Robust Optimization in Electric Energy Systems

Chapter 1: Introduction and motivation

1.1.       Introduction

1.2.       Motivation

1.3.       Optimization vs. complementarity problems: definition and characterization

1.4.       Illustrative examples: of an optimization problem, equilibrium problem, optimization problem with equilibrium constraints and equilibrium problem with equilibrium constraints.

1.5.       Exercises

 

Chapter 2: Optimality conditions

2.1.       KKT conditions

2.2.       Constraint qualifications for necessary conditions

2.3.       Sufficiency conditions

2.4.       Simple optimization problem (analytically solvable)

2.5.       Simple optimization problem solved as an LCP or NLCP

2.6.       Simple equilibrium problem (analytically solvable)

2.7.       Simple MPEC (analytically solvable)

2.8.       Simple EPEC (analytically solvable)

2.9.       Nonconvex problems

2.10.   Exercises

 

Chapter 3: Introductory microeconomic principles relevant for complementarity problems and market equilibria

3.1.       Basics

1.1.   Supply curves

1.2.   Demand curves

1.3.   Notion of equilibrium as intersection of supply and demand curves

3.2.       Social Welfare Maximization

2.1.   Definition of social welfare and associated optimization problem

2.2.   Maximization of consumers' + producers' surpluses

3.3.       Modeling individual players

3.1.   Profit-maximization problem as paradigm

3.2.   Perfect vs. imperfect competition

3.2.1.      Price-taking producers

3.2.2.      Monopoly

3.2.3.      Oligopoly (Nash-Cournot, Bertrand games)

3.2.4.      Cartel

3.4.       Multi-level games

4.1.   Stackelberg leader follower games (MPECs)

4.2.   Multi-leader games (EPECs).

4.3.   Nash vs. Generalized Nash equilibria

3.5.       Exercises

 

Chapter 4: Equilibria as complementarity problems

4.1.       Equilibria 

4.2.       Conditions involving primal and dual variables

4.3.       Combination of equations and KKT conditions

4.4.       LCP and Mixed LCP

4.5.       NLCP and Mixed NLCP

4.6.       Stochastic equilibrium problems

4.7.       Formulation issues

4.8.       Example: Electricity market equilibrium

4.9.       Example: Gas market equilibrium

4.10.   Exercises

 

Chapter 5: Variational Inequality problems

5.1.       Variational Inequality (VI) formulation

5.2.       VI vs. complementarity problems

5.3.       Example of electricity/gas equilibrium