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Socially Optimal Energy Usage via Adaptive Pricing

Jiayi Li, Matthew Motoki, Baosen Zhang

TL;DR

The paper addresses coordinating electricity usage under privacy constraints by introducing a two-time-scale mechanism in which an operator updates a price vector $\mathbf{p}$ based on the marginal social cost $\mathbf{e}(\mathbf{x}) = \nabla g(\sum_i \mathbf{x}_i)$ and users respond by solving $\min_{\mathbf{x}_i \in \mathcal{X}_i} f_i(\mathbf{x}_i) + \mathbf{p}^\top \mathbf{x}_i$. It proves that the iterative price updates converge to a unique $\mathbf{p}^{*}$ and the associated user actions $\mathbf{x}^{*}(\mathbf{p}^{*})$ solve the global social welfare problem $\min_{\mathbf{x}} \sum_i f_i(\mathbf{x}_i) + g\left(\sum_i \mathbf{x}_i\right)$, even when $f_i$ are nonconvex or when $T>1$, using continuous-time Lyapunov analyses. Convergence is guaranteed under two sufficient conditions: (i) a quadratic system cost $g(\mathbf{z}) = \tfrac{1}{2} \mathbf{z}^T \mathbf{B} \mathbf{z}$ or (ii) strictly convex differentiable $f_i$ with strictly convex differentiable $g$, with proofs relying on either a quadratic Lyapunov function or a Bregman-divergence based Lyapunov function. Simulations on single-period and multi-period load scenarios, including Q-learning based load control for water heaters, demonstrate rapid convergence and meaningful reductions in both social cost and peak demand, suggesting practical privacy-preserving demand response via adaptive pricing.

Abstract

A central challenge in using price signals to coordinate the electricity consumption of a group of users is the operator's lack of knowledge of the users due to privacy concerns. In this paper, we develop a two-time-scale incentive mechanism that alternately updates between the users and a system operator. As long as the users can optimize their own consumption subject to a given price, the operator does not need to know or attempt to learn any private information of the users for price design. Users adjust their consumption following the price and the system redesigns the price based on the users' consumption. We show that under mild assumptions, this iterative process converges to the social welfare solution. In particular, the cost of the users need not always be convex and its consumption can be the output of a machine learning-based load control algorithm.

Socially Optimal Energy Usage via Adaptive Pricing

TL;DR

The paper addresses coordinating electricity usage under privacy constraints by introducing a two-time-scale mechanism in which an operator updates a price vector based on the marginal social cost and users respond by solving . It proves that the iterative price updates converge to a unique and the associated user actions solve the global social welfare problem , even when are nonconvex or when , using continuous-time Lyapunov analyses. Convergence is guaranteed under two sufficient conditions: (i) a quadratic system cost or (ii) strictly convex differentiable with strictly convex differentiable , with proofs relying on either a quadratic Lyapunov function or a Bregman-divergence based Lyapunov function. Simulations on single-period and multi-period load scenarios, including Q-learning based load control for water heaters, demonstrate rapid convergence and meaningful reductions in both social cost and peak demand, suggesting practical privacy-preserving demand response via adaptive pricing.

Abstract

A central challenge in using price signals to coordinate the electricity consumption of a group of users is the operator's lack of knowledge of the users due to privacy concerns. In this paper, we develop a two-time-scale incentive mechanism that alternately updates between the users and a system operator. As long as the users can optimize their own consumption subject to a given price, the operator does not need to know or attempt to learn any private information of the users for price design. Users adjust their consumption following the price and the system redesigns the price based on the users' consumption. We show that under mild assumptions, this iterative process converges to the social welfare solution. In particular, the cost of the users need not always be convex and its consumption can be the output of a machine learning-based load control algorithm.
Paper Structure (14 sections, 3 theorems, 18 equations, 4 figures, 1 algorithm)

This paper contains 14 sections, 3 theorems, 18 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Pricing Updates
  3. Global Optimization Problem
  4. User's Optimization Problem
  5. Price Update
  6. Theoretical Results
  7. Single time-period case (T = 1)
  8. Multi-Time Period Case (T > 1)
  9. Simulation Results
  10. Single Time-Period
  11. Peak Pricing of Multiple Time-Periods
  12. Water Heater Load Optimization with Q-Learning
  13. Conclusion
  14. Proof of Theorem \ref{['thm:t_multi']}

Key Result

Theorem 1

Suppose Assumption assump:local_unique holds and $x_i^{*}(p)$ is decreasing for each $i$ and $g$ is convex and differentiable. Then the scalar dynamical system eqn:continuous_dynamics is globally asymptotically stable and has a unique fixed point $p^{*}$. The solutions $x_i^{*}(p^{*})$ solve the glo

Figures (4)

  • Figure 1: Convergence of user actions and price incentive for a system with 5 users and a single time-period. Both the actions and the price converge quickly.
  • Figure 2: The top figure shows the convergence in the sum of the users' actions at each time period. The bottom figure shows the convergence of price.
  • Figure 3: The initial and converged price and demand profiles demonstrate that this adaptive pricing framework is effective in reducing system peak.
  • Figure 4: Simulation results for water heater optimization. There are 10 users minimizing discomfort using a Q-learning algorithm motokiwaterheater, and the system operator tries to minimize the peak load. The initial price is chosen to be flat, which leads to a profile with high peaks. After several iterations, the price becomes uneven, and the final load has much lower peaks.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof