Download Transformer Models

IEEE PES General Meeting
July 13 - 17, 2003, Toronto
Transformer Modeling for
Simulation of Low Frequency
Transients
J.A. MARTINEZMARTINEZ-VELASCO
Univ. Politècnica Catalunya
Barcelona, Spain
B.A MORK
Michigan Tech.
Tech. Univ.
Houghton,
Houghton, USA
Introduction
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Large number of core designs
Some of transformer parameters are both
nonlinear and frequency dependent
Physical attributes whose behavior may
need to be correctly represented
‹ core and coil configurations
‹ selfself- and mutual inductances
‹ leakage fluxes
‹ skin effect and
between coils
proximity effect in coils
hysteresis
‹ magnetic core saturation and
‹ eddy current losses in core
‹ capacitive
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effects
Aim of this presentation
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Transformer Models
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Matrix representation
‹ BCTRAN
model
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Saturable Transformer Component (STC)
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Topology-based models
‹ Duality
based models
‹ Geometric
models
Transformer Models
Matrix representation (BCTRAN model)
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Branch impedance matrix of a multi-phase multiwinding transformer
‹
Steady state equations
‹
Transient equations
[V ] = [ Z ] [ I ]
[ v] = [ R] [i ] + [ L] [ di / dt ]
[R] and jω[L] are the real and the imaginary part of [Z],
whose elements can be derived from excitation tests
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The approach includes phase-to-phase couplings,
models terminal characteristics, but does not consider differences in core or winding topology
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Transformer Models
Saturable Transformer Component (STC model)
L1
R1
Lm
N1 : N2
λ
i
R2
L2
Rm
ideal
N1 : NN RN
.
.
.
LN
ideal
StarStar-circuit representation of singlesingle-phase NNwinding transformers
Transformer Models
Models derived using duality
Core design
Equivalent circuit
DualityDuality-derived model for a singlesingle-phase shellshellform transformer
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MODEL
EQUATIONS
• [R] − [ωL] option
Matrix Representation
(BCTRAN model)
[ v ] = [ R] [ i ] + [ L] [ di / dt ]
• [A] − [R] option
[di / dt ] = [ L]−1[v ] − [ L]−1[ R][i ]
−1
Saturable Transformer
Component (STC model)
Topology-based models
−1
[ L] [v ] = [ L] [ R][i ] + [di / dt ]
CHARACTERISTICS
• These models include all phase-to-phase
coupling and terminal characteristics.
• Only linear models can be represented.
• Excitation may be attached externally at
the terminals in the form of non-linear
elements.
• They are reasonable accurate for frequencies below 1 kHz.
• It cannot be used for more than 3
windings.
• The magnetising inductance is connected
to the star point.
• Numerical instability can be produced with
3-winding models.
• Duality-based models : They are de- • Duality-based models include the effects of
saturation in each individual leg of the
rived using a circuit-based approach
core, interphase magnetic coupling, and
without a mathematical description
leakage effects.
• Geometric models
• The mathematical formulation of geome[v ] = [ R][i ] + [dλ / dt ]
tric models is based on the magnetic
equations and their coupling to the
electrical equations, which is made taking
into account the core topology. Models
differ from each other in the way in which
the magnetic equations are derived.
Nonlinear and Frequency-Dependent
Parameters
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Some transformer parameters are nonlinear
and/or frequency-dependent due to
‹ saturation
‹ hysteresis
‹ eddy
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currents
Saturation and hysteresis introduce distortion in waveforms
Hysteresis and eddy currents originate losses
Saturation is predominant in power transformers, but eddy current and hysteresis effects
can play an important role in some transients
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Nonlinear and Frequency-Dependent
Parameters
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Modeling of iron cores
‹ Iron
core behavior represented by a relationship between the magnetic flux density
B and the magnetic field intensity H
‹ Each magnetic field value is related to an
infinity of possible magnetizations depending on the history of the sample
‹ To characterize the material behavior fully,
a model has to be able to plot
• major and minor hysteresis loops
(minor loops can be symmetric or asymmetric)
Nonlinear and Frequency-Dependent
Parameters
B
Initial curve
Anhysteretic curve
Major loop
Symmetric minor loop
H
Asymmetric minor loop
Magnetization curves and hysteresis loops
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Modeling of iron cores
+
z
i
Equivalent circuit for reprerepresenting a nonlinear inductor V
R
I
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Hysteresis loops have a negligible influence on the
magnitude of the magnetizing current
‹
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Hysteresis losses can have some influence on some
transients; the residual flux has a major influence on the
magnitude of inrush currents
The saturation characteristic can be modeled by a
piecewise linear inductance with two slopes, except
in some cases, e.g. ferroresonance
Eddy current effects
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Excitation losses are mostly iron-core
losses
‹ hysteresis
and eddy current losses
‹ they cannot be separated
‹ hysteresis losses are much smaller than
eddy current losses
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Eddy current models for
‹ transformer
windings
‹ iron laminated cores
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Eddy current effects
Models for windings
R1
R2
RN
L1
L2
LN
R0
Series Foster equivalent circuit
Eddy current effects
Models for iron laminated cores
R1
L1
R2
L2
L1
R1
RN
L2
R2
Standard Cauer
equivalent
LN
LN
RN
Dual Cauer
equivalent
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Transformer Models
Matrix representation (BCTRAN model)
A
B C
a
b c
Winding Leakages
BCTRAN Model
ZL
ZY
ZL
ZL
ZY
Transformer Core Equivalent
BCTRAN model for a threethree-phase threethree-legged
stacked core transformer
Parameter Determination
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Data usually available for any power transformer
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
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power rating
voltage rating
excitation current
excitation voltage
excitation losses
shortshort-circuit current
shortshort-circuit voltage
shortshort-circuit losses
saturation curve
capacitances between terminals and between windings
Excitation and shortshort-circuit currents, voltages and
losses must be provided from both direct and
homopolar measurements
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Parameter Determination
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An accurate representation for threethree-phase core
transformers should be based on the core topology,
include eddy current effects and saturation/hystere
saturation/hystere-sis representation
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A very careful representation and calculation of
leakage inductances is usually required
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CoilCoil-capacitances have to be included for an accurate
simulation of some transients
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Since no standard procedures have been developed,
a parameter estimation seems to be required regardregardless of the selected model
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Temperature influence should not be neglected
Modeling - Case 1
Three-legged stacked-core transformer
Cross section of core and winding assembly
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Modeling - Case 1
L0
Rh
Rl
Ll
R0
L0
Rh
Lm
Rm
Lm
Rm
Lm
Ly
Ry
Ll
R0
Rm
Rl
Ly
Ry
L0
Rh
Rl
Ll
R0
Three-legged stacked-core transformer
Duality-based equivalent circuit
Modeling - Case 1
U
Three-legged stacked-core transformer
ATP implementation
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Modeling - Case 1
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Phase A
Phase B
Phase C
Current (A)
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0
-5
-10
100
120
140
160
Time (ms)
180
200
Three-legged stacked-core transformer
Excitation currents
Modeling - Case 1
Phase A
Phase B
Phase C
180
120
60
0
-60
-120
180
120
60
0
-60
-120
180
120
60
0
-60
-120
0
100
200
Time (ms)
300
400
500
Three-legged stacked-core transformer
Inrush currents
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Ferroresonance - Case 2
CORE
U
SRC
SRCX
X1
Rw
Leak
Rc
98_8Seg
Magnetic Saturation - Case 2
0.8
0.8
Fluxlinked [Wb-T]
Fluxlinked [Wb-T]
0.6
0.6
0.4
0.4
0.2
0.2
I [A]
0.0
I [A]
0.0
0.0
62.7
125.3
188.0
250.7
0.0
62.7
125.3
188.0
250.7
8-Segment Curve vs. 2-Segment Curve
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Steady-State Excitation - Case 2
Excitation at Rated Voltage – 8-Segment Curve
200
2.0
[V]
[A]
150
1.5
100
1.0
50
0.5
0
0.0
-50
-0.5
-100
-1.0
-150
-1.5
-200
0
10
20
(file FR_Mart.pl4; x-var t) v:XFMR
30
40
[ms]
-2.0
50
c:SRC -XFMR
Ferroresonance - Case 2A
Ferroresonance: 15uF, 8-Segment Magnetization Curve
400
12.0
[V]
[A]
300
6.8
200
100
1.6
0
-3.6
-100
-200
-8.8
-300
-400
0.00
11.11
(file FR_Mart.pl4; x-var t) v:X1
22.22
v:SRCX
33.33
44.44
55.56
[ms]
-14.0
66.67
c:SRC -SRCX
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Ferroresonance - Case 2B
Ferroresonance: 15uF, 2-Segment Magnetization Curve
350.0
50
[V]
[A]
262.5
35
175.0
20
87.5
0.0
5
-87.5
-10
-175.0
-25
-262.5
-350.0
0.00
11.11
(file FR_Mart.pl4; x-var t) v:X1
22.22
v:SRCX
33.33
44.44
55.56
[ms]
-40
66.67
c:SRC -SRCX
Effect of Mag Curve Representation
400
[V]
300
Ferroresonant Voltage: Red(2A): 8-Seg; Blue(2B): 2-Seg @ 1.0A; Green: 2-Seg @ 1.5A
200
100
0
-100
-200
-300
-400
0.00
11.11
22.22
33.33
44.44
55.56
[ms]
66.67
FR_Mart.pl4:v:X1
FR_MART1p5.pl4:v:X1
FR_MART1p0.pl4:v:X1
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Effect of Mag Curve Representation
Ferroresonant Current: Red(2A): 8-Seg; Blue(2B): 2-Seg @ 1.0A; Green: 2-Seg @ 1.5A
45
[A]
30
15
0
-15
-30
-45
0
10
20
30
40
[ms]
50
FR_Mart.pl4:c:SRC -SRCX
FR_MART1p5.pl4:c:SRC -SRCX
FR_MART1p0.pl4:c:SRC -SRCX
Conclusions
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There is no agreement on the most adequate
model
Modeling difficulties
‹ great
variety of core designs
‹ nonlinear and frequency dependent parameters
‹ inadequacy for acquisition and determination of
some transformer parameters
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Several modeling levels could be considered since not all parameters have the same
influence on all transient phenomena
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