Modeling power transformers to support the interpretation of frequency-response
analysis
by Steven D. M
itchell, James S. Welsh
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right © 2011 IEEE.
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Delivery, Vol. 26, Issue 4, p. 2705-2717.
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1
Modelling Power Transformers to Support the
Interpretation of Frequency Response Analysis
Steven.D.Mitchell, Member, IEEE, James.S.Welsh, Member, IEEE.
Abstract—A power transformer will yield a frequency response
which is unique to its mechanical geometry and electrical
properties. Changes in the frequency response of a transformer
can be potential indicators of winding deformation, as well as
other structural and electrical problems. A diagnostic tool which
leverages this knowledge in order to detect such changes is
Frequency Response Analysis (or FRA). To date, FRA has been
used to identify changes in a transformer’s frequency response
but with limited insight into the underlying cause of the change.
However there is now a growing research interest in specifically
identifying the structural change in a transformer directly from
its FRA signature. The aim of this paper is to support FRA
interpretation through the development of wideband three phase
transformer models which are based on three types of FRA
test. The resulting models can be used as a flexible test bed for
parameter sensitivity analysis, leading to greater insight into the
effects geometric change can have on transformer FRA. The pa-
per will demonstrate the applicability of this modelling approach
by simultaneously fitting each model to the corresponding FRA
data sets without a priori knowledge of the transformer’s internal
dimensions, and then quantitatively assessing the accuracy of key
model parameters.
Index Terms—power, transformer, frequency, response, anal-
ysis, FRA, wideband, model, interpretation, geometry, deforma-
tion, sensitivity.
I. INTRODUCTION
T
HE average age of the world’s population of power
transformers is increasing [1]. Due to high replacement
costs, research into maximizing their longevity is a high
priority. To successfully accomplish this in a safe and efficient
manner, their condition must be regularly monitored in order
to schedule the appropriate maintenance and repair.
When subjected to fault currents, tremendous forces can
be placed upon the transformer windings and mechanical
structure [2]. Such high levels of mechanical stress can
lead to winding deformation which could ultimately lead to
transformer failure. The mechanical and electrical integrity
of a transformer can also be compromised through poor
workmanship and during transportation [3].
Frequency Response Analysis (FRA) is a commonly used
diagnostic approach which looks for changes in the frequency
response of a transformer. Each FRA test will yield a signature
unique to the transformer’s mechanical geometry from the
perspective of the input and output measurement terminal
positions [4]. Changes in the frequency response signature
can be indicative of winding deformation. However, to date,
there is relatively little understanding of how to interpret the
Steven.D.Mitchell and James.S.Welsh are with the School of Electrical
and Computer Engineering, University of Newcastle, Callaghan, N.S.W 2308,
Australia.
Manuscript received January 2011
underlying cause of the change directly from the frequency
response [5].
Transformer Frequency Response Analysis was initially
proposed by Dick and Erven in 1978 [6]. Since its introduction
there has been an interest in advancing the diagnostic benefits
associated with FRA by improving our ability to interpret any
observed changes. One approach which has been adopted by
many researchers has been the use of a transformer model
based on its geometric parameters [4], [7], [8]. The rationale
behind this approach is that a change in the geometry of
a transformer will affect the parameters represented in the
model. A sensitivity analysis of the model parameters could
therefore be used to assist in determining the root cause behind
any change in a transformer’s frequency response.
A number of researchers have made significant contributions
to this area in recent years. In 2003 Rahimpour et al proposed
a detailed single phase transformer model in order to diagnose
axial displacement and radial deformation [7]. In 2005 Bjerkan
made further advances by proposing a detailed three phase
transformer model for the sensitivity analysis of FRA [4]. Ar-
ticles by Abeywickrama et al between 2005 and 2008 further
extended the research area by providing a comprehensive three
phase transformer model which included the high frequency
behaviour of the transformer’s core [8]–[10]. An article by
Ragavan et al in 2008 presented an impedance function for
a single phase transformer based upon FRA measurement
[3]. This work utilised the FRA data to determine model
parameters for a simple ladder network directly from the pole
and zero locations of the frequency response. In 2009 a three
phase transformer model was proposed by Shintemirov et al
that utilised genetic algorithms to interprete FRA results at
low frequencies [11]. Shintemirov’s approach facilitated the
determination of a transformer core’s parameters. A model
proposed by Sofian et al (2010) was used to conduct simulated
sensitivity studies on a large three phase autotransformer [12].
Sofian et al investigated the influence that changes to the
transformer’s structure would have on its frequency response.
However despite these advances in the research area, the
interpretation of FRA remains subjective and typically based
on an experienced technician’s assessment [13]. A report from
the CIGRE WG A2.26 has called for further research to
be conducted in the area of transformer modelling based on
geometric parameters in order to support the interpretation of
FRA [5].
As highlighted above, researchers have developed compre-
hensive three phase transformer models for use with FRA
[4], [8], [12]. However in each of these cases parameter
determination was made via FEA and analytical techniques.
This approach requires a priori knowledge of the transformer’s
2
internal dimensions and material properties. Such informa-
tion is typically unavailable outside of the laboratory. Other
researchers have utilized estimation techniques in order to
determine their model parameters, however in the case of
[3], [7], [14], [15], they have only utilised a single phase
transformer topology which will limit the model’s practical ap-
plication, or a reduced parameter circuit as in the case of [16].
Research by Shintemirov et al estimates parameters directly
from FRA and is based on a three phase topology, however
it is only aimed at core identification and is not applicable
for the entire FRA spectrum [11]. Our research extends on
this previous work through the development of a wideband
frequency, three phase transformer model, whose parameters
are determined directly from FRA. Parameter determination
is facilitated by the application of a constrained nonlinear
optimisation algorithm to the data sets of three different types
of FRA. This approach significantly increases the accuracy
of our parameter prediction, and the authors believe that this
approach is novel in its application.
To demonstrate the applicability of the FRA modelling
approach, FRA data was recorded from an air cooled 1.3MVA
11kV/433V distribution transformer. Since three different
types of FRA tests are being considered and each test has
three terminal permutations, this results in nine unique FRA
data sets. To maximise the accuracy of the modelling approach,
the FRA models are simultaneously fitted to all nine sets of
data. The accuracy of the approach is verified by quantitatively
assessing a number of key parameters against their physically
measured counterparts.
In this paper we assume that the transformer is of core type
construction with concentric windings. The leakage inductance
is considered to be restricted to the axial path between the high
and low voltage windings [17].
This paper is structured in the following manner. Section
II develops a generic phase model that is used as a building
block in later sections. Section III adopts a layered modelling
approach in order to build models for three different types of
FRA test. Section IV develops the FRA model transfer func-
tions and the corresponding estimation algorithm. In Section V
each of the FRA models is fitted to FRA data and the accuracy
of the estimated model parameters is assessed. A sensitivity
analysis example is then presented. Concluding remarks are
given in Section VI.
II. G
ENERIC PHASE MODEL
The origins of transformer modelling can be traced back to a
1902 paper by Thomas [18]. In this paper Thomas highlighted
the need for modelling in order to determine the surge voltage
distribution across a transformer’s windings. The original
ladder network model of a transformer is generally associated
with Blume and Boyajian’s work of 1919 [19]. In their paper
they considered the influence of a transformer’s mutual and
leakage inductance, the capacitance between adjacent coils,
and the capacitance between the winding and ground. Since
these foundation papers there have been many significant
contributions to the area of power transformer modelling
including [20]–[25]. We build upon this previous research
L
X1
R
x1
L
x1
R
x1
Y
1
Z
1
C
SX1
C
Sx1
L
X2
R
x2
L
x2
R
x2
C
Xx2
Y
2
Z
2
C
SX2
C
XY2
C
ZX2
C
gX2
C
Sx2
C
gx2
L
X3
R
x3
L
x3
R
x3
C
Xx3
Y
3
Z
3
C
SX3
C
XY3
C
ZX3
C
gX3
C
Sx3
C
gx3
L
xn
R
xn
L
xn
R
xn
C
Xxn
Y
n
Z
n
C
SXn
C
XYn
C
ZXn
C
gXn
C
sxn
C
gxn
C
xx(n+1)
C
Xx4
C
gx4
Y
(n+1)
Z
(n+1)
C
XY(n+1)
C
ZX(n+1)
C
gX(n+1)
Y
4
Z
4
C
XY4
C
ZX4
C
gX4
X
1
X
(n+1)
x
1
x
(n+1)
C
gx1
C
Xx1
C
XY1
C
ZX1
C
gX1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
C
gx(n+1)
2
1
2
1
Figure 1. Transformer model for generic phase X
implementing a generic n section lumped parameter three
phase transformer model.
There are three terminal permutations associated with each
FRA test. For example, for the Capacitive Interwinding test
[5], the three terminal combinations are the high voltage
terminal A to the low voltage terminal a, the high voltage
terminal B to the low voltage terminal b, and the high voltage
terminal C to the low voltage terminal c. To represent each
of the terminal combinations for this example in a model,
it is convenient to have generic phase references for the
injection, measurement and open circuited terminals. This has
been accommodated in our modelling approach through the
use of generic phase referencing. Throughout this paper the
generic high voltage terminals are designated X-Y -Z, and the
corresponding low voltage terminals are x-y-z. The n section
lumped parameter model for generic phase X is given in
Figure 1.
Each section of the high and low voltage windings consists
of the series combination of an inductive element L and a
resistive element R. To account for the capacitance between
these windings, a capacitive element C
Xx
couples each equiv-
3
R
Y
R
Y
R
E
l
Y
l
E
F
a
F
A
R
L
R
E
F
b
F
B
R
E
F
c
F
C
R
L
R
L
Figure 2. Magnetic equivalent circuit of a three phase two winding core
type transformer
alent winding section. The capacitance between turns and
adjacent discs is modelled with the addition of C
SX
and C
Sx
,
for the high and low voltage windings respectively. The capac-
itance between the low voltage winding and ground is given
by C
gx
and the capacitance between the high voltage winding
and the transformer tank walls is given by C
gX
. Finally, the
capacitances C
XY
and C
ZX
represent the capacitance between
adjacent high voltage windings (i.e. A to B and B to C). Note
that the magnitude of all shunt capacitances at the winding
ends are at half their normal value to reflect their relative
distribution.
Each of the inductive, resistive and capacitive elements
shown in Figure 1 will now be discussed in detail.
A. Inductive Element
The use of a reluctance model is a convenient method for
modelling flux paths within a transformer. This modelling ap-
proach is a physical representation, utilising a magnetic circuit
based on the transformer’s core geometry. In the magnetic
circuit, each winding is replaced with a magnetomotive source
and each flux path, including those representative of leakage,
with a corresponding reluctance.
The proposed magnetic circuit for an FRA injection test
is presented in Figure 2. In this figure, F
A
represents the
magnetomotive force (mmf) due to the high voltage winding
of phase A, F
a
the mmf due to the low voltage winding on
phase A and so forth for the other phases. R
E
is the transformer
core limb reluctance and R
Y
is the transformer core yoke
reluctance. R
L
is the winding leakage flux reluctance. The
linear dimensions of the core are l
E
for the mean core limb
length and l
Y
for the mean core yoke width. In order to reduce
the level of complexity associated with the model’s magnetic
circuit, the influence of core joints will not be considered [8].
The core reluctance can be defined in terms of the mean
path length l, core cross sectional area A
CS
, and the core
Table I
T
RANSFORMER FERROMAGNETIC INDUCTANCE MATRIX
L A
j
B
j
C
j
a
j
b
j
c
j
A
i
¯
L
¯
L
2Γ
¯
L
Λ
¯
L
˜a
¯
L
2Γ˜a
¯
L
Λ˜a
B
i
¯
L
2Γ
¯
L
Γ
¯
L
2Γ
¯
L
2Γ˜a
¯
L
Γ˜a
¯
L
2Γ˜a
C
i
¯
L
Λ
¯
L
2Γ
¯
L
¯
L
Λ˜a
¯
L
2Γ˜a
¯
L
˜a
a
i
¯
L
˜a
¯
L
2Γ˜a
¯
L
Λ˜a
¯
L
˜a
2
¯
L
2Γ˜a
2
¯
L
Λ˜a
2
b
i
¯
L
2Γ˜a
¯
L
Γ˜a
¯
L
2Γ˜a
¯
L
2Γ˜a
2
¯
L
Γ˜a
2
¯
L
2Γ˜a
2
c
i
¯
L
Λ˜a
¯
L
2Γ˜a
¯
L
˜a
¯
L
Λ˜a
2
¯
L
2Γ˜a
2
¯
L
˜a
2
permeability μ such that,
R
E
=
1
μA
CS
l
E
(1)
R
Y
=
1
μA
CS
l
Y
, (2)
where R
E
and R
Y
are directly proportional to the limb and
yoke length. By definition, inductance is proportional to the
winding flux linkage relative to the current in the winding for
self inductance, or the current in another winding for mutual
inductance. From (1) and (2), the inductive relationships that
exist between winding sections on the three phase transformer
core will therefore be dependent upon the core dimensions.
It is convenient to specify the self and mutual inductance
relationships that exist for each winding section in terms of a
common base inductance
1
,
¯
L =
μA
CS
N
X
n
2
(2l
E
+ l
Y
)
(l
E
+ l
Y
)(3l
E
+ l
Y
)
, (3)
and two core dimension constants,
Γ=
2l
E
+ l
Y
2(l
E
+ l
Y
)
(4)
Λ=
2l
E
+ l
Y
l
E
, (5)
where n is the number of lumped parameter sections within
the transformer model and N
X
is the number of turns on each
high voltage winding. A matrix that represents each of the
respective self and mutual inductances can then be generated.
This matrix is presented in Table I where i and j are the
lumped parameter section numbers and ˜a is the turns ratio.
The model also incorporates a leakage inductance between
winding sections. For example, the leakage inductance be-
tween sections i and j of generic winding X is given by
L
LXij
.
1
The base inductance relationship is based on the self inductance of an
outer limb high voltage winding section (refer Table I).
4
The permeability of the transformer core as specified in the
base inductance relationship of (3) is a complex frequency
dependent term. Under low field conditions and the wide
frequency spectrum of an FRA test, μ can be defined to be
[26],
μ = μ
0
μ
s
= μ
0
μ
s
− jμ
s
=
kμ
0
μ
i
γb
tanh (γb) . (6)
In this relationship μ
0
is the permeability of free space, μ
s
is the effective permeability, μ
i
is the initial permeability of
the core material, k is the lamination stacking factor (which
approaches unity) and b is the core lamination half thickness.
The propagation constant, γ, is defined as,
γ =
jωσμ
0
μ
i
, (7)
where w is the angular frequency in radians per second, and
σ is the lamination material conductivity. As a result of the
complex permeability of the core in (6), a winding’s induc-
tance and its associated magnetic losses can be represented
as an impedance. This impedance can be modelled as the
series combination of a frequency dependent inductance, L
,
and resistance R
[27],
Z = jwL
0
(μ
s
− jμ
s
) (8)
= jwL
+ R
, (9)
where R
can be considered to be the magnetic loss resistance
and L
0
is the inductance with unity permeability.
An inductive element L
Xi
(ω) is now proposed that takes
into account the self and mutual inductance relationships
described in Table I and the real and imaginary components
of (9). L
Xi
(ω) represents the frequency dependent inductance
from section i of generic winding X and is shown in Figure
3(a).
B. Resistive Element
The transformer winding can be considered to have both a
DC and a frequency dependent AC resistance. The DC resis-
tance is directly proportional to the resistivity of the conductor
and inversely proportional to the winding conductor cross
sectional area. The AC resistance is due to the induction of
eddy currents within the windings. These induced resistances
can be classified into two categories, skin and proximity effect.
An analytical estimate for skin effect can be made using
the Dowell Method [28]. If it is assumed that the conductors
are closely packed, each layer of a winding will approximate
the geometry of a conductor foil. The problem can then be
reduced to a one dimensional model [29], i.e.
R
S
=
R
DC
ξ
2
sinh ξ + sin ξ
cosh ξ − cos ξ
, (10)
where
ξ =
d
√
π
2δ
, (11)
d is the conductor diameter,
δ =
1
√
πfμσ
, (12)
is the skin depth, with f the frequency in Hz, and the
permeability and conductivity of the conductor material are
given by μ and σ respectively.
Assuming once again that the conductors are closely packed
and that each layer of a winding will approximate the geometry
of a conductor foil, the AC resistance due to proximity effect
for the m
th
winding layer [29] is,
R
P
=
R
DC
ξ
2
(2m − 1)
2
sinh ξ − sin ξ
cosh ξ + cos ξ
. (13)
Assuming that the magnetic field from the other conductors
is uniform across the conductor cross section, an orthogonal
relationship exists between the skin and proximity effect [30].
The two effects can be decoupled and an estimate for the total
eddy current losses can be determined through the addition
of both effects (10) and (13). By combining both the AC and
DC winding resistances and considering them to be sectionally
distributed, the series resistance element, R
Xi
, is given by,
R
Xi
(ω)=R
PXi
(ω)+R
SXi
(ω)+R
DCXi
, (14)
where ω is the angular frequency. R
Xi
is presented in Figure
3(c).
C. Capacitive Element
In addition to the displacement current in a capacitor, a
dielectric material will also experience losses through con-
duction and material polarization [8]. For the non-ideal ca-
pacitance C, the circuit admittance is given by [8] [31],
Y = jωC
= G + jωC . (15)
It is well known that the admittance of a non-ideal capac-
itance can be represented as the parallel combination of a
conductance G and an ideal capacitance C [17]. The lumped
parameter circuit element used to represent each of the non-
ideal capacitors used in the transformer model takes the form
presented in Figure 3(b). For mathematical convenience an
equivalent resistance is substituted for conductance.
Each of the model’s capacitance and dielectric loss terms
are estimated directly by the algorithm and are not devolved
into lower level parameters.
III. M
ODELLING FOR FRA
The main FRA test types as classified by CIGRE are the
End to End Open Circuit, End to End Short Circuit, Capacitive
Interwinding, and the Inductive Interwinding test [5]. The End
to End Open Circuit test inputs a signal into one end of a
winding and measures the response at the other end. It is the
most commonly used topology due to its simplicity and its
ability to examine individual windings separately. The End to
End Short Circuit test is similar to the Open Circuit version,
but with a winding of the same phase short circuited. This
topology enables the influence of the magnetising inductance
to be removed such that the leakage inductance will dominate
the low frequency response. At high frequencies the response
will be similar to the End to End Open Circuit test [5]. The
