Formulating the geometric foundation of Clarke, Park, and FBD transformations by means of Clifford's geometric algebra

Several of the most fundamental transformations widely used by the power engineering community are strongly based on geometrical considerations. Clifford's geometric algebra (GA) is the natural language for describing concepts in Euclidean geometry. In this work, we show how Clarke, Park, and D...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2022-05, Vol.45 (8), p.4252-4277
Main Authors: Montoya, Francisco G., Eid, Ahmad H.
Format: Article
Language:English
Subjects:
Clifford algebra
Complex numbers
Euclidean geometry
Euclidean space
geometric algebra
Mathematical analysis
Matrix algebra
Orthogonality
power transformations
time domain power theory
Transformations (mathematics)
Vectors (mathematics)
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Summary:Several of the most fundamental transformations widely used by the power engineering community are strongly based on geometrical considerations. Clifford's geometric algebra (GA) is the natural language for describing concepts in Euclidean geometry. In this work, we show how Clarke, Park, and Depenbrock's FBD transformations can be derived by imposing orthogonality on the voltage and current vectors defined in a Euclidean space by using GA. This paper presents these transformations as spatial‐like rotations and projections by means of the use of special algebraic objects named GA rotors. We prove that there is no need to use complex numbers nor matrices to perform the mentioned transformations and to provide geometrical intuition for electrical quantities and their transformations. Furthermore, power properties can be described using GA terminology by means of the proposed geometric power. The manipulation of the geometric power allows a useful current decomposition for a variety of applications such active filtering, frequency estimation, or electrical machinery. We provide in this paper an alternative approach focused on power systems under the paradigm of GA.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.8038