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QCQP-Net: Reliably Learning Feasible Alternating Current Optimal Power Flow Solutions Under Constraints

Sihan Zeng, Youngdae Kim, Yuxuan Ren, Kibaek Kim

TL;DR

This work tackles the challenge of reliably solving the ACOPF problem in real time by formulating ACOPF as a non-convex $QCQP$ and introducing QCQP-Net, which maps load inputs to ACOPF control variables with a neural network and derives state variables via a relaxed $PF$ activated by a differentiable QCQP. The authors develop a KKT-based framework to compute gradients through the QCQP activation, enabling a bi-level loss that penalizes constraint violations while preserving data fidelity. They demonstrate differentiable, closed-form gradient expressions (or subgradients) for the QCQP parameters, ensuring backpropagation even when feasibility is violated in the learned control. Empirical results on IEEE 30-, 118-, and 300-bus systems show high feasibility and low generation costs, including in scenarios with infeasible power-flow instances, where traditional learning-based methods struggle.

Abstract

At the heart of power system operations, alternating current optimal power flow (ACOPF) studies the generation of electric power in the most economical way under network-wide load requirement, and can be formulated as a highly structured non-convex quadratically constrained quadratic program (QCQP). Optimization-based solutions to ACOPF (such as ADMM or interior-point method), as the classic approach, require large amount of computation and cannot meet the need to repeatedly solve the problem as load requirement frequently changes. On the other hand, learning-based methods that directly predict the ACOPF solution given the load input incur little computational cost but often generates infeasible solutions (i.e. violate the constraints of ACOPF). In this work, we combine the best of both worlds -- we propose an innovated framework for learning ACOPF, where the input load is mapped to the ACOPF solution through a neural network in a computationally efficient and reliable manner. Key to our innovation is a specific-purpose "activation function" defined implicitly by a QCQP and a novel loss, which enforce constraint satisfaction. We show through numerical simulations that our proposed method achieves superior feasibility rate and generation cost in situations where the existing learning-based approaches fail.

QCQP-Net: Reliably Learning Feasible Alternating Current Optimal Power Flow Solutions Under Constraints

TL;DR

This work tackles the challenge of reliably solving the ACOPF problem in real time by formulating ACOPF as a non-convex and introducing QCQP-Net, which maps load inputs to ACOPF control variables with a neural network and derives state variables via a relaxed activated by a differentiable QCQP. The authors develop a KKT-based framework to compute gradients through the QCQP activation, enabling a bi-level loss that penalizes constraint violations while preserving data fidelity. They demonstrate differentiable, closed-form gradient expressions (or subgradients) for the QCQP parameters, ensuring backpropagation even when feasibility is violated in the learned control. Empirical results on IEEE 30-, 118-, and 300-bus systems show high feasibility and low generation costs, including in scenarios with infeasible power-flow instances, where traditional learning-based methods struggle.

Abstract

At the heart of power system operations, alternating current optimal power flow (ACOPF) studies the generation of electric power in the most economical way under network-wide load requirement, and can be formulated as a highly structured non-convex quadratically constrained quadratic program (QCQP). Optimization-based solutions to ACOPF (such as ADMM or interior-point method), as the classic approach, require large amount of computation and cannot meet the need to repeatedly solve the problem as load requirement frequently changes. On the other hand, learning-based methods that directly predict the ACOPF solution given the load input incur little computational cost but often generates infeasible solutions (i.e. violate the constraints of ACOPF). In this work, we combine the best of both worlds -- we propose an innovated framework for learning ACOPF, where the input load is mapped to the ACOPF solution through a neural network in a computationally efficient and reliable manner. Key to our innovation is a specific-purpose "activation function" defined implicitly by a QCQP and a novel loss, which enforce constraint satisfaction. We show through numerical simulations that our proposed method achieves superior feasibility rate and generation cost in situations where the existing learning-based approaches fail.
Paper Structure (14 sections, 1 theorem, 24 equations, 3 figures, 1 table)

This paper contains 14 sections, 1 theorem, 24 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Works
  4. ACOPF Formulation
  5. A Constrained Machine Learning Approach to ACOPF
  6. ML Approach for Control Prediction
  7. Infeasible Power Flow System
  8. Differentiable QCQP
  9. Numerical Experiments
  10. Experiment Settings
  11. Small Test Cases
  12. Large Test Case with Infeasible Power Flow
  13. Concluding Remarks
  14. Appendix - Detailed Derivation of Results in Section \ref{['sec:differentiable-qcqp']} & Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

Suppose that strict complementary slackness, linear constraint qualification and second-order sufficient conditions hold at $(x^*,\nu^*,\lambda^*)$. Then, the matrix $M$ is invertible and $\ell$ is a differentiable function of the QCQP parameters. Given $\frac{\partial \ell}{\partial z^*}$, we have where $d_z\in\mathbb{R}^{k}, d_{\nu}\in\mathbb{R}^{m_I}, d_{\lambda}\in\mathbb{R}^{m_E}$ are the so

Figures (3)

  • Figure 1: QCQP-Net Architecture. Computation path in red only taken in training phase.
  • Figure 2: Performances on small grid networks (IEEE 30- and 118-bus systems)
  • Figure 3: Training and testing performances on the IEEE 300-bus network

Theorems & Definitions (1)

  • Theorem 1