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Decentralized and Privacy-Preserving Learning of Approximate Stackelberg Solutions in Energy Trading Games with Demand Response Aggregators

Styliani I. Kampezidou, Justin Romberg, Kyriakos G. Vamvoudakis, Dimitri N. Mavris

TL;DR

A novel Stackelberg game theoretic framework is proposed for trading the energy bidirectionally between the demand-response aggregator and the prosumers (distributed load) and the decentralized privacy-preserving algorithm is proposed to find approximate equilibria with online sampling and learning of the prosumers’ cumulative best response.

Abstract

In this work, a novel Stackelberg game theoretic framework is proposed for trading energy bidirectionally between the demand-response (DR) aggregator and the prosumers. This formulation allows for flexible energy arbitrage and additional monetary rewards while ensuring that the prosumers' desired daily energy demand is met. Then, a scalable (linear with the number of prosumers), decentralized, privacy-preserving algorithm is proposed to find approximate equilibria with online sampling and learning of the prosumers' cumulative best response, which finds applications beyond this energy game. Moreover, cost bounds are provided on the quality of the approximate equilibrium solution. Finally, real data from the California day-ahead market and the UC Davis campus building energy demands are utilized to demonstrate the efficacy of the proposed framework and algorithm.

Decentralized and Privacy-Preserving Learning of Approximate Stackelberg Solutions in Energy Trading Games with Demand Response Aggregators

TL;DR

A novel Stackelberg game theoretic framework is proposed for trading the energy bidirectionally between the demand-response aggregator and the prosumers (distributed load) and the decentralized privacy-preserving algorithm is proposed to find approximate equilibria with online sampling and learning of the prosumers’ cumulative best response.

Abstract

In this work, a novel Stackelberg game theoretic framework is proposed for trading energy bidirectionally between the demand-response (DR) aggregator and the prosumers. This formulation allows for flexible energy arbitrage and additional monetary rewards while ensuring that the prosumers' desired daily energy demand is met. Then, a scalable (linear with the number of prosumers), decentralized, privacy-preserving algorithm is proposed to find approximate equilibria with online sampling and learning of the prosumers' cumulative best response, which finds applications beyond this energy game. Moreover, cost bounds are provided on the quality of the approximate equilibrium solution. Finally, real data from the California day-ahead market and the UC Davis campus building energy demands are utilized to demonstrate the efficacy of the proposed framework and algorithm.
Paper Structure (11 sections, 4 theorems, 49 equations, 7 figures, 1 algorithm)

This paper contains 11 sections, 4 theorems, 49 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Stackelberg Game Formulation
  3. The prosumer's problem
  4. The aggregator's problem
  5. Scalable Decentralized Privacy-Preserving Equilibrium Solution Learning
  6. Experimental Results
  7. Conclusion and Future Work
  8. Appendix
  9. Proof of Theorem \ref{['thm:UUB_PE_with_wo_appr_error']}
  10. Part 1: Without Approximation Errors
  11. Part 2: With Approximation Errors

Key Result

Lemma 1

Consider the basis $\bm{\phi}(\mathbf{p}) \in \mathbb{R}^K$, $K {<} \infty$ and assume that $\mathbf{d}^{\star}(\mathbf{p}) {\notin} \textrm{span}\{\bm{\phi}(\mathbf{p})\}$. Then, the estimation problem, has a bounded approximation error $\bm{\epsilon}_m(\mathbf{p}) {\in} \mathbb{R}^T$, i.e., $\|\bm{\epsilon}_m(\mathbf{p})\|_2 {\leq} \epsilon_m^\mathrm{max} {<} \infty$, $\forall \mathbf{p} \in [0

Figures (7)

  • Figure 1: Original (\ref{['agg_learning_inf']}) and perturbed (\ref{['agg_learning_appr']}) DR-aggregator's problems for different parameter values for Theorem \ref{['thm:model_mismatch']}.
  • Figure 2: Results for $a=1$ and $Q_i=0.01\cdot W_{i}$, for May 1st-31st, 2022.
  • Figure 3: Results for $a=2$ and $Q_i=0.01\cdot W_{i}$, for May 1st-31st, 2022.
  • Figure 4: Player utilities for $a{=}2$, $Q_i{=}0.01\cdot W_{i}$, averaged over May 2022. Prosumers' utilities are averaged over all prosumers.
  • Figure 5: Results for $a=2$ and $Q_i=0$, for May 1st-31st, 2022.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Remark 3
  • Remark 4