Supply chain network design involves location decisions for production facilities and distribution centers. We consider a make-to-order supply chain environment where distribution centers serve as crossdocking terminals. Long waiting times may occur at a cross-docking terminal, unless sucient handling capacity is installed. In this study, we deal with a facility location problem with congestion eects at distribution centers. Along with location decisions, we make capacity allocation (service rate) and demand allocation decisions so that the total cost, including facility opening, transportation and congestion costs, is minimized. Response time to customer orders is a critical performance measure for a supply chain network. The decisions like where the plants and distribution centers are located aect the response time of the system. Response time is more sensitive to these decisions in a make-to-order business environment. In a distribution network where distribution centers function as cross-docking terminals, capacity or the service rate decisions also aect the response time performance. This study is closely related to a recent work Vidyarthi et al. (2009) which models distribution centers asM/G/1 queuing systems. They use the average waiting time formula ofM/G/1 queuing model. Thus, the average waiting time at a distribution center is a nonlinear function of the demand rate allocated to and the service rate available at the distribution center. The authors Vidyarthi et al. (2009) propose a linear approximation approach and a Lagrangian based heuristic for the problem. Dierent than the solution approach proposed in Vidyarthi et al. (2009), we propose a closed form formulation for the problem. In particular, we show that the waiting time function derived from M/G/1 queuing model can be represented via second order conic inequalities. Then, the problem becomes a mixed integer second order cone programming problem which can be solved by using commercial branch-and-bound software such as IBM ILOG CPLEX. Our computational tests show that proposed reformulation can be solved in reasonable CPU times for practical size instances.

The dissertation research investigates estimating of power system static and dynamic states (e.g. rotor angle, rotor speed, mechanical power, voltage magnitude, voltage phase angle, mechanical reference point) as well as identification of synchronous generator parameters. The research has two focuses: i. Synchronous generator dynamic model states and parameters estimation using real-time PMU data. ii.Integrate PMU data and conventional measurements to carry out static state estimation. The first part of the work focuses on Phasor Measurement Unit (PMU) data-based synchronous generator states and parameters estimation. In completed work, PMU data-based synchronous generator model identification is carried out using Unscented Kalman Filter (UKF). The identification not only gives the states and parameters related to a synchronous generator swing dynamics but also gives the states and parameters related to turbine-governor and primary and secondary frequency control. PMU measurements of active power and voltage magnitude, are treated as the inputs to the system while voltage phasor angle, reactive power, and frequency measurements are treated as the outputs. UKF-based estimation can be carried out at real-time. Validation is achieved through event play back to compare the outputs of the simplified simulation model and the PMU measurements, given the same input data. Case studies are conducted not only for measurements collected from a simulation model, but also for a set of real-world PMU data. The research results have been disseminated in one published article. In the second part of the research, new state estimation algorithm is designed for static state estimation. The algorithm contains a new solving strategy together with simultaneous bad data detection. The primary challenge in state estimation solvers relates to the inherent non-linearity and non-convexity of measurement functions which requires using of Interior Point algorithm with no guarantee for a global optimum solution and higher computational time. Such inherent non-linearity and non-convexity of measurement functions come from the nature of power flow equations in power systems. The second major challenge in static state estimation relates to the bad data detection algorithm. In traditional algorithms, Largest Normalized Residue Test (LNRT) has been used to identify bad data in static state estimation. Traditional bad data detection algorithm only can be applied to state estimation. Therefore, in a case of finding any bad datum, the SE algorithm have to rerun again with eliminating found bad data. Therefore, new simultaneous and robust algorithm is designed for static state estimation and bad data identification. In the second part of the research, Second Order Cone Programming (SOCP) is used to improve solving technique for power system state estimator. However, the non-convex feasible constraints in SOCP based estimator forces the use of local solver such as IPM (interior point method) with no guarantee for quality answers. Therefore, cycle based SOCP relaxation is applied to the state estimator and a least square estimation (LSE) based method is implemented to generate positive semi-definite programming (SDP) cuts. With this approach, we are able to strengthen the state estimator (SE) with SOCP relaxation. Since SDP relaxation leads the power flow problem to the solution of higher quality, adding SDP cuts to the SOCP relaxation makes Problem’s feasible region close to the SDP feasible region while saving us from computational difficulty associated with SDP solvers. The improved solver is effective to reduce the feasible region and get rid of unwanted solutions violate cycle constraints. Different Case studies are carried out to demonstrate the effectiveness and robustness of the method. After introducing the new solving technique, a novel co-optimization algorithm for simultaneous nonlinear state estimation and bad data detection is introduced in this dissertation. ${\ell}_1$-Norm optimization of the sparse residuals is used as a constraint for the state estimation problem to make the co-optimization algorithm possible. Numerical case studies demonstrate more accurate results in SOCP relaxed state estimation, successful implementation of the algorithm for the simultaneous state estimation and bad data detection, and better state estimation recovery against single and multiple Gaussian bad data compare to the traditional LNRT algorithm.

[pt] Este trabalho tem por objetivo a avaliação da estabilidade de taludes de solo quando sometidos a processos de infiltração de chuva, utilizando conceitos de Análise Limite e Análise de Confiabilidade. Primeiramente, determina-se a variação da sução no solo, para isto, emprega-se o Método dos Elementos Finitos e o Método de diferenças finitas na solução da equação de Richards. O modelo de Van Genuchten (1980) é utilizado para a curva característica. Na solução da nãolinearidade, emprega-se o método Picard Modificado. A instabilidade de taludes é estudada mediante o método de Análise Limite Numérica com base no Método de Elementos Finitos e o critério de Mohr Coulomb como critério de escoamento. A solução do problema matemático será realizada no espaço cônico quadrático com o objetivo de tornar a solução mais computacionalmente eficiente. Considerando as propriedades do solo como variáveis aleatórias foi incluída a determinação do Índice de Confiabilidade utilizando as formulações dos métodos de Monte Carlo e FORM (first order reliability method). Inicialmente são introduzidos conceitos básicos associados ao fluxo saturado-não saturado. A seguir são apresentados alguns conceitos. Sobre Análise Limite e sua formulação pelo Método de Elementos Finitos. Finalmente são introduzidos os fundamentos da Análise de Confiabilidade. Análises de confiabilidade das encostas de Coos Bay no estado de Oregon nos Estados Unidos e da Vista Chinesa no Rio de Janeiro Brasil, são apresentadas devido a que estes taludes sofreram colapso quando submetidos a processos de infiltração de água de chuva. Os resultados deste trabalho mostram que a falha das encostas ocorre quando o índice de confiabilidade atinge um valor perto de dois. / [en] This thesis aims to perform a reliability analysis of the stability of 2D soil slopes when they are submitted to water infiltration due to the rains.The time variation of the soil matric suctions is calculated first. The Finite Element Method is used to transform the Richards differential equation into a system of nonlinear first order equations. The nonlinearity of the problem is due to the use of the characteristic curve proposed by van Genuchten (1980). The Modified Picard Method is applied to solve de time-dependent nonlinear equation system. The responses of the flux-problem are transferred to the stability problem in some instants using the same time-interval (normally days).To estimate the stability of the slopes, limit analysis is used. The limit analyses are performed based on the Inferior Limit Theorem of the Plasticity Theory. The problem is defined as an optimization problem where the load factor is maximized. The equilibrium equations are obtained via Finite Element discretization and the strength criterion of Mohr-Couomb is written in the conic quadratic space. Therefore, a SOCP (Second Order Conic Programming) problem is generated. The problem is solved using an interior point algorithm of the code Mosek.Since the soil properties are random variables a reliability analysis can be performed at each instant of the time-dependent problem. In order to perform the reliability analyses, Response Surfaces for the failure function of the slope are generated. In this work, the Stochastic Collocation Method is used to generate Response Surfaces. The Simulation Monte Carlo Method and the FORM (First Order Reliability Method) are used to obtain both the reliability index and the probability of failure of the slopes.Reliability analyses of the Coos Bay Slope in the state of Oregon in USA and in the Vista Chinesa Slope in Rio de Janeiro, Brazil, are presented because they collapse due to rainfall infiltration. The results show that the soil slope fails when the related reliability index is close to two.

Orientador: Jose Roberto Sanches Mantovani / Resumo: Neste trabalho são propostos modelos matemáticos determinístico e estocástico de programação cônica de segunda ordem em coordenadas retangulares para o problema de fluxo de potência ótimo de geração e controle de potência reativa no sistemas elétricos de potência, considerando as minimização dos custos de geração de energia, perdas ativas da rede e emissão de poluentes no meio ambiente. Os modelos contemplam as principais características físicas e econômicas do problema estudado, assim como os limites operacionais do sistema elétrico. Os modelos são programados em linguagem AMPL e suas soluções são obtidas através do solver comercial CPLEX. Os sistemas testes IEEE30, IEEE118 e ACTIVSg200 são utilizados nas simulações computacionais dos modelos propostos. Os resultados obtidos pelo modelo determinístico desenvolvido são validados através de comparações com os resultados fornecidos pelo software MATPOWER , onde ambos consideram apenas a existência de gerações termoelétricas. No modelo estocástico utiliza-se a técnica de geração de cenários e considera-se um período de um ano (8760 horas), e geradores que utilizam fontes de geração renováveis e não renováveis. / Abstract: In this work we propose deterministic and stochastic mathematical models of second order conical programming in rectangular coordinates for the optimal power flow problem of reactive power generation and control in electric power systems, considering the minimization of energy generation costs, losses networks and emission of pollutants into the environment. The models contemplate the main physical and economic characteristics of the studied problem, as well as the operational limits of the electric system. The models are programmed in AMPL language and their solutions are obtained through the commercial solver CPLEX. The IEEE30, IEEE118 and ACTIVSg200 test systems are used in the computer simulations of the proposed models. The results obtained by the deterministic model developed are validated through comparisons with the results provided by the software MATPOWERR , where both consider only the existence of thermoelectric generations. The stochastic model uses the scenario generation technique and considers a period of one year (8760 hours), and generators using renewable and non-renewable generation sources. / Mestre

The research on optimal reinsurance design dated back to the 1960’s. For nearly half a century, the quest for optimal reinsurance designs has remained a fascinating subject, drawing significant interests from both academicians and practitioners. Its fascination lies in its potential as an effective risk management tool for the insurers. There are many ways of formulating the optimal design of reinsurance, depending on the chosen objective and constraints. In this thesis, we address the problem of optimal reinsurance designs from an insurer’s perspective. For an insurer, an appropriate use of the reinsurance helps to reduce the adverse risk exposure and improve the overall viability of the underlying business. On the other hand, reinsurance incurs additional cost to the insurer in the form of reinsurance premium. This implies a classical risk and reward tradeoff faced by the insurer. The primary objective of the thesis is to develop theoretically sound and yet practical solution in the quest for optimal reinsurance designs. In order to achieve such an objective, this thesis is divided into two parts. In the first part, a number of reinsurance models are developed and their optimal reinsurance treaties are derived explicitly. This part focuses on the risk measure minimization reinsurance models and discusses the optimal reinsurance treaties by exploiting two of the most common risk measures known as the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE). Some additional important economic factors such as the reinsurance premium budget, the insurer’s profitability are also considered. The second part proposes an innovative method in formulating the reinsurance models, which we refer as the empirical approach since it exploits explicitly the insurer’s empirical loss data. The empirical approach has the advantage that it is practical and intuitively appealing. This approach is motivated by the difficulty that the reinsurance models are often infinite dimensional optimization problems and hence the explicit solutions are achievable only in some special cases. The empirical approach effectively reformulates the optimal reinsurance problem into a finite dimensional optimization problem. Furthermore, we demonstrate that the second-order conic programming can be used to obtain the optimal solutions for a wide range of reinsurance models formulated by the empirical approach.

The research on optimal reinsurance design dated back to the 1960’s. For nearly half a century, the quest for optimal reinsurance designs has remained a fascinating subject, drawing significant interests from both academicians and practitioners. Its fascination lies in its potential as an effective risk management tool for the insurers. There are many ways of formulating the optimal design of reinsurance, depending on the chosen objective and constraints. In this thesis, we address the problem of optimal reinsurance designs from an insurer’s perspective. For an insurer, an appropriate use of the reinsurance helps to reduce the adverse risk exposure and improve the overall viability of the underlying business. On the other hand, reinsurance incurs additional cost to the insurer in the form of reinsurance premium. This implies a classical risk and reward tradeoff faced by the insurer. The primary objective of the thesis is to develop theoretically sound and yet practical solution in the quest for optimal reinsurance designs. In order to achieve such an objective, this thesis is divided into two parts. In the first part, a number of reinsurance models are developed and their optimal reinsurance treaties are derived explicitly. This part focuses on the risk measure minimization reinsurance models and discusses the optimal reinsurance treaties by exploiting two of the most common risk measures known as the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE). Some additional important economic factors such as the reinsurance premium budget, the insurer’s profitability are also considered. The second part proposes an innovative method in formulating the reinsurance models, which we refer as the empirical approach since it exploits explicitly the insurer’s empirical loss data. The empirical approach has the advantage that it is practical and intuitively appealing. This approach is motivated by the difficulty that the reinsurance models are often infinite dimensional optimization problems and hence the explicit solutions are achievable only in some special cases. The empirical approach effectively reformulates the optimal reinsurance problem into a finite dimensional optimization problem. Furthermore, we demonstrate that the second-order conic programming can be used to obtain the optimal solutions for a wide range of reinsurance models formulated by the empirical approach.

[pt] Esse trabalho propõe uma implementação numérica para otimização de forma em problemas bi-dimensionais de elasticidade. O objetivo principal é propor uma metodologia eficiente e robusta para solução de problemas de otimização de forma considerando a minimização de concentração de tensões. Na implementação proposta, a análise estrutural é realizada pelo Método dos Elementos de Contorno Simétrico de Galerkin (MECSG), evitando-se assim a dispendiosa etapa de geração da malha. A avaliação das tensões no contorno é obtida por meio de um método preciso, ideal para problemas com concentrações de tensões. Outro aspecto relevante na implementação é a adequada partição das equações do MECSG de forma a reduzir, consideravelmente, o esforço computacional associado à etapa da análise estrutural. O problema de otimização é resolvido utilizando-se um método de otimização moderno, conhecido como Programação Cônica de Segunda Orderm (PCSO). Especificamente, busca-se a reposta do problema de otimização não linear por meio da solução de uma sequência de subproblemas de PCSO. / [en] In this work a numerical implementation of shape optimization in two-dimensional linear elasticity problems is proposed. The main goal is to propose a robust and efficient methodology for the solution of shape optimization problems regarding the minimization of stress concentration effects. In the proposed implementation, the structural analysis is performed by the Symmetric Galerkin Boundary Element Method (SGBEM), thus disposing of the mesh generation burden. The boundary stress evaluation is carried out by an accurate approach which is ideally suited for problems with stress concentrations. Another relevant feature of the proposed implementation is a suitable partition of the SGBEM equations which aims at reducing the computational effort associated with the structural analysis stage. The solution for the optimization problem is obtained by means of a modern numerical optimization method, the so-called Second Order Conic Programming (SOCP). Specifically, the solution for the non-linear optimization is sought by solving a sequence of SOCP subproblems.

Orientador: José Roberto Sanches Mantovani / Resumo: Neste trabalho propõem-se formulações matemáticas e metodologias para resolver o problema de planejamento da expansão e operação de sistemas de distribuição de energia elétrica de longo prazo com instalação de geração distribuída despachável, renovável e dispositivos armazenadores de energia, considerando as incertezas nos parâmetros e variáveis envolvidas no comportamento do sistema. No modelo de otimização desenvolvido considera- se uma formulação com espaço de busca convexo como um problema de programação cônica inteira de segunda ordem. Como primeira metodologia de solução para o modelo matemático proposto, usam-se solvers de otimização comerciais através de linguagem de programação matemática. Em segundo lugar é proposta a técnica de otimização meta-heurística VND combinada com um solver de otimização para resolver o modelo de otimização desenvolvido. Os algoritmos e modelos matemáticos de otimização usados para resolver o planejamento de sistemas de distribuição são implementados em AMPL e testados em sistemas presentes na literatura. Finalmente são comparadas as metodologias segundo a solução obtida e desempenho em tempo computacional. / Abstract: This work proposes mathematical formulations and methodologies to solve the long-term electric power distribution system operation and expansion planning with distributed renewable energy sources and energy storage devices, considering the uncertainties in the involved parameters and variables in the system behavior. In the developed optimization model, a convex formulation is considered as integer second-order conic programming problem. The first solution methodology for the proposed mathematical model, the commercial optimization solvers that uses mathematical modelling language is used. In the second way, the VND meta-heuristic optimization technique is proposed combined with the optimization solver to analyze the obtained solutions of the search through optimal neighborhoods. The mathematical optimization model and the proposed algorithm used to solver the planning of distribution systems are implemented in AMPL and tested in literature’s systems. Finally, the methodologies according to the obtained solution and computational time performance are compared. / Doutor

[pt] O presente trabalho tem como principal objetivo mostrar a aplicabilidade prática da análise limite pelo método de elementos finitos na avaliação de problemas de estabilidade de talude, sendo este colocado como um problema de programação matemática, no qual se precisa realizar um processo de otimização para a solução do problema. Apresenta-se um método para obter a solução do problema de estabilidade de taludes utilizando para isso a programação matemática, e fazendo ênfase na utilidade da programação cônica da segunda ordem (SOCP). Inicialmente faz uma revisão das formulações da análise limite, via o método de elementos finitos, encontradas na literatura existente. A seguir é descrita a formulação da análise limite numérica partindo do principio do trabalho virtual para sua formulação, e utilizando a ferramenta dos elementos finitos para realizar a implementação numérica. São propostas diferentes formas de trabalhar com o critério de resistência do material, sendo a de melhor desempenho, em termos de tempo de processamento a forma cônica quadrática que permite acoplar a programação cônica da segunda ordem (SOCP) na ferramenta numérica. É acoplada a técnica da redução dos parâmetros de resistência do material com a finalidade de encontrar o fator de segurança da estrutura do talude (FS). Finalmente são apresentados exemplos de validação e aplicação, os quais permitem visualizar a eficiência da ferramenta desenvolvida em termos de tempo de processamento ao utilizar a programação cônica da segunda ordem (SOCP). Os resultados sugerem viabilidade da utilização da técnica estudada na solução de problemas relacionada à estabilidade de taludes. / [en] The main objective of this work is to show the practical applicability of limit analysis by finite element method in the evaluation of slope stability problems, and this placed as a mathematical programming problem, which you need to perform an optimization process to solve the problem. We present a method to obtain the solution of the problem of slope stability using for this mathematical programming, and making emphasis on the usefulness of the second order conic programming (SOCP). Initially, a review of formulations Limit Analysis via Finite Element Method, found in the existing literature. Then is described the Numerical Limit Analysis formulation starting from virtual work principle their formulation, and using Finite Element Method as a tool to carry out the numerical implementation. We propose different ways of working with the yield criterion of the material, being the best performing in terms of processing time the conic quadratic form that allows to coupling to the second order conic programming (SOCP) in numerical implementation. It is coupled to the technique of reducing the strength parameters of the material in order to find the safety factor of the slope of the structure (FS). Finally, examples are presented for validation and application, which allow you to view the efficiency of the developed implementation in terms of processing time with the use of second order conic programming (SOCP). The results suggest the feasibility of using the technique studied in the solution of problems related to Slope Stability.