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Mean-Field Learning for Storage Aggregation

Jingguan Liu, Cong Chen, Xiaomeng Ai, Jiakun Fang, Jinsong Wang, Jinyu Wen

TL;DR

This work introduces a mean-field learning framework to aggregate large, heterogeneous storage populations into a convex mean-field limit that is tractable for power-system operations and wholesale markets. By interpreting aggregation as the expectation of random sets, it proves the existence of a unique convex mean-field limit and bounds the MF approximation error using price–response data. A convex surrogate model is learned via a gradient-based inverse-optimization approach that fits historical price–response observations, yielding high data efficiency and accurate market-relevant behavior. Case studies show significant improvements in approximation accuracy and profit outcomes compared to model-based surrogates, supporting practical deployment in grid operations and market clearing.

Abstract

Distributed energy storage devices can be pooled and coordinated by aggregators to participate in power system operations and market clearings. This requires representing a massive device population as a single, tractable surrogate that is computationally efficient, accurate, and compatible with market participation requirements. However, surrogate identification is challenging due to heterogeneity, nonconvexity, and high dimensionality of storage devices. To address these challenges, this paper develops a mean-field learning framework for storage aggregation. We interpret aggregation as the average behavior of a large storage population and show that, as the population grows, aggregate performance converges to a unique, convex mean-field limit, enabling tractable population-level modeling. This convexity further yields a price-responsive characterization of aggregate storage behavior and allows us to bound the mean-field approximation error. Leveraging these results, we construct a convex surrogate model that approximates the aggregate behavior of large storage populations and can be embedded directly into power system operations and market clearing. Surrogate parameter identification is formulated as an optimization problem using historical market price-response data, and we adopt a gradient-based algorithm for efficient learning procedure. Case studies validate the theoretical findings and demonstrate the effectiveness of the proposed framework in approximation accuracy, data efficiency, and profit outcomes.

Mean-Field Learning for Storage Aggregation

TL;DR

This work introduces a mean-field learning framework to aggregate large, heterogeneous storage populations into a convex mean-field limit that is tractable for power-system operations and wholesale markets. By interpreting aggregation as the expectation of random sets, it proves the existence of a unique convex mean-field limit and bounds the MF approximation error using price–response data. A convex surrogate model is learned via a gradient-based inverse-optimization approach that fits historical price–response observations, yielding high data efficiency and accurate market-relevant behavior. Case studies show significant improvements in approximation accuracy and profit outcomes compared to model-based surrogates, supporting practical deployment in grid operations and market clearing.

Abstract

Distributed energy storage devices can be pooled and coordinated by aggregators to participate in power system operations and market clearings. This requires representing a massive device population as a single, tractable surrogate that is computationally efficient, accurate, and compatible with market participation requirements. However, surrogate identification is challenging due to heterogeneity, nonconvexity, and high dimensionality of storage devices. To address these challenges, this paper develops a mean-field learning framework for storage aggregation. We interpret aggregation as the average behavior of a large storage population and show that, as the population grows, aggregate performance converges to a unique, convex mean-field limit, enabling tractable population-level modeling. This convexity further yields a price-responsive characterization of aggregate storage behavior and allows us to bound the mean-field approximation error. Leveraging these results, we construct a convex surrogate model that approximates the aggregate behavior of large storage populations and can be embedded directly into power system operations and market clearing. Surrogate parameter identification is formulated as an optimization problem using historical market price-response data, and we adopt a gradient-based algorithm for efficient learning procedure. Case studies validate the theoretical findings and demonstrate the effectiveness of the proposed framework in approximation accuracy, data efficiency, and profit outcomes.
Paper Structure (29 sections, 4 theorems, 89 equations, 6 figures, 2 tables)

This paper contains 29 sections, 4 theorems, 89 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Main Contributions
  4. Paper Organization and Notations
  5. Problem Formulation
  6. Individual Formulation
  7. Aggregate Formulation
  8. Mean-Field Performance Analysis
  9. Preliminaries on Random Sets
  10. Existence and Convexity of Mean-Field Limits
  11. Mean-Field Approximation with Price-Response Bounds
  12. Mean-Field Learning Framework
  13. Framework Overview
  14. Surrogate Model for Mean-Field Limit Approximation
  15. Parameter Identification for Surrogate Model
  16. ...and 14 more sections

Key Result

Lemma 1

Let $\mathcal{X}_{i}\subset\mathbb{R}^n, \forall i \in {1,...,I}$ be i.i.d. random sets defined on a non-atomic probability space and assume that $\mathcal{X}_{i}$ is integrable, i.e., $\mathbb{E}\!\left[\|\mathcal{X}_{i}\|\right]<\infty$. Then where $d_{\mathrm H}$ denotes the Hausdorff distance, $\operatorname{Conv}(\cdot)$ denotes the convex hull operator, and $\bigoplus$ denotes the Minkowski

Figures (6)

  • Figure 1: Conceptual relations across the main sections.
  • Figure 2: Flowchart of the proposed learning framework.
  • Figure 3: Visualization of mean-field flexibility set.
  • Figure 4: Impact of price number and device number.
  • Figure 5: Learning results of different methods.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition 1: Random Set molchanov2005theory
  • Definition 2: Expectation of Random Set aumann1965integrals
  • Lemma 1: Strong Law of Large Numbers for Random Sets artstein1975strong
  • Proposition 1: Existence and Convexity of Mean-Field Limits
  • Proposition 2: Approximation Error Bound
  • Proposition 3: Consistence of Optimal Responses