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Regression Equilibrium in Electricity Markets

Vladimir Dvorkin

TL;DR

This work investigates how renewable generators should choose forecast models in a two-stage electricity market to maximize profits, acknowledging that private forecast choices affect other participants and overall welfare. It introduces the concept of regression equilibrium, where wind producers optimize decision-focused forecasts against day-ahead and real-time prices, and shows that this equilibrium exists and is unique under reasonable assumptions by formulating the problem as a variational inequality. The authors provide two computational avenues—a centralized regularized optimization and an ADMM-based distributed algorithm—to compute the equilibrium and demonstrate its efficacy on the IEEE 24-bus RTS: the equilibrium improves profits and reduces dispatch costs, particularly in tail-risk scenarios, while revealing that optimal forecast features depend on wind-farm locations. The findings suggest that market participants can collectively achieve near-social-optimal outcomes without altering market structure, informing design of decision-focused forecasting and the role of ML in electricity markets.

Abstract

In two-stage electricity markets, renewable power producers enter the day-ahead market with a forecast of future power generation and then reconcile any forecast deviation in the real-time market at a penalty. The choice of the forecast model is thus an important strategy decision for renewable power producers as it affects financial performance. In electricity markets with large shares of renewable generation, the choice of the forecast model impacts not only individual performance but also outcomes for other producers. In this paper, we argue for the existence of a competitive regression equilibrium in two-stage electricity markets in terms of the parameters of private forecast models informing the participation strategies of renewable power producers. In our model, renewables optimize the forecast against the day-ahead and real-time prices, thereby maximizing the average profits across the day-ahead and real-time markets. By doing so, they also implicitly enhance the temporal cost coordination of day-ahead and real-time markets. We base the equilibrium analysis on the theory of variational inequalities, providing results on the existence and uniqueness of regression equilibrium in energy-only markets. We also devise two methods to compute regression equilibrium: centralized optimization and a decentralized ADMM-based algorithm.

Regression Equilibrium in Electricity Markets

TL;DR

This work investigates how renewable generators should choose forecast models in a two-stage electricity market to maximize profits, acknowledging that private forecast choices affect other participants and overall welfare. It introduces the concept of regression equilibrium, where wind producers optimize decision-focused forecasts against day-ahead and real-time prices, and shows that this equilibrium exists and is unique under reasonable assumptions by formulating the problem as a variational inequality. The authors provide two computational avenues—a centralized regularized optimization and an ADMM-based distributed algorithm—to compute the equilibrium and demonstrate its efficacy on the IEEE 24-bus RTS: the equilibrium improves profits and reduces dispatch costs, particularly in tail-risk scenarios, while revealing that optimal forecast features depend on wind-farm locations. The findings suggest that market participants can collectively achieve near-social-optimal outcomes without altering market structure, informing design of decision-focused forecasting and the role of ML in electricity markets.

Abstract

In two-stage electricity markets, renewable power producers enter the day-ahead market with a forecast of future power generation and then reconcile any forecast deviation in the real-time market at a penalty. The choice of the forecast model is thus an important strategy decision for renewable power producers as it affects financial performance. In electricity markets with large shares of renewable generation, the choice of the forecast model impacts not only individual performance but also outcomes for other producers. In this paper, we argue for the existence of a competitive regression equilibrium in two-stage electricity markets in terms of the parameters of private forecast models informing the participation strategies of renewable power producers. In our model, renewables optimize the forecast against the day-ahead and real-time prices, thereby maximizing the average profits across the day-ahead and real-time markets. By doing so, they also implicitly enhance the temporal cost coordination of day-ahead and real-time markets. We base the equilibrium analysis on the theory of variational inequalities, providing results on the existence and uniqueness of regression equilibrium in energy-only markets. We also devise two methods to compute regression equilibrium: centralized optimization and a decentralized ADMM-based algorithm.
Paper Structure (21 sections, 2 theorems, 29 equations, 2 figures, 2 tables)

This paper contains 21 sections, 2 theorems, 29 equations, 2 figures, 2 tables.

Table of Contents

  1. Optimization Parameters
  2. Optimization Variables
  3. Special Notation
  4. Introduction
  5. Electricity Market Preliminaries
  6. Market Setting
  7. Revenue-Maximizing Wind Power Forecasting
  8. Regression Equilibrium
  9. Regression Equilibrium Model
  10. Wind Power Producers
  11. Conventional Generators
  12. Loads
  13. Equilibrium Conditions
  14. The Model
  15. Existence and Uniqueness of Regression Equilibrium
  16. ...and 6 more sections

Key Result

Theorem 1

For the NE problem NE--comp_cond resting on Assumptions ass:1--ass:4, suppose that Assumption ass:5 also holds. Then there exists a unique equilibrium profile $(\boldsymbol{\Theta}^{\star},\mathbf{G}^{\star},\mathbf{S}^{\star})$, such that for every wind power producer $j=1,\dots,b$, we have:

Figures (2)

  • Figure 1: Baseline and equilibrium forecasts as functions of wind speed at six wind farms installed in the IEEE 24-Bus RTS.
  • Figure 2: The regression weights assigned to each of the 30 kernel feature transformations (15 kernels for wind speed and 15 kernels for wind direction) under the baseline method (red) and in equilibrium (green).

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Corollary 1