General Reference Frame Identification and Transformation in Unbalanced Power Systems

Abstract

Coordinate transformations provide dimensional reduction benefits across power system analysis, electric machine modeling, and power electronic converter control. This paper introduces a novel transformation based on Geometric Algebra that directly identifies the plane containing unbalanced quantity loci through bivector analysis. The method provides a direct transformation valid for any degree of unbalance in $n$-phase, $(n+1)$-wire sinusoidal systems, requiring only two voltage or current measurements at different time instants. Through pure geometric reasoning, we demonstrate that our approach generalizes existing techniques while extending naturally to multi-dimensional systems. Experimental validation using real-time digital simulation and physical laboratory testing confirms the method's effectiveness under realistic conditions. Power electronics converter control implementation demonstrates significant practical advantages, eliminating zero component oscillations present in Clarke transformation under unbalanced conditions and enabling more effective control architectures. The combination of computational efficiency, robustness, and practical applicability represents a significant advancement for power system control applications.