Equilibrium, uncertainty and risk in hydro-thermal electricity systems

Abstract

The correspondence of competitive partial equilibrium with a social optimum is well documented in the welfare theorems of economics. These theorems can be applied to single-period electricity pool auctions in which price-taking agents maximize profits at competitive prices, and extend naturally to standard models with locational marginal prices. In hydro-thermal markets where the auctions are repeated over many periods, agents seek to optimize their current and future profit accounting for future prices that depend on uncertain inflows. This makes the agent problems multistage stochastic optimization models, but perfectly competitive partial equilibrium still corresponds to a social optimum when all agents are risk neutral and share common knowledge of the probability distribution governing future inflows. The situation is complicated when agents are risk averse. In this setting we show under mild conditions that a social optimum corresponds to a competitive market equilibrium if agents have time-consistent dynamic coherent risk measures and there are enough traded market instruments to hedge inflow uncertainty. We illustrate some of the consequences of risk aversion on market outcomes using a simple two-stage competitive equilibrium model with three agents.

Appendix: Extended mathematical programming

All of the computational results in this paper are solved using the extended mathematical programming (EMP) [3, 7, 9] features of the GAMS modeling system [4]. The EMP framework exists to enable formulations of problems that fall outside the standard framework within the modeling system. A high-level description of these extended models, along with tools to automatically create the different realizations or formulations possible, pass them on to the appropriate solvers, and interpret the results in the context of the original model, makes it possible to model more easily, to conduct experiments with formulations otherwise too time-consuming to consider, and to avoid errors that can make results meaningless or worse.

Multiple optimization problems with equilibrium constraints (MOPEC) involves a collection of agents $\mathcal {A}$ that determine their decisions $x_\mathcal {A} = (x_a,\,a\in \mathcal {A})$ by solving, independently, an optimization problem,

$$\begin{aligned} x_a \in \mathrm {argmax}_{x\in {\mathbb {R}}^{n_a}} f_a(p,x,x_{- a}), \quad a\in \mathcal {A}, \end{aligned}$$

(15)

where $f_a(p,\cdot ,x_{- a}):{\mathbb {R}}^{n_a}\rightarrow \overline{{\mathbb {R}}} = {\mathbb {R}} \cup \{-\infty , +\infty \}$ is their criterion function, with $x_{- a} = (x_o, o\in \mathcal {A}\setminus \{a\})$ representing the decision of the other agents and $p \in {\mathbb {R}}^d$ being a parameter that may refer to prices in an economic application, stresses in mechanical systems, and environmental conditions in numerous other applications. This parameter and the decisions $x_\mathcal {A}$ satisfy a global equilibrium constraint, formulated as a geometric variational inequality,

$$\begin{aligned} F(p,x_\mathcal {A}) \in N_C(p), \end{aligned}$$

(16)

with $N_C(p)$ the normal cone to C at p. We refer to (15)–(16) as a MOPEC, whose solution is a pair $(p,x_\mathcal {A})$ that satisfies the preceding inclusions. Even though (15) omits an explicit expression of constraints on $x_\mathcal {A}$, that possibility is handled herein by extended-value functions.

EMP provides the ability to describe a variational inequality within a modeling system. We annotate existing equations in the model, detailing which ones provide the function F, and which ones are part of the description of the underlying feasible set C. Note that there is no requirement that C is polyhedral, and the format generalizes both nonlinear equations and nonlinear complementarity systems. The main formulation of interest here is MOPEC, for example the problem described via (15) and (16). In this setting that variables $x_a$ and p, and the functions $f_a$ and F are defined with the usual model system (along with C being defined by equations defC), but an additional annotation is provided of the form:

This describes a problem involving k agents, each of which solve an optimization problem whose objective function involves not only variables $x_k$ but also other agents variables, and the price vector p. Similarly to above, the VI involving F and p nails down the values of p. In the GAMS implementation, the VI is converted into its KKT form, and then solved using the PATH solver [5, 8]—this allows problems of hundreds or thousands of variables to be processed.

Other features of EMP include stochastic programming and risk measures, hierarchical optimization, such as bilevel programming, extended nonlinear programming and disjunctive programming.