Power Flow Geometry and Approximation

Journal Article · Fri Sep 19 00:00:00 EDT 2025 · IEEE Transactions on Power Systems

DOI:https://doi.org/10.1109/TPWRS.2025.3612220· OSTI ID:3011880

Goodwin, Ariel ^[1]; Maack, Jonathan ^[2]; Sigler, Devon ^[2]

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1. Cornell University
2. National Renewable Energy Lab., Golden, CO (United States)

The power flow equations are important in numerous power systems problems of practical interest which consider alternating current power flow (ACPF) physics. Perhaps the most well studied being the alternating current optimal power flow problem (ACOPF), seeking to optimize the operation of an electric power system. Due to their non-linearity, problems which include the power flow equations are typically challenging, particularly in optimization. Interestingly, the set of solutions to the power flow equations forms a smooth manifold. As a result, differential geometry can be used to describe and analyze this set of equations. This approach has proven effective in several engineering applications (e.g., solving ACOPF and analyzing the solution space boundary). Central to the success of this approach is an understanding of the power flow manifold's geometry. In this work, we develop the geometric and topological properties of this manifold using concepts from differential geometry. After demonstrating the convenience of this manifold's representation as a function's graph, computational methods are emphasized: we develop retractions, error bounds for linear approximation, and formulas for evaluating the Riemannian metric (including associated objects such as geodesics and the curvature tensor). Scalar curvature and the second fundamental form play a new role in quantifying the quality of linear approximations, like the popular direct current approximation. All functions are implemented in Julia and available in an online repository. Proofs are included for completeness.

• Research Organization: National Renewable Energy Laboratory (NREL), Golden, CO (United States)
• Sponsoring Organization: USDOE National Renewable Energy Laboratory (NREL), Laboratory Directed Research and Development (LDRD) Program
• DOE Contract Number: AC36-08GO28308
• OSTI ID: 3011880
• Report Number(s): NREL/JA-2C00-88682
• Journal Information: IEEE Transactions on Power Systems, Journal Name: IEEE Transactions on Power Systems
• Country of Publication: United States
• Language: English

Related Subjects

24 POWER TRANSMISSION AND DISTRIBUTION
Riemannian geometry
curvature
manifolds
optimization methods
power systems