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A Framework for Eliminating Paradoxical Orders in European Day-Ahead Electricity Markets through Mixed-Integer Linear Programming Strong Duality

Zhen Wang, Mohammad Reza Hesamzadeh, Shudian Zhao, Jan Kronqvist

Abstract

The presence of integer variables in the European day-ahead electricity market renders the social welfare maximization problem non-convex and non-differentiable, making classical marginal pricing theoretically inconsistent. Existing pricing mechanisms often struggle to balance revenue adequacy with incentive compatibility, typically relying on discriminatory uplift payments or suffering from weak duality. Leveraging the Augmented Lagrangian Duality (ALD) framework, which establishes strong duality for Mixed-Integer Linear Programming (MILP), this paper proposes a novel ALD pricing mechanism. We analytically prove that this mechanism is inherently incentive-compatible, aligning centralized dispatch with individual incentives without requiring side payments. Notably, we demonstrate that the ALD price signals intrinsically eliminate Paradoxically Rejected Orders (PROs) and Paradoxically Accepted Orders (PAOs) for supply orders. For the demand side, a sufficient condition is introduced to further guarantee revenue adequacy, resulting in a transparent and financially consistent settlement system. To ensure computational tractability, we modify the Surrogate Absolute-Value Lagrangian Relaxation (SAVLR) method to efficiently compute the exact penalty coefficients and optimal Lagrangian multipliers. Numerical experiments on illustrative examples and the Nordic 12-area electricity market model confirm the superior economic properties of the ALD pricing mechanism and the tractability of the modified SAVLR algorithm.

A Framework for Eliminating Paradoxical Orders in European Day-Ahead Electricity Markets through Mixed-Integer Linear Programming Strong Duality

Abstract

The presence of integer variables in the European day-ahead electricity market renders the social welfare maximization problem non-convex and non-differentiable, making classical marginal pricing theoretically inconsistent. Existing pricing mechanisms often struggle to balance revenue adequacy with incentive compatibility, typically relying on discriminatory uplift payments or suffering from weak duality. Leveraging the Augmented Lagrangian Duality (ALD) framework, which establishes strong duality for Mixed-Integer Linear Programming (MILP), this paper proposes a novel ALD pricing mechanism. We analytically prove that this mechanism is inherently incentive-compatible, aligning centralized dispatch with individual incentives without requiring side payments. Notably, we demonstrate that the ALD price signals intrinsically eliminate Paradoxically Rejected Orders (PROs) and Paradoxically Accepted Orders (PAOs) for supply orders. For the demand side, a sufficient condition is introduced to further guarantee revenue adequacy, resulting in a transparent and financially consistent settlement system. To ensure computational tractability, we modify the Surrogate Absolute-Value Lagrangian Relaxation (SAVLR) method to efficiently compute the exact penalty coefficients and optimal Lagrangian multipliers. Numerical experiments on illustrative examples and the Nordic 12-area electricity market model confirm the superior economic properties of the ALD pricing mechanism and the tractability of the modified SAVLR algorithm.

Paper Structure

This paper contains 43 sections, 9 theorems, 39 equations, 6 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Pricing Mechanisms and the Uplift Problem
  4. Convex Hull Pricing (CHP) and Long-Run Impacts
  5. Paradoxical Orders and Investment Incentives
  6. Paper organization and the conceptual roadmap
  7. MILP model of the European day-ahead electricity market
  8. Notations
  9. The electricity market model and the discrete optimization problem
  10. Interpretations of the optimization problem \ref{['MILP:original:one']}
  11. The ALD-based pricing model
  12. Decomposed Variable and the Feasible Set
  13. Compact formulation
  14. ALD Formulation
  15. Strong duality of the ALD approach
  16. ...and 28 more sections

Key Result

Lemma 1

The dual optimization problem $\max\limits_{\lambda \in \mathbb{R}^{|\mathcal{A}|\times |\mathcal{T}|}} {z_\rho^{ALD}(\lambda)}$ of mix:integer:linear:program:ALD is concave, where the objective function is piecewise linear and sub-differentiable.

Figures (6)

  • Figure 1: Conceptual roadmap linking the paper's organization to the contributions.
  • Figure 2: The ALD approach to 1-area case
  • Figure 3: The ALD approach to the 2-area case
  • Figure 4: The Nordic day-ahead electricity market topology.
  • Figure 5: LMPs of DK1 FI NO4, and SE3.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Lemma 1: Characterization of the MILP Dual
  • Theorem 1: Exact Penalty Coefficient
  • Definition 1: ALD Price Signals
  • Definition 2: Decentralized Optimization
  • Definition 3: Incentive Compatibility
  • Theorem 2: Incentive Compatibility of ALD
  • proof
  • Proposition 1: Aggregate Congestion Rent
  • proof
  • Proposition 2: Revenue Adequacy for TSO of Radial Networks
  • ...and 13 more