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496
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
Finite Element Based Transformer Operational Modelfor Dynamic Simulations
O. A. Mohammed
1
,
Z. Liu
1
,
S. Liu
1
,
N. Y. Abed
1
,
and L. J. Petersen
21
Florida International University, U.S.A.
2
Oﬃce of Naval Research, U.S.A.
Abstract
The transformer model proposed in this paper considers the eﬀects of the excitation levels as well as theperiodic ﬂuctuations of ac excitation on the winding self and mutual inductances during dynamic operation. Theinductance proﬁle is obtained from sequential FE solutions covering a complete ac cycle at various excitationlevels. The values are then used by table-look-up technique. The technical details, the creation of the inductancetables as well as the Simulink implementation, are explained. Simulation results show that the established modelis capable of restoring the nonlinear magnetization phenomena of transformer iron core. The signiﬁcance of this model is due to its accuracy and its applicability for dynamic simulation of interconnected components ina power system.
Introduction
Accurate electromagnetic transient studies, such as, harmonic load–ﬂow require accuratemodeling of networkelements and their components. The modeling of iron-core transformer plays an important role in the dynamicsimulation of power system transients such as inrush currents, short circuits, and fault conditions.The key point of iron core transformer modeling is the representation of nonlinear magnetization. Twocommonly used methods are the piece-wise linear curve and the simple saturated reluctance function [1-3].These approachesconsider the eﬀects of averageexcitation level on the ﬂux/inductance but ignore the ﬂuctuatingeﬀects of the ac excitation itself. For cases requiring high-precision modeling, this is not adequate.Reference [4] developed an FE based method for determining the saturated transformer inductances utilizingenergy perturbation. Reference [5] studied the transformer inductance variations with respect to the averageexcitation level and the periodic ﬂuctuations of ac excitation. The 2D proﬁles are used to describe each induc-tance. Making use of such an inductance deﬁnition, we proposed our new transformer model. As an example,a 187.8kW, 288/232V three-phase power transformer is studied. The transformer equation, inductance calcu-lation and 2D inductance table establishment, Simulink implementation in addition to simulation results arepresented.
Transformer’s Equation and Inductances Calculation
A. Basic equation The voltage and ﬂux linkage equations of the three-phase transformer are as follows:
u
abc
u
ABC
=
R
1
00
R
2
i
abc
i
ABC
+
ddt
ψ
abc
ψ
ABC
(1)
ψ
abc
ψ
ABC
=
L
1
M
12
M
21
L
2
i
abc
i
ABC
(2)Where,
u
abc
,
i
abc
,
R
1
,
L
1
, and,
ψ
abc
are the voltage, current, resistance, self inductance, and the ﬂux linkageof the primary winding.
u
ABC
,
i
ABC
,
R
2
,
L
2
, and
ψ
ABC
are the corresponding parameters of the secondarywinding.
M
12
and
M
21
are the mutual inductances between the primary and secondary windings.The inductances
L
1
,
L
2
,
M
12
, and
M
21
are considered as magnetization status dependent so as to accuratelyrepresent the nonlinear magnetization property of iron core. They are determined in terms of the maximumvalue and the phase angle of ac excitation during a complete electrical cycle.B. Inductance calculation and inductance tableInductances are evaluated using the energy perturbation mehtod [4]. While performing the energy perturba-tion algorithm, the magnetizing currents of the primary winding are assigned to
i
abc
; zero currents are assigned
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
497
Figure 1: waveform of rated magnetizing current Figure 2: inductance proﬁle of Laato
i
ABC
. The energy of the transformer is calculated based on the nonlinear FE magnetic ﬁeld analysis of thetransformer. The magnetizing currents of the primary winding is determined through circuit-FE diect coupling,while the primary winding is fed with sinusodal voltage source and the secondary winding is open-circuited.Fig. 1 shows the obtained rated excitation current waveform of the 187.8kW, 288/232V three-phase powertransformer.The excitation level is represented by the magnitude of
ψ
m
. The determination of
ψ
m
is as follows:
ψ
m
=
(
ψ
2
α
+
ψ
2
β
) (3)Where,
ψ
α
=
(
u
α
−
R
1
i
α
)
dt
,
ψ
β
=
(
u
β
−
R
1
i
β
)
dt
.
u
α
,
u
β
,
i
α
, and
i
β
are obtained by transferring thesinusoidal voltage and magnetizing current of the primary winding from
a
−
b
−
c
coordinate system to
α
−
β
coordinate system. While building the 2D inductance table, the excitation level is adjusted by changing themagnitude of the sinusoidal voltage source.The phase angle of the ac excitation during a complete ac cycle is identiﬁed by
θ
. It is calculated uding theformulation below:
θ
=
tg
−
1
(
ψ
β
/
ψ
α
) (4)Table 1: 2-dimentional inductance table
θψ
m
1
0
2
0
3
0
· · ·
358
0
359
0
360
0
25% 0.0426 0.0425 0.0423
· · ·
0.0427 0.0428 0.043150% 0.0430 0.0428 0.0426
· · ·
0.0423 0.0425 0.0426100% 0.0424 0.0420 0.0417
· · ·
0.0417 0.0421 0.0425150% 0.0412 0.0410 0.0405
· · ·
0.0416 0.0417 0.0419Using the primary winding self inductance
L
aa
as an example, Table I gives the structure of the 2D inductancetable and Fig.2 shows the inductance proﬁle.
Simulink Implementation and Simulation Results
In our previous work, two procedures were proposed to build the machine model in Simulink; equation-based and circuit component-based [6-7]. For the implementation of the transformer equation (1), the circuitcomponent-based model is adopted to allow arbitrary connection (Wye or Delta) of the transformer threephase winding. In the circuit component-based model in reference [7], an adjustable inductance component wasdeveloped to represent the rotor position dependent self inductances. Here, a new procedure is proposed torepresent the magnetization status dependent self inductances.A constant inductance term is separated from each varying self inductance, as shown in Fig.3. The constantterm is used to apply the initial condition of inductance. The varying term is used to reﬂect the inductancevariation with the iron core magnetization status. Equation (5) gives the ﬂux equation rewritten in terms of the separated inductances. Where
L
′
aa
,L
′
bb
,
· · ·
are the constant inductance terms.
498
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26Figure 3: inductance separation Figure 4: circuit diagram of phase a winding
ψ
a
ψ
b
ψ
c
ψ
A
ψ
B
ψ
C
=
L
aa
L
ab
L
ac
M
aA
M
aB
M
aC
L
ba
L
bb
L
bc
M
bA
M
bB
M
bC
L
ca
L
cb
L
cc
M
cA
M
cB
M
cC
M
Aa
M
Ab
M
Ac
L
AA
L
AB
L
AC
M
Ba
M
Bb
M
Bc
L
BA
L
BB
L
BC
M
Ca
M
cB
M
cC
L
CA
L
CB
L
CC
i
a
i
b
i
c
i
A
i
B
i
C
=
L
′
aa
0 0 0 0 00
L
′
bb
0 0 0 00 0
L
′
cc
0 0 00 0 0
L
′
AA
0 00 0 0 0
L
′
BB
00 0 0 0 0
L
′
CC
i
a
i
b
i
c
i
A
i
B
i
C
+
L
′′
aa
L
ab
L
ac
M
aA
M
aB
M
aC
L
ba
L
′′
bb
L
bc
M
bA
M
bB
M
bC
L
ca
L
cb
L
′′
cc
M
cA
M
cB
M
cC
M
Aa
M
Ab
M
Ac
L
′′
AA
L
AB
L
AC
M
Ba
M
Bb
M
Bc
L
BA
L
′′
BB
L
BC
M
Ca
M
cB
M
cC
L
CA
L
CB
L
′′
CC
i
a
i
b
i
c
i
A
i
B
i
C
(5)For simplicity, the back EMF of phase “
a
” is given below as an example:
e
a
=
dψ
a
dt
=
L
′
aa
di
a
dt
+
e
a
′′
(6)
Figure 5: Block diagram of the transformer model in Simulink
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
499
e
′′
a
=
ddt
L
′′
aa
i
a
+
L
ab
i
b
+
L
ac
i
c
+
M
aA
i
A
+
M
aB
i
B
+
M
aC
i
C
(7)The circuit diagram of phase “
a
” winding is shown in Fig. 4. The controlled voltage source is used torepresent the back EMF term
e
′′
a
.Fig.5 is the circuit diagram of the transformer model. Subsystems 1 and 2 are used to calculate the magnitudeof
ψ
m
and the phase angle
θ
. According to
ψ
m
and
θ
, the inductances are picked up from the 2D tables storedin blocks
L
1,
M
12,
L
2, and
M
21. The ﬂux linkage and back EMF of the primary and secondary windings arecalculated using equations (1) and (5).Table 2
L
aa
=43.9
L
ab
=-22.1
L
ac
=-21.4
M
aA
=35.4
M
aB
=-17.8
M
aC
=-17.3
L
ba
=-22.1
L
bb
=44.4
L
bc
=-22.1
M
bA
=-17.8
M
bB
=35.9
M
bC
=-17.8
L
ca
=-21.4
L
cb
=-22.1
L
cc
=43.8
M
cA
=-17.3
M
cB
=-17.8
M
cC
=35.4
M
Aa
=35.4
M
Ab
=-17.8
M
Ac
=-17.3
L
AA
=28.6
L
AB
=-14.4
L
AC
=-13.9
M
Ba
=-17.8
M
Bb
=35.8
M
Bc
=-17.8
L
BA
=-14.4
L
BB
=29.0
L
BC
=-14.4
M
Ca
=-17.3
M
Cb
=-17.8
M
Cc
=35.4
L
CA
=-13.9
L
CB
=-14.4
L
CC
=28.6
Time s
C u r r e n t ( A )
(a) Time s
C u r r e n t ( A )
(b)
Figure 6: Magnetizing current waveform obtained by, (a) using inductances in Table 1, (b) using inductancesin Table 2For comparison purpose, the mean values of the transformer winding inductances are calculated also, whichare given in Table 2. Then, the no load experiment is performed using the inductances in Table 1 and Table 2respectively.Fig.6 shows the magnetizing current waveform obtained from simulation. Comparison of Fig.6(a) andFig.6(b) indicates that the proposed transformer model restores the nonlinear magnetization phenomenon of the iron core. Comparison of Fig.6(a) with Fig. 1 shows that the proposed FE based transformer model can beconsidered as accurate as the full FE model. In addition, the FE based transformer model supports very fastsimulation speed, while the full FE model is computational cumbersome.
Conclusion
An accurate transformer model is proposed for dynamic simulation purposes. It uses the magnetizationstatus dependent inductances to restore the nonlinear magnetization behavior of the transformer iron core. Theinductance variations due to the excitation level and the periodic ﬂuctuations of ac excitation are considered,which are obtained from sequential FE solutions. The deﬁnition of 2D inductance table is given and its im-plementation in Simulink is studied. Veriﬁcation examples show the correctness and validity of the developedtransformer model. Compared with the conventional transformer models, the proposed model provides anaccurate description of the iron core magnetization behavior and its applicability to dynamic simulations.
REFERENCES
1. leon, F. de and A. Semlyen, “Complete Transformer Model for Electromagnetic Transients,”
IEEE Trans.Power Delivery
, Vol. 9, No. 1, 231-239, 1994.

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