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Stochastic Mean Field Game for Strategic Bidding of Consumers in Congested Distribution Networks

Amirreza Silani, Simon H. Tindemans

TL;DR

The paper tackles congestion in distribution networks caused by distributed energy resources by formulating a two-stage day-ahead and redispatch market and modeling strategic bidding as a stochastic mean-field game with a reverse Stackelberg structure. It shows that in large populations, agent strategies are ordered by utility and proves the existence and uniqueness of a Nash equilibrium under uncertainty; it also characterizes the optimal day-ahead bidding and redispatch responses, linking congestion prices to the population distribution. The approach provides insight into how anticipation of redispatch affects welfare and congestion, and it highlights conditions under which unique equilibrium exists versus cases with potential multiplicity when congestion is certain or day-ahead prices are fixed. Practically, the framework offers a tractable method to analyze and design congestion-management markets in congested distribution networks with high penetrations of DERs.

Abstract

The rapid increase of photovoltaic cells, batteries, and Electric Vehicles (EVs) in electric grids can result in congested distribution networks. An alternative to enhancing network capacity is a redispatch market, allowing Distribution System Operators (DSOs) to alleviate congested networks by asking energy consumers to change their consumption schedules. However, energy consumers can anticipate the redispatch market outcomes and strategically adjust their bids in the day-ahead market. This behaviour, known as increase-decrease gaming, can result in the exacerbation of congestion and enable energy consumers to gain windfall profits from the DSO. In this paper, we consider a two-stage problem consisting of the day-ahead market (first stage) and redispatch market (second stage). Then, we model the increase-decrease game for large populations of energy consumers in power networks using a stochastic mean field game approach. The agents (energy consumers) maximize their individual welfare in the day-ahead market with anticipation of the redispatch market. We show that all the agent strategies are ordered along their utilities and there exists a unique Nash equilibrium for this game.

Stochastic Mean Field Game for Strategic Bidding of Consumers in Congested Distribution Networks

TL;DR

The paper tackles congestion in distribution networks caused by distributed energy resources by formulating a two-stage day-ahead and redispatch market and modeling strategic bidding as a stochastic mean-field game with a reverse Stackelberg structure. It shows that in large populations, agent strategies are ordered by utility and proves the existence and uniqueness of a Nash equilibrium under uncertainty; it also characterizes the optimal day-ahead bidding and redispatch responses, linking congestion prices to the population distribution. The approach provides insight into how anticipation of redispatch affects welfare and congestion, and it highlights conditions under which unique equilibrium exists versus cases with potential multiplicity when congestion is certain or day-ahead prices are fixed. Practically, the framework offers a tractable method to analyze and design congestion-management markets in congested distribution networks with high penetrations of DERs.

Abstract

The rapid increase of photovoltaic cells, batteries, and Electric Vehicles (EVs) in electric grids can result in congested distribution networks. An alternative to enhancing network capacity is a redispatch market, allowing Distribution System Operators (DSOs) to alleviate congested networks by asking energy consumers to change their consumption schedules. However, energy consumers can anticipate the redispatch market outcomes and strategically adjust their bids in the day-ahead market. This behaviour, known as increase-decrease gaming, can result in the exacerbation of congestion and enable energy consumers to gain windfall profits from the DSO. In this paper, we consider a two-stage problem consisting of the day-ahead market (first stage) and redispatch market (second stage). Then, we model the increase-decrease game for large populations of energy consumers in power networks using a stochastic mean field game approach. The agents (energy consumers) maximize their individual welfare in the day-ahead market with anticipation of the redispatch market. We show that all the agent strategies are ordered along their utilities and there exists a unique Nash equilibrium for this game.
Paper Structure (10 sections, 12 theorems, 28 equations, 6 figures, 1 table)

This paper contains 10 sections, 12 theorems, 28 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Model description
  4. Mean field game appraoch
  5. Individual agent strategy
  6. Ordering of agent strategies
  7. Stochastic mean field game
  8. Fixed day-ahead price and certain congestion occurring
  9. Simulation Results
  10. Conclusions and Future Work

Key Result

Lemma 1

(Optimal second stage strategy). Let the purchased consumption schedule $e_i^\mathrm{d}$ in the day-ahead and price $\pi^\mathrm{r}$ in the redispatch market be given. Then, the optimal second stage strategy is given by The optimal second stage strategy of agent $i\in\mathcal{N}$ is indeterminate if

Figures (6)

  • Figure 1: Market model.
  • Figure 2: The agent outcomes based on their utilities in the Nash equilibrium.
  • Figure 3: Day-ahead market with medium network capacity and $D\neq 0$: (a) supply and demand curves; (b) normalized expected welfare when the energy consumers can and cannot anticipate the redispatch market; and (c) probabilities of trading on day-ahead and redispatch market.
  • Figure 4: Supply and demand curves: (a) high network capacity; (b) low network capacity; and (c) fixed day-ahead price and no uncertainty.
  • Figure 5: The total expected welfare, utility, redispatch revenue and day-ahead cost at time $t = 1$ with medium network capacity and $D\neq 0$: (a) energy consumers cannot anticipate the redispatch market and (b) energy consumers can anticipate the redispatch market.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Definition 1
  • Lemma 4
  • ...and 16 more