabalho é o de propor um modelo de Programação Linear para a programação ou produção e alocação de recursos necessários. Optimizar uma quantidade física designada função objectivo, tendo em conta um conjunto de condicionalismos endógenas às actividades em gestão. O objectivo crucial é dispor um modelo de apoio à gestão contribuindo assim para afectação eficiente de recursos escassos à disposição da unidade económica. Com o trabalho desenvolvido ficou patente a importância da abordagem quantitativa como recurso imprescindível de apoio ao processo de decisão. The operational research has proven to be a valuable management tool today we live in an increasingly competitive market. Through Linear Programming can be mathematically reproduce a problem of maximizing performance or minimizing production costs in order to assist managers in decision making. The Linear Programming is a mathematical method in which the objective function and constraints are linear features, with several applications in the control of management, usually involving problems of resource use are available subject to limitations imposed by the production process or the market. The overall objective of this work is to propose a Linear Programming model for scheduling or production and allocation of necessary resources. Optimizing a physical quantity called the objective function, given a set of endogenous constraints on management thus contributing to efficient allocation of scarce resources available to the economic unit. With the work has demonstrated the importance of the quantitative approach as essential resource to support the decision process.

Implementing interactive computing in an object-oriented environment

Statistical computing when input/output is driven by a Graphical User Interface is considered. A proposal is made for automatic control ofcomputational flow to ensure that only strictly required computationsare actually carried on. The computational flow is modeled by a directed graph for implementation in any object-oriented programming language with symbolic manipulation capabilities. A complete implementation example is presented to compute and display frequency based piecewise linear density estimators such as histograms or frequency polygons.

Lagrangian-informed mixed integer programming reformulations

La programmation linéaire en nombres entiers est une approche robuste qui permet de résoudre rapidement de grandes instances de problèmes d'optimisation discrète. Toutefois, les problèmes gagnent constamment en complexité et imposent parfois de fortes limites sur le temps de calcul. Il devient alors nécessaire de développer des méthodes spécialisées afin de résoudre approximativement ces problèmes, tout en calculant des bornes sur leurs valeurs optimales afin de prouver la qualité des solutions obtenues. Nous proposons d'explorer une approche de reformulation en nombres entiers guidée par la relaxation lagrangienne. Après l'identification d'une forte relaxation lagrangienne, un processus systématique permet d'obtenir une seconde formulation en nombres entiers. Cette reformulation, plus compacte que celle de Dantzig et Wolfe, comporte exactement les mêmes solutions entières que la formulation initiale, mais en améliore la borne linéaire: elle devient égale à la borne lagrangienne. L'approche de reformulation permet d'unifier et de généraliser des formulations et des méthodes de borne connues. De plus, elle offre une manière simple d'obtenir des reformulations de moins grandes tailles en contrepartie de bornes plus faibles. Ces reformulations demeurent de grandes tailles. C'est pourquoi nous décrivons aussi des méthodes spécialisées pour en résoudre les relaxations linéaires. Finalement, nous appliquons l'approche de reformulation à deux problèmes de localisation. Cela nous mène à de nouvelles formulations pour ces problèmes; certaines sont de très grandes tailles, mais nos méthodes de résolution spécialisées les rendent pratiques.

Modeling of hybrid systems by piecewise decomposition

Piecewise linear models systems arise as mathematical models of systems in many practical applications, often from linearization for nonlinear systems. There are two main approaches of dealing with these systems according to their continuous or discrete-time aspects. We propose an approach which is based on the state transformation, more particularly the partition of the phase portrait in different regions where each subregion is modeled as a two-dimensional linear time invariant system. Then the Takagi-Sugeno model, which is a combination of local model is calculated. The simulation results show that the Alpha partition is well-suited for dealing with such a system

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.

Modeling small-scale dairy farms in central Mexico using multi-criteria programming

Milk supply from Mexican dairy farms does not meet demand and small-scale farms can contribute toward closing the gap. Two multi-criteria programming techniques, goal programming and compromise programming, were used in a study of small-scale dairy farms in central Mexico. To build the goal and compromise programming models, 4 ordinary linear programming models were also developed, which had objective functions to maximize metabolizable energy for milk production, to maximize margin of income over feed costs, to maximize metabolizable protein for milk production, and to minimize purchased feedstuffs. Neither multicriteria approach was significantly better than the other; however, by applying both models it was possible to perform a more comprehensive analysis of these small-scale dairy systems. The multi-criteria programming models affirm findings from previous work and suggest that a forage strategy based on alfalfa, rye-grass, and corn silage would meet nutrient requirements of the herd. Both models suggested that there is an economic advantage in rescheduling the calving season to the second and third calendar quarters to better synchronize higher demand for nutrients with the period of high forage availability.

Computational study of the step size parameter of the subgradient optimization method

The subgradient optimization method is a simple and flexible linear programming iterative algorithm. It is much simpler than Newton's method and can be applied to a wider variety of problems. It also converges when the objective function is non-differentiable. Since an efficient algorithm will not only produce a good solution but also take less computing time, we always prefer a simpler algorithm with high quality. In this study a series of step size parameters in the subgradient equation is studied. The performance is compared for a general piecewise function and a specific p-median problem. We examine how the quality of solution changes by setting five forms of step size parameter.

Utilização de modelos de programação não-linear para sistematização de terras para irrigação em áreas regulares

Objetivou-se com este trabalho, desenvolver modelos de programação não-linear para sistematização de terras, aplicáveis para áreas com formato regular e que minimizem a movimentação de terra, utilizando o software GAMS para o cálculo. Esses modelos foram comparados com o Método dos Quadrados Mínimos Generalizado, desenvolvido por Scaloppi & Willardson (1986), sendo o parâmetro de avaliação o volume de terra movimentado. Concluiu-se que, ambos os modelos de programação não-linear desenvolvidos nesta pesquisa mostraram-se adequados para aplicação em áreas regulares e forneceram menores valores de movimentação de terra quando comparados com o método dos quadrados mínimos.

Modelo de programação linear para seleção de pulverizadores agrícolas de barras

A seleção de pulverizadores agrícolas que se adaptem às necessidades da propriedade, é um processo trabalhoso, sendo uma das etapas mais importantes dentro do processo produtivo. O objetivo do presente trabalho foi o de desenvolver e utilizar um modelo de programação linear para auxiliar na seleção de pulverizadores agrícolas de barras, baseado no menor custo horário do equipamento. Foram utilizadas as informações técnicas referentes a 20 modelos de pulverizadores disponíveis no mercado, sendo quatro autopropelidos, oito de arrasto e oito do tipo montado. A análise de sensibilidade dos componentes dos custos operacionais mostrou que as taxas de reparo e depreciação foram os fatores que mais interferiram na variação do custo horário do conjunto trator-pulverizador. O modelo matemático desenvolvido facilitou a realização da análise de sensibilidade que foi processada em um tempo muito pequeno.

Distribuição da carga hidráulica em linhas de derivação otimizadas por programação linear

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)