Clarke Transformation Solution of Asymmetrical Transients in Three-Phase Circuits

Abstract

This work deals with the use of the Clarke transformation for the theoretical derivation of circuit models for the analysis of asymmetrical transients in three-phase circuits. Asymmetrical transients occur when only one or two phases of a three-phase power system are involved in a switch operation. Such a condition is critical from a theoretical viewpoint since the Clarke transformation is based on the assumption of circuit symmetry between the three phases. If the symmetry assumption is not met, the equivalent circuits in the transformed variables α, β, and 0 are not uncoupled. The literature concerning numerical approaches for asymmetrical transient analysis is very rich, but a comprehensive and rigorous analytical investigation of circuit models within the framework of the Clarke transformation is still lacking. Contrary to numerical approaches, analytical solutions provide deeper insight into the phenomenon and allow for theoretical interpretation and better understanding of the transient behavior. The proposed circuit models show that the β variables are always uncoupled with α and 0 variables, whereas coupling between α and 0 variables can be properly represented by an ideal transformer. Moreover, in the case of single-line switching, the β variables have no transient, i.e., they keep the steady-state behavior. Transient properties can be readily and effectively observed by representing the trajectory of the space vector on the complex plane. All the analytical results have been checked numerically through the Simulink (Matlab R2020a, The MathWorks, Inc., Natick, MA, USA) implementation of a specific three-phase circuit introduced to illustrate the main theoretical issues.

Differential Analysis of Fault Currents in a Power Distribution Feeder Using abc, αβ0, and dq0 Reference Frames

“…According to [45], this technique can be applied in situations where the microgrid topology is variable. Clarke transform is used to find a methodology for the analytical solution of asymmetric transients [46]. This requires specific theoretical work, as the direct application of the Clarke transform does not produce the diagonalization of the system matrices.…”

Space-Vector Transient Analysis of Delta-Connected ThreePhase Loads

“…Let us consider a set of phase voltages in the time domain. The Clarke transformation of the phase voltages is defined as [19,20,21,22,23,24,25,26]:…”

“…In this paper, a further theoretical contribution is presented to extend the Author's results concerning a general approach for the transient analysis of three-phase networks based on the Clarke transformation and space vectors [23,24,25,26]. In particular, in this work the special case of delta-connected three-phase loads is considered.…”

Hardware-in-the-Loop Implementation and Performance Evaluation of Three-Phase Hybrid Shunt Active Power Filter for Power Quality Improvement

“…(3) Applications of Clarke transformation: Clarke transformation assumes symmetry in a three-phase circuit, so Clarke's simple application does not produce matrices' diagonalization. In Bellan's study [28], a time-domain analysis due to line switching in a three-phase system is presented based on mathematical modeling. Clarke transformation is applied across each triplet of phase voltages and currents that deal with the asymmetry in three-phase systems.…”