Frequency Generalization via Darboux Bivector and Electrical Curves in Multi-Phase Power Systems

This paper investigates the concept of frequency in arbitrary multi-phase systems based on geometrical principles. The proposed approach relies on state-of-the-art mathematical techniques such as differential geometry and geometric algebra in $\bm{n}$ dimensions. By analyzing the generalized Frénet-Serret frame, we derive how the Darboux bivector can accurately express the rotation of this frame as a rigid body in space. It is shown how the concept of frequency in power grids can be intimately linked to spatial rotations. New insights are presented based on the comparison with other recently published works. It is also concluded that the application to single-phase systems cannot always be accommodated by spatial curves. Several examples are used to illustrate the findings of this paper.