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

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.