[Paper Review] A Learning-boosted Quasi-Newton Method for AC Optimal Power Flow

Kyri Baker|arXiv (Cornell University)|Jul 12, 2020

Power System Optimization and Stability19 references16 citations

TL;DR

This paper proposes a learning-boosted Quasi-Newton method for AC Optimal Power Flow (OPF) that replaces Newton-Raphson’s Jacobian inversion with a deep neural network to accelerate convergence. The model learns iterative descent directions from training data, enabling fast, stable solutions without computing Jacobians, and achieves solutions up to 22.5x faster than state-of-the-art solvers on large systems like 1,354-bus networks, with low optimality gaps despite moderate feasibility errors.

ABSTRACT

Power grid operators typically solve large-scale, nonconvex optimal power flow (OPF) problems throughout the day to determine optimal setpoints for generators while adhering to physical constraints. Despite being at the heart of many OPF solvers, Newton-Raphson can be slow and numerically unstable. To reduce the computational burden associated with calculating the full Jacobian and its inverse, many Quasi-Newton methods attempt to find a solution to the optimality conditions by leveraging an approximate Jacobian matrix. In this paper, a Quasi-Newton method based on machine learning is presented which performs iterative updates for candidate optimal solutions without having to calculate a Jacobian or approximate Jacobian matrix. The proposed learning-based algorithm utilizes a deep neural network with feedback. With proper choice of weights and activation functions, the model becomes a contraction mapping and convergence can be guaranteed. Results shown for networks up to 1,354 buses indicate the proposed method is capable of finding approximate solutions to AC OPF very quickly.

Motivation & Objective

• To address the computational burden and numerical instability of Newton-Raphson methods in large-scale AC OPF due to expensive Jacobian calculations and ill-conditioned matrices.
• To develop a fast, real-time capable OPF solver that avoids matrix inversion while maintaining solution quality for large power systems.
• To leverage machine learning to learn descent directions in the optimization process, bypassing explicit Jacobian construction.
• To ensure convergence through contraction mapping properties by proper design of network weights and activation functions.
• To evaluate performance on realistic systems, including tracking dynamic load changes at one-second intervals.

Proposed method

• A fully connected three-layer feedforward neural network $F_R$ is trained to map the current candidate solution $\mathbf{x}^k$ to the next iterate $\mathbf{x}^{k+1}$, replacing the Newton-Raphson update step.
• The network is trained on data generated from Newton-Raphson iterations, learning to imitate the descent direction without computing the Jacobian or its inverse.
• The model uses feedback connections, functioning like a recurrent network, to iteratively refine the solution until convergence $||\mathbf{x}^{k+1} - \mathbf{x}^k|| \leq \epsilon$.
• Convergence is guaranteed if the network is designed as a contraction mapping via appropriate weight initialization and activation functions.
• Variable bounds are enforced via thresholding on the output layer, and feasibility is assessed post-inference.
• The method avoids ill-conditioned Jacobian issues by eliminating matrix inversion entirely, relying solely on learned dynamics.

Experimental results

Research questions

• RQ1 Can a deep neural network effectively learn and replace the Newton-Raphson update step in AC OPF without explicit Jacobian computation?
• RQ2 Does the proposed learning-boosted method achieve faster convergence than traditional solvers on large-scale AC OPF problems?
• RQ3 Can the model maintain solution quality (optimality gap) while significantly reducing computational time and avoiding numerical instability?
• RQ4 How well does the method track dynamic changes in system conditions, such as one-second load updates, compared to conventional solvers?
• RQ5 What is the trade-off between solution speed and feasibility, particularly for large systems like the 1,354-bus case?

Key findings

• The learning-boosted method achieved solutions up to 22.5x faster than the MIPS solver on the 1,354-bus system, with a mean optimality gap of 1.16% across test cases.
• For the 1,354-bus system, the mean constraint violation for power flow equations was 9.95 pu, indicating a need for more training data or increased model complexity.
• Despite being an approximation, the method tracked real-time load changes every second more accurately than MATPOWER, which required nearly three seconds per solution.
• The 500-bus system showed that the learning-boosted method had lower computational time variance, making it more reliable for regular interval deployment.
• The 30-bus system demonstrated that the method could produce solutions with lower mean constraint violations than MIPS in some cases, likely due to tighter training tolerance.
• Convergence was guaranteed in the proposed method due to the network's design as a contraction mapping, ensuring iterative stability.