ETEP zy On the Meaning of the Park Power Components z in Three-Phase Systems under Non-Sinusoidal Conditions A. Ferrero, A. P. Morando, R. Ottoboni, G. Superti-Furga Abstract zyxwvutsrqp The definition and measurement of the power components in three-phase systems under non-sinusoidal con- ditions is becoming more and more an up-to-date problem. Some theories have afready been proposed to face this problem, but the proposed solutions are not completely satisfactory, in particular when unsymmetrical, unbalanced systems are concerned. The study of three-phase systems in terms of the Park transformation seems to be more attractive than other proposed methods. The present paper, after a short review of the. properties of the Park transformation, discusses the physical meaning of the power components defined em- ploying it, proposes a measuring equipment for the measurement of the defined power components and shows the results of some experimental work. 1 Introduction zyxwvuts The problem of defining the power components in three-phase systems under non-sinusoidal conditions is still far from a satisfactory solution. Some attempts have been made to extend the definitions given for sin- gle-phase systems to three-phase systems. However, a great difficulty is found in extending these definitions to unsymmetrical and unbalanced systems. A further difficulty arises when trying to adopt the same defini- tions for three-wire and four-wire systems. Even Czar- necki's approach zyxwvutsrq [ 1,2], which is one of the most rigor- ous and formally correct ones, considers only three- phase systems with unbalanced load, but does not con- sider unsymmetrical supply voltages and does not state clearly whether the given definitions are valid for both three-wire and four-wire systems or not. Moreover, the extension of the power definitions given for single-phase systems to the three-phase ones leads to non-univocal definitions of the apparent power (and consequently of the power factor) as clearly re- ported by Filipski [3]. These problems can be overcome if the Park trans- formation is employed: it represents a powerful mathe- matical tool to describe the behaviour of three-phase systems in any possible working condition (unsym- metrical, unbalanced, non-sinusoidal, .. .). This approach was first introduced by Akagi zyxwvu [4] and then extended by the authors of zyxwvut [5, 61 who derived general validity power definitions in three-phase sys- tems under non-sinusoidal conditions. This paper will briefly recall the Park transforma- tion theory and the relationship between the Park com- ponents and the symmetrical components. Then the power definitions will be reconsidered in order to in- vestigate their meaning and the possibility of being used in power compensation. At last a general purpose measuring equipment for the measurement of the Park quantities will be proposed and the result of some experimental work will be dis- cussed. 2 Three-Phase System Representation in Terms of Park Components Let y,(t), Yb(t) and zyxw y,(t) represent the signals (volt- ages, currents, ...) of a three-phase system. If the Park transformation [7] is applied to these signals, the Park components yd(c) (direct component), y,(t) (quadrature component) and yo(') (zero-sequence component) are obtained: [TI being the orthogonal matrix: rm -I/& -l/&l L J The Park vector can be then defined in the d-q plane as the complex quantity: thus changing the original quantities y,(t). Yb(t) and y, (t) into the Park vector y (t) plus the zero-sequence component yo('), both depending on time t. The pro- posed transformation is a p d c u l a r case - sometimes referred to as the Clarke transformation-of a more ETEP Vol. 3, No. I, JanuarylFebruary 1993 33