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1
Analytical Nonlinear Reluctance Model of a Single
Phase Saturated Core Fault Current Limiter
Philip A. Commins, Jeffrey W. Moscrop Member, IEEE
Abstract—A saturated core Fault Current Limiter (FCL) is a
device that is designed to limit the fault currents in electrical
energy networks and consequently, protect existing network
equipment from damage. Due to complex nonlinear magnetic
properties, the performance of saturated core FCLs has largely
been characterised through experimentation and Finite Element
Analysis (FEA) simulations. Although both of these techniques
are quite accurate, they are time consuming and do not describe
the behaviour of FCLs in actual electrical networks. This has led
to increasing demand for an accurate analytical model that is
suitable for transient network analyses. This paper presents the
development of an analytical model of a single-phase open-core
FCL, which accurately describes the nonlinear magnetic prop-
erties of the FCL through a reduced reluctance approach. The
extension of this model to other saturated core FCL arrangements
(such as closed-core) is also discussed.
Index Terms—Analytical models, fault current limiters, mag-
netic circuits, magnetic flux, nonlinear magnetics, power system
protection
I. INTRODUCTION
T
HE increasing demands placed on modern power sys-
tems [1] have not only led to increased occurrences
of fault currents in networks, but also to increases in both
fault current levels and equipment sensitivity to fault currents.
Although new technologies and improvements in infrastructure
continue to emerge, Fault Current Limiting (FCL) devices
are becoming an increasingly necessary technology to pro-
tect existing infrastructure and improve network availability
for consumers. There are several different FCL technologies
that are currently attracting worldwide attention from both
researchers and commercial engineering companies. This paper
focuses on one particular technology, the saturated core FCL.
The saturated core FCL utilises the change in permeability
between saturated and unsaturated states of the core material
to provide both a low steady-state insertion impedance (so
as not to load the network during normal operation) and a
high transient impedance during fault conditions (for current
limiting). The major advantages of this technology are that
it provides instantaneous reaction to a fault event and instan-
taneous recovery. A DC current carrying winding is used to
initially drive the cores into saturation, with separate windings
The research presented in this paper was supported under the Australian
Research Council’s (ARC) Linkage Projects funding scheme (project number
LP100200461). Grateful acknowledgment is given to the ARC, along with the
project partner organisations Zenergy Power and Powerlink Queensland, for
their support.
P. A. Commins is with the Faculty of Engineering, University
of Wollongong, Wollongong, NSW 2522, AUSTRALIA (e-mail: pcom-
mins@uow.edu.au).
J. W. Moscrop is with the Faculty of Engineering, University of Wollon-
gong, Wollongong, NSW 2522, AUSTRALIA (e-mail: jeffm@uow.edu.au).
used to carry the AC load current of the circuit to be protected.
Under normal steady-state load conditions the AC current is
not large enough to drive the cores out of saturation, hence
the AC coils behave like standard air core reactors. However,
during a fault event the high AC fault currents drive the
cores out of saturation, resulting in a significant increase in
impedance that effectively limits the current. In a single-
phase device, two AC coils and cores are necessary to limit
both the positive and negative half cycles of the current. A
more thorough description of the principles of operation of a
saturated core FCL is given in [2].
As FCL technology becomes more viable for electrical
utilities, there is an increasing demand for accurate simula-
tions that can demonstrate the effects an FCL has on other
network equipment. Although there have already been several
experimental and Finite Element Analysis (FEA) studies on
the performance characteristics of saturated core FCLs [2]–
[5], network simulation packages (such as PSCAD and other
ElectroMagnetic Transient Programs) cannot easily be coupled
to an electromagnetic FEA simulator. Hence, there is growing
need for an accurate analytical model of the FCL, which
can be easily incorporated into transient network simulation
packages. The development of an accurate analytical model has
already received some attention [6], [7]; however, the models
developed to date do not include the AC to DC coupling
effects of the device. It has been found that a “transformer”
coupling effect exists between the AC and DC windings
during a fault [5], which can have a significant impact on the
performance of the FCL.
This paper presents the development of an analytical “reluc-
tance” model of a saturated core FCL that describes the full
nonlinear range of magnetic operation as well as the AC to
DC coupling effects of the device. The model is based on the
magnetic circuit concept [8], with the geometry of the FCL
represented by an equivalent magnetic circuit that includes all
significant flux paths. A simple single-phase air-core geometry
is initially examined, with the concept then extended to an
open-core arrangement (i.e. the AC coils enclose separate iron
cores that do not have a return path). Further extension of
the model to other arrangements, such as closed-core, is also
discussed. The model is validated against FEA simulations and
shown to provide an accurate representation of a saturated core
FCL, which with further development can be integrated with
transient network simulation packages.
II. A
IR-CORE EQUIVALENT MODEL
The initial development of the analytical reluctance model
was carried out on a simple single-phase air-core arrangement,
2
as shown in Fig. 1. This arrangement consists of two AC
current carrying windings (AC coils) that are placed side
by side and encompassed by a single DC current carrying
winding (DC coil). Although this arrangement is not useful
as a saturated core FCL (i.e. the AC coils are equivalent to
air-core reactors and the DC coil serves no practical purpose),
it is beneficial for model development as the significant flux
paths are identical to the iron-core cases yet the associated
reluctances are constant. Hence, the magnetic circuit described
in this section can easily be extended to include the nonlinear
reluctances associated with other iron-core arrangements (as
shown in Section III).
Fig. 1. Single-Phase Air-Core Arrangement
The significant flux paths (and associated reluctances) for
the arrangement of Fig. 1 are identified in Fig. 2, where
c
represents the reluctance of the flux paths inside the AC
coils (i.e. the air-cores),
i
represents the reluctance of the
remaining flux paths inside the DC coil (identical on each
side),
o
represents the reluctance of the flux paths outside the
DC coil (identical on each side),
y
represents the reluctance
of the flux paths between the two AC coils (top & bottom),
and
a
represents the reluctance of the flux paths that link
the inner AC loop with the other paths (top and bottom and
identical on each side). The equivalent magnetic circuit for
this arrangement is shown in Fig. 3, where NI
dc
, NI
ac1
and
NI
ac2
represent mmf due to the DC coil and two AC coils
respectively. Note also that the
y
and
a
reluctances can be
lumped in the equivalent circuit (as shown in Fig. 3).
One important feature of the magnetic circuit shown in
Fig. 3 is that it is symmetrical, with
a
,
i
,
o
and NI
dc
identical on the left and right sides of the circuit. In particular,
NI
dc
on each side represents the same physical mmf. The rea-
son that both sides are required is so that the flux due to each
AC coil is accurately described – although the circuit elements
are the same on each side, the flux in each corresponding path
is only the same when NI
ac1
= NI
ac2
(which is not the case
during normal operation). Hence, the total flux linking the DC
coil is the sum of the flux through
o
on the left side and the
flux through
o
on the right side.
The values of the reluctances shown in Fig. 3 can be calcu-
lated using standard circuit analysis techniques. In general, all
of the reluctances can be determined via flux measurements
(using either FEA or experimentation) under three different
o
i
c
c
i
o
a
y
a
a
y
a
Fig. 2. Equivalent Reluctance Paths
−
+
NI
dc
o
i
a
c
−
+
NI
ac1
y
c
−
+
NI
ac2
a
i
o
−
+
NI
dc
Fig. 3. Equivalent Magnetic Circuit
test conditions:
• Test 1: NI
dc
=0and NI
ac1
= NI
ac2
=0;
• Test 2: NI
dc
=0and NI
ac1
= NI
ac2
=0;
• Test 3: NI
dc
=0and NI
ac1
= −NI
ac2
=0.
Test 1 and Test 2 are used to determine the values of
i
and
o
, along with the series combination of
c
+
a
(=
ca
).
Test 3 is then used to determine the individual values of
c
and
a
, along with
y
.
A. Test 1
The only source of mmf in this test is NI
dc
, which results in
a completely balanced circuit. Hence, the flux paths between
the two AC coils are no longer significant and the equiva-
lent circuit reduces to two separate sides that have identical
reluctance and flux in each corresponding path (as shown in
Fig. 4). The nominal direction of each flux path is as indicated
in Fig. 4. The flux linkage of each coil is measured during the
test.
Through simple circuit analysis:
i
(φ
o1
− φ
c1
)=NI
dc
−
o
φ
o1
(1)
and
ca
=
i
(φ
o1
− φ
c1
)
φ
c1
(2)
Note also that φ
o1
is equal to one half of the measured flux
linkage of the DC coil and φ
c1
is equal to the measured flux
linkage of either AC coil.
3
−
+
NI
dc
o
φ
o1
i
φ
i1
a
c
φ
c1
c
φ
c1
a
i
φ
i1
o
φ
o1
−
+
NI
dc
Fig. 4. Equivalent Circuit for Test 1
B. Test 2
In this test the sources of mmf are restricted to the AC
coils, with NI
ac1
= NI
ac2
. The circuit is again completely
balanced, with the flux paths between the two AC coils no
longer significant. The equivalent circuit for the test is shown in
Fig. 5. The nominal direction of each flux path is as indicated
in Fig. 5. The flux linkage of each coil is again measured
during the test.
o
φ
o2
i
φ
i2
a
c
φ
c2
−
+
NI
ac1
−
+
NI
ac2
c
φ
c2
a
i
φ
i2
o
φ
o2
Fig. 5. Equivalent Circuit for Test 2
Through simple circuit analysis:
i
(φ
o2
− φ
c2
)=−
o
φ
o2
(3)
and
ca
=
i
(φ
o2
− φ
c2
)+NI
ac1
φ
c2
(4)
Again φ
o2
is equal to one half of the measured flux linkage
of the DC coil and φ
c2
is equal to the measured flux linkage
of either AC coil.
Solving (1) and (3) simultaneously results in:
i
=
NI
dc
φ
o1
− φ
c1
−
φ
o1
φ
o2
(φ
o2
− φ
c2
)
(5)
o
=
NI
dc
φ
o1
− φ
o2
φ
o1
−φ
c1
φ
o2
−φ
c2
(6)
Note that all of the elements in (5) and (6) are either
measured flux linkage values or an applied mmf. Hence,
i
and
o
can be completely determined from Tests 1 and 2. The
series reluctance
ca
can be calculated by substituting
i
from
(5) into either (2) or (4).
C. Test 3
In Test 3 the sources of mmf are again restricted to the
AC coils; however, this time with NI
ac1
= −NI
ac2
.For
this case the flux paths between the two AC coils, along with
the associated reluctance
y
, are significant. Hence, the flux
between the two AC coils must also be measured, along with
the flux linkage of each coil. In FEA the flux through
y
can
be determined by integrating the flux density across the entire
plane between the two coils. The equivalent circuit for this
test is shown in Fig. 6. The nominal direction of each flux
path is as indicated in Fig. 6. Note that the directions of the
corresponding flux paths (on each side of the circuit) are now
opposite to each other. This is important when considering
the flux linkage of the DC coil, as the net flux linkage will
be zero; however, the flux linking each side of the coil is
non-zero, but equal and opposite (i.e. |φ
o3l
| = |φ
o3r
| and
φ
dc
= φ
o3l
+ φ
o3r
=0).
o
φ
o3l
i
φ
i3
a
φ
a3
c
φ
c3
−
+
NI
ac1
y
y
φ
y
c
φ
c3
−
+
NI
ac2
a
φ
a3
i
φ
i3
o
φ
o3r
Fig. 6. Equivalent circuit for Test 3
To simplify the circuit of Fig. 6,
y
has been split into
two equal reluctances in series (each denoted as
y
). Since
the circuit is symmetrical, this allows for the two sides of the
circuit to be separated – this is possible because the mid point
of
y
is at the same potential as the common node between
NI
ac1
and NI
ac2
(as illustrated by the dashed line in Fig. 6).
Hence, the analysis can be confined to a single side of the
circuit (as was the case with Tests 1 and 2). A simplified
version of the left hand side of Fig. 6 is shown in Fig. 7.
a
+
p
φ
a3
y
φ
y3
c
φ
c3
−
+
NI
ac1
Fig. 7. Simplified Circuit for Test 3
Note that
p
in Fig. 7 is the parallel combination of
i
and
o
(which were both determined via Tests 1 and 2):
p
=
i
o
i
+
o
(7)
Through simple circuit analysis:
a
=
NI
ac1
−
c
φ
c3
−
p
(φ
c3
− φ
y3
)
(φ
c3
− φ
y3
)
(8)
