Abstract
This study presents an advanced framework for harmonic analysis in three-phase systems based on differential geometry principles. The proposed method employs the Frenet frame to geometrically represent voltage and current waveforms as spatial curves, establishing a consistent foundation for power computation under unbalanced and dynamically changing conditions. Compared to conventional approaches such as Clarke and Park transforms, this framework demonstrates improved accuracy in handling asymmetrical operating scenarios. The methodology is validated through extensive simulation using ERPI-DOE test signals, confirming its robustness and computational efficiency. The outcomes highlight the potential of the proposed approach to enhance power quality assessment and strengthen control strategies in modern power converter systems.
Data availability
The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.
Abbreviations
- r(t):
-
Position vector
- v(t):
-
Velocity vector
- a(t):
-
Acceleration vector
- J(t):
-
Jerk vector
- s(t):
-
Arc length
- r′(t):
-
First derivative of r(t)
- r′′(t):
-
Second derivative r(t)
- r′′′(t):
-
Third derivative of r(t)
- T:
-
Tangent vector of Frenet frame
- N:
-
Normal vector of Frenet frame
- B:
-
Binormal vector of Frenet frame
- \(\:\dot{p}\) :
-
velocity vector
- ω:
-
Angular velocity
- κ:
-
The curvature of a curve
- τ:
-
Torsion of a curve
- \(\:v\kappa\:\) :
-
Rate of Binormal vector
- \(\:v{\uptau\:}\) :
-
Speed of Tangent vector
- κv :
-
Curvature associated with voltage
- κi :
-
Curvature associated with current
- τv :
-
Torque associated with voltage
- PQ:
-
Power quality
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Dr. Nitin Sundriyal: Data curation, Conceptualization, Investigation, validation and Writing of the original draft.Dr. Padmanabh Thakur: Supervision, Conceptualization, Visualization, Data curation.Mr. Ashutosh Dixit: Visualization, Data curation, Validation.Dr. Sandeep Gupta: Data curation, Formal analysis, Validation.Dr. Mukesh Kumar: Visualization, Formal analysis.
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Sundriyal, N., Thakur, P., Dixit, A. et al. Differential geometry-based harmonic analysis of three-phase systems. Sci Rep (2026). https://doi.org/10.1038/s41598-026-40101-9
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DOI: https://doi.org/10.1038/s41598-026-40101-9
