Almost Symmetries and the Unit Commitment Problem

This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation

Optimization Methods for Day Ahead Unit Commitment

This work examines a variety of optimization techniques to better solve the day ahead unit commitment problem. The first method looks at the impact of almost identical generators on the problem and how to exploit that fact for computational gain. The second work seeks to improve the fidelity of the problem by better modeling the impact of pumped storage hydropower. Lastly, the relationship between the length of the planning horizon and the quality of the solutions is investigated

Novel Mixed Integer Programming Approaches to Unit Commitment and Tool Switching Problems

In the first two chapters, we discuss mixed integer programming formulations in Unit Commitment Problem. First, we present a new reformulation to capture the uncertainty associated with renewable energy. Then, the symmetrical property of UC is exploited to develop new methods to improve the computational time by reducing redundancy in the search space. In the third chapter, we focus on the Tool Switching and Sequencing Problem. Similar to UC, we analyze its symmetrical nature and present a new reformulation and symmetry-breaking cuts which lead to a significant improvement in the solution time. In chapter one, we use convex hull pricing to explicitly price the risk associated with uncertainty in large power systems scheduling problems. The uncertainty associated with renewable generation (e.g. solar and wind) is highlighting the need for changes in how power production is scheduled. It is known that symmetry in the integer programming formulations can slow down the solution process due to the redundancy in the search space caused by permutations. In the second chapter, we show that having symmetry in the unit commitment problem caused by having identical generating units could lead to a computational burden even for a small-scale problem. We present an effective method to exploit symmetry in the formulation introduced by identical (often co-located) generators. We propose a cut-generation approach coupled with aggregation method to remove symmetry without sacrificing feasibility or optimality. In the third chapter, we focus on the Job Sequencing and Tool Switching Problem (SSP), which is a well-known combinatorial optimization problem in the domain of Flexible Manufacturing Systems (FMS). We propose a new integer linear programming approach with symmetry-breaking and tightening cuts that provably outperformed the existing methodology described in the literature

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Topology control algorithms in power systems

This research focuses on improving the efficiency of power market operations by providing system operators additional tools for managing the costs of supplying and delivering electricity. A transmission topology control (TC) framework for production cost reduction based on a shift factor (SF) representation of branch and breaker flows is proposed. The framework models topology changes endogenously while maintaining linearity in the overall Mixed Integer Linear Programming (MILP) formulation. This work develops the DC lossless, and loss-adjusted TC formulations that can be used in a Day Ahead or intra-day market framework as well as an AC-based model that can be used in operational settings. Practical implementation choices for the Shift Factor formulation are discussed as well as the locational marginal prices (LMPs) under the TC MIP setting and their relation to LMPs without TC. Compared to the standard B-theta alternative used so far in TC research, the shift factor framework has significant computational complexity advantages, particularly when a tractably small switchable set is optimized under a representative set of contingency constraints. These claims are supported and elaborated by numerical results

Branching strategies for mixed-integer programs containing logical constraints and decomposable structure

Decision-making optimisation problems can include discrete selections, e.g. selecting a route, arranging non-overlapping items or designing a network of items. Branch-and-bound (B&B), a widely applied divide-and-conquer framework, often solves such problems by considering a continuous approximation, e.g. replacing discrete variable domains by a continuous superset. Such approximations weaken the logical relations, e.g. for discrete variables corresponding to Boolean variables. Branching in B&B reintroduces logical relations by dividing the search space. This thesis studies designing B&B branching strategies, i.e. how to divide the search space, for optimisation problems that contain both a logical and a continuous structure. We begin our study with a large-scale, industrially-relevant optimisation problem where the objective consists of machine-learnt gradient-boosted trees (GBTs) and convex penalty functions. GBT functions contain if-then queries which introduces a logical structure to this problem. We propose decomposition-based rigorous bounding strategies and an iterative heuristic that can be embedded into a B&B algorithm. We approach branching with two strategies: a pseudocost initialisation and strong branching that target the structure of GBT and convex penalty aspects of the optimisation objective, respectively. Computational tests show that our B&B approach outperforms state-of-the-art solvers in deriving rigorous bounds on optimality. Our second project investigates how satisfiability modulo theories (SMT) derived unsatisfiable cores may be utilised in a B&B context. Unsatisfiable cores are subsets of constraints that explain an infeasible result. We study two-dimensional bin packing (2BP) and develop a B&B algorithm that branches on SMT unsatisfiable cores. We use the unsatisfiable cores to derive cuts that break 2BP symmetries. Computational results show that our B&B algorithm solves 20% more instances when compared with commercial solvers on the tested instances. Finally, we study convex generalized disjunctive programming (GDP), a framework that supports logical variables and operators. Convex GDP includes disjunctions of mathematical constraints, which motivate branching by partitioning the disjunctions. We investigate separation by branching, i.e. eliminating solutions that prevent rigorous bound improvement, and propose a greedy algorithm for building the branches. We propose three scoring methods for selecting the next branching disjunction. We also analyse how to leverage infeasibility to expedite the B&B search. Computational results show that our scoring methods can reduce the number of explored B&B nodes by an order of magnitude when compared with scoring methods proposed in literature. Our infeasibility analysis further reduces the number of explored nodes.Open Acces