2007 - azevedo - transformer mechanical stress caused by external short-circuit. a time domain...
Post on 10-Feb-2016
220 Views
Preview:
DESCRIPTION
TRANSCRIPT
Abstract-- This paper aims at presenting investigation results
on the study of electromechanical forces in transformers due to
short circuit currents. A time domain transformer model using
magnetomotive forces and magnetic reluctances approach is
described. This software allows for simulating the transformer
transient and steady state behavior regarding electric, magnetic
and mechanical variables. To highlight the method potentiality
studies are carried out using a typical power transformer
operating under distinct conditions. Due to the lack of mechanical
stress reference values in the literature, a comparative analysis is
performed by comparing the time domain approach
computational results to similar ones derived from a finite element
program.
Index Terms-- Electromagnetic force, finite element method,
short-circuit currents, transformer modeling.
I. INTRODUCTION
URING normal lifetime, transformers are submitted to a
variety of electrical, mechanical and other stresses. One
of the most critical situations is related to the occurrence of
external short circuits. Under these circumstances, high
currents will circulate throughout the transformer windings and
high internal forces are produced. This mechanism can
strongly arise as a potential source for mechanically damaging
transformers as described in [1]. Techniques to mitigate the
impacts of these forces are also considered in this reference.
According to [2], solutions to overcome the high forces
occurring at the transformer windings during short circuit
conditions depend on the material selection and careful
structural design.
D
Focusing this subject, this paper is concentrated on the
development of a time domain transformer model using
magnetomotive forces and magnetic reluctances approach to
model transformers to evaluate the referred forces. This
method shows to be a powerful mean to achieve a
comprehensive view of the overall transformer magnetic and
electrical behavior under normal and disturbance conditions.
Using commercial 15kVA three-phase transformer
parameters, computational investigations are carried out to
A. C. de Azevedo, A. C. Delaiba and J. C. de Oliveira are with Federal
University of Uberlandia, Minas Gerais, Brazil (+55-34-3239-4701, E-mails:
anaclaudia.ene@gmail.com, delaiba@ufu.br, jcoliveira@ufu.br).
B. C. Carvalho is with Federal University of Mato Grosso, Mato Grosso,
Brazil (+55-65-3615-8732, E-mail: bcc@ufmt.br).
H. de S. Bronzeado is with the Companhia Hidro Elétrica do São
Francisco -CHESF, Brazil, (+55-81-3229-4105, E-mail:
hebron@chesf.gov.br).
illustrate the method application. Due to difficulties associated
to the lack of well established and accepted reference values,
the results are compared to corresponding analytical results
extracted from a traditional and well-accepted finite element
approach. The chosen package is a very well known program
known as Finite Element Method Magnetics (FEMM) which is
dedicated to the resolution of electromagnetic problems using
2D domain [3].
II. ELECTROMAGNETIC FORCES IN TRANSFORMERS
It is known that the electromagnetic forces occurring at the
transformer windings are generated by the interaction between
current density and leakage flux density. These forces can be
calculated using (1) [4].
f J X B→ → →
= (1)
In the above equation f
is the force density vector, J
is
the current density vector and B
is the leakage flux density
vector.
Fig. 1 is illustrates a typical leakage flux distribution within
the transformer windings. The principal flux density showed in
the figure is the axial component. Near to the winding ends, the
leakage flux bends towards to the core leg so as to make
shorter its return path. Therefore, at the top and bottom ends of
the windings it is possible to recognize the existence of both
flux axial and radial components.
Br
Ba
Ba
B
Fig. 1. Transformer flux distribution for concentric windings.
By taking into account a transformer with concentric
windings, the axial component of leakage flux density (Ba)
interacts with the current in the windings, producing a radial
force (Fr). This is a well established phenomenon and it is
responsible for the mutual repulsion between the inner and
Ana C. de AZEVEDO, MSc, Antônio C. DELAIBA, DSc, José C. de OLIVEIRA, PhD, Bismarck C.
CARVALHO, DsC and Herivelto de S. BRONZEADO, MSc, IEEE Member
Transformer mechanical stress caused by
external short-circuit: a time domain approach
1
outer windings. Similarly, the radial flux component (Br)
interacts with the corresponding current, producing an axial
force (Fa) which acts towards the compression or the expansion
of the windings [2].
With the transformer operating under normal conditions, the
electromagnetic forces in the windings are small due to fact
that the currents and leakage magnetic fluxes are associated to
rated values. However, during external fault situations, the
currents and the fluxes are quite high. This produces new and
much higher electromagnetic axial a radial forces in the
windings.
As the leakage flux can be expressed as a function of the
current in the windings, the resultant force, as given in (1), will
be proportional to the squared current, independently of the
type of transformer windings arrangement [5].
Since the maximum forces are produced by three-phase
short circuits, it is usual to design transformers to withstand the
corresponding maximum peak using values derived from short-
circuited transformers connected to an infinite busbar [5]. In
this context, the equation to determine the highest short circuit
current level (Icc) is given in [6] and represented by expression
(2). The variable k is related to the asymmetry factor, Sn is the
transformer rated power in MVA, V is the rated voltage and Z
the transformer pu impedance.
62 10
3
ncc
k Si A
VZ= (2)
A. Radial Electromagnetic Forces for Concentric Windings
Template
The radial force components in a transformer with
concentric windings are calculated in accordance with the
procedures described in [6]. Fig. 2 taken from this reference
illustrates a transformer cross section with two concentric
windings. The resultant forces in the inner and outer windings
are also shown. In addition to the forces, it is also shown the
axial field density (Ba). This field is taken as constant all over
the extension comprising the outer and inner windings in their
internal area. Outside this region the flux is decreased until it
becomes null. Hence, the force, in per unit of length,
throughout the length of the coils, remains practically constant.
Ba
Distribution of
axial flux
Fr
Fr
h
Core
Fig. 2. Cross section of a transformer with concentric windings - axial field
density (Ba) and radial force (Fr).
By neglecting the flux at the winding ends, the
instantaneous ampere-turns for each winding (ni) is responsible
for producing the leakage field density (Ba) given by (3).
4
4 ( )
10a
niB T
π= (3)
This flux density will interact with the current and the
following mean radial force (Fr) will occur:
272 ( )
10mr
ni DF N
h
π −= (4)
In equation i is the current in transformer winding, n is the
number of turns, h is the length of the winding and Dm is the
winding mean diameter.
This force will act in such a way to produce a hoop stress
upon the outer windings and another effect towards the axis of
the limb belonging to the inner winding (compressive stress).
These forces are illustrated in Fig. 3.
Fig. 3. Forces producing hoop stress and compressive stress in concentric
windings.
Both, the mean hoop and compressive stress in the
concentric windings are calculated considering that the
winding has n turns with a cross section ac by (5) [6].
2
2
rmean
c
FN m
naσ = (5)
B. Axial Electromagnetic Forces for Concentric Windings
The analytical calculation of the radial leakage field, and
therefore, the axial force, is not as simple as the axial field
calculations [6]. Nevertheless, a well-accepted approach for
the above calculation is the Residual Ampere-turns Method.
This is based on the principle that any arrangement of
concentric windings can be split into two groups, each one
having balanced ampere-turns relationship. One produces the
axial field and the other the radial field [5].
Fig. 4 shows the radial field distribution and the axial forces
in an arrangement with asymmetrical windings. It can be noted
the height of the outer winding is shorter than that of the inner
winding. This asymmetry causes a great radial flux density in
the region where the imbalanced of ampere-turns occurs [7].
2
heff
Core
Br
Fa
BrFa
BrBr
Fa Fa
Radial
flux
Fig. 4. Radial field density and axial force [6].
To calculate the axial compression force by the mentioned
method, it is necessary to known the following parameters: heff
is the effective path length of the radial flux and depends on
the arrangement of taping; Br corresponds to the average radial
flux density; (1/2)a(ni) is the average value of ampere-turns,
being a the length of tap section expressed as a fraction of the
total length of the winding.
The average radial flux density is given by:
( )4
4
10 2r
eff
a niB T
h
π= (6)
To determine the axial force (Fa) for a transformer with tap
section in one end of the external winding, the following
equation is to be used:
( )2
7
2
10
ma
eff
a ni DF N
h
π π= (7)
By taking into account the physical arrangement given in
Fig. 4, the value of heff is 0.222h.
Reference [6] provides expressions to calculate
electromagnetic forces for other specific tap arrangements.
The axial force cause bending between radial spacers and in
this case the axial stress is given by (8) [7].
22
22
amean
F LN m
tbσ = (8)
where: Fax is distributed axial force - N/m
L is distance between stampings - m
b conductor axial dimension - m
t conductor radial dimension - m
III. TIME DOMAIN COMPUTATIONAL MODEL
Different approaches have been traditionally used to
computationally obtain time domain transformer models.
These methods are based on: electric equations (duality);
electric/magnetic equations and, through magnetomotive force
(mmf)/reluctance models [8]. Concerning this paper purposes,
the representation of three-phase core type transformer will be
carried throughout the mmf/reluctance model and a time
domain simulator. The choice of the mmf/reluctance method
for transformer modeling is particularly advantageous and
necessary, so it allows that the interactions between the
magnetic fluxes of different phases are taken in account, as
well as the distinct connections of the windings can easily be
accomplished. Using this software, a 100 MVA transformer
having four concentric windings per phase was implemented.
Table I shows the geometric, electric and magnetic
characteristics for the tested transformer.
TABLE I
CORE-TYPE TRANSFORMER CHARACTERISTICS: 15 KVA, THREE-PHASE
Leakage impedance 9.32%
Core loss 53.35kW
Frequency 60 Hz
VoltageOuter winding Inner winding
230 kV 138 V
Windings 2 4 1 3
Turns 756 64 394 98
Copper resistance 0.6883 Ω 0.0608 Ω 0.132 Ω 0.0438 Ω
External diameter 1290 m 1574 m 988 m 1460 m
Internal diameter 1112 m 1510 m 818 m 1410 m
Leg Yoke
Area 0.3775 m2 0.3775 m2
Medium magnetic
path length2.46 m 1.607 m
Magnetic flux density 1.61 tesla 1.614 tesla
Following the software procedures, Fig. 5 shows the three-
phase equivalent circuit.
mmF4
Ra
irR
Ra
irT
Rco
reR
Rco
reS
Rco
reT
RuRS RuST
RbRS RbST
Rle
ak
2
Rle
ak
2
Rle
ak
2
Ra
irS
mmF4 mmF4
mmF2 mmF2 mmF2
mmF1 mmF1 mmF1
mmF3 mmF3 mmF3
Rle
ak
1
Rle
ak
1
Rle
ak
1
Rle
ak
3
Rle
ak
3
Rle
ak
3
Fig. 5. Transformer equivalent electromagnetic model.
In Fig. 5: ℜcoreR, ℜcoreS and ℜcoreT are the non linear
reluctances for the left, central and right leg, respectively; ℜairR,
ℜairS and ℜairT are the linear reluctances of the air path flux
between the core and internal winding; ℜleak1, ℜleak2 and ℜleak3
represent the leakage linear reluctances associated to leakage
flux between inner and outer windings; ℜtRS, ℜtST, ℜbRS, and ℜ
bRS correspond to the non-linear reluctances associated with
flux in top and bottom yokes; mmf2 and mmf4 express the
magnetomotive forces associated to the outer windings
and;mmf1 and mmf3 are the magnetomotive forces produced
by the inner windings.
The described methodology requires the leakage fluxes for
calculating both the axial and radial electromagnetic forces. To
obtain these fluxes the corresponding magnetic area related to
the leakage effect. Fig. 6 illustrates the physical parameters
required to evaluate the necessary area (ALeak) [10].
Using the physical construction information illustrated in
Fig. 6, the required area between the outer and inner windings
can be calculated throughout (7).
3
X
X
X
X
X
X
X
X
X
X
Leakage
flux
Outer
windingInner
winding
d1 d0 d2
x
lm
y
H
i i
Fig. 6. Concentric windings physical parameters.
21 20
3 3Leak m
d dA l d m
= + +
(9)
Where: lm is the average length of the winding
circumference; d1 and d2 are the thickness of the outer and the
inner windings, respectively and; d0 is the space thickness
between windings.
IV. TIME DOMAIN AND FEMM RESULTS
The studies selected to be presented in this paper comprise
two situations concerning transformer operation: normal
functioning with rated loading and short circuit conditions.
Results obtained from the time-domain model are firstly given
and these are later compared to the Finite Element Method
Magnetics (FEMM) approach.
Case A. Time-Domain Results
The results achieved with normal operating conditions are
given in Figs. 7 and 8. Due to the fact there are a large number
of variables to be visualized; only those at the outer winding
are illustrated.
Fig. 7(a) shows the supply three-phase voltages and Fig.
7(b) the corresponding transformer magnetic flux density. The
maximum flux density value found is of 1.6 T and this is in
agreement with expected values.
0.2 0.22 0.24 0.26 0.28 0.3-200
-150
-100
-50
0
50
100
150
200x 10
3 vTvSvR
time (s)
vo
ltage
(V)
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time (s)
mag
net
ic i
nduct
ion (T
)
BTBSBR
(a) (b)
Fig.7. Transformer normal operation: (a) Supplied voltage; (b) Magnetic field
density.
Fig. 8(a) gives the load currents and Fig. 8(b) highlights the
leakage magnetic flux in the three spaces between the four
windings. The maximum value is low and it has reached a peak
value of about 194 mT.
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2
-600
-400
-200
0
200
400
600
time (s)
curr
ent (A
)
iTiSiR
RMS: 416,06
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
time (s)
leak
age
mag
neti
c f
lux d
ensi
ty (
T)
Bleak_12 =195 mT
Bleak_23 =31 mTBleak_34 =20 mT
(a) (b)
Fig.8. Transformer normal operation: (a) Current waveform (b) Leakage flux
density.
Once the short circuit is established a new set of results are
obtained. Fig. 9(a) provides the currents waveform and shows
that the phenomenon occurs at 200 ms. The related leakage
flux is given in Fig. 9(b). It can be noticed that the maximum
value for this variable is 3.5 T. Again, this is within standard
values during short-circuit conditions.
0.15 0.2 0.25 0.3 -15000
-10000
-5000
0
5000
10000
time (s)
curr
ent
(A)
0.35
iR(peak)= -11061 A
iS (peak)=
8.103.3 A
i T(peak)= 8.311.2 A
0.15 0.2 0.25 0.3
time (s)
0.35-2
-1
0
1
2
3
4
leak
age
mag
net
ic f
lux d
ensi
ty (T
)
Bleak_12 =3,5 T
Bleak_34 =360 mT
Bleak_23 =920 mT
(a) (b)
Fig. 9. Short circuit condition: (a) currents waveform; (b) leakage flux density.
By knowing the time domain current, the radial force can be
founded throughout expression (4). The final result associated
to the highest current phase, for both the normal and short-
circuit conditions are given in Figs. 10 and 11, respectively for
the external and internal winding.
0.1 0.11 0.12 0.13 0.14 0.150
5000
10000
15000
20000
25000
30000
35000
40000
time (s)
rad
ial
forc
e (N
)
36397 N
0.1 0.11 0.12 0.13 0.14 0.15 0
5000
10000
15000
20000
25000
time (s)
rad
ial
forc
e (N
)
20344 N
(a) (b)
Fig.10. Radial force at normal operation. (a) external winding - 2; (b) internal
winding - 1.
0.15 0.2 0.25 0.30
2
4
6
8
10
12
14
time (s)
radia
l fo
rce
(N)
(x106)
12,3 M N
0.35 0.15 0.2 0.25 0.30
1
2
3
4
5
6
7
8
time (s)
rad
ial
forc
e (N
)
0.35
(x106)
7,0 M N
(a) (b)
4
Fig.11. Radial force in short-circuit: (a) external winding; (b) internal
winding.
A comparison between the results clearly shows that the
radial force occurring at the windings, during short circuit
situation, is increased severely with regard to the normal value.
The mechanical stresses associated to the radial force are
shown in Fig. 12 for the phase that present highest value.
0.2 0.22 0.24 0.26 0.28 0.30
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time (s)
stre
ss (
N/c
m2)
Inner winding (1):
5927.0 N/cm2
Outer winding (2):
9172.7 N/cm2
Fig.12. Mechanical stress during short-circuit: phase R
The stress mechanical was showed only for short-circuit
condition, since for normal condition it had a small value.
Case B. FEMM Results
To verify the previous results accuracy to obtain
transformers internal forces, the solution would be a
comparison with experimental results, however, no reference
publication concerning the tested transformer or others were
found. In this way, the analysis was carried out throughout two
software performance. One being the mentioned time domain
program and the other the well accepted finite element based
FEMM. Due to the fact the latter is a commercial program no
further discussions are made in relation to its potentiality and
use. This can be found in [3].
Fig. 13 provides the magnetic flux density distribution
during normal transformer operation for a particular instant.
Fig. 13. Magnetic flux density with normal transformer conditions.
Following standard procedures, the rated load current peak
values of 355A in the HV e 590 A in the LV were injected in
the windings. As it should be expected, with normal operating
conditions, the magnetic flux stays predominantly in the
ferromagnetic material.
On the other hand, the effect of applying an external short-
circuit, represented by an injection of 6510 A in the HV and
11061 in the LV windings can be seen in Fig. 14. The flux
density pattern can be easily seen. The leakage flux has been
largely increased due to the short-circuit conditions. The short
circuit current level was determined in accordance with (2)
using an asymmetry factor of 1.6. To provide means for a
comparative analysis, the chosen instant to represent the
magnetic situation of Fig. 14 is the same as with normal
conditions.
Fig.14. Magnetic flux density with short circuit conditions.
So as to have a better understanding about the leakage flux
distribution along the winding extension, Fig. 15 the flux
behavior in relation to the physical turns dimension. By
comparing the normal to the short-circuit situations, it can be
concluded that the leakage flux of about 200 mT went up to
about 3.6 T.
0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
1 1600height winding (mm)
leak
age
flu
x d
ensi
ty (
T)
normal condition short-ciricuit
800
Fig.15. Leakage flux density against winding height: normal and short circuit
conditions.
As the required information to feed equation (4) is known,
the forces can be readily calculated. These are given in the
following section.
Table II synthesizes the main results obtained from the
Time Domain and the FEMM approach.
TABLE II
RESULTS SUMMARY WITH NORMAL AND SHORT-CIRCUIT CONDITIONS
Normal Operation
Time domain FEMM
Leakage magnetic flux density [T] 195x10-3 200x10-3
Magnetic flux density [T] 1.6 1.6
Radial force [N]
Outer winding (2) 36397 36890
Inner winding (1) 20344 20580
Short Circuit Condition
Leakage magnetic flux density [T] 3.5 3.6
Magnetic flux density [T] - 1.32
Radial force [N]
Outer winding (2) 12300x103 13470x103
Inner winding (1) 7000x103 7540x103
5
The results derived from the two methods are in good
agreement. Therefore, by taking the FEMM calculations as
reference value, the time domain software performance has
shown to be appropriated for this paper purposes, i.e, to
calculate internal forces and further estimation of equipment
mechanical stress due to short-circuit conditions.
Mechanical stress was estimated only using the time domain.
Nevertheless, if the forces calculated by two numerical
methods are in agreement, hence the mechanical stresses also
they are. The maximum value founded for this variable was
9173 N/cm2. This is closer of hoop stress is closer of the
maximum hoop stress permissible in the copper which is 9806
N/cm2.
V. CONCLUSIONS
Using mmf and reluctance time domain transformer
modeling strategy, this paper has proposed a way to calculate
mechanical stresses occurring in the windings with short-
circuit conditions. Due to the absence of real site values, the
results were compared to as well established software based on
finite elements. The calculated values have demonstrated that
the transformer performance with normal and short-circuit
conditions are in close agreement. Therefore, as for the present
comparison, the time domain approach can be assumed as a
good procedure to achieve transformer internal forces and
subsequently stress evaluation. However, it must be stressed
the investigations were carried out for radial forces and
stresses. The paper did not took into account the axial force
and nor the stresses caused for them. However, this matter will
be investigated in the next paper.
VI. REFERENCES
[1] Kulkarni, S. V., Khaparde, S. A., “Transformer Engineering - Design
and Practice”, Marcel Dekker, Inc, New York, 2004.
[2] IEEE Guide for Failure Investigation, Documentation and Analysis for
Power Transformers and Shunt Reactors, IEEE Standard C57.125, 1991.
[3] Meeker, D., (September 2006) “Finite Element Method Magnetics -
User's Manual Version 3.4” [online]. Available: http://femm.berlios.de.
[4] Yun-Qiu, T., Jing-Qiu, Q., Zi-Hong, X., “Numerical Calculation of
Short Circuit Electromagnetic Forces on the Transformer Winding”,
IEEE Transaction on Magnetic vol. 26, No.2, March, 1990.
[5] Heathcote, J. Martin, “J&P Transformer Book”, 12th ed., Oxford,
Elsevier Science Ltd, 1998.
[6] Waters, M., “The Short-Circuit Strength of Power Transformers”,
McDonald & Co. Ltd, London, 1966.
[7] The Short Circuit Performance of Power Transformers, Brochure CIGRE
WG 12.19, 2002.
[8] Apolônio, R., Oliveira, J. C., Bronzeado, H. S., Vasconcellos, A. B.,
“The Use of Saber Simulator for Three-Phase Non-Linear Magnetic
devices Simulations: Steady-State Analysis”, The 7th Brazilian Power
Electronic Conference, Fortaleza, Ceará, Brazil, September, 2003, pp
524-529.
[9] Yacamini, R., Bronzeado, H. S., “Transformer Inrush Calculations using
a Coupled Electromagnetic Model”, IEE Proc. Sci Meas. Technol. Vol.
141, no. 6, November, 1994, pp 491-498.
[10] Slemon, G. R., “Magnetoelectric Devices: Transducers, Transformers
and Machines”, 5th ed, John Wiley and Sons, 1966.
VII. BIOGRAPHIES
Ana Claudia de Azevedo was born in São Gonçalo
do Abaeté, Brazil, in 1972. She received her BSc
from the Federal University of Mato Grosso, Brazil
in 1999, and her MSc degree from the Federal
University of Uberlandia, Brazil, in 2002. She is
currently working towards her PhD degree. Her
research interest areas are: Electromechanical Stress
in Power Transformers.
Antônio Carlos Delaiba was born in Botucatu,
Brazil, in 1954. He graduated from Federal
University of Barretos, Brazil in 1979. He received
his MSc degrees from of University of São Paulo,
Brazil in 1987 and his PhD degree from the Federal
University of Uberlandia, Brazil in 1997. He is
currently a professor and a researcher at the Faculty
of Electrical Engineering, Federal University of
Uberlândia. His research interest areas are Power
Quality, Transmission, and Distribution of Energy.
José Carlos de Oliveira was born in Itajuba, MG,
Brazil. He received his BSc and MSc degrees from
the Federal University of Itajubá–Brazil, and his
PhD degree from the University of Manchester -
Institute of Science Technology – Manchester-UK.
He is currently a professor and a researcher at the
Faculty of Electrical Engineering, Federal
University of Uberlandia - Brazil. He has taught
and published in a variety of subjects related to
Electrical Power Systems and Power Quality.
Bismarck Castillo Carvalho was born in Roboré,
Santa Cruz, Bolívia, in 1957. He received his BSc
from the Federal University of Mato Grosso, Brazil
in 1981 and his MSc and PhD degrees from the
Federal University of Uberlandia, Brazil. His
research interest areas are: Electrical Power System,
Power Quality and Renewable Energy concerned to
Wind Power.
Herivelto de Souza Bronzeado (M’ 97) was born
in Remígio, Paraiba, Brazil, on April 2, 1953. He
was graduated by the Universidade Federal da
Paraíba, Brazil, in July 1975, and since then he
works for the Companhia Hidro Elétrica do São
Francisco – CHESF, being responsible for Power
Quality and R&D Projects. He received his MSc
degree in Power System Engineering, in 1993, from
the University of Aberdeen, Scotland (UK). He is
also the chairperson of the Study Committee C4 of
Cigré-Brasil and the IEEE Joint Chapter
PES/IAS/PELS, Northeast 1, Section Bahia, Brazil.
His research interests include transformer modeling
for transient studies and Power Quality problems.
6
top related