Elsevier

Energy Economics

Volume 103, November 2021, 105493
Energy Economics

Deregulated electricity market, a stochastic variational approach

https://doi.org/10.1016/j.eneco.2021.105493Get rights and content

Highlights

  • We propose an electricity market model in a continuum state of nature and multiple trading dates.

  • Agents consume, produce, and sign contracts.

  • There is uncertainty on the state which will realize at each stage t.

  • Uncertainty on future possible stages leads to consider a filtered probability space.

Abstract

This paper focuses on the study of an electricity market model, which evolves in T+1 stages, so that, at each stage, a continuum state of nature is possible. The decision-making framework of large consumers is considered: agents consume, they have the opportunity to produce, and the capability of signing contracts. Uncertainty on future possible stages leads to the problem being located in a filtered probability space. The aim is to reformulate the problem as a stochastic quasi-variational inequality, in order to obtain an existence result of equilibrium solutions.

Introduction

Since 1996, in many countries, the electric power industry has undergone a transformation from a government-regulated to a competitive regime. This fact motivated the growing interest in the study of electricity market models. The main features of these models are the central role of information, the possibility of highly uncertain market prices that cause volatility of profits and costs, the presence of tools to hedge against these risks, in terms of future markets.

We focus on the decision-making framework of large consumers, such as petrochemical industries, aluminum production complex, or vehicle-assembling facilities.

In the economy under study, agents consume, have the opportunity to produce, and the capability of signing contracts. The final aim of each agent is to either minimize its procurement cost or to maximize the utility it obtains from electricity usage. The problem has a strategic nature. Indeed, each agent is allowed to sign opportune contracts, before the uncertainty is resolved and for each future times, as tools to reduces her exposure to risk associated with the volatility of market prices that open at each time after the uncertainty is revealed; however, at the same time, the reduction of the risks associated with the volatility of procurement cost usually comes at the cost of high average prices for the signed contracts. Hence, an optimal mix among the different sources is needed. To deal with the complications that this case involves, we need to formalize the structure of time-uncertainty-information by introducing a suitable functional setting relative to an opportunely built filtered probability space.

The starting point of our model is the decision-problem of multistage energy procurement for large consumers under uncertainty introduced in Chapter 9 of Conejo et al. (2010). The authors consider the decision-problem of each consumer in a finite-dimensional space, under a given price system. Instead, in our model, the market prices are determined by the market by introducing opportune market-clearing conditions. These assumptions allow us to frame the model into an equilibrium problem of plans, prices, and price expectations due to  Radner (1972). Radner generalized the Debreu equilibrium: the agents have the possibility to transfer wealth among all possible future times and to trade at each possible contingency after that the uncertainty is revealed and the market reopens. Two different market structures, forward and spot markets, are so considered. The model is studied into a general stochastic framework by following Mas-Colell and Zame (1996), who introduced a Radner equilibrium problem with a continuum of the states of nature and multiple trading dates. Hence, we set an economy with multiple trading dates and a continuum of states in order to be as much as close to the realistic case.

We point out that these complications are introduced only to be as much as close to the realistic case where most real-world phenomena vary with continuity and influence, in a relevant manner, the decision process of the decision-makers. For instance, one could imagine the variability of weather conditions that influence the power generation from renewable resources. In this setting, decisions are so classified on the basis of when they are made and, then, of the information available: we distinguish between here-and-now and wait-and-see decisions.

We study the equilibrium problem for the deregulated electricity market by using the variational inequalities theory. Introduced by Stampacchia and Fichera in 1964, it provides powerful and handy tools to perform quantitative and qualitative studies in relation to optimization problems, equilibrium problems, systems of equations, etc. Nowadays, the variational theory unifies a large range of applications (see, e.g., Donato et al., 2009, Donato et al., 2018, Jofré et al., 2007). However, although some practical problems involve only deterministic data, in most real-world applications there are many important examples where problem data contain some uncertainty and randomness, as for instance in economics, finance, and management. Consequently, to reflect and capture these aspects, stochastic variational inequality problems are been recently introduced and studied as a natural extension of deterministic variational inequalities models. In particular, we refer to the multistage formulation introduced by  Rockafellar and Wets (2016): it provides innovative and flexible tools to study real-life problems complicated by time, uncertainty, and risk, and to efficiently solve them by means of parallel algorithms. This approach is able to study situations where the decisions have to interact dynamically with the availability of information. Indeed, with this approach, it is possible to capture the dynamics that are essential to stochastic decision processes in response to an increasing level of information. The key concept of this new formulation turns out to be that relating to particular constraints, called nonanticipativity, that have to be included in the formulation of the problem in question.

The paper is organized as follows. Section 2 is devoted to the introduction of the electricity market model, and the relative time-uncertainty-information structure. The chosen probabilistic setting allows us to investigate how the information influences the decision processes of the agents and how these choices evolve over time. In addition, opportune equilibrium conditions for the considered economy are involved, following the philosophy of the Radner scheme. On this basis, in Section 3, the resulting equilibrium for the electricity market is studied by means of a suitable stochastic quasi-variational problem. Finally, a section of the Conclusion is given.

Section snippets

Electricity market model

In this section, we describe an electricity market model with a continuum of the states of nature and multiple trading dates. The economy begins at the initial date 0 and extends over T stages: T1,,t,,T and T0{0}T are the sets of stages. There is uncertainty on the state which will realize at each stage tT: ΩtR is the set of the possible alternatives at t; we pose ΩΩ1×Ω2××ΩT and each ω=(ω1,ω2,,ωT)Ω describes all evolution of the market. If ωΩ, when tT, ω[t](ω1,ω2,,ωt) represents

Stochastic variational approach

This section deals with the connection between an equilibrium vector for E and the solution of a suitable stochastic quasi-variational problem. We follow the approaches used in Rockafellar and Wets (2016) and Rockafellar and Wets (1978). We assume the following assumptions on utility ui, for all iI.

Assumptions A

  • (A.1.)

    ui(ω,,) is continuous, directionally differentiable and there exists a function Z(ω), with expected value well defined and finite, such that ui(ω,xi,bi)Z(ω) a.e.ωΩ for all (xi,bi);

  • (A.2.)

    u

Conclusions

In this paper, we made use of a recent variational approach, introduced by  Rockafellar and Wets, 2016, Rockafellar and Wets, 2015, to study an electricity market problem from the decision-making framework of large consumers. In such a model, all agents simultaneously maximize their utility under the budget constraint set, which depends on the equilibrium price system; in particular, spot and forward prices are determined by the market by introducing the market-clearing conditions. Hence, we

Acknowledgments

We would like to thanks the referees for their insightful comments which led to an improved version of the present paper. Research of M. Milasi is partially supported by PRIN 2017 “Nonlinear Differential Problems via Variational, Topological and Set-valued Methods ”(Grant Number: 2017AYM8XW).

References (17)

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    Indeed, to capture the dynamics of stochastic decision processes in response to an increasing level of information, in [16], Rockafellar and Wets introduced a multistage extension of a stochastic variational inequality problem. This formulation provides innovative and flexible tools to study real-world problems complicated by time, uncertainty, and risk and allows us to capture the role of information in the recursive decision processes; see, e.g., [21,22]. The key of this multistage formulation turns out to be the nonanticipativity: some constraints have to be included in the formulation of the multistage stochastic variational inequality problem to take into account the partial information progressively revealed.

1

The authors have equally contributed to the manuscript.

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