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Learning with Adaptive Conservativeness for Distributionally Robust Optimization: Incentive Design for Voltage Regulation

Zhirui Liang, Qi Li, Joshua Comden, Andrey Bernstein, Yury Dvorkin

TL;DR

The paper tackles incentive design for voltage regulation under information asymmetry between the DSO and DERAs by formulating a Stackelberg game and introducing a model-based online learning approach to infer the incentive–DERA response mapping. It advances a distributionally robust incentive design that uses a Wasserstein- metric ambiguity set and yields a convex reformulation, enabling tractable optimization. A gradient-based method adaptively tunes the conservativeness level based on historical performance, supported by convergence analysis and numerical experiments. The work provides a principled framework for robust, learning-enabled incentives in distribution networks, with practical implications for reliable voltage support under uncertainty.

Abstract

Information asymmetry between the Distribution System Operator (DSO) and Distributed Energy Resource Aggregators (DERAs) obstructs designing effective incentives for voltage regulation. To capture this effect, we employ a Stackelberg game-theoretic framework, where the DSO seeks to overcome the information asymmetry and refine its incentive strategies by learning from DERA behavior over multiple iterations. We introduce a model-based online learning algorithm for the DSO, aimed at inferring the relationship between incentives and DERA responses. Given the uncertain nature of these responses, we also propose a distributionally robust incentive design model to control the probability of voltage regulation failure and then reformulate it into a convex problem. This model allows the DSO to periodically revise distribution assumptions on uncertain parameters in the decision model of the DERA. Finally, we present a gradient-based method that permits the DSO to adaptively modify its conservativeness level, measured by the size of a Wasserstein metric-based ambiguity set, according to historical voltage regulation performance. The effectiveness of our proposed method is demonstrated through numerical experiments.

Learning with Adaptive Conservativeness for Distributionally Robust Optimization: Incentive Design for Voltage Regulation

TL;DR

The paper tackles incentive design for voltage regulation under information asymmetry between the DSO and DERAs by formulating a Stackelberg game and introducing a model-based online learning approach to infer the incentive–DERA response mapping. It advances a distributionally robust incentive design that uses a Wasserstein- metric ambiguity set and yields a convex reformulation, enabling tractable optimization. A gradient-based method adaptively tunes the conservativeness level based on historical performance, supported by convergence analysis and numerical experiments. The work provides a principled framework for robust, learning-enabled incentives in distribution networks, with practical implications for reliable voltage support under uncertainty.

Abstract

Information asymmetry between the Distribution System Operator (DSO) and Distributed Energy Resource Aggregators (DERAs) obstructs designing effective incentives for voltage regulation. To capture this effect, we employ a Stackelberg game-theoretic framework, where the DSO seeks to overcome the information asymmetry and refine its incentive strategies by learning from DERA behavior over multiple iterations. We introduce a model-based online learning algorithm for the DSO, aimed at inferring the relationship between incentives and DERA responses. Given the uncertain nature of these responses, we also propose a distributionally robust incentive design model to control the probability of voltage regulation failure and then reformulate it into a convex problem. This model allows the DSO to periodically revise distribution assumptions on uncertain parameters in the decision model of the DERA. Finally, we present a gradient-based method that permits the DSO to adaptively modify its conservativeness level, measured by the size of a Wasserstein metric-based ambiguity set, according to historical voltage regulation performance. The effectiveness of our proposed method is demonstrated through numerical experiments.
Paper Structure (19 sections, 2 theorems, 24 equations, 3 figures, 1 algorithm)

This paper contains 19 sections, 2 theorems, 24 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of Incentive Design Problem
  3. Incentive-based Voltage Regulation
  4. Leader's Problem: Optimal Incentive Design
  5. Follower's Problem: DER Redispatch
  6. Learning for Incentive Design
  7. Learning with Surrogate Model
  8. Properties of Surrogate Model
  9. Distributionally Robust Incentive Design
  10. Distributionally Robust Incentive Design Model
  11. Wasserstein Metric-based Ambiguity Set
  12. Convex Formulation of Distributionally Robust Model
  13. Conservativeness Update in Incentive Design
  14. Adequacy of Current Conservativeness Level
  15. Conservativeness Update Algorithm
  16. ...and 4 more sections

Key Result

Proposition 1

(Convergence of $\epsilon_T$) For each iteration $T \le T^\text{out}$, Algorithm alg:adaptive will return a near optimal value of $\epsilon_T$, under the condition that $count^{\max}$ is adequately large, and both the learning rate $\chi$ and the convergence threshold $\Delta \epsilon^{\min}$ are se

Figures (3)

  • Figure 1: Incentive-based voltage regulation modeled as a Stackelberg game between the DSO (leader) and DERAs (followers).
  • Figure 2: Distributionally robust incentive design for voltage regulation with learning and adaptive conservativeness level.
  • Figure 3: Convergence trajectory of optimal conservativeness level.

Theorems & Definitions (6)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1