A mixed-integer approximation of robust optimization problems with mixed-integer adjustments

In the present article we propose a mixed-integer approximation of adjustable-robust optimization (ARO) problems, that have both, continuous and discrete variables on the lowest level. As these trilevel problems are notoriously hard to solve, we restrict ourselves to weakly-connected instances. Our approach allows us to approximate, and in some cases exactly represent, the trilevel problem as a single-level mixed-integer problem. This allows us to leverage the computational efficiency of state-of-the-art mixed-integer programming solvers. We demonstrate the value of this approach by applying it to the optimization of power systems, particularly to the control of smart converters.Comment: 17 pages, 4 figure

Novel Optimization Approaches for Integrated Design and Operation of Smart Manufacturing and Energy Systems

This dissertation contributes novel theoretical results that enable the use of efficient optimization algorithms for the design of energy and manufacturing systems with high operational flexibility. Operational flexibility is a central theme of the smart grid and smart manufacturing paradigms because it enables systems to optimally adapt to highly dynamic and uncertain operating environments. Such environments are increasingly prevalent in the energy and manufacturing industries due to factors such as the increasing use of variable renewable energy resources (e.g., wind and solar) and the potential benefits of responding quickly to variations in product demands, real-time electricity markets, etc. For systems such as microgrids, combined heat and power plants, multiproduct chemical plants, and biorefineries, such flexibility has the potential to provide huge economic and environmental benefits. However, it also requires systems to make substantial changes in their operating conditions over very short-time scales, including discrete changes in their operating modes of process equipment (e.g., on/off) or the portfolio of products being produced. Designing systems with such operational flexibility requires consideration of the short-term operational details (e.g., minutes to hours) and future uncertainties that will affect system\u27s performance over its entire lifetime (e.g., decades). This gives rise to a complex optimization problem called integrated design and operation under uncertainty. This problem is complex mainly because the long-term design decisions of interest are tightly coupled with a very large number of short-term operational decisions that must be made over many operational periods and under significant uncertainty. Moreover, these operational decision are mixed-integer decisions, which are particularly challenging for optimization, because they are used to model both discrete and continuous changes in operations. Unfortunately, such problems cannot be solved both accurately and efficiently by standard mathematical programming approaches without major simplifications. At the same time, simplifications that are computationally tractable significantly reduce the level of operational detail that can be captured by the optimization model, which often result in system designs that are sub-optimal or even infeasible for real operations. An alternative approach, which we refer to as the simulation-based optimization (SO) approach, is to evaluate candidate system designs using a stochastic simulation of the system’s operations over all operational periods and in multiple uncertain scenarios. The design problem is then solved by optimizing the output of this simulation with respect to the design decisions. This approach is scalable to models with much more operational detail in terms of the number of operational periods and the number of uncertain scenarios considered, both of which are essential for representing operational flexibility. However, this approach results in highly complex and discontinuous optimization problems due to the discrete decisions that are made within the simulation to represent short-term operations. Hence, solving this formulation usually requires heuristic gradient-free optimization algorithms that are extremely inefficient for high-dimensional problems and offer no theoretical guarantee of finding an optimal design. To address these challenges, this dissertation presents novel theoretical results that enable the SO formulation to be solved much more efficiently using gradient-based local optimization algorithms. In contrast to the common practice of approximating the cost function as a finite sum of costs associated with discrete uncertain scenarios (i.e., sample-average approximation), we instead model the cost as the true expected value over all possible scenarios described by a continuous probability distribution. In this context, our key insight is that averaging over uncertain scenarios is a smoothing operation, and hence this expected cost can be a smooth function of the design decisions despite the fact that sample average approximations are discontinuous. When this is true, the SO formulation can be solved efficiently using gradient-based optimization methods. In Chapter 2, we develop this approach assuming that the operational decisions within the simulation are made with a logical control policy that is specified a priori. Specifically, we consider a type of controller called an energy management policy that is in common use in microgrid simulations. We then derive and rigorously prove two sets of sufficient conditions on the energy management policy under which the expected cost of the simulation is smooth. We demonstrate that these conditions are easily verifiable and often satisfied in practical applications. Finally, we implement different gradient-based algorithms, including a custom-made stochastic gradient descent algorithm, to solve the SO formulation for a representative example problem and show that this approach significantly outperforms derivative-free algorithms in both computational speed and solution quality. In Chapter 3, we extend this approach to address a much more general mathematical programming formulation of the integrated design and operation problem called multistage stochastic programming (MSP). We argue that this general MSP formulation can be accurately approximated by making all operational decisions using a parameterized mixed-integer decision rule, which reduces the MSP to an SO problem that can be solved efficiently as in Chapter 2. We then extend the smoothness conditions developed in Chapter 2. To develop this approach, we first propose a very general class of mixed-integer decision rules that is flexible enough to approximate near-optimal operational decisions for general MSPs, and then extend the sufficient conditions developed in Chapter 2 to rigorously establish smoothness of the resulting SO approximation. The resulting sufficient conditions are significantly more general than those in Chapter 2, and therefore apply to a much larger class of problems. We then show that these conditions are often satisfied in practice, and that they can always be made to hold by randomizing the decision rule. Finally, we implement different gradient-based algorithms to solve the SO approximation for a representative example problem and show that this approach significantly outperforms derivative-free algorithms in both computational speed and solution quality. Overall, the novel theoretical results developed in this dissertation are shown to enable efficient solution of significantly larger integrated design and operation problems than could be solved by existing approaches

Advances in Data-Driven Modeling and Global Optimization of Constrained Grey-Box Computational Systems

The effort to mimic a chemical plant’s operations or to design and operate a completely new technology in silico is a highly studied research field under process systems engineering. As the rising computation power allows us to simulate and model systems in greater detail through careful consideration of the underlying phenomena, the increasing use of complex simulation software and generation of multi-scale models that spans over multiple length and time scales calls for computationally efficient solution strategies that can handle problems with different complexities and characteristics. This work presents theoretical and algorithmic advancements for a range of challenging classes of mathematical programming problems through introducing new data-driven hybrid modeling and optimization strategies. First, theoretical and algorithmic advances for bi-level programming, multi-objective optimization, problems containing stiff differential algebraic equations, and nonlinear programming problems are presented. Each advancement is accompanied with an application from the grand challenges faced in the engineering domain including, food-energy-water nexus considerations, energy systems design with economic and environmental considerations, thermal cracking of natural gas liquids, and oil production optimization. Second, key modeling challenges in environmental and biomedical systems are addressed through employing advanced data analysis techniques. Chemical contaminants created during environmental emergencies, such as hurricanes, pose environmental and health related risks for exposure. The goal of this work is to alleviate challenges associated with understanding contaminant characteristics, their redistribution, and their biological potential through the use of data analytics