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Towards a Socially Acceptable Competitive Equilibrium in Energy Markets

Koorosh Shomalzadeh, Nima Monshizadeh

TL;DR

This work addresses fairness in electricity markets by combining decentralized primal-dual gradient dynamics with controllable agent utilities to achieve a Socially Acceptable Competitive Equilibrium (SCE). A price cap and an optimization of modest utility adjustments yield the optimal SCE, and a decentralized controller ensures convergence to the optimal triple ($x^*$, $lambda^*$, $u^*$) without revealing private parameters. The authors show a modified dual formulation preserves the SWSA solution and establish global asymptotic stability of the closed-loop system using a Lyapunov function, supported by a case study. The results demonstrate that the SCE reallocates energy toward urgent users while keeping prices within a socially acceptable threshold, indicating practical potential for fairness-aware energy sharing in smart grids.

Abstract

This paper addresses the problem of energy sharing between a population of price-taking agents who adopt decentralized primal-dual gradient dynamics to find the Competitive Equilibrium (CE). Although the CE is efficient, it does not ensure fairness and can potentially lead to high prices. As the agents and market operator share a social responsibility to keep the price below a certain socially acceptable threshold, we propose an approach where the agents modify their utility functions in a decentralized way. We introduce a dynamic feedback controller for the primal-dual dynamics to steer the agents to a Socially acceptable Competitive Equilibrium (SCE). We demonstrate our theoretical findings in a case study.

Towards a Socially Acceptable Competitive Equilibrium in Energy Markets

TL;DR

This work addresses fairness in electricity markets by combining decentralized primal-dual gradient dynamics with controllable agent utilities to achieve a Socially Acceptable Competitive Equilibrium (SCE). A price cap and an optimization of modest utility adjustments yield the optimal SCE, and a decentralized controller ensures convergence to the optimal triple (, , ) without revealing private parameters. The authors show a modified dual formulation preserves the SWSA solution and establish global asymptotic stability of the closed-loop system using a Lyapunov function, supported by a case study. The results demonstrate that the SCE reallocates energy toward urgent users while keeping prices within a socially acceptable threshold, indicating practical potential for fairness-aware energy sharing in smart grids.

Abstract

This paper addresses the problem of energy sharing between a population of price-taking agents who adopt decentralized primal-dual gradient dynamics to find the Competitive Equilibrium (CE). Although the CE is efficient, it does not ensure fairness and can potentially lead to high prices. As the agents and market operator share a social responsibility to keep the price below a certain socially acceptable threshold, we propose an approach where the agents modify their utility functions in a decentralized way. We introduce a dynamic feedback controller for the primal-dual dynamics to steer the agents to a Socially acceptable Competitive Equilibrium (SCE). We demonstrate our theoretical findings in a case study.
Paper Structure (10 sections, 7 theorems, 38 equations, 1 figure, 1 table)

This paper contains 10 sections, 7 theorems, 38 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement and Formulation
  3. Market model
  4. Decentralized ce seeking
  5. A fair ce
  6. Main Results
  7. Modified dual problem
  8. Controller design
  9. Case study
  10. Conclusions

Key Result

Lemma 1

Suppose that $f_i(x_i)$ is strictly concave for all $i \in \mathcal{N}$. Then, the ce coincides with the social welfare equilibrium and is unique.

Figures (1)

  • Figure 1: Trajectory of the closed-loop system \ref{['dyn:pd2']} and \ref{['dyn:cont']}, demonstrating convergence to the sce with price $\lambda^{\max}=4$.

Theorems & Definitions (17)

  • Definition 1: ce
  • Definition 2: Social welfare equilibrium
  • Lemma 1: li2015demand
  • Definition 3: sce
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • ...and 7 more