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Generalized Vector Locus Transformation for Unbalanced Three-Phase Systems

Maitraya Avadhut Desai, Francisco Escobar, Gabriela Hug

TL;DR

The paper addresses unbalanced three-phase four-wire systems where classical Clarke and $dq0$ transformations fail to maintain a null $0$-axis and constant $dq0$ signals. It introduces a generalized vector locus transformation that reparametrizes the three-phase signals as two unit-amplitude sinusoids in quadrature using a basis aligned with the elliptical space-vector locus, followed by a standard $dq0$ transform to yield constant $d$ and $q$ components. In the balanced case, this transformation reduces to a scaled Clarke transform, ensuring backward compatibility. Numerical validation on unbalanced signals demonstrates unit-amplitude, orthogonal $v_1$ and $v_2$ with $v_3=0$, and constant $v_d,v_q$ after $dq0$, illustrating practical viability for real-time three-phase control.

Abstract

Coordinate transformations significantly simplify power systems computations. Most notably, the classical Clarke and dq0 transformations are widely used in three-phase systems, as together they transform balanced abc quantities into constant-valued signals. However, during unbalanced operation, the utility of these transformations diminishes, since a null 0 coordinate cannot be ensured and oscillating signals emerge. While recently proposed transformations ensure a null 0 coordinate, they still do not lead to constant-valued signals in the dq0 domain. In this letter, we propose a generalized vector locus transformation that ensures both a null 0 coordinate and constant-valued signals. Moreover, we show that, in the balanced case, the classical amplitude-invariant Clarke transformation is an instance of the proposed transformation.

Generalized Vector Locus Transformation for Unbalanced Three-Phase Systems

TL;DR

The paper addresses unbalanced three-phase four-wire systems where classical Clarke and transformations fail to maintain a null -axis and constant signals. It introduces a generalized vector locus transformation that reparametrizes the three-phase signals as two unit-amplitude sinusoids in quadrature using a basis aligned with the elliptical space-vector locus, followed by a standard transform to yield constant and components. In the balanced case, this transformation reduces to a scaled Clarke transform, ensuring backward compatibility. Numerical validation on unbalanced signals demonstrates unit-amplitude, orthogonal and with , and constant after , illustrating practical viability for real-time three-phase control.

Abstract

Coordinate transformations significantly simplify power systems computations. Most notably, the classical Clarke and dq0 transformations are widely used in three-phase systems, as together they transform balanced abc quantities into constant-valued signals. However, during unbalanced operation, the utility of these transformations diminishes, since a null 0 coordinate cannot be ensured and oscillating signals emerge. While recently proposed transformations ensure a null 0 coordinate, they still do not lead to constant-valued signals in the dq0 domain. In this letter, we propose a generalized vector locus transformation that ensures both a null 0 coordinate and constant-valued signals. Moreover, we show that, in the balanced case, the classical amplitude-invariant Clarke transformation is an instance of the proposed transformation.
Paper Structure (4 sections, 9 equations, 3 figures)

This paper contains 4 sections, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Vector Locus Transformation
  3. Numerical Validation
  4. Conclusion

Figures (3)

  • Figure 1: Clarke and $dq0$ transformations applied to three-phase quantities. (a) Coordinates of $\mathbf{v}_{}$ in $abc$ during balanced and unbalanced operation. (b) Corresponding loci of $\mathbf{v}_{}$ in $\mathbb{R}$ and basis vectors of both $abc$ and $\alpha\beta0$. (c) Coordinates of $\mathbf{v}_{}$ in $\alpha\beta0$. (d) Coordinates of $\mathbf{v}_{}$ in $dq0$ for axes that rotate at the (synchronous) electrical frequency.
  • Figure 2: Special choices of the basis vectors, found by measuring $\mathbf{v}_{}$ when $v_{ a}$ reaches its peak ($\diamond$) or $\left\lVert \mathbf{v}_{} \right\rVert$ is maximized ($\star$). The vector $\mathbf{e}_{3}$, being orthogonal to the ellipse and having length $\sqrt 3$, is independent of these choices.
  • Figure 3: Proposed transformation applied to an unbalanced three-phase system. (a) Coordinates of $\mathbf{v}_{}$ in $abc$. (b) Choices of the basis vectors. (c) Corresponding coordinates of $\mathbf{v}_{}$ under the new transformation. (d) Coordinates of $\mathbf{v}_{}$ in $dq0$ for axes that rotate at the (synchronous) electrical frequency.