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All-Pass Fractional OPF: A Solver-Friendly, Physics-Preserving Approximation of AC OPF

Milad Hasanzadeh, Amin Kargarian, Javad Lavaei

TL;DR

The paper tackles the nonconvex AC OPF problem caused by trigonometric coupling between bus voltages, which yields oscillatory derivatives that hinder interior-point methods. It proposes an all-pass fractional (APF) approximation that replaces cos(δ) and sin(δ) with smooth surrogates r_cos(Δ) and r_sin(Δ) derived from a first-order all-pass kernel r(δ) = (1 + j a δ)/(1 - j a δ), together with a pre-rotation based on a DCOPF/DC PF solution to center the approximation in the operating range. This APF preserves unit magnitude and the symmetry of the original kernel, improves the conditioning of the Jacobian and Hessian, and remains solver-friendly while maintaining physical loss behavior. Extensive experiments on more than 25 IEEE and PGLib networks, up to 10,480 buses, show near-identical objective values and feasible regions compared with classical AC OPF, while achieving substantial reductions in solver time and iteration counts; the code is publicly available for reproducibility.

Abstract

This paper presents a fractional approximation of the AC optimal power flow (AC OPF) problem based on an all-pass approximation of the exponential power flow kernel. The classical AC OPF relies on trigonometric coupling between bus voltage phasors, which yields a nonconvex program with oscillatory derivatives that can slow, or in some cases destabilize, interior-point methods. We replace the trigonometric terms with an all-pass fractional (APF) approximation whose real and imaginary components act as smooth surrogates for the cosine and sine functions, and we introduce a pre-rotation to shift the argument of the approximation toward its most accurate region, ensuring that the reformulated power flow model preserves physical loss behavior, maintains the symmetry of the classical kernels, and improves the conditioning of the Jacobian and Hessian matrices. The proposed APF OPF formulation remains nonconvex, as in the classical model, but it eliminates trigonometric evaluations and empirically produces larger and more stable Newton steps under standard interior-point solvers. Numerical results on more than 25 IEEE and PGLib test systems ranging from 9 to 10{,}000 buses demonstrate that the APF OPF model achieves solutions with accuracy comparable to that of the classical formulation while reducing solver times, indicating a more solver-friendly nonconvex representation of AC OPF. All code, functions, verification scripts, and generated results are publicly available on \href{https://github.com/LSU-RAISE-LAB/APF-OPF}{GitHub}, along with a README describing how to run and reproduce the experiments.

All-Pass Fractional OPF: A Solver-Friendly, Physics-Preserving Approximation of AC OPF

TL;DR

The paper tackles the nonconvex AC OPF problem caused by trigonometric coupling between bus voltages, which yields oscillatory derivatives that hinder interior-point methods. It proposes an all-pass fractional (APF) approximation that replaces cos(δ) and sin(δ) with smooth surrogates r_cos(Δ) and r_sin(Δ) derived from a first-order all-pass kernel r(δ) = (1 + j a δ)/(1 - j a δ), together with a pre-rotation based on a DCOPF/DC PF solution to center the approximation in the operating range. This APF preserves unit magnitude and the symmetry of the original kernel, improves the conditioning of the Jacobian and Hessian, and remains solver-friendly while maintaining physical loss behavior. Extensive experiments on more than 25 IEEE and PGLib networks, up to 10,480 buses, show near-identical objective values and feasible regions compared with classical AC OPF, while achieving substantial reductions in solver time and iteration counts; the code is publicly available for reproducibility.

Abstract

This paper presents a fractional approximation of the AC optimal power flow (AC OPF) problem based on an all-pass approximation of the exponential power flow kernel. The classical AC OPF relies on trigonometric coupling between bus voltage phasors, which yields a nonconvex program with oscillatory derivatives that can slow, or in some cases destabilize, interior-point methods. We replace the trigonometric terms with an all-pass fractional (APF) approximation whose real and imaginary components act as smooth surrogates for the cosine and sine functions, and we introduce a pre-rotation to shift the argument of the approximation toward its most accurate region, ensuring that the reformulated power flow model preserves physical loss behavior, maintains the symmetry of the classical kernels, and improves the conditioning of the Jacobian and Hessian matrices. The proposed APF OPF formulation remains nonconvex, as in the classical model, but it eliminates trigonometric evaluations and empirically produces larger and more stable Newton steps under standard interior-point solvers. Numerical results on more than 25 IEEE and PGLib test systems ranging from 9 to 10{,}000 buses demonstrate that the APF OPF model achieves solutions with accuracy comparable to that of the classical formulation while reducing solver times, indicating a more solver-friendly nonconvex representation of AC OPF. All code, functions, verification scripts, and generated results are publicly available on \href{https://github.com/LSU-RAISE-LAB/APF-OPF}{GitHub}, along with a README describing how to run and reproduce the experiments.
Paper Structure (41 sections, 39 equations, 5 figures, 2 tables)

This paper contains 41 sections, 39 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. AC OPF: Modeling and Interior-Point Methods
  3. AC OPF Model
  4. Interior-Point Solution
  5. Proposed APF OPF and Its Properties
  6. All-Pass Kernel Approximation
  7. Pre-Rotation
  8. APF OPF Model
  9. Properties of APF OPF Formulation
  10. Unit-Magnitude Property and Preservation of Power Losses
  11. Symmetry Preservation of Trigonometric Structure
  12. Role of APF Parameter and Numerical Smoothness
  13. Derivative Structure and Improved Conditioning
  14. Computational Efficiency of Fractional Evaluations
  15. Impact on Newton Steps and Solver Convergence
  16. ...and 26 more sections

Figures (5)

  • Figure 2: Comparison of the rate of variation of the trigonometric functions and their all-pass approximations over $[-180^\circ, 180^\circ]$ with no pre-rotation
  • Figure 3: Effect of applying different shifts to the argument of the all-pass kernels
  • Figure : (a) $\sin(\delta)$ vs $r_{\sin}(\delta)$
  • Figure : (a) $\sin(\delta)$ vs $r_{\sin}(\delta)$
  • Figure : (b) $\cos(\delta)$ vs $r_{\cos}(\delta)$