Three-Phase Distortion Analysis based on Space-Vector Locus
Diagrams
DIEGO BELLAN
Department of Electronics, Information and Bioengineering,
Politecnico di Milano,
Piazza Leonardo da Vinci 32, 20133 Milan,
ITALY
Abstract: - This work deals with the use of the space vector concept to characterize the harmonic content of a
three-phase voltage/current. It is shown that the shape of the trajectory of the space vector on the complex plane
(i.e., the locus diagram) provides information about its harmonic content. In particular, it is shown that each
harmonic contributes to the locus diagram with a number of lobes depending on the relative angular frequency
between the harmonic and the fundamental component. To this aim, the different contributions of positive-
sequence and negative-sequence harmonics is explained and put into evidence with specific examples. The
expressions for the magnitude and phase of the space vector as functions of the harmonics are derived
analytically. Numerical examples are provided to show how the locus diagram can represent a three-phase
quantity with positive-sequence and negative-sequence harmonics.
Key-Words: - harmonic analysis, power quality, power system analysis, space vector, symmetrical component
transformation, three-phase variables.
Received: April 11, 2023. Revised: October 26, 2023. Accepted: November 25, 2023. Published: December 31, 2023.
1 Introduction
Power quality is an issue of paramount importance
in modern electrical systems. Many aspects, ranging
from voltage/current magnitude and frequency
variations, fluctuation, unbalance and harmonics
have been extensively investigated in the relevant
literature in the past decades, [1], [2], [3], [4], [5],
[6], [7], [8], [9], [10], [11], [12]. One of the most
interesting mathematical approaches for power
quality assessment, emerged in recent years, is the
use of the space vector concept in a three-phase
system, [13], [14], [15], [16], [17], [18], [19], [20],
[21], [22], [23]. A space vector is a time-domain
complex-valued function representing a set of three
phase variables (i.e., phase voltages or currents).
Under ideal sinusoidal conditions the trajectory of
the space vector on the complex plane (i.e., the
locus diagram of the space vector) is a circle. In
case of an event like a single-phase or a double-
phase fault the locus diagram takes an elliptical
shape. The geometrical characteristics of the space
vector ellipse allow the detection and classification
of the fault.
The use of the space-vector locus diagram for
the distortion (i.e., harmonic) analysis of a three-
phase system, however, is a new approach not yet
exploited in the relevant literature. In fact, to the
Author’s knowledge, the idea of locus diagrams for
harmonic analysis and power quality
characterization can be found only in one paper in a
very early stage, [24]. In that paper, the Authors
propose the general idea that each harmonic can be
identified by a specific shape of the locus diagram.
However, that idea was not furtherly investigated,
and no analytical details were provided.
In this work, the impact of harmonics on the
locus diagram of a space vector is investigated
analytically. In particular, by resorting to the series
expansion of space vectors, the analytical
expressions for the magnitude and the phase of the
space vector are derived for each harmonic
component. It is shown that the effect on the locus
diagram is not related only to the harmonic order,
but also to the positive or negative sequence of each
harmonic, according to the well-known Symmetrical
Component Transformation (SCT). Thus, the locus
diagram of a space vector can provide information
about the harmonic content in a very concise way.
This is a unique property because, unlike a phasor
diagram, a single locus diagram can fully represent
a three-phase quantity with harmonics. From this
viewpoint, the results derived in the paper provide
an original contribution to power quality analysis of
distorted three-phase variables.
The paper is organized as follows. In Section 2
the space vector definition and series expansion are
WSEAS TRANSACTIONS on POWER SYSTEMS
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E-ISSN: 2224-350X
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recalled. In Section 3 the expressions for the
magnitude and phase of the space vector are
analytically derived as functions of the harmonics of
the three-phase quantity represented by the space
vector. In Section 4 some numerical examples are
shown in order to show typical shapes of locus
diagrams for specific harmonics. Conclusions are
presented in Section 5.
2 Space-Vector Series Expansion
Let us consider the time-domain phase currents
at a specific point of a given three-phase
system (the following derivations hold in the case
where the phase voltages were
considered instead of the currents).
The Clarke transformation of the phase currents
is defined as, [25], [26]:
(1)
where the transformation matrix T is orthogonal,
i.e., , such that (1) is power invariant.
In the relevant literature it has been shown that
under the assumption of a symmetrical three-phase
system (i.e., a three-phase system consisting of
components with equal phases and equal coupling
between phases) the α and β variables defined
through the Clarke transformation (1) fulfil the same
equations. Thus, a new complex variable (i.e., the
space vector) combining the α and β variables can
be defined as:
(2)
where . Therefore, the space vector is
defined as a complex valued function whose real
and imaginary parts are given by the α and β
components, respectively, provided by the Clarke
transformation (1).
Under sinusoidal steady-state conditions it can
be shown that the space vector (2) can be written as:
(3)
where asterisk denotes complex conjugate, whereas
and are the positive- and negative-sequence
phasor components, respectively, according to the
well-known Symmetrical Component
Transformation (SCT) in the phasor domain, [27],
[28]:
(4)
where
.
Thus, under sinusoidal conditions, according to
(3) the space vector (2) is given by the sum of two
rotating phasors: the positive-sequence phasor
rotating with positive angular frequency , and
the complex conjugate of the negative-sequence
phasor rotating with negative angular frequency
. The corresponding trajectory of the space
vector on the complex plane is elliptical, with semi-
major and semi-minor axes given by and
, respectively.
Under distorted conditions, i.e., periodic non-
sinusoidal waveforms, the space vector (2) can be
expanded in series, [23]. By adopting the well-
known complex form of the Fourier series, we can
write:
(5)
where
(6)
Notice that, by generalizing (3), each frequency
component results in the sum of two
components in the Fourier series (5), i.e.,
(7)
where the first component is the positive-sequence
SCT component rotating with positive angular
frequency , whereas the second component is
the negative-sequence (complex conjugate) SCT
component rotating at negative angular frequency
.
Notice that, since the space vector is a complex
valued function, the two Fourier coefficients and
are not related by complex conjugation, i.e.,
.
Thus, the Fourier series expansion (5) provides
a double decomposition of the space vector. The
first is the conventional frequency decomposition in
harmonic frequencies . The second
decomposition provides, for each harmonic
component, the sequence decomposition in a
positive sequence component rotating with positive
angular frequency , and a negative-sequence
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component rotating with negative angular frequency
.
In the general case, all the frequency and
sequence components can be found in the series
expansion (5). In many applications, however, due
to the phase symmetry of the three-phase system,
the phase variables are characterized by the
same distortion with
time shift, i.e.,
, and
. It means
that the waveforms and are exact replicas
of the distorted waveform provided that a
translation is introduced.
Under such assumption, with simple algebra it
can be shown that the series expansion (5) can
include only the frequency components with
frequency index with
[29], [30]. Thus, according to the
double decomposition outlined above, the allowed
positive-sequence harmonics are given by the list
, whereas the allowed negative-
sequence harmonics are given by
.
3 Space-Vector Locus Diagrams
The space vector defined in (2) is a complex valued
function of time whose trajectory on the complex
plane, according to (5), can provide information
about the harmonic content and the sequence
decomposition. In this Section the main properties
of such trajectories, called locus diagrams, will be
investigated under the assumption of symmetrical
phase distortion introduced at the end of Section II.
The simplest case is the sinusoidal non-distorted
case (see (3)). Indeed, when only the fundamental
component is present it means that only the
positive-sequence component at
fundamental frequency is present. The
corresponding locus diagram of the space vector
rotating with positive angular frequency is a
circle with radius .
According to the assumption of symmetrical
phase distortion, the next harmonic component is
, i.e., a negative-sequence (complex conjugate)
phasor rotating with negative angular frequency
. The resulting space vector is the sum of the
two rotating phasors. Notice that, in terms of
relative rotation, the second harmonic phasor is
rotating with angular frequency with respect to
the fundamental frequency phasor. The two rotating
phasors and the resulting space vector are depicted
in Figure 1. For the sake of simplicity, we assumed
that the two phasors were aligned at . By using
the well-known cosine rule the magnitude of the
space vector can be readily evaluated:
(8)
By introducing normalization with respect to the
fundamental component we obtain:
(9)
where . Notice the term in (9)
due to the relative angular frequency between the
rotating phasors and . Indeed, as far as the next
harmonic is considered, i.e., the rotating positive-
sequence phasor , from Figure 2 and by using the
cosine rule we obtain a n expression similar to (9):
(10)
where .
The above results can be generalized to the
contribution of any harmonic. Indeed, by
considering that the allowed harmonic indices are
given by , the corresponding
normalized absolute value of the space vector is
given by:
(11)
Notice that since , the
same behavior is obtained for each positive and
negative value of m, i.e., for two rotating phasors
(one positive-sequence and one negative-sequence
phasor). Thus, the two phasors and (
) rotating at the same relative angular frequency
with respect to , the two phasors and
() rotating at the same relative angular
frequency with respect to , and so on.
Although the time behavior of the space vector
magnitude (9) and (10) is the same, from Figure 1
and Figure 2 it is apparent that the space vector
phase is different in the two cases.
From Figure 1, by using the sine rule:
(12)
we obtain:
WSEAS TRANSACTIONS on POWER SYSTEMS
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(13)
where is given by (9).
Similarly, for the case in Figure 2 we obtain:
(14)
Fig. 1: Rotating phasors and space vector in the case
of a negative-sequence second harmonic
Fig. 2: Rotating phasors and space vector in the case
of a positive-sequence fourth harmonic
where is given by (10).
Notice the negative sign in (13) related to the
phasor rotating with negative angular frequency,
whereas the positive sign in (14) related to the
phasor rotating with positive angular frequency.
Thus, in the general case we can write:
(15)
where provides the right sign
in the equation.
Finally, the space-vector locus diagram is the
diagram of the complex valued function with
magnitude and phase .
4 Numerical Validation and
Investigation
In this Section, (11) and (15) are used to draw locus
diagrams of space vectors under distorted
conditions. Typical shapes of locus diagrams will be
obtained and discussed.
As a first example, let us consider the following
phase currents including fundamental and second
harmonic components:
,
. Notice the large relative magnitude of
the second harmonic (i.e., 30%) to the aim of
highlighting its effects on the locus diagram. By
using (9) and (13) (i.e., the general results (11) and
(15) specialized to the second harmonic) the locus
diagram represented by the black line in Figure 3
was obtained. Notice the three lobes due to the
relative angular frequency between the
fundamental and the negative-sequence second
harmonic. Moreover, notice the sharp shape of the
three lobes due to the fact that the two rotating
phasors (Figure 1) rotate in the opposite direction.
The blue line in Figure 3 shows the case of the
fourth harmonic, i.e.,
,
. Also in this case we obtain three lobes
because the relative angular frequency between the
fundamental and the fourth harmonic is still .
However, notice that the shape of the lobes in this
case is round, due to the fact that the two rotating
phasors rotate in the same direction (Figure 2). The
red line in Figure 3 shows the case where both the
second and the fourth harmonics are present with
equal magnitude, i.e.,
,
. The resulting
locus diagram has an intermediate behavior between
the black and the blue lines.
The second example, represented in Figure 4, is
related to the case of fifth and seventh harmonics. In
particular, the black line in Figure 4 corresponds to
the case of fifth harmonic, i.e., it was assumed
,
. Notice that,
since in this case the relative angular frequency
between the fundamental and the negative-sequence
fifth harmonic is , the number of lobes is six.
Also in this case the sharp shape of the lobes can be
observed due to the fact that the two rotating
phasors rotate in the opposite direction. On the
contrary, when the seventh harmonic is considered
(blue line), i.e.,
,
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, the six lobes have round shape
because the two rotating phasors rotate in the same
direction. When both the fifth and the seventh
harmonics are present, i.e.,
,
, the
corresponding behavior (red line) is intermediate
between the black and the blue lines.
Figure 5 shows the case of fifth and seventh
harmonics with phases different from zero. In this
specific case, the phase of the fundamental was
assumed
, whereas for the fifth and the seventh
harmonics the phases were assumed
and
,
respectively. Notice that the locus diagrams show
same shapes and a simple rotation with respect to
Figure 4 due to the phases different from zero.
Figure 6 shows an example of mixed harmonics,
i.e., rotating phasors with different relative angular
frequency with respect to the fundamental. In this
case it was assumed
,
. The second harmonic is
responsible of three lobes as in Figure 3, whereas
the fifth harmonic is responsible of six lobes as in
Figure 4. Notice that the characteristic marks of
both the harmonics can be still observed in Figure 6.
Of course, in the more general case where many
harmonics are present, it is expected that the
characteristic shape of each harmonic cannot be
clearly identified. Thus, the analysis derived in this
work is mainly devoted to the case where the
frequency spectrum has one dominant harmonic.
5 Conclusion
The locus diagram of a space vector has been used
to provide a concise representation of a three-phase
variable with harmonics. It has been shown that the
number of lobes of the diagram is related to the
harmonic order, whereas the shape of the lobes is
related to the positive or negative sequence of the
harmonic. In fact, the number of lobes depends on
the relative angular frequency between the harmonic
and the fundamental components. Thus, for
example, it was made clear that a positive-sequence
fourth harmonic provides the same number of lobes
(i.e., three) as a negative-sequence second harmonic
because both space vectors rotate with angular
frequency equal to three times the angular frequency
of the fundamental component. Similar
considerations can be extended to higher order
harmonics.
Future work will be devoted to further
investigate the analytical properties of locus
diagrams, with particular focus on the case of
superposition of harmonics with similar magnitude.
Indeed, the results proposed in this paper hold in the
case of one dominant harmonic. However, in many
practical cases the concurrent contribution of several
significant harmonics must be evaluated.
Fig. 3: Locus diagram of the space vector in case of
second harmonic (black line), fourth harmonic (blue
line), and both the harmonics (red line)
Fig. 4: Locus diagram of the space vector in case of
fifth harmonic (black line), seventh harmonic (blue
line), and both the harmonics (red line)
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Fig. 5: Locus diagram of the space vector in case of
fifth harmonic (black line) and seventh harmonic
(blue line) with phases different from zero. A
change in the phase values results in diagram
rotation with respect to Figure 4
Fig. 6: Locus diagram of the space vector in case of
both second and fifth harmonics. The three lobes
corresponding to the second harmonic (Figure 3)
and the six lobes corresponding to the fifth
harmonic (Figure 4) can be still observed
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
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