Variational Inequality Formulations for Oligopolistic Electricity Models with Marketable CO₂ Emission Permits

Abstract

In this paper, we first provide an introduction of the electricity market and its major players. Then, equilibrium models of the CO₂ emission permits market are discussed. We consider a variational inequality (VI) approach to model an oligopolistic competition in the market of CO₂ emission permits and electric power. A Cournot model of electricity market is described and a variational inequality framework is developed for oligopolistic electricity models with marketable CO₂ permits. Properties of the model are provided and some solution methods are discussed. A simple example is presented and implemented in a PC and solved by a built-in solver powered by Newton’s method of VI. Discussions of possible extensions of the model are given, like using conjectured supply function approach and including non-electric power participants of CO₂ trading.

Appendix: A Brief Introduction to Variational Inequalities

Some key concepts of variational inequality representations are provided to readers who are familiar with constrained convex optimization models. More complete discussions can be found in various sources – e.g., Nagurney [21], Harker and Pang [12], or Facchinei and Pang [8].

Let K be a closed convex set in ${{\mathbf{R}}^n}$, and f be a continuously differentiable, convex function defined on K. Consider the optimization problem

$$ \mathbf{OPT}(\,{\Pi}, \mathbf{K}): minimize f(z) $$

$$ {\hbox{subject}}\;{\hbox{to:}}\;z \in K $$

Consider also the variational inequality problem $VI(\nabla f,K)$: find ${z^{*}} \in K$ such that $\nabla f{({z^{*}})^T}(z - {z^{*}}) \geq 0$ for all $z \in K$. Nagurney [21] shows (Propositions 1, 2 and 1.3) that OPT( f , K) and $VI(\nabla f,K)$ are equivalent, i.e., that ${z^{*}}$ is a solution to OPT( f , K) if and only if ${z^{*}}$ is a solution to $VI(\nabla f,K)$. This can be understood intuitively by examining the condition for ${z^{*}}$ to be a solution to $VI(\nabla f,K)$, i.e., $\nabla f{({z^{*}})^T}(z - {z^{*}}) \geq 0$ for all $z \in K$.

This condition says that, to a first order approximation, using the gradient of f at ${z^{*}}$, any change, $(z - {z^{*}})$ away from ${z^{*}}$ causes either an increase or no change in f.

The general form of a variational inequality may be written VI(G, K): find ${z^{*}} \in K$ such that $G{({z^{*}})^T}(z - {z^{*}}) \geq 0$ for all $z \in K$. In the general case, it is not required that the vector function G be the gradient of a single valued function, which means that it may not be possible to rewrite VI(G, K) as an optimization model in the straightforward manner above.