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Distributionally Robust Joint Chance-Constrained Optimal Power Flow using Relative Entropy

Eli Brock, Haixiang Zhang, Javad Lavaei, Somayeh Sojoudi

TL;DR

The paper addresses robust joint chance-constrained optimal power flow (CCOPF) under uncertainty from variable renewable energy. It introduces a distributionally robust optimization framework using a relative-entropy ambiguity set to obtain an exact reformulation of joint chance constraints, along with strong out-of-sample guarantees and a least-conservative optimality property among all robust solutions. The approach is instantiated for both DC and AC power-flow models and demonstrated on IEEE 14- and 300-bus systems, where it achieves competitive generation costs with improved reliability and faster computation relative to state-of-the-art methods. The work offers a scalable, parameter-free methodology that preserves feasibility guarantees and improves robustness for large-scale power systems facing forecast errors.

Abstract

Designing robust algorithms for the optimal power flow (OPF) problem is critical for the control of large-scale power systems under uncertainty. The chance-constrained OPF (CCOPF) problem provides a natural formulation of the trade-off between the operating cost and the constraint satisfaction rate. In this work, we propose a new data-driven algorithm for the CCOPF problem, based on distributionally robust optimization (DRO). \revise{We show that the proposed reformulation of the distributionally robust chance constraints is exact, whereas other approaches in the CCOPF literature rely on conservative approximations. We establish out-of-sample robustness guarantees for the distributionally robust solution and prove that the solution is the most efficient among all approaches enjoying the same guarantees.} We apply the proposed algorithm to the the CCOPF problem and compare the performance of our approach with existing methods using simulations on IEEE benchmark power systems.

Distributionally Robust Joint Chance-Constrained Optimal Power Flow using Relative Entropy

TL;DR

The paper addresses robust joint chance-constrained optimal power flow (CCOPF) under uncertainty from variable renewable energy. It introduces a distributionally robust optimization framework using a relative-entropy ambiguity set to obtain an exact reformulation of joint chance constraints, along with strong out-of-sample guarantees and a least-conservative optimality property among all robust solutions. The approach is instantiated for both DC and AC power-flow models and demonstrated on IEEE 14- and 300-bus systems, where it achieves competitive generation costs with improved reliability and faster computation relative to state-of-the-art methods. The work offers a scalable, parameter-free methodology that preserves feasibility guarantees and improves robustness for large-scale power systems facing forecast errors.

Abstract

Designing robust algorithms for the optimal power flow (OPF) problem is critical for the control of large-scale power systems under uncertainty. The chance-constrained OPF (CCOPF) problem provides a natural formulation of the trade-off between the operating cost and the constraint satisfaction rate. In this work, we propose a new data-driven algorithm for the CCOPF problem, based on distributionally robust optimization (DRO). \revise{We show that the proposed reformulation of the distributionally robust chance constraints is exact, whereas other approaches in the CCOPF literature rely on conservative approximations. We establish out-of-sample robustness guarantees for the distributionally robust solution and prove that the solution is the most efficient among all approaches enjoying the same guarantees.} We apply the proposed algorithm to the the CCOPF problem and compare the performance of our approach with existing methods using simulations on IEEE benchmark power systems.
Paper Structure (22 sections, 6 theorems, 72 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 6 theorems, 72 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Distributionally Robust Optimization Approach
  3. Chance-constrained Optimal Power Flow
  4. Nomenclature
  5. Formulation
  6. Approximate linear uncertainty response
  7. DC power flow formulation
  8. AC power flow formulation
  9. Fixed-point algorithm
  10. Convex relaxation
  11. Demonstration on IEEE Test Cases
  12. Tutorial
  13. Overview of benchmark methods
  14. Experimental setup
  15. Comparison with OPT-CVAR and M-OBA on 14-bus DC model
  16. ...and 7 more sections

Key Result

Theorem 1

For all $\epsilon\in(0,1]$ and $r > 0$, if $\mathbf{X}\in\mathbb{R}^d$ satisfies constraint (eqn:dr-cc), then it holds that where $\mathbb{P}_\infty$ is the probability measure of the sample path space of $\xi$ under distribution $\mathbb{P}_0$.

Figures (5)

  • Figure 1: The 100 historical samples with the two worst-case samples in red.
  • Figure 2: The output distribution of generator 2 under the proposed method compared to the zero-error dispatch.
  • Figure 3: Performance on the 14-bus DC network.
  • Figure 4: Performance on the 14-bus AC network.
  • Figure 5: Performance on 300-bus DC network.

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Remark 3
  • Corollary 4
  • Definition 1: Distributionally Robust Predictor
  • Lemma 5: Restatemeant of Lemma \ref{['lem:drpred']}
  • proof
  • ...and 5 more