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Prescribing Decision Conservativeness in Two-Stage Power Markets: A Distributionally Robust End-to-End Approach

Zhirui Liang, Qi Li, Anqi Liu, Yury Dvorkin

TL;DR

The paper addresses wind-integrated two-stage power markets by jointly calibrating wind forecast parameters and decision conservativeness within a distributionally robust, end-to-end framework. It introduces a convex Wasserstein DRO reformulation of the DR-OPF and uses the Envelope Theorem to enable differentiable training, including a distributed calibration variant for privacy. Its cost-oriented calibration optimizes forecast accuracy and reserve conservativeness to minimize total look-ahead and real-time costs, demonstrated on an IEEE 5-bus test system where calibrated $\Theta$ and $\epsilon$ adapt to forecast-error uncertainty. This approach promises improved operational efficiency and reliability in wind power integration by aligning uncertainty quantification with actual system costs.

Abstract

This paper presents an end-to-end framework for calibrating wind power forecast models to minimize operational costs in two-stage power markets, where the first stage involves a distributionally robust optimal power flow (DR-OPF) model. Unlike traditional methods that adjust forecast parameters and uncertainty quantification (UQ) separately, this framework jointly optimizes both the forecast model parameters and the decision conservativeness, which determines the size of the ambiguity set in the DR-OPF model. The framework aligns UQ with actual uncertainty realizations by directly optimizing downstream operational costs, a process referred to as cost-oriented calibration. The calibration is achieved using a gradient descent approach. To enable efficient differentiation, the DR-OPF problem is reformulated into a convex form, and the Envelope Theorem is leveraged to simplify gradient derivation in the two-stage setting. Additionally, the framework supports distributed implementation, enhancing data privacy and reducing computational overhead. By proactively calibrating forecast parameters and prescribing optimal decision conservativeness, the framework significantly enhances cost efficiency and reliability in power system operations. Numerical experiments on an IEEE 5-bus system demonstrate the effectiveness and efficiency of the proposed approach.

Prescribing Decision Conservativeness in Two-Stage Power Markets: A Distributionally Robust End-to-End Approach

TL;DR

The paper addresses wind-integrated two-stage power markets by jointly calibrating wind forecast parameters and decision conservativeness within a distributionally robust, end-to-end framework. It introduces a convex Wasserstein DRO reformulation of the DR-OPF and uses the Envelope Theorem to enable differentiable training, including a distributed calibration variant for privacy. Its cost-oriented calibration optimizes forecast accuracy and reserve conservativeness to minimize total look-ahead and real-time costs, demonstrated on an IEEE 5-bus test system where calibrated and adapt to forecast-error uncertainty. This approach promises improved operational efficiency and reliability in wind power integration by aligning uncertainty quantification with actual system costs.

Abstract

This paper presents an end-to-end framework for calibrating wind power forecast models to minimize operational costs in two-stage power markets, where the first stage involves a distributionally robust optimal power flow (DR-OPF) model. Unlike traditional methods that adjust forecast parameters and uncertainty quantification (UQ) separately, this framework jointly optimizes both the forecast model parameters and the decision conservativeness, which determines the size of the ambiguity set in the DR-OPF model. The framework aligns UQ with actual uncertainty realizations by directly optimizing downstream operational costs, a process referred to as cost-oriented calibration. The calibration is achieved using a gradient descent approach. To enable efficient differentiation, the DR-OPF problem is reformulated into a convex form, and the Envelope Theorem is leveraged to simplify gradient derivation in the two-stage setting. Additionally, the framework supports distributed implementation, enhancing data privacy and reducing computational overhead. By proactively calibrating forecast parameters and prescribing optimal decision conservativeness, the framework significantly enhances cost efficiency and reliability in power system operations. Numerical experiments on an IEEE 5-bus system demonstrate the effectiveness and efficiency of the proposed approach.

Paper Structure

This paper contains 13 sections, 1 theorem, 19 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Operations of Two-Stage Power Markets
  3. Wind Power Forecasting and Uncertainty Quantification
  4. Reserve Procurement in Look-Ahead Scheduling
  5. Reserve Activation in Real-Time Dispatch
  6. Cost-Oriented Forecast Model Calibration
  7. Overview of the Proposed Framework
  8. Objective Function Construction
  9. Gradient Descent Approach
  10. Distributed Forecast Model Calibration
  11. Numerical Experiments
  12. Conclusion and Future Work
  13. Envelop Theorem

Key Result

Theorem 1

Consider the following parametric optimization problem, where $f(x, \alpha)$ and $g_j(x, \alpha), j=1,...,m$ are real-valued functions that are continuously differentiable with respect to both $x$ and $\alpha$: Let $x^*(\alpha)$ and $\lambda_j(\alpha)$ be the optimal primal and dual solutions of def:: Envelop Theorem opt and define the value function as $V(\alpha) = f(x^{*}(\alpha), \alpha)$. The

Figures (3)

  • Figure 1: End-to-End Forecast Model Calibration Framework.
  • Figure 2: Forecast Model Calibration in a Distributed Manner.
  • Figure 3: Convergence Analysis of Loss Functions for Different Weights $\eta$

Theorems & Definitions (1)

  • Theorem 1