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Measurement and FEM analysis of DC/GIC effects on transformer magnetization
parameters
Conference Paper · June 2019
DOI: 10.1109/PTC.2019.8810423
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Measurement and FEM analysis of DC/GIC effects on
transformer magnetization parameters
Hilary K. Chisepo and C.T. Gaunt
Leslie D. Borrill
Department of Electrical Engineering
University of Cape Town
South Africa
chshil001@myuct.ac.za; ct.gaunt@uct.ac.za
Koeberg Nuclear Power Station
Eskom Holdings SOC
Cape Town, South Africa
BorrilLD@eskom.co.za
Abstract— Topologically derived equivalent circuit models for
transformers can be improved by more accurate
parameterization of the magnetization characteristics. We
examined the changes in transformer magnetization parameters
by measuring the effects of dc on a scaled down model singlephase four-limb (1p4L) transformer and confirmed them with
FEM modelling. The results define the non-linear character of the
transformer inductance when partial saturation occurs, such as
in over-excited transformers or with half-wave saturation caused
by leakage dc or geomagnetically induced currents. The analysis
clarifies the difference between the constant air inductance of a
winding (Lair) and the instantaneously changing saturation
inductance (Lterminal) used in low-frequency transformer
equivalent circuit models and shows how to choose a
representative value for the equivalent circuit parameter. The
tests lead to a more accurate graphical depiction of related
parameters during transformer half-cycle saturation than usually
presented.
Index Terms-- flux, inductance, GIC, FEM, measurements
I.
INTRODUCTION
Geomagnetically induced currents (GIC) or other sources of
dc excitation in the presence of ac energization can disturb the
normal operation of power transformers. GIC/dc components
of current cause transformers to operate under half-cycle
saturation and the response depends on the core structure and
magnetization characteristics. Half-cycle saturation generates
several unwanted conditions, including the generation of even
and odd harmonics, overheating, draw of reactive/non-active
power, and audible noise [1-5]. In severe cases, it may lead to
power transformer damage, power system instability, and
even blackouts [6], [7].
The derivation of representative electromagnetic transients
(EMT) transformer models has been widely researched with
the aim of improving the accuracy for mid- to low-frequency
studies [8-13]. Several studies depict the transformer halfcycle saturation diagrammatically by relating various
parameters to the flux B and rms magnetic field H [14-18].
Amidst the numerous efforts to enhance transformer
This work was supported in part by Eskom Holdings under the EPPEI
program, Royal Smit Transformatoren and a grant from the Open
Philanthropy Project. Mentor, a Siemens Business provided the Simcenter
MAGNETTM academic license for the FEM simulations.
modelling for GIC (and inrush currents and ferroresonance
[9]), the understanding of the complex and different
transformer responses is incomplete [10] or contradictory.
A simplified illustration [12] suggests the terminal
inductance of a transformer in deep saturation, Lterminal, “may
or may not” differ from the inductance of the same
transformer’s windings without its core, Lair. (The two
inductances are considered equal in some GIC studies [9], [1921].) The uncertainty arises from the difficulty of measuring
Lterminal in an industrial testing facility/laboratory, and because
manufacturers do not readily provide the data. Despite the
uncertainty, Lterminal (or Lair) is probably the most important
parameter for accurate transformer modelling for GIC and
other studies involving deep core saturation [22].
By measurement and FEM simulation, this study shows a
clear difference between Lterminal and Lair thus consolidating the
understanding of the suggested differences in [12]. This leads
to a new composite depiction of the important magnetization
parameters during half-cycle saturation derived from real BH
properties of power transformer core steel.
II.
THEORY AND MEASUREMENTS
GICs are variable very low frequency currents (mHz range,
so quasi-dc relative to the power frequency) driven by an
induced earth surface potential between the neutrals of widely
separated transmission transformers. An offset in the magnetic
flux B(t) is introduced in the transformer core and, depending
on the size of the GIC, may shift B(t) into the saturation region
in one half of the ac cycle (usually depicted on a B-H curve).
At the same time, a corresponding Imag is drawn by the
transformer to meet the partially saturated core’s H
requirements. (Since the Imag lags the applied voltage by 90°,
the reactive/non-active power demand in the transformer also
increases.) Fig. 1, based on widely presented illustrations,
shows the interacting magnetization parameters. A significant
question is “what is the inductance between ‘a’ and ‘b’ in Fig
1”?
Ideal BH curve of core steel
b
a
Developed
mean flux
‘Normal’ permeability
H
Imag(t)
TIME
B(t)
TIME
Fig. 1. Transformer core half-cycle saturation due to
GIC/dc, adapted from various reports [14-18, 23].
A method proposed by de León et al. [22] to measure
Lterminal was applied to a model single-phase four-limb (1p4L)
transformer (at the time of this preliminary work [22] it was
common belief that a saturated transformer was in “air core
inductance” operation, however the actual measurements here
represent Lterminal). The model was fabricated with two parallelconnected winding assemblies (each consisting of two
windings with subtractive polarity) and the same grainoriented electrical core steel as an actual 1p4L generator
transformer. The model had no other materials of
electromagnetic significance, so Lterminal is a function of the
core and windings only. The rating of the model transformer
was defined at the limit of linear magnetization (no overexcitation) to be 110/206.2 V, 4.4 kVA. Each pair of primary
(outer) windings had 80 turns and secondary (inner) windings
had 150 turns, referred to as 80t and 150t hereafter.
A. Description of Lterminal
De León’s three-phase measurement circuit [22] was
adapted as shown in Fig. 2 to measure Lterminal using a singlephase circuit (only a single-phase rectifier was available). This
circuit used supplemental batteries to drive the transformer
into deep saturation with sufficient dc Ampère-turns and a
small ac ripple supplied by the rectifier sufficient to determine
the transformer inductance in saturation). One phase of a 0380 V, 60 A, 50 Hz three-phase variac supplied the circuit and
all measurements were recorded with an IEC 76-1 (1993)
compliant WT1800 Yokogawa Power Analyzer.
the incremental flux while removing winding and source
resistances from the calculations, thus eliminating possible
error caused by temperature-dependent changes in resistance.
Fast Fourier Transform values of the voltages and currents
were then substituted in (1) [22]:
!"#$%&'() =
*+,-_.
(1)
/01. '234_.
where k is the harmonic order, Vout_k is the amplitude of the
secondary voltage of the dominant harmonic, fk is the dominant
harmonic frequency, n is the transformer ratio and I in_k is the
amplitude of the dominant harmonic of the primary current.
While other harmonics were considered and documented,
separately, yielding good correlation suggested by [22] using
a three-phase circuit, the dominant saturating harmonic of the
distorted input current (see Fig. 3) was chosen in this study
using a single-phase circuit to calculate Lterminal with increasing
dc up to its final value.
90
Voltages (Volts), Input Current (Amps)
B
Result of dc component
40
-10 0
10
20
30
40
50
-60
-110
-160
V out
V in
I in
Time (ms)
Fig. 3. Measured input and output voltages (black) and resultant input current
with a dc component of 60 A dc. Input current has a dominant 2nd harmonic.
B.
Description of Lair measurements
After completing the Lterminal measurements, the core was
removed without disturbing its winding positions, which was
possible with the construction method of the transformer.
Fig. 4 shows the “air core” transformer for measuring Lair.
Winding assemblies
Fig. 2. Schematic circuit for measuring the saturation inductance of a 1p4L
transformer by supplementing the dc offset with batteries.
The 80t and 150t windings were investigated separately,
controlling the current in the test winding and measuring the
voltage across the other open circuited windings. This captured
Fig. 4. 1p4L test transformer without the core for the Lair measurements
relative to the removed core (indicated by the white dotted lines).
Lair was measured with ac only, using a 0-240 V, 20 A
single-phase variac in series with a current limiting variable
resistor. We have not found any description of measuring L air
with which to compare this approach.
The linear or constant characteristic of an “air core” was
verified by injecting currents ranging from 3-10 A rms and
then calculating using (2):
!(&$ =
5
/012 6
runs of parametric analyses) and maintain accuracy, a oneeighth symmetric mirror model was used, illustrated in Fig. 5.
After each simulation, the full model solution was
recalculated.
(2)
/
(2) is derived from the expression78 = 9 :(&$ , where 8 is the
reactive power, I is the line current, f is the supply frequency
of 50 Hz and :(&$ is the air core reactance.
These two measurements of Lterminal and Lair were then
compared with the results of FEM simulations.
III.
FEM SIMULATION PROTOCOL
A FEM model of the 1p4L transformer had already been
validated in a parallel study [24] using FEM models and
physical measurement with ac-dc excitation. The core nonlinear characteristic used in the FEM is detailed in that study
of the core joints. The purpose of the further FEM simulation
here is to investigate Lterminal and Lair beyond the limits of
measurement of the physical model.
A.
FEM calculations for Lterminal
The FEM model included core nonlinearity effects on the
magnetizing currents and leakage/stray flux with simultaneous
ac and dc excitation in the transient analysis domain. The
winding excitations were performed using a field-circuit
coupled approach with an external circuit adapted from the
schematic in Fig. 2. Unlike the laboratory procedure in Chapter
II that relied on the dominant saturating harmonics to
determine a final value for Lterminal, post-processing of the
global quantities in multiple FEM solutions measured the
change of Lterminal with current.
From the FEM solution, Lterminal is determined from the
magnetic stored energy W when a current i flows in the
windings, given by (3):
/;
(3)
!"#$%&'() = 6 <
&
W is the integral over the whole 3D volume of the magnetic
energy density derived from Poynting’s theorem [25] and is
expressed as (4):
> = 7 ?* @ AB
(4)
where w is the corresponding magnetic energy volume density
relating to the area under the BH curve. When the magnetic
flux density is B1 the magnetic energy volume is given by (5):
H
@ = ?H I CDEF G AE
(5)
J
where H(B) is the magnetic field intensity corresponding to a
given B for the non-linear core.
A 3D model with stranded winding cylinders was required
for the Lterminal calculation to capture the magnetic spatial
distributions and inductances in the third dimension. Although
2D models solve more quickly, accuracy is sacrificed. Instead,
to shorten simulation time (which was necessary for multiple
Fig. 5. A 1/8th symmetric 3D model with explicitly modelled laminations close
to the core surface at normal ac only excitation
For the voltage driven coupled field-circuit model, the
current density J for the full FEM model is defined by:
L*
(6)
K= <
MN
where A is the cross-sectional area of the windings as a
function of the winding height, V is the applied voltage, N is
the number of turns, R is the resistance of the windings given
by
)M6
O=
(7)
PL
and l and s are the windings’ sweep distance and conductivity,
respectively. Keeping J constant for the symmetric and full
models, it can be shown that the symmetric model requires an
applied ac voltage Vsym that is a quarter of the equivalent full
model’s voltage (8).
BPQ% =
B.
TU
RSG 6 G V W
X
6
=
SMN
YL
*
= <
Y
(8)
FEM calculations Lair
A FEM geometry for the windings was based on the actual
“air core” transformer of Fig. 4. Because excluding the core
avoided long calculations of the nonlinearities and core joint
details, the problem became linear and could be solved in the
magnetostatic domain with dc only.
The primary and secondary windings separately, with the
other side open circuited. For each pair of windings (which are
connected in parallel with opposing polarity in the actual
laboratory transformer) a dc current of 0.5x A is injected into
Winding Assembly A and a reverse dc current of -0.5x A is
simultaneously injected in the Winding Assembly B.
This means that if x=1 then the total winding current for
substitution into (3) is 1 A. Since it is a linear problem, the
calculated Lair is independent of the magnitude of the current.
This approach was repeated with ac in a time-harmonic domain
(50 Hz) to check for consistency with the global solutions
derived from the preceding magnetostatic solutions.
IV.
RESULTS
A.
Measured and simulated results
In the presence of an applied voltage of 60 V rms and with
increasing levels of dc, the 2nd harmonic component of the
input current was found to be dominant for both the 150t and
80t windings. The measured Lterminal appeared to converge at
798 μH for the 150t windings, and 275 μH for the 80t
windings. The measured Lair was constant over the full range
of the applied ac voltage, as expected, and yielded 694 μH and
355 μH for 150t and 80t, respectively. (The analytical
calculation L=N2μoA/l using the geometry of the windings
yielded 664 μH and 353 μH for Lair, corresponding closely to
the measurements.)
The FEM simulation protocol for Lterminal differed from the
practical measurements in the sense that saturation was not
achieved by injecting harmonics and a dc component with a
non-ideal rectifier. Instead, as is allowable in the FEM domain,
increasing levels of dc (much higher than experimental dc
levels) were added to an applied voltage of 60 V rms. The
simulated Lterminal initially appeared to converge at 722 μH and
380 μH for the 150t and 80t windings, respectively. The
simulated Lair was a constant 654 μH (150t windings) and
380 μH (80t windings).
B.
Multiple parameter simulations for Lterminal
The FEM simulation experiments were extended to
investigate why the measured 80t Lterminal (275 μH) was lower
than the measured and simulated Lair, (355 μH and 340 μH).
In Table I, experiments (Exp) 1 and 4 are the results
corresponding to the physical model in subsection A. Without
the constraint of the physical model, Exp 2 and 5 investigated
the extent to which mutual inductances between the windings
affected the simulated Lterminal by removing from the simulation
the winding not under investigation. FEM Exp. 3 and 6
represent the effect of swopping the positions of the 80t and
150t windings while keeping the winding not under
investigation open circuited.
In a further experiment, not shown in the table, the effect of
drastically reducing the applied ac (to approximately 5 V) in
the presence of the same high dc was shown to have no effect
on the value of Lterminal.
Exp 1 and 2 show the effect of mutual inductance Lm
between the 80t and 150t windings (an increase of 16 μH when
the 150t winding is removed) at Lterminal. Similarly, Exp 4 and
5 show that removing the outer 80t winding reduces the 150t
Lterminal by Lm =20 µH.
Changing the positions of the windings, the outer 80t
winding (Exp 1) to the inner position (Exp 3), or the inner 150t
(Exp 4) to the outer position (Exp 6), returns a higher
inductance for the larger diameter winding, as expected.
The measured and FEM modelled Lterminal of Exp 4 are
comparable (10% difference). By contrast, the measured
Lterminal of Exp 1 is much lower than the simulated Lterminal of
Exp 1 and 2 and the measured and simulated Lair, especially
considering the FEM model’s consistency with other measured
parameters.
C.
Further Lterminal, Lair comparisons in the FEM with
high dc
Consideration was given to the possibility of measurement
error for the 80t inner windings. Looking at the final dc levels
measured in the laboratory to derive Lterminal it became clear that
the very large dc input needed to reach deep core saturation in
the 80t windings exceeded the limits of accuracy of the
Yokogawa power meter. Lterminal for the 150t windings was
reached at a dc input of 71 A, whereas the fewer turns of the
80t winding required 117 A dc. Even using shunt resistors to
measure high dc, the interpretation of the power meter resulted
in errors at high dc levels.
Therefore, the Lterminal and Lair parameters were compared
directly in the FEM analysis with very high dc up to 10 kA.
Lterminal with a dc of 10 kA was reached at 341 μH and 657 μH
for the 80t and 150t respectively, shown in Fig. 6. These values
are only a few µH higher than the FEMair core solutions (340 μH
and 654 μH, respectively), and suggest that the FEM model’s
core is very close to (complete) “air core” saturation.
It should be noted, though, that the unrealistic dc in the kA
range far exceeds maximum real GIC [~100 A/phase]. It was
used only to test how much dc would be needed to reach an Lair
of 340 μH and show that the result is much higher than the
erroneous measurement value of 275 μH.
2000
TABLE I.
FURTHER SIMULATIONS PERFORMED TO INVESTIGATE DISCREPANCY IN
MEASURED 80 TURN TERMINAL SATURATION INDUCTANCE
Exp 1
Exp 2
Exp 3
Exp 4
Exp 5
Exp 6
Description of FEM Exp
80t outer, 60 V ac + dc, 150t
inner open circuit
80t outer, 60 Vac + dc,
without 150t inner winding
80t inner, 60 V ac + dc, 150t
outer open circuit
150t inner, 60 Vac + dc, 80t
outer open circuit
150t inner, 60 V ac + dc,
without 80t outer winding
150t outer, 60 V ac + dc, 80t
inner open circuit
FEM
Lterminal
(μH)
Measured
Lterminal
(μH)
380
275
396
-
270
-
722
798
702
1133
L air 150t inner
1600
L (μH)
FEM
Exp
L terminal 150t inner
1800
1400
L terminal 80t outer
1200
L air 80t outer
1000
800
600
400
200
0
0
2
4
6
dc current (kA)
Fig. 6. Further FEM simulations with extremely high dc
8
10
In this study, then, the difference between measured and
modelled Lterminal for the 80t windings is attributed to
limitations in the resolution of measurement equipment at very
high distorted currents and should be replaced with the FEM
derived value of 380 μH.
V.
PARAMETRIC HALF-CYCLE SATURATION DEPICTION
This study’s results lead to an improved depiction of halfcycle saturation. Fig. 7 shows some of the multi-parameter
relationships under steady state GIC or dc offset of the ac.
The widely used [14-18] half-cycle saturation depiction of
the BH curve is adapted. The ac flux B(t) and dc component
interaction is represented by the black waveform on the left of
Fig. 7. The dc offset creates a ‘displaced zero’ axis above the
mid-point of the linear part of the BH curve.
A simplified piecewise inductance model, as used in EMTP
software [26], defines the knee-point at the flux density Bn at
rated voltage, with two linearized portions at 1.15B n and
1.25Bn between the linear ideal operating BH curve and a
linear air core characteristic, but such models often lack
accuracy in saturation studies.
When the transformer is driven beyond the knee of the BH
curve, the relatively small (exciting) magnetizing current,
shown by the green line in the bottom right part of Fig. 7,
changes with dc offset to the Imag (solid red) curve for a singlephase transformer. (The shape and displacement of the red Imag
curve is derived from separate FEM simulations with the same
typical core steel [27] used for power transformers.)
McLyman [23] defined the knee point of the BH curve
where the tangent from the origin identifies the maximum
‘normal’ permeability (Z7 B/H), also called amplitude
permeability [27].
L=k1μr,
B(t)
B
[H
B(t) with dc μr=k2
[\
Real BH curve of core steel
1.8
1.6
1.4
Developed
mean flux
1.2
B =1 pu
1
Instantaneous
Lterminal due to
dc
0.8
L, μr
0.6
0.4
Lair
0.2
0
0
100
200
TIME
H "∞
B(t) with
ac only
Imag(t)
Imag(t) with dc
Nominal Imag(t)
TIME
‘Displaced
zero’ axis
caused by dc
Fig. 7. Parametric relationships during half-cycle saturation showing a real
BH curve, relative permeability and transformer inductance.
The permeability μ and relative permeability μ r as defined
by the local gradient (dB/dH) at any ‘point’ on the BH curve
have a different maximum magnitude and location from that
identified by the amplitude permeability. The instantaneous (or
incremental or differential) inductance L Z7 dφ/di is directly
proportional to μ and μr. Therefore, L, μ and μr follow the same
shape with different scaling constants. μr tends to 1 and L tends
to Lair as H tends to infinity, but these values are not reached
even with high practical values of dc or GIC.
The incremental or differential μr is directly related to the
BH curve, requiring complete consistency between the
measurements of L and BH core steel data. The real BH curve
(shown in blue in Fig. 7) was determined by linearizing the
portion of the BH curve [27] below 0.4 T and compensating
for the corresponding H to incorporate hysteresis, and then
performing a ‘moving’ calculation for dB/dH for the entire BH
curve to yield μr
The shape and slope of μr and L were verified using several
grades of power transformer electrical core steel. As a result,
the real BH curve shows the saturation transition beyond the
knee in the region of a-b of Fig. 1. The instantaneous Lterminal
in Fig. 7 corresponds to the peak of the Imag as a result of the dc
component.
Further, the calculated instantaneous inductance and ‘point’
permeability before the knee are shown without the straight
line simplification used in other models [14].
VI.
CONCLUSIONS
This paper identified the need for more representative EMT
transformer models for GIC and related studies. The most
important parameter for improved accuracy requires the
interpretation of the saturation inductance in topological
models. The measurements and FEM simulations have
clarified the difference between Lterminal and Lair for the 1p4L
laboratory transformer without a tank and these results support
the suggestions from the literature [22].
The main improvement of Fig. 7 over Fig. 1 is the avoidance
of misleading simplification of the BH curve, enabling a
consistent depiction of real BH data and the instantaneous L
and μr and Imag with dc. Also, it clarifies that the maximum
instantaneous permeability is not coincident with the knee
point of the BH curve.
The Fig. 7 (time domain) model answers the question about
the shape and slope of the inductance and relative permeability
between ‘a’ and ‘b’ in Fig 1. The instantaneous inductance L
falls sharply as the gradient of the BH curve changes, and
Lterminal tends towards Lair but Lair is not reached even with the
highest measured GICs.
In the context of GIC, the suitable parameter for input into
any single-phase topological model is Lterminal and it does not
have a single value because it is a function of the GIC or dc as
shown in Fig. 6.
The implications of the improved model are better accuracy
in power system modelling with GIC and dc, and even to
partial saturation caused by over-excitation.
It is seldom reported that the transformer Q-GIC
characteristic is not strictly linear moving from partial to deep
saturation [28]. Figs. 6 and 7 from this paper could be used to
further understand the non-linearity of L, its effect on Q-GIC,
and implications for power system stability studies.
The results have been derived and tested using single-phase
transformers. Ongoing studies are extending the investigation
to transformers with tanks and with three-phase core structures.
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