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On the Data-Driven Modeling of Price-Responsive Flexible Loads: Formulation and Algorithm

Mingji Chen, Shuai Lu, Wei Gu, Zhaoyang Dong, Yijun Xu, Jiayi Ding

TL;DR

This paper addresses the challenge of accurately identifying price-responsive flexible loads (PRFLs) by developing a data-driven, noise-aware inverse-optimization framework. It formulates forward static and dynamic response models for PRFL aggregates, extends them with inner costs, and embeds them into a bilevel inverse problem to estimate the aggregate-model parameters under noisy observations. A Bayesian optimization-based algorithm with a Gaussian-process surrogate and a block-Cholesky acceleration is proposed, delivering scalability to large samples and providing posterior identifiability certificates. Theoretical noise analysis (Theorem 1) and numerical results demonstrate improved accuracy over Newton-based methods and highlight the importance of price signaling and data diversity for robust identifiability. The approach holds promise for more reliable integration of PRFLs into power-system operations and market mechanisms, enabling better demand flexibility management under uncertainty.

Abstract

The flexible loads in power systems, such as interruptible and transferable loads, are critical flexibility resources for mitigating power imbalances. Despite their potential, accurate modeling of these loads is a challenging work and has not received enough attention, limiting their integration into operational frameworks. To bridge this gap, this paper develops a data-driven identification theory and algorithm for price-responsive flexible loads (PRFLs). First, we introduce PRFL models that capture both static and dynamic decision mechanisms governing their response to electricity price variations. Second, We develop a data-driven identification framework that explicitly incorporates forecast and measurement errors. Particularly, we give a theoretical analysis to quantify the statistical impact of such noise on parameter estimation. Third, leveraging the bilevel structure of the identification problem, we propose a Bayesian optimization-based algorithm that features the scalability to large sample sizes and the ability to offer posterior differentiability certificates as byproducts. Numerical tests demonstrate the effectiveness and superiority of the proposed approach.

On the Data-Driven Modeling of Price-Responsive Flexible Loads: Formulation and Algorithm

TL;DR

This paper addresses the challenge of accurately identifying price-responsive flexible loads (PRFLs) by developing a data-driven, noise-aware inverse-optimization framework. It formulates forward static and dynamic response models for PRFL aggregates, extends them with inner costs, and embeds them into a bilevel inverse problem to estimate the aggregate-model parameters under noisy observations. A Bayesian optimization-based algorithm with a Gaussian-process surrogate and a block-Cholesky acceleration is proposed, delivering scalability to large samples and providing posterior identifiability certificates. Theoretical noise analysis (Theorem 1) and numerical results demonstrate improved accuracy over Newton-based methods and highlight the importance of price signaling and data diversity for robust identifiability. The approach holds promise for more reliable integration of PRFLs into power-system operations and market mechanisms, enabling better demand flexibility management under uncertainty.

Abstract

The flexible loads in power systems, such as interruptible and transferable loads, are critical flexibility resources for mitigating power imbalances. Despite their potential, accurate modeling of these loads is a challenging work and has not received enough attention, limiting their integration into operational frameworks. To bridge this gap, this paper develops a data-driven identification theory and algorithm for price-responsive flexible loads (PRFLs). First, we introduce PRFL models that capture both static and dynamic decision mechanisms governing their response to electricity price variations. Second, We develop a data-driven identification framework that explicitly incorporates forecast and measurement errors. Particularly, we give a theoretical analysis to quantify the statistical impact of such noise on parameter estimation. Third, leveraging the bilevel structure of the identification problem, we propose a Bayesian optimization-based algorithm that features the scalability to large sample sizes and the ability to offer posterior differentiability certificates as byproducts. Numerical tests demonstrate the effectiveness and superiority of the proposed approach.

Paper Structure

This paper contains 26 sections, 2 theorems, 59 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Descriptions
  3. Physical Model of PRFL
  4. Components of PRFLs
  5. Aggregate model of PRFLs
  6. Identification Problem of PRFLs
  7. Forward Problem: Response Model
  8. Static Response Model
  9. Dynamic Response Model
  10. Extending to Integrating Inner Costs
  11. Inverse Problem: Identification Model
  12. Identification Model
  13. Impact of Noise
  14. Bayesian Identification Algorithm
  15. Solution Method
  16. ...and 11 more sections

Key Result

Theorem 1

If the $l^2$-norm is used for $f_{nd}(\theta)$ in model obj_noise and $\forall n\in \mathbf{N}, \varepsilon^n_{agg}- \varepsilon_{fix}^n\sim\mathcal{N}\left(0,\Sigma_P\right) (\Sigma_P\in\mathbb{R}^{T\times T}$ is the covariance matrix), we have: wherein the ${\hat{\theta}}_{nd}$ and ${\hat{\theta}}_{nf}$ represent the optima for models obj_noise and obj_nf respectively, and $\mathrm{tr}(\cdot)$

Figures (9)

  • Figure 1: Data-driven modeling framework of PRFLs.
  • Figure 2: Algorithm framework based on Bayesian loop.
  • Figure 3: Performance of static model with noise data.
  • Figure 4: Performance of extended static model with noise data.
  • Figure 5: NRMSE under different number of energy storages: (a) (Extended) Static case; (b) (Extended) Dynamic case.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • proof
  • proof