Recent Advances in Electrical and Electronic Engineering
Coupling of Three-Phase Sequence Circuits Due to Line and Load
Asymmetries
DIEGO BELLAN
Department of Electronics, Information and Bioengineering
Politecnico di Milano
Piazza Leonardo da Vinci, 32, 20133 Milano
ITALY
diego.bellan@polimi.it
Abstract: - In this paper a rigorous circuit representation is derived for the coupling between positive- and
negative-sequence circuits due to asymmetrical loads and/or lines in three-phase systems. It is shown that each
asymmetrical impedance results in a two-port network connecting the two circuits. An effective circuit
representation is derived in the paper for such two-port networks, based on the definition of an ideal
transformer with complex turns ratio. The proposed equivalent circuit is useful to get deeper insight into the
fundamental mechanisms generating negative-sequence voltages/currents from system asymmetries. Moreover,
the proposed circuit representation can be implemented into software for circuit simulation of complex systems.
Key-Words: -power system analysis; symmetrical components transformation; asymmetrical three-phase loads
show a different behavior due to the effects of
unpredictable parasitic elements.
Since sequence circuits are widely used in the
analysis of three-phase systems, in this work the
impact of asymmetries on sequence circuits is
investigated. It is shown that each impedance
asymmetry results in coupling between positive- and
negative-sequence circuits which can be effectively
represented by an ideal transformer with complex
turns ratio. The equivalent circuit derived in the
paper can be useful to get deeper insight into the
fundamental mechanism of asymmetry effects, and
for the implementation in software for circuit
simulation of complex systems.
1 Introduction
Three-phase power systems under steady state
conditions are usually analyzed in the frequency
(phasor) domain by means of the symmetrical
components transformation [1]. Other similar
transformations are available, operating in the
frequency or in the time domain [2]-[3]. The main
assumption underlying such methods is the
symmetry of the three-phase loads and/or the line,
i.e., the three impedances must be equal and with
the same coupling coefficient (if any). The reason is
clearly due to the fact that, under such assumption,
the transformations mentioned above decouple the
modal circuits (e.g., the positive, negative, and zero
sequence circuits in the symmetrical components
transformation), allowing a much easier and direct
analysis. If the assumption of symmetrical
load/lineis not met, the transformations provide
coupled modal circuits, leading to a much more
complicated analysis. To the Author’s knowledge,
no much work has been done to evaluate the impact
of the load/line asymmetry on modal circuits [4]-[6].
This is an interesting issue because asymmetry can
be a feature of modern power systems for several
reasons. First, geometric arrangement of nearby
power systems or components can result in
asymmetrical behavior of lines and/or loads.
Second, the analysis of power systems under
distorted conditions is of increasing importance. At
harmonic frequencies, each electric component can
ISBN: 978-960-474-399-5
2 Background
Three-phase power systems with symmetrical loads
can be conveniently analyzed by resorting to the
well-known
symmetrical
components
transformation. Indeed, even in the case of mutual
coupling between the phases, the assumption of
symmetrical system results in three uncoupled
sequence circuits. Solving each sequence circuit is
much simpler than solving the system as a whole.
The transformation matrix, in its rational form, is
defined as
1 α
1 
2
S=
1 α
3
1 1
127
α2 

α
1 
(1)
Recent Advances in Electrical and Electronic Engineering
sequence circuits when the transformation is applied
to the whole three-phase system.
where
α=e
2
j π
3
=−
1
3
+ j
2
2
3 Asymmetrical Line/Load
(2)
Let us consider an asymmetrical three-phaseload
consisting in three uncoupled impedances
,
,
taking values possibly different with respect
and
to a nominal value Z.Notice that such model can be
actually used to represent either a load or an
asymmetrical line resulting from a non-ideal
transposition of the conductors. We assume,
therefore
and
. The transformation matrix is a
.
Hermitian matrix, i.e.,
The symmetrical components transformation
when applied to phasor voltages provides
V+ 
Va 
V  = S ⋅ V 
 −
 b
V0 
Vc 
(3)
(9a)
(9b)
where , , and
are the positive, negative, and
zero sequence voltages. Of course, the same
transformation applies to the currents.
Symmetrical three-phase loads can be described
in terms of an impedance matrix with the following
structure
Z
Z = Z m

 Z m
Zm
Z
Zm
Zm 
Zm 

Z 
(9c)
where the relative deviations
,
, and
are
complex quantities where both the real and the
imaginary parts can take positive or negative values.
The asymmetrical impedance matrix can be
written
(4)
By defining the column vectors
V+ 
I + 
Va 
I a 






Vs = V− , I s = I − , V = Vb , I =  I b 
 
 
 
 
V0 
 I 0 
Vc 
 I c 
(5)
the transformed current/voltage relationship for a
symmetrical load can be written
Vs = SZS −1I s = Z s I s
(6)
Z +
Zs =  0

 0
(7)
(10)
By applying the transformation (6) to (10), the
nominal impedance matrix Z remains unchanged
) since its structure is the same as (4)
(i.e.,
with out-of-diagonal elements equal to zero.
Therefore, only the transformed version ofmatrices
,
, and
must be analyzed. After some
algebra, the following expressions can be obtained:
where
0
Z−
0
0
0

Z 0 
and
(11)
Z+ = Z− = Z − Zm
(8a)
Z 0 = Z + 2Z m
(8b)
(12)
The diagonal form of the sequence impedance
matrix (7) leads to the above-mentioned uncoupled
ISBN: 978-960-474-399-5
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Recent Advances in Electrical and Electronic Engineering
representation, however, can be derived by resorting
to the definition of a new component consisting in
an ideal transformer with complex turns ratio (see
Appendix).
(13)
In this work, the analysis is performed for power
systems with three wires. Therefore, the zerosequence circuit corresponding to the zero-sequence
voltage and current in (6) is not defined. Thus, by
taking into account (11)-(13), the positive- and
negative-sequence components of the sequence
provide the following
impedance matrix
relationship between positive- and negativesequence variables:
I−
I+
zs
δsa
V−
V+
δsb
(14)
From a circuit point of view, the four additive
impedance terms in (14)
δsc
(15a)
Fig. 1. Series connection of two-port networks representing
nominal impedances and coupling due to impedance
asymmetries.
(15b)
I+
(15c)
I−
Va+
Zδa/3
Va−
(15d)
Fig. 2. Two-port network representing asymmetry in Za.
correspond to a series connection of four two-port
networks(see Fig. 1) whose explicit circuit
representations will be provided in the following
Section.
α2
I+
Vb+ Zδb/3
4 Circuit Representation of
Asymmetries
Vb−
Fig. 3. Two-port network representing asymmetry in Zb.
In this Section the circuit representation of the four
two-port networks (15a)-(15d) are provided.
The two-port network corresponding to (15a) is
simply a pair of uncoupled impedances Z, i.e., the
nominal impedance of the load.
The circuit representation of (15b) consists in an
connected in parallel with the
impedance
two ports (see Fig. 2).
The two-port networks (15c) and (15d) are
similar each other. Notice that since the two
matrices in (15c) and (15d) are not symmetrical, the
equivalent circuit cannot be given in terms of
reciprocal
components
only.
A
simple
ISBN: 978-960-474-399-5
I−
α
I+
Vc+ Zδc/3
I−
Vc−
Fig. 4. Two-port network representing asymmetry in Zc.
In fact, for the two-port network (15c) it can be
easily shown that the equivalent circuit (see Fig. 3)
consists in an ideal transformer with complex turns
ratio given by
, and an impedance
129
Recent Advances in Electrical and Electronic Engineering
connected in parallel to one of the two ports (by
reporting the impedance through the ideal
transformer its value does not change since the
).
scaling coefficient is
In this work a rigorous circuit representation of
coupling between positive- and negative-sequence
circuits due to asymmetries in three-phase line
and/or loads has been derived. Simple equivalent
circuits have been obtained thanks to the definition
of the ideal transformer with complex turns ratio.
The proposed equivalent circuit allows deeper
insight into the analysis of the effects of three-phase
asymmetries. Moreover, the proposed equivalent
circuit is suited for implementation into the software
for circuit simulation. Future work will be devoted
to numerical quantification of asymmetries effects
and to the analysis of specific cases of practical
interest.
Ι−
I+
Z
Z
Appendix
Zδa/3
V+
An ideal transformer with complex turns ratio kis
defined by (passive sign convention at the two
ports):
α2
V−
(A1)
(A2)
Zδb/3
Notice the complex conjugate of k in(A2). Such
definition leads to conservation of complex power:
α
(A3)
Moreover, the scaling coefficient for reporting
impedances from the secondary to the primary side
is
. In fact, an impedance
on the secondary
side results in a primary side equivalent impedance
Zδc/3
Fig. 5. Complete two-port network representation of coupling
between positive- and negative-sequence circuits due to
asymmetries in impedances.
(A4)
References:
[1] C. L. Fortescue, "Method of symmetrical
coordinates applied to the solution of polyphase
networks," Trans. AIEE, 1918, pp. 1027-1140.
[2] G. C. Paap, "Symmetrical components in the
time domain and their application to power
network calculations," IEEE Trans. on Power
Systems, vol. 15, no. 2, May 2000, pp. 522-528.
[3] G. Chicco, P. Postolache, and C. Toader,
"Analysis of three-phase systems with neutral
under distorted and unbalanced conditions in
the symmetrical component-based framework,"
IEEE Trans. on Power Delivery, vol. 22, no.1,
Jan. 2007, pp. 674-683.
[4] P. Paranavithana, S. Perera, R. Koch, and Z.
Emin, "Global voltage unbalance in MV
networks due to line asymmetries," IEEE
Trans. on Power Delivery, Oct. 2009, pp. 23532360.
For the two-port network (15d) the circuit
representation is similar to (15c), where the ideal
transformer has complex turns ratio instead of ,
and the impedance, of course, is
(see Fig. 4).
The complete circuit representation of coupling
between positive- and negative-sequence circuits is
shown in Fig. 5. It is worth noticing that in many
cases of practical interest only one impedance
deviates significantly from its nominal value (e.g.,
in an untransposed line one conductor is
geometrically asymmetrical with respect to the other
two conductors). In such cases, the circuit coupling
is due only to the contribution of the asymmetrical
impedance, whereas the other two result in shortcircuited two-port networks in Fig. 5.
5 Conclusion
ISBN: 978-960-474-399-5
130
Recent Advances in Electrical and Electronic Engineering
[5] Z. Emin and D. S. Crisford, "Negative phasesequence voltages on E&W transmission
system," IEEE Trans. on Power Delivery, vol.
21, no. 3, July 2006, pp. 1607-1612.
[6] “Electromagnetic
compatibility
(EMC)—
limits—Assessment of emissionlimits for the
connection of unbalanced installations to MV,
HVand EHV power systems,” IEC Tech. Rep.
61000-3-13, 2008.
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