In practice, the presence of uncertain parameters in optimization problems introduces new challenges in modeling and solvability to operations research. There are three main paradigms proposed for optimization problems under uncertainty. These include stochastic programming, robust optimization, and sensitivity analysis. In this thesis, we examine, improve, and combine the latter two paradigms in several relevant models and applications. In the second chapter, we study a two-stage adjustable robust linear optimization problem in which the right-hand sides are uncertain and belong to a compact, convex, and tractable uncertainty set. Under standard and simple assumptions, we reformulate the two-stage problem as a copositive optimization program, which in turns leads to a class of tractable semidefinite-based approximations that are at least as strong as the affine policy, which is a well studied tractable approximation in the literature. We examine our approach over several examples from the literature and the results demonstrate that our tractable approximations significantly improve the affine policy. In particular, our approach recovers the optimal values of a class of instances of increasing size for which the affine policy admits an arbitrary large gap. In the third chapter, we leverage the concept of robust optimization to conduct sensitivity analysis of the optimal value of linear programming (LP). In particular, we propose a framework for sensitivity analysis of LP problems, allowing for simultaneous perturbations in the objective coefficients and right-hand sides, where the perturbations are modeled in a compact, convex, and tractable uncertainty set. This framework unifies and extends multiple approaches for LP sensitivity analysis in the literature and has close ties to worst-case LP and two-stage adjustable linear programming. We define the best-case and worst-case LP optimal values over the uncertainty set. As the concept aligns well with the general spirit of robust optimization, we denote our approach as robust sensitivity analysis. While the best-case and worst-case optimal values are difficult to compute in general, we prove that they equal the optimal values of two separate, but related, copositive programs. We then develop tight, tractable conic relaxations to provide bounds on the best-case and worst case optimal values, respectively. We also develop techniques to assess the quality of the bounds, and we validate our approach computationally on several examples from—and inspired by—the literature. We find that the bounds are very strong in practice and, in particular, are at least as strong as known results for specific cases from the literature. In the fourth chapter of this thesis, we study the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing all distributions that could have generated the observed samples with a high confidence. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumptions, we reformulate the problem into a copositive programming problem, which naturally leads to a tractable semidefinite-based approximation. We compare our approach with a moment-based approach from the literature for two applications. The numerical results illustrate the effectiveness of our approach. Finally, we conclude the thesis with remarks on some interesting open questions in the field of optimization under uncertainty. In particular, we point out that some interesting topics that can be potentially studied by copositive programming techniques.

One of the classical problems in finance is that of an economic unit who aims at maximizing his expected life-time utility from consumption and/or terminal wealth by an effective asset-liability management. The purpose of this thesis is to determine the optimal investment strategies , from the point of view of their economic utility, for individual and institutional investors such pension funds.

Demand fluctuations will bias the measurement of productivity and efficiency. This dissertation described three ways to characterize the effect of demand fluctuations. First, a two-dimensional efficiency decomposition (2DED) of profitability is proposed for manufacturing, service, or hybrid production systems to account for the demand effect. The first dimension identifies four components of efficiency: capacity design, demand generation, operations, and demand consumption, using Network Data Envelopment Analysis (Network DEA). The second dimension decomposes the efficiency measures and integrates them into a profitability efficiency framework. Thus, each component's profitability change can be analyzed based on technical efficiency change, scale efficiency change and allocative efficiency change. Second, this study proposes a proactive DEA model to account for demand fluctuations and proposes input or output adjustments to maximize effective production. Demand fluctuations lead to variations in the output levels affecting measures of technical efficiency. In the short-run, firms can adjust their variable resources to address the demand fluctuates and perform more efficiently. Proactive DEA is a short-run capacity planning method, proposed to provide decision support to a firm interested in improving the effectiveness of a production system under demand uncertainty using a stochastic programming DEA (SPDEA) approach. This method improves the decision making related to short-run capacity expansion and estimates the expected value of effectiveness given demand. In the third part of the dissertation, a Nash-Cournot equilibrium is identified for an oligopolistic market. The standard assumption in the efficiency literature that firms desire to produce on the production frontier may not hold in an oligopolistic market where the production decisions of all firms will determine the market price, i.e. an increase in a firm's output level leads to a lower market clearing price and potentially-lower profits. Models for both the production possibility set and the inverse demand function are used to identify a Nash-Cournot equilibrium and improvement targets which may not be on the strongly efficient production frontier. This behavior is referred to as rational inefficiency because the firm reduces its productivity levels in order to increase profits.

During the last two decades, the increase in electricity demand and environmental concern resulted in fast growth of power production from renewable sources. Wind power is one of the most efficient alternatives. Due to the rapid development of wind turbine technology and increasing size of wind farms, wind power plays a significant part in the power production mix of Germany, Spain, Denmark, and some other countries. The best conditions for the development of wind farms are in remote, open areas with low population density. The transmission system in such areas might not be dimensioned to accommodate additional large-scale power infeed. Furthermore a part of the existing transmission capacity might already be reserved for conventional power plants situated in the same area. In this thesis four alternatives for large-scale wind power integration in areas with transmission bottlenecks are considered. The first possibility is to revise the methods for calculation of available transmission capacity. The second solution for large-scale integration of wind power in such areas is to reinforce the network. This alternative, however, may be expensive and time consuming. As wind power production depends on the wind speed, the full load hours of wind turbine generator are only 2000-4000 hours per year. Therefore reinforcing a transmission network in order to remove a bottleneck completely is often not economically justified. Wind energy curtailments during congestion situations is then the third solution for large-scale wind power integration with less or no grid reinforcement. The fourth solution is to store excess wind energy. Pumped hydro storage or battery storage for the large-scale wind farms are still rather expensive options, but existing conventional power plants with fast production control capabilities and sufficient storage capacity, e.g., hydro power plants, could be used for this purpose. As there is a lot of research work on the first two alternatives, the thesis provides a review and summarizes the main conclusions from the existing work. The thesis is then directed towards the development of the methods for estimation of wind energy curtailments, evaluation of wind energy storage possibility in hydro reservoirs and development of short term hydro power production planning methods, considering coordination with wind power. Additionally in the thesis the strategy that minimizes imbalance costs of a wind power utility, trading wind power on the short term power market is elaborated and analyzed. / QC 20100608

This research deals with urban water reuse planning and management modeling in the context of sustainable development. Rapid urbanization and population growth have presented a great challenge to urban water resources management. As water reuse may alleviate pollution loads and enhance water supply sources, water reuse is being recognized as a sustainable urban water management strategy and is becoming increasingly attractive in urban water resources management. An efficient water reuse planning and management model is of significance in promoting water reuse practices. This thesis introduces an urban water reuse management and planning model using optimization methods with an emphasis on modeling uncertainty issues associated with water demand and water quality. The model is developed in conjunction with the overall urban water system with considerations over water supply, water demand, water distribution, water quality, and wastewater treatment and discharge. The objective of the model is to minimize the overall cost of the system subject to technological, societal and environmental constraints. Uncertainty issues associated with water demand and treatment quality are modeled by introducing stochastic programming methods, namely, two-stage stochastic recourse programming and chance-constraint programming. The model is capable of identifying and evaluating water reuse in urban water systems to optimize the allocation of urban water resources with regard to uncertainties. It thus provides essential information in planning and managing urban water reuse systems towards a more sustainable urban water resources management. An application was presented in order to demonstrate the modeling process and to analyze the impact of uncertainties.

This research deals with urban water reuse planning and management modeling in the context of sustainable development. Rapid urbanization and population growth have presented a great challenge to urban water resources management. As water reuse may alleviate pollution loads and enhance water supply sources, water reuse is being recognized as a sustainable urban water management strategy and is becoming increasingly attractive in urban water resources management. An efficient water reuse planning and management model is of significance in promoting water reuse practices. This thesis introduces an urban water reuse management and planning model using optimization methods with an emphasis on modeling uncertainty issues associated with water demand and water quality. The model is developed in conjunction with the overall urban water system with considerations over water supply, water demand, water distribution, water quality, and wastewater treatment and discharge. The objective of the model is to minimize the overall cost of the system subject to technological, societal and environmental constraints. Uncertainty issues associated with water demand and treatment quality are modeled by introducing stochastic programming methods, namely, two-stage stochastic recourse programming and chance-constraint programming. The model is capable of identifying and evaluating water reuse in urban water systems to optimize the allocation of urban water resources with regard to uncertainties. It thus provides essential information in planning and managing urban water reuse systems towards a more sustainable urban water resources management. An application was presented in order to demonstrate the modeling process and to analyze the impact of uncertainties.

The subject of this thesis is to study approximation algorithms for some network design problems in face of uncertainty. We consider two widely studied models of handling uncertainties - Robust Optimization and Stochastic Optimization. We study a robust version of the well studied Uncapacitated Facility Location Problem (UFLP). In this version, once the set of facilities to be opened is decided, an adversary may close at most β facilities. The clients must then be assigned to the remaining open facilities. The performance of a solution is measured by the worst possible set of facilities that the adversary may close. We introduce a novel LP for the problem, and provide an LP rounding algorithm when all facilities have same opening costs. We also study the 2-stage Stochastic version of the Steiner Tree Problem. In this version, the set of terminals to be covered is not known in advance. Instead, a probability distribution over the possible sets of terminals is known. One is allowed to build a partial solution in the first stage a low cost, and when the exact scenario to be covered becomes known in the second stage, one is allowed to extend the solution by building a recourse network, albeit at higher cost. The aim is to construct a solution of low cost in expectation. We provide an LP rounding algorithm for this problem that beats the current best known LP rounding based approximation algorithm.

<p>Unforeseen changes in production or consumption in power systems lead to changes in grid frequency. This can cause damages to the system, or to frequency sensitive equipment at the consumers. The system operator (SO) is the responsible for balancing production and consumption in the system. The regulating market is the market place where the SO can sell or purchase electricity in order to balance unforeseen events. Producers acting on the regulating market must be able to change their production levels fast (within minutes) when required. Hydropower is therefore suitable for trading on the regulating market because of its flexibility in power production. This thesis describes models that hydropower owners can use to generate optimal bidding strategies when the regulating market is considered.</p><p>When planning for trading on the market, the prices are not known. Therefore, the prices are considered as stochastic variables. The planning problems in this thesis are based on multi-stage stochastic optimization, where the uncertain power prices are represented by scenario trees. The scenario trees are generated by simulation of price scenarios, which is achieved by using a model based on ARIMA and Markov processes. Two optimization models are presented in this thesis:</p><p>* Model for generation of optimal bidding strategies for the regulating market.</p><p>* Model for generation of optimal bidding strategies for the spot market when trading on the regulating market is considered.</p><p>The described models are applied in a case study with real data from the Nordic power system.</p><p>Conclusions of the thesis are that the proposed approaches of modelling prices and generation of bidding strategies are possible to use, and that the models produces reasonable data when applied to real data.</p> / <p>Oväntade produktions- eller konsumtionsändringar i kraftsystem leder till ändringar i nätfrekvens. Detta kan orsaka skador på systemet eller på frekvenskänslig utrustning hos konsumenterna. Systemoperatören (SO) är den ansvarige för att balansera produktion och konsumtion i kraftsystemet. Till sin hjälp har SO reglermarknaden, som är den handelsplats där SO köper eller säljer el för att balansera oväntade händelser i systemet. Producenter som agerar på reglermarknaden måste snabbt (inom minuter) kunna ändra sina produktionsnivåer om så behövs. Vattenkraft är därför lämplig för handel på reglermarknaden på grund av dess flexibilitet i kraftproduktion. Denna avhandling beskriver modeller som vattenkraftägare kan använda för generering av optimala budstrategier då reglermarknaden beaktas.</p><p>När en producents planering för handel på marknaden utförs är marknadspriserna okända. Dessa är därför betraktade som stokastiska variabler. Planeringmodellerna som presenteras i denna avhandling är baserade på multi-periodisk stokastisk programmering, där de osäkra marknadspriserna är representerade av ett scenarieträd. Scenarierna i trädet genereras genom simulering av marknadspriser. En prismodell, baserad på ARIMA- och Markovprocesser, har därför utvecklats. Två olika optimeringsmodeller presenteras i denna avhandling:</p><p>* Model för generering av optimala budstrategier för reglermarknaden.</p><p>* Model för generering av optimala budstrategier för spotmarknaden då handel på reglermarknaden beaktas.</p><p>Modellerna tillämpas i en studie där data från den nordiska elmarknaden appliceras. Slutsatser i avhandlingen är att de föreslagna ansatserna för modellering av priser och generering av budstrategier är möjliga att anvÄanda, samt att modellerna producerar rimliga resultat när applicerade på verkliga data.</p>

We consider a hydrothermal scheduling problem with a mid-term horizon(HTSPM) modeled as a large-scale multistage stochastic program with stochastic monthly inflows of water to each hydro generator. In the HTSPM we seek an operating policy to minimize the sum of present and expected future costs, which include thermal generation costs and load curtailment costs. In addition to various simple bounds, problem constraints involve water balance, demand satisfaction and power interchanges. Sampling-based decomposition algorithms (SBDAs) have been used in the literature to solve HTSPM. SBDAs can be used to approximately solve problem instances with many time stages and with inflows that exhibit interstage dependence. Such dependence requires care in computing valid cuts for the decomposition algorithm. In order to help maintain tractability, we employ an aggregate reservoir representation (ARR). In an ARR all the hydro generators inside a specific region are grouped to effectively form one hydro plant with reservoir storage and generation capacity proportional to the parameters of the hydro plants used to form that aggregate reservoir. The ARR has been used in the literature with energy balance constraints, rather than water balance constraints, coupled with time series forecasts of energy inflows. Instead, we prefer as a model primitive to have the time series model forecast water inflows. This, in turn, requires that we extend existing methods to compute valid cuts for the decomposition method under the resulting form of interstage dependence. We form a sample average approximation of the original problem and then solve this problem by these special-purpose algorithms. And, we assess the quality of the resulting policy for operating the system. In our analysis, we compute a confidence interval on the optimality gap of a policy generated by solving an approximation on a sampled scenario tree. We present computational results on test problems with 24 monthly stages in which the inter-stage dependency of hydro inflows is modeled using a dynamic linear model. We further develop a parallel implementation of an SBDA. We apply SBDA to solve the HTSPM for the Brazilian power system that has 150 hydro generators, 151 thermal generators and 4 regions that each characterize an aggregate reservoir. We create and solve four different HTSPM instances where we change the input parameters with respect to generation capacity, transmission capacity and load in order to analyze the difference in the total expected cost. / text

This dissertation proposes new methods to improve the efficiency of electricity markets with respect to market monitoring and reserve allocation. We first present new approaches to monitor the level of competition in electricity markets, a critical task for helping the markets function smoothly. The proposed approaches are based on economic principles and a faithful representation of transmission constraints. The effectiveness of the new approaches is demonstrated by examples based on medium- and large-scale electric power systems. We then propose a new system-operation model using stochastic optimization to systematically allocate reserves under uncertainty. This model aims to overcome the difficulties in both system and market operations caused by the integration of wind power, which results in a higher degree of supply uncertainty. The numerical examples suggest that the proposed model significantly lower the operation costs, especially under high levels of wind penetration. / text