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\begin{document}

\title{Frequency Generalization via Darboux Bivector and Electrical Curves in
Multi-Phase Power Systems}



\author[1]{Francisco G. Montoya }%
\author[2]{Jorge Ventura }%
\author[2]{Francisco
M. Arrabal-Campos }%
\author[2]{alfredo alcayde }%
\author[2]{Ahmad H. Eid }%
\affil[1]{Universidad de Almeria}%
\affil[2]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 28, 2024}


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\begin{abstract}
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
\$\textbackslash{}bm\{n\}\$ dimensions. By analyzing the generalized
Fr\selectlanguage{ngerman}énet-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.%
\end{abstract}\selectlanguage{ngerman}%



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