Clarke Circuit Analysis for Space-Vector Ellipse Characterization of
Phase-to-Ground Faults in Three-Phase Radial Systems
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 technique for fault detection and classification in three-phase power
systems based on the elliptical trajectory of the voltage space vector on the complex plane. A new approach is
presented leading to the derivation of equivalent circuits directly in the Clarke domain where the space vector is
defined. A specific methodology is introduced to manage the asymmetrical behavior of single-phase and
double-phase faults. In particular, an in-depth analysis is presented for the single-phase-to-ground fault. The
proposed equivalent circuits allow straightforward derivation and interpretation of the voltage ellipse for fault
characterization. The analytical results are validated through the simulation of a three-phase radial system.
Key-Words: - Clarke transformation, fault analysis, power quality, power system analysis, space vectors,
voltage sags.
Received: February 21, 2023. Revised: November 29, 2023. Accepted: December 15, 2023. Published: December 31, 2023.
1 Introduction
Power quality is a major issue in modern power
systems. Extensive literature is available about
disturbance analysis and classification techniques,
[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11],
[12]. In recent years, one of the most interesting and
promising methodologies for disturbance detection
and classification is the approach based on the
analysis of space vector trajectories on the complex
plane, [13], [14], [15], [16], [17], [18], [19], [20],
[21], [22], [23]. Although the space vector technique
has been proposed for several kinds of disturbances,
the straightforward application of such technique is
in the detection and classification of voltage sags
due to fault conditions. Indeed, the basic principle of
the space vector methodology consists of the
observation that, under fault conditions, the voltage
space vector traces an elliptical trajectory on the
complex plane. The inclination angle of such an
ellipse allows the classification of the fault. Figure 1
shows the characteristic inclination angle of the
space vector ellipse for different kinds of single-
phase (Sa, Sb, Sc) and double-phase (Dab, Dbc,
Dac) faults, [23].
In the relevant literature the relationship
between the faulted three-phase system and the
inclination of the voltage space-vector ellipse is
obtained in two steps, [20], [22]. First, a simplified
three-phase faulted circuit is solved through
conventional techniques in the original abc domain
of voltage/current variables. Second, the Clarke
transformation is used to obtain analytical
expressions for the voltage space-vector
components on the complex plane. Such an
approach leads to correct results, but since the
circuits are solved in the original abc domain they
cannot provide direct insight into the space vector
components because the space vector is defined on
the Clarke transformed variables α and β. In other
terms, the conventional approach does not provide
equivalent circuits in the Clarke transformed
variables, thus the circuit interpretation of the space
vector trajectory is prevented.
In this paper, a circuit methodology directly
based on the Clarke transformed variables is
proposed. In particular, the methodology is derived
in detail for single-phase grounded faults.
Equivalent circuits are derived directly in the Clarke
variables αβ0 domain. A specific approach is
described to manage the asymmetrical behavior of a
single-phase fault. The proposed approach allows
the straightforward determination of the voltage
alpha and beta components leading to the definition
of the space vector. Thus, a direct link is established
between equivalent circuits and space vector
properties. Notice that this paper provides a
methodological contribution, whereas analytical
evaluations concerning the single-phase fault can be
equivalently performed through the conventional
approach proposed in [16].
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The paper is organized as follows. In Section II
the Clarke transformation and its conventional use
in the analysis of symmetrical circuits is recalled. In
Section III the use of the Clarke transformation is
extended to the case of an asymmetrical component
such as a single-phase fault. Specific equivalent
circuits in the Clarke domain are derived to take into
account such asymmetrical behavior. In Section IV
the proposed methodology is used to analyze fault
conditions in a simple radial system, and some
properties of the space vector ellipses are put in
relation with the derived equivalent circuits in the
Clarke domain. Finally, discussion and concluding
remarks are drawn in Section V.
Fig. 1: Inclination angle of the voltage space-vector
ellipse for different kinds of single-phase (Sa, Sb, Sc)
and double-phase (Dab, Dbc, Dac) faults
2 Topological Properties of Clarke
Transformed Three-Phase Circuits
Let us consider a column vector
of phase voltages across a three-phase component in
the time domain. The Clarke transformation of is
defined as, [24]:
(1)
Notice that the transformation matrix T is
defined in its rational form, i.e., with the coefficient
which guarantees the orthogonality property
. It can be readily shown that the
orthogonality property results in conservation of
power from the abc domain to the transformed
domain. This is a crucial point in view of deriving
coherent equivalent circuits in the domain.
When a basic and symmetrical three-phase
component is considered, the Clarke transformation
operates the diagonalization of the matrix
component. For example, by considering an
inductive symmetrical component (i.e., a component
with equal self-inductances , and equal mutual
inductances ), the Clarke transformation
provides:
(2)
Since , the two scalar equations in the
and variables can be combined in only one
equation:
(3)
where , and the voltage/current space
vectors and have been defined as complex-
valued time-domain functions. Similar derivations
can be readily obtained for resistive and capacitive
components.
As far as a three-phase sinusoidal voltage source
is considered, by using the Clarke transformation for
the source phase voltages we
can readily express the corresponding space vector
as:
(4)
where and are the phasors of the positive and
the negative sequence components according to the
Symmetrical Component Transformation (SCT)
operating in the phasor domain, [23]:
(5)
where
, the transformation matrix
S is in its rational form such that , and the
asterisk denotes complex conjugation.
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In the general case, the space vector defined in
(4) has an elliptical trajectory on the complex plane
characterized by the following semi-major axis ,
semi-minor axis , and inclination angle [23]:
(6)
(7)
(8)
In the special case of null negative-sequence
component, i.e., , the space vector trajectory
becomes a circle with radius .
In order to obtain a complete circuit
characterization in the domain, the connections
of the three-phase components terminals must be
considered. In particular, the star connection (or
wye connection) must be investigated as the most
common in three-phase power systems. Two kinds
of star connections can be recognized. First, a star
connection with star center not accessible. Second, a
star connection with accessible star center used to
interface a single-phase network.
2.1 Star Connection with Non-Accessible
Center
This kind of star connection is shown in Figure 2.
Voltages are defined with respect to the reference
(possibly fictitious) terminal of the whole three-
phase system characterized by null total current.
Thus, the star connection in Figure 2 can be treated
as a three-port network whose independent voltage-
current relationships can be written as:
(9)
From the Clarke transformation (1), when
voltages are considered we readily obtain:
(10)
i.e., the star connection with a non-accessible
center can be treated as a short circuit in both the
and domains. Moreover, when currents are
considered, from (1) and (9) we readily obtain
(11)
i.e., the star connection with a non-accessible
center can be treated as an open circuit in the 0
domain.
2.2 Star Connection with Accessible Center
This kind of star connection is depicted in Figure 3.
It is typically used to interconnect a three-phase
system (left side) to a single-phase network (right
side). In this case the number of ports is four,
therefore four independent voltage-current
relationships must be written:
(12)
As far as voltages are considered, from (1) and
(12) we readily obtain the same result as (10), i.e.,
the star connection with an accessible center can be
treated as a short circuit in both the and
domains. Moreover, for the voltage zero component,
we obtain:
(13)
As far as the currents are considered, from (1)
and (12) we obtain:
(14)
Fig. 2: Star connection with non-accessible center
Fig. 3: Star connection with accessible center
Fig. 4: Zero-component equivalent circuit of a star
connection with accessible center
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Therefore, from (13) and (14) we obtain that the
star connection with accessible center can be
represented as an ideal transformer (Figure 4) with
turn ratio between the zero components of the
three-phase system (primary side) and the single-
phase network (secondary side). This equivalent
representation is useful in applications since all the
well-known properties of an ideal transformer can
be exploited.
3 Asymmetrical Faults in the Clarke
Domain
An asymmetrical fault in a three-phase system can
be modeled as a three-phase switch where only one
or two switches are operated. A simplified scheme
is represented in Figure 5 where a three-phase
switch is connected to a generic three-phase system
(possibly including single-phase networks).
Since the three-phase switch is operated
asymmetrically, however, the general methodology
outlined in Section II cannot be used in a
straightforward way. A modified methodology can
be introduced by resorting to the well-known
Thevenin theorem, [24]. Indeed, when the three-
phase switch is detached, the remaining circuit is a
conventional symmetrical three-phase system that
can be treated with the standard technique derived in
Section II. Thus, three Thevenin equivalents can be
derived in the , and 0 domains (Figure 6).
On the other hand, the three-phase switch
(Figure 7) can be treated by using the Clarke
transformation on the switch variables by setting the
constraints of the specific fault under analysis. For
example, a faulted phase a (i.e., shorted switch a)
can be described by the constraints
. Similarly for the other single-phase or
double-phase faults. Such constraints will result in
equivalent constraints in the ,,0 variables, leading
to corresponding interconnections of the ,,0
Thevenin equivalents.
The proposed modified methodology is detailed
in the next Subsection for single-phase faults.
Fig. 5: Three-phase switch connected to a generic
three-phase system
(a)
(b)
(c)
Fig. 6: Thevenin equivalents ( and 0) of the
three-phase system connected to the three-phase
switch
Fig. 7: Three-phase switch and related variables
3.1 Single-phase Faults
A faulted phase a can be modeled by setting the
constraints in the three-phase
switch represented in Figure 7. As far as the
voltages are considered, by setting in the
Clarke transformation (1) we obtain:
(15)
From (15) we obtain the following relationship
between Clarke voltages:
(16)
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As far as the currents are considered, by setting
in the Clarke transformation we obtain:
(17)
From (17) we obtain the following relationships
for the Clarke variables:
(18)
(19)
Thus, from (16) and (18) we obtain that the fault
constraints can be represented as an ideal
transformer with turn ratio
between the and
0 circuits. Moreover, from (19) we obtain that the
circuit must be set as an open circuit. Thus, the
single-phase fault (line a) can be represented and
analyzed through the equivalent circuits depicted in
Figure 8. Similar results can be obtained in the cases
where the faulted phase is b or c instead of a.
Indeed, a change in the faulted phase results in a
shift in the phase of the space vectors.
(a)
(b)
Fig. 8: Equivalent circuits in the Clarke domain
taking into account the single-phase fault on phase a
4 Validation for a Radial System
Let us consider the simplified radial system
represented in Figure 9. At the Point of Common
Coupling (PCC) the source is represented by the
equivalent voltage sources and the
equivalent impedances . The source is grounded
through the impedance . The line is characterized
by the impedance , and the fault location is
such that the line left side has impedance
whereas the right side has the impedance
, where . The load is characterized
by the impedance . According to the methodology
conventionally used to study grounded faults,
however, the load current in the following analysis
is neglected, i.e., we assume . The grounded
fault is characterized by the fault impedance .
As far as the single-phase fault (phase a) to
ground is considered, according to the results
derived in Section III the equivalent circuits in the
Clarke domain are shown in Figure 10. In particular,
Figure 10a shows the and 0 circuits coupled
through an ideal transformer with turn ratio
.
On the other hand, Figure 10b shows the circuit
which is uncoupled to the other circuits and
unloaded (i.e., open circuit). The objective of the
analysis is the evaluation of the and voltages at
the PCC. To this aim, the phasor voltages and
in Figure 10a and Figure 10b can be readily
calculated:
(20)
(21)
where
,
, and
is the phasor of the phase voltage source a in Figure
9. It can be readily shown that (20)-(21) are
equivalent to the analytical results derived in [16]
through a different methodology based on the circuit
solution in the natural abc domain.
The sine waves and corresponding
to the phasors (20)-(21) allow the definition of the
voltage space vector
(22)
whose trajectory on the complex plane is an
ellipse. According to Figure 1, the inclination of the
ellipse in case of faulted phase a to ground should
be close to .
To assess this point, the following data are
assumed to simulate the radial system in Figure 9:
, ,
, line length = 5
km, =0. The location x of the faulted phase
a was treated as a parameter ranging from 0.1 to 1.
The corresponding elliptical trajectories of the space
vector (22) are represented in Figure 11. Figure 12
shows the same space vectors by assuming 15 km
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instead of 5 km for the line length. In both the
figures it can be noticed that the ellipse inclination
is very close to . In Figure 12, however, the
spread of the elliptical shapes is much larger as a
consequence of a longer line. Moreover, notice that
the changes in Figure 11 and Figure 12 are along the
real axes (i.e., the real part of the space vector),
whereas along the imaginary axes (i.e., the
imaginary part of the space vector) the behavior
appears insensitive to the line impedance changes.
This is in agreement with (20)-(21) from which it is
clear that only the component is affected by the
circuit impedances, whereas the component is not
affected by the circuit impedances because the
circuit is open.
Fig. 9: Radial system used to validate the proposed
approach
(a)
(b)
Fig. 10: Equivalent circuits in the Clarke domain for
the radial system in Figure 9 in case of grounded
fault of phase a
Figure 13 shows the effect of ground and fault
impedances, i.e., and were
assumed in the case of line length 5 km. By
comparing Figure 13 and Figure 11 a slight change
in the ellipse inclination can be noticed as the effect
of ground and fault impedances.
Finally, it can be observed that if the faulted
phase was b instead of a, the related space vector
was simply obtained by rotating by 120° the space
vector of the faulted phase a, i.e.,
.
Similarly, if the faulted phase was c, the related
space vector was obtained by rotating by 120° the
space vector of the faulted phase a, i.e.,
. As a result, the corresponding ellipse
inclinations approach 210° and 30°, respectively,
according to Figure 1. The cases of faulted phase b
and c are represented in Figure 14 by assuming line
length 5 km, and .
Fig. 11: Trajectory of the voltage space vector for
faulted phase a and different locations of the fault.
The assumed line length is 5 km. Ground and fault
impedances are zero
Fig. 12: Trajectory of the voltage space vector for
faulted phase a and different locations of the fault.
The assumed line length is 15 km
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Fig. 13: Trajectory of the voltage space vector for
faulted phase a and different locations of the fault.
The assumed line length is 5 km. Ground and fault
impedances are different from zero
Fig. 14: Trajectory of the voltage space vectors for
faulted phases b and c, and different locations of the
fault. The assumed line length is 5 km
5 Conclusion
The single-phase to ground fault in a three-phase
system has been investigated by introducing
equivalent circuits directly in the Clarke domain.
Since the derived equivalent circuits allow the direct
evaluation of the α and β components of the space
vector, the proposed approach allows
straightforward interpretation of the properties of
the space vector ellipse used to detect and classify
the fault.
In particular, the main findings of the paper can
be summarized as follows:
Equivalent circuits have been derived directly in
the transformed Clarke variables α, β, 0.
As a result of the constraints representing the
single-phase fault, the α and 0 equivalent
circuits are coupled through an ideal
transformer. Notice that such a result was
obtained thanks to the adopted assumption of
Clarke transformation in its power invariant
form.
As far as the β circuit is considered, the fault
constraints result in an open circuit. Thus, the β
component of the space vector is not involved in
the fault.
The proposed methodology has been derived in
detail for the case of a single-phase-to-ground fault.
Future work will be devoted to extend the results to
double-phase faults.
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Contribution of Individual Authors to the
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The authors equally contributed in the present
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problem to the final findings and solution.
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Conflict of Interest
The authors have no conflicts of interest to declare.
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