SOLVING TWO-LEVEL OPTIMIZATION PROBLEMS WITH APPLICATIONS TO ROBUST DESIGN AND ENERGY MARKETS

This dissertation provides efficient techniques to solve two-level optimization problems. Three specific types of problems are considered. The first problem is robust optimization, which has direct applications to engineering design. Traditionally robust optimization problems have been solved using an inner-outer structure, which can be computationally expensive. This dissertation provides a method to decompose and solve this two-level structure using a modified Benders decomposition. This gradient-based technique is applicable to robust optimization problems with quasiconvex constraints and provides approximate solutions to problems with nonlinear constraints. The second types of two-level problems considered are mathematical and equilibrium programs with equilibrium constraints. Their two-level structure is simplified using Schur's decomposition and reformulation schemes for absolute value functions. The resulting formulations are applicable to game theory problems in operations research and economics. The third type of two-level problem studied is discretely-constrained mixed linear complementarity problems. These are first formulated into a two-level mathematical program with equilibrium constraints and then solved using the aforementioned technique for mathematical and equilibrium programs with equilibrium constraints. The techniques for all three problems help simplify the two-level structure into one level, which helps gain numerical and application insights. The computational effort for solving these problems is greatly reduced using the techniques in this dissertation. Finally, a host of numerical examples are presented to verify the approaches. Diverse applications to economics, operations research, and engineering design motivate the relevance of the novel methods developed in this dissertation

Applications of Optimization Under Uncertainty Methods on Power System Planning Problems

This dissertation consists of two published journal paper, both on transmission expansion planning, and a report on distribution network hardening. We first discuss our studies of two optimization criteria for the transmission planning problem with a simplified representation of load and the forecast generation investment additions within the robust optimization paradigm. The objective is to determine either the minimum of the maximum investment requirement or the maximum regret with all sources of uncertainty explicitly represented. In this way, transmission planners can determine optimal planning decisions that are robust against all sources of uncertainty. We use a two layer algorithm to solve the resulting trilevel optimization problems. We also construct a new robust transmission planning model that considers generation investment more realistically to improve the quantification and visualization of uncertainty and the impacts of environmental policies. With this model, we can explore the effect of uncertainty in both the size and the location of candidate generation additions. The corresponding algorithm we develop takes advantage of the structural characteristics of the model so as to obtain a computationally efficient methodology. The two robust optimization tools provide new capabilities to transmission planners for the development of strategies that explicitly account for various sources of uncertainty. We illustrate the application of the two optimization models and solution schemes on a set of representative case studies. These studies give a good idea of the usefulness of these tools and show their practical worth in the assessment of ``what if\u27\u27 cases. We compare the performance of the minimax cost approach and the minimax regret approach under different characterizations of uncertain parameters. In addition, we also present extensive numerical studies on an IEEE 118-bus test system and the WECC 240-bus system to illustrate the effectiveness of the proposed decision support methods. The case study results are particularly useful to understand the impacts of each individual investment plan on the power system\u27s overall transmission adequacy in meeting the demand of the trade with the power output units without violation of the physical limits of the grid. In the report on distribution network hardening, a two-stage stochastic optimization model is proposed. Transmission and distribution networks are essential infrastructures to modern society. In the United States alone, there are there are more than 200,000 miles of high voltage transmission lines and numerous distribution lines. The power network spans the whole country. Such vast networks are vulnerable to disruptions caused by natural disasters. Hardening of distribution lines could significantly reduce the impact of natural disasters on the operation of power systems. However, due to the limited budget, it is impossible to upgrade the whole power network. Thus, intelligent allocation of resources is crucial. Optimal allocation of limited budget between different hardening methods on different distribution lines is explored

Models, Theoretical Properties, and Solution Approaches for Stochastic Programming with Endogenous Uncertainty

In a typical optimization problem, uncertainty does not depend on the decisions being made in the optimization routine. But, in many application areas, decisions affect underlying uncertainty (endogenous uncertainty), either altering the probability distributions or the timing at which the uncertainty is resolved. Stochastic programming is a widely used method in optimization under uncertainty. Though plenty of research exists on stochastic programming where decisions affect the timing at which uncertainty is resolved, much less work has been done on stochastic programming where decisions alter probability distributions of uncertain parameters. Therefore, we propose methodologies for the latter category of optimization under endogenous uncertainty and demonstrate their benefits in some application areas. First, we develop a data-driven stochastic program (integrates a supervised machine learning algorithm to estimate probability distributions of uncertain parameters) for a wildfire risk reduction problem, where resource allocation decisions probabilistically affect uncertain human behavior. The nonconvex model is linearized using a reformulation approach. To solve a realistic-sized problem, we introduce a simulation program to efficiently compute the recourse objective value for a large number of scenarios. We present managerial insights derived from the results obtained based on Santa Fe National Forest data. Second, we develop a data-driven stochastic program with both endogenous and exogenous uncertainties with an application to combined infrastructure protection and network design problem. In the proposed model, some first-stage decision variables affect probability distributions, whereas others do not. We propose an exact reformulation for linearizing the nonconvex model and provide a theoretical justification of it. We designed an accelerated L-shaped decomposition algorithm to solve the linearized model. Results obtained using transportation networks created based on the southeastern U.S. provide several key insights for practitioners in using this proposed methodology. Finally, we study submodular optimization under endogenous uncertainty with an application to complex system reliability. Specifically, we prove that our stochastic program\u27s reliability maximization objective function is submodular under some probability distributions commonly used in reliability literature. Utilizing the submodularity, we implement a continuous approximation algorithm capable of solving large-scale problems. We conduct a case study demonstrating the computational efficiency of the algorithm and providing insights

Unit Commitment Problem in Electrical Power System: A Literature Review

Unit commitment (UC) is a popular problem in electric power system that aims at minimizing the total cost of power generation in a specific period, by defining an adequate scheduling of the generating units. The UC solution must respect many operational constraints. In the past half century, there was several researches treated the UC problem. Many works have proposed new formulations to the UC problem, others have offered several methodologies and techniques to solve the problem. This paper gives a literature review of UC problem, its mathematical formulation, methods for solving it and Different approaches developed for addressing renewable energy effects and uncertainties