A Bezier Curve Based Approach to the Convexification of the AC Optimal Power Flow Problem

Abstract

The Alternating Current Optimal Power Flow (ACOPF) problem remains one of the most fundamental yet computationally challenging tasks in power systems operation and planning due to its nonconvex, nonlinear, and multimodal nature. This paper proposes a convex reformulation of the AC power flow problem by introducing auxiliary variables to isolate nonlinear terms, applying logarithmic transformations to exploit product-sum properties, and approximating with Bezier curves using a novel convexifying butterfly shaped function. This model is intended for assessing and operating weak power systems that face challenges with reactive power supply and overall network robustness. Its formulation closely mirrors the AC formulation, particularly regarding active and reactive power dispatch and network voltage levels. The proposed model achieves convergence on large test systems (e.g., IEEE 118 bus) in seconds and is validated against exact AC solutions. This convex formulation stands out not only for its mathematical transparency and intuitive structure but also for its ease of validation and implementation, making it an accessible and reliable tool for researchers and system operators for energy planning. The numerical analysis conducted on the IEEE 118 bus system yielded average percentage errors in the state variables specifically, the magnitudes and angles of nodal voltages of just 0.0008 percentage and 0.014 degree, respectively, when compared with the precise AC formulation. These results underscore the high accuracy and reliability of the proposed methodology.