This book covers robust optimization theory and applications in the electricity sector. The advantage of robust optimization with respect to other methodologies for decision making under uncertainty are first discussed. Then, the robust optimization theory is covered in a friendly and tutorial manner. Finally, a number of insightful short- and long-term applications pertaining to the electricity sector are considered.
Specifically, the book includes: robust set characterization, robust optimization, adaptive robust optimization, hybrid robust-stochastic optimization, applications to short- and medium-term operations problems in the electricity sector, and applications to long-term investment problems in the electricity sector. Each chapter contains end-of-chapter problems, making it suitable for use as a text.
The purpose of the book is to provide a self-contained overview of robust optimization techniques for decision making under uncertainty in the electricity sector. The targeted audience includes industrial and power engineering students and practitioners in energy fields. The young field of robust optimization is reaching maturity in many respects. It is also useful for practitioners, as it provides a number of electricity industry applications described up to working algorithms (in JuliaOpt).
Author(s): Xu Andy Sun, Antonio J. Conejo
Series: International Series in Operations Research & Management Science, 313
Publisher: Springer
Year: 2021
Language: English
Pages: 339
City: Cham
Preface
Contents
1 Decision Making Under Uncertainty in the Power Sector
1.1 Introduction
1.2 Stochastic Programming
1.2.1 Single-Stage Stochastic Programming
1.2.2 Two-Stage Stochastic Programming
1.2.3 Multi-stage Stochastic Programming
1.3 Robust Optimization
1.3.1 Single-Stage Robust Optimization
1.3.2 Adaptive Robust Optimization
1.3.3 Adaptive Robust Stochastic Optimization
1.3.4 Adaptive Distributionally Robust Optimization
1.4 Uncertainty Modeling
1.4.1 Scenarios
1.4.2 Robust Sets
1.4.3 Uncertainty Budget
1.5 Why Robust Optimization?
1.6 Concluding Remarks
1.7 End-of-Chapter Exercises
1.8 Appendix: Acronyms
1.9 Appendix: GAMS Code
References
2 Static Robust Optimization
2.1 Introduction
2.1.1 Preliminary
2.1.2 Summary and Organization
2.2 Robust Linear Optimization
2.2.1 Preliminary
2.2.2 Reformulations
2.2.2.1 Polyhedral Uncertainty Sets
2.2.2.2 Linear Conic Uncertainty Sets
2.3 Robust Convex Conic Optimization
2.3.1 Preliminary
2.3.2 Robust Second-Order Conic Constraints
2.3.2.1 Convex Hull Uncertainty Set
2.3.2.2 Interval Uncertainty Set
2.3.2.3 Simple Ellipsoidal Uncertainty Set
2.3.2.4 Ellipsoidal Uncertainty Set for Robust Quadratic Constraints
2.3.3 Robust Semidefinite Constraints
2.3.3.1 Convex Hull Uncertainty Set
2.3.3.2 Rank-2 Ellipsoidal Uncertainty Set
2.4 Approximations of Robust Convex Constraints
2.4.1 Safe Approximations of Robust Second-Order Conic Optimization
2.4.1.1 The Concept of Structured Uncertainty
2.4.1.2 Safe Approximation of Robust SOC with Ustr
2.4.2 Safe Approximation of Robust SDP Constraints
2.4.3 A Quick Summary
2.4.4 A General Principle for Deriving Safe Approximation
2.4.4.1 Basic Model and Assumption
2.4.4.2 Safe Approximation
2.4.4.3 Reformulation
2.4.4.4 Applications
2.5 End-of-Chapter Exercises
References
3 Adaptive Robust Optimization
3.1 Introduction
3.2 Two-Stage Robust Optimization
3.2.1 Two-Stage Robust Linear Optimization Formulations
3.2.1.1 Functional Formulation of 2S-RLO
3.2.1.2 Nested Formulation for 2S-RLO
3.2.1.3 Value-Function Formulation of 2S-RLO
3.2.2 Structural Properties of Two-Stage Robust Linear Optimization
3.2.2.1 Properties of the Cost-to-Go Function
3.2.2.2 Properties of the Value Function
3.2.2.3 Properties of the 2S-RLO
3.2.3 Computational Complexity of Two-Stage Robust Linear Optimization
3.2.4 The Concepts of Oracles and Oracle-Based Algorithms
3.2.5 Benders Decomposition
3.2.6 Column-and-Constraint Generation
3.3 Multistage Robust Optimization
3.3.1 Multistage Robust Linear Optimization Formulations
3.3.1.1 A Functional Formulation of MS-RLO
3.3.1.2 Nested Formulation of MS-RLO
3.3.1.3 Value-Function Formulation of MS-RLO and DP Recursion
3.3.2 Structural Properties of Multistage Robust Linear Optimization
3.3.2.1 Uncertainty Sets with Stagewise Independence
3.3.2.2 Convexity with Stagewise Independent Uncertainty
3.3.2.3 What Could Go Wrong Without Stagewise Independence
3.3.3 Nested Benders Decomposition
3.3.4 Linear Decision Rules and a Cutting Plane Method
3.3.4.1 Linear Decision Rules in Functional Form MS-RLO
3.3.4.2 Reformulation by Dualization
3.3.4.3 A Constraint Generation Method
3.4 End-of-Chapter Exercises
References
4 Distributionally Robust Optimization
4.1 Introduction
4.2 Static Distributionally Robust Convex Optimization
4.2.1 Moment-Based Ambiguity Sets and Reformulations
4.2.1.1 Motivation
4.2.1.2 The Principle of Reformulation Through a Simple Example
4.2.1.3 A More Sophisticated Example with Cross Moments
4.2.2 ϕ-Divergence Based Ambiguity Sets and Reformulations
4.2.2.1 Motivation
4.2.2.2 Kullback-Leibler Divergence
4.2.2.3 General Definition of ϕ-Divergence and Examples
4.2.2.4 Equivalent Reformulation of General Form
4.2.2.5 Examples of Equivalent Reformulation
4.2.3 Wasserstein-Distance Based Ambiguity Sets and Reforulations
4.2.3.1 The Optimal Transportation Problem and Wasserstein Distance
4.2.3.2 Comparison of Wasserstein Distance and ϕ-Divergence
4.2.3.3 Equivalent Reformulations of DRO with Wasserstein-Distance Ambiguity Sets
4.2.3.4 Further Reformulation to Finite Convex Program
4.2.3.5 The Worst-Case Distribution and Relation to Robust Optimization
4.2.4 Statistical Approaches for Determining the Size of Ambiguity Sets
4.2.4.1 ϕ-Divergence Based Ambiguity Sets
4.2.4.2 Wasserstein-Distance Based Ambiguity Sets
4.3 Adaptive Distributionally Robust Optimization
4.3.1 Two-Stage DRO and Reformulations
4.3.1.1 Two-Stage DRO with Moment-Based Ambiguity Sets
4.3.1.2 Two-Stage DRO with ϕ-Divergence Based Ambiguity Sets
4.3.1.3 Two-Stage DRO with Wasserstein Ambiguity Sets
4.4 Distributionally Robust Chance Constrained Programs
4.4.1 Motivation
4.4.2 Exact Reformulations of DRCC with Moment Ambiguity Sets
4.4.2.1 Equivalence Between DRCC and CVaR
4.4.2.2 Tractable Reformulations
4.4.3 Exact Reformulations of DRCC with ϕ-Divergence Ambiguity Sets
4.4.4 Exact Reformulations of DRCC with Wasserstein Ambiguity Sets
4.4.4.1 Exact Reformulation of an Individual DRCC as a CVaR Constraint
4.4.4.2 Mixed Integer Representation
4.4.5 Theory of Safe Convex Approximations of DRCC
4.4.5.1 Chance Constraints and CVaR Approximation
4.4.5.2 Bernstein Approximation
4.4.5.3 Connections to Robust Optimization
4.4.5.4 Generalizations to DRCC
4.5 End-of-Chapter Exercises
References
5 Hybrid Adaptive Robust Optimization Models
5.1 Introduction
5.2 Long- and Short-Term Uncertainty
5.3 Formulation and Solution
5.4 Transmission Expansion Planning (TEP)
5.4.1 Notation
5.4.2 Formulation
5.5 TEP: Case Study
5.5.1 Introduction
5.5.2 Formulation
5.5.3 Formulation Restructuring
5.5.3.1 Linearized Third-Level Problem
5.5.3.2 Dual of the Third-Level Problem
5.5.3.3 Subproblem: Merged Second- and Third-Level Problem
5.5.4 Solution via Decomposition
5.5.4.1 Master Problem
5.5.4.2 Algorithm
5.5.5 Actual Solution
5.5.5.1 First Master Problem
5.5.5.2 First Subproblem
5.5.5.3 Second Master Problem
5.5.5.4 Second Subproblem
5.5.5.5 Convergence Achieved
5.6 End-of-Chapter Exercises
5.7 Appendix: GAMS Code
References
6 Robust Optimization in Short-Term Power System Operations
6.1 Introduction
6.2 Two-Stage Robust Unit Commitment
6.2.1 A Deterministic SCUC Model
6.2.2 Two-Stage Robust UC Model
6.2.2.1 Two-Stage Robust UC Model
6.2.3 Computational Experiments
6.2.3.1 Reduction of Average Cost and Budget Level
6.2.3.2 Reduction of Volatility and Penalty Costs
6.3 Multistage Robust UC Model
6.3.1 The Need for Multistage Robust UC Model
6.3.2 Multistage Robust UC Model
6.3.3 Affine Multistage Robust UC
6.3.4 A Constraint Generation Algorithm and Speed-Ups
6.3.4.1 Algorithmic Speed-Ups
6.3.5 Computational Experiments
6.3.5.1 Optimality Gap for Simplified Affine Policies
6.3.5.2 Worst-Case Performance Analysis
6.3.5.3 Average Performance
6.4 Rolling-Horizon Robust Economic Dispatch with Dynamic Uncertainty Sets
6.4.1 Dynamic Uncertainty Sets
6.4.1.1 Definition of Dynamic Uncertainty Sets
6.4.1.2 Constructing Dynamic Uncertainty Sets for Wind Power
6.4.2 A Rolling Horizon Robust Dispatch Model
6.4.3 Simulation Platform and Evaluation Metrics
6.4.4 Computational Experiments
6.4.4.1 Robust ED Versus Look-Ahead ED
6.4.4.2 Dynamic Uncertainty Sets Versus Static Uncertainty Sets
6.4.4.3 Impact of System Ramping Capacity
6.4.4.4 Tests on 118-Bus System
6.5 Robust Economic Dispatch with AC Optimal Power Flow
6.5.1 Multi-Period AC-OPF Models and Convex Relaxations
6.5.2 Robust AC-OPF
6.5.3 Robust Conic-OPF Models
6.5.4 Uncertainty Sets for Load and Renewables
6.5.5 Computational Experiments
6.6 End-of-Chapter Exercises
References
7 Medium-Term Planning Models
7.1 Introduction
7.2 Grid Maintenance Scheduling
7.2.1 Simple Example
7.2.2 General Formulation
7.3 Fuel Procurement
7.3.1 Simple Example
7.3.2 General Formulation
7.4 Medium-Term Hydro Scheduling
7.4.1 Simple Example
7.4.2 General Formulation
7.5 Selling Electricity via Contracts
7.5.1 Simple Example
7.5.2 General Formulation
7.6 Consumer Electricity Procurement
7.6.1 Simple Example
7.6.2 General Formulation
7.7 Concluding Remarks
7.8 End-of-Chapter Exercises
References
8 Long-Term Planning Models
8.1 Introduction
8.2 Generation Expansion Planning
8.2.1 Simple Example
8.2.2 General Formulation
8.3 Generation and Transmission Expansion Planning
8.3.1 Simple Example
8.3.2 General Formulation
8.4 Transmission Expansion Planning
8.4.1 Simple Example
8.4.2 General Formulation
8.5 Long-Term Hydroelectric Scheduling
8.5.1 Simple Example
8.5.2 General Formulation
8.6 Concluding Remarks
8.7 End-of-Chapter Exercises
References
Index
