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Demand Response Under Stochastic, Price-Dependent User Behavior

Guido Cavraro, Andrey Bernstein, Emiliano Dall'Anese

Abstract

This paper focuses on price-based residential demand response implemented through dynamic adjustments of electricity prices during DR events. It extends existing DR models to a stochastic framework in which customer response is represented by price-dependent random variables, leveraging models and tools from the theory of stochastic optimization with decision-dependent distributions. The inherent epistemic uncertainty in the customers' responses renders open-loop, model-based DR strategies impractical. To address this challenge, the paper proposes to employ stochastic, feedback-based pricing strategies to compensate for estimation errors and uncertainty in customer response. The paper then establishes theoretical results demonstrating the stability and near-optimality of the proposed approach and validates its effectiveness through numerical simulations.

Demand Response Under Stochastic, Price-Dependent User Behavior

Abstract

This paper focuses on price-based residential demand response implemented through dynamic adjustments of electricity prices during DR events. It extends existing DR models to a stochastic framework in which customer response is represented by price-dependent random variables, leveraging models and tools from the theory of stochastic optimization with decision-dependent distributions. The inherent epistemic uncertainty in the customers' responses renders open-loop, model-based DR strategies impractical. To address this challenge, the paper proposes to employ stochastic, feedback-based pricing strategies to compensate for estimation errors and uncertainty in customer response. The paper then establishes theoretical results demonstrating the stability and near-optimality of the proposed approach and validates its effectiveness through numerical simulations.
Paper Structure (11 sections, 2 theorems, 45 equations, 8 figures, 1 algorithm)

This paper contains 11 sections, 2 theorems, 45 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and Demand Response Events
  3. A Demand Response Problem Formulation
  4. Online Feedback Optimization Algorithm
  5. Online stochastic algorithm
  6. Discussion
  7. Convergence
  8. Numerical Examples
  9. Constant Load Scenario
  10. Time-varying Load Scenario
  11. Conclusion

Key Result

Proposition 1

Consider the stochastic algorithm eq:PD_In_st, and let $\mathbf{z}(t)$, $t \in \mathbb{N}$, be the sequence of prices and dual variables generated by eq:PD_In_st, starting from $\mathbf{z}(0) \in \mathcal{Z}$. Let Assumption as:boundederror hold. Let $\epsilon$ be such that $\epsilon < 2\nu/L^2$. Th where $c(\epsilon) := (1- 2 \epsilon \nu + \epsilon^2 L^2)^\frac{1}{2} < 1$ and $B := 2 X + |\pi -

Figures (8)

  • Figure 1: Diagram illustrating the steps of Algorithm \ref{['alg:inpo']}. This is a scenario where $p_0(t)$ represents a measurement of the power at the point of common coupling.
  • Figure 2: IEEE 37-bus feeder. Bus 0 represents the grid substation.
  • Figure 3: Average power exchanged with the external network.
  • Figure 4: Average cost of operating the power grid during the DRE, computed according to \ref{['eq:opt_probl_x']}.
  • Figure 5: Distance between the price vector $\mathbf{x}(t)$ and the optimal prices $\mathbf{x}^*$. The distance increases with larger estimation errors affecting $\hat{\boldsymbol{\beta}}$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1: Convergence with static nonflexible loads
  • Proposition 2: Tracking of prices with dynamic nonflexible loads