Transactions
on
Power Delivery, Vol. 7 No.1,
January
1992
REDUCED ORDER
MODEL
FOR TRANSFORMER TRANSIENTS
Francisco de Leon Adam Semlyen
Department
of
Electrical Engineering
University
of
Toronto
Toronto, Ontario, Canada,
M5S
1A4
Abstract.
-
A complete model for transformers is derived on the base of
very efficiently calculated elementary (turn-to-turn) parameters.
A
high-order turn-to-turn model is constructed for the windings. This
model is reduced to a lower order by operating on the resulting matrices.
An electric equivalent circuit for the core is obtained from the principle
of duality. By the use of test turns the winding model is interfaced with
the iron-core. For validation, the frequency response of the model has
been compared with test results. The model for the calculation of tran-
sients has the form of a Norton equivalent circuit and it can easily be
incorporated in a power system transients program such
as
the
Em.
Examples of calculated transients are given for illustration and further
validation.
Keywords:
Transformer modeling, Electromagnetic transients.
INTRODUCTION
The progress in transformer modeling for the calculation of elec-
tromagnetic transients has not kept pace with the advances in the model-
ing of most other major power system components.
This
is probably due
to the complexity of the physical phenomena that take place in the
transformer. There has been much work reported in the literature but no
power transformer model for a wide range of frequencies is yet available.
In this paper we present a model for the transformer windings that is suit-
able for a wide frequency range.
In
reference [l] we computed the
parameters (leakage inductances and capacitances) in a very efficient
way
on a turn-to-turn basis. Now we use that information to form a
model for the windings. Then we combine the windings and iron-core
models to form a complete transformer model.
There are three main approaches followed in the modeling of
transformers:
a)
Modeling based on the principle
of
duality.
In references [2] and
[3] it was shown that an equivalent electric circuit for a transformer
can be derived from its magnetic circuit by applying the principle
of duality. Therefore, the leakage flux (rather than leakage induc-
tance) is used for modeling the phenomena in the air.
This
may
lead to inaccurate terminal response, which depends on the leakage
inductance and not on the leakage flux. This problem was solved
by the use of negative inductances; see references [4] and
[5].
The
parameters of these (duality based) models are calculated assuming
that the magnetic field is in axial direction in cylindrical geometry.
As shown
in
reference
[
11,
this
leads
to
a greater error than our pro-
posed image method when predicting the terminal leakage induc-
tance. However, a model based on the principle of duality reflects
properly the flux balance and thus the nonlinear iron-core can be
represented accurately. Recently, this approach has been used in
the modeling of transformers in highly saturated conditions; see
reference [6].
91
WM
126-1
PWRD
A paper recommended and approved
by the IEEE Transformers Committee of the IEEE
Power Engineering Society
for
presentation at the
IEEE/PES 1991 Winter Meeting, New York, New York,
February
3-7,
1991. Manuscript submitted
August 31, 1990; made available for printing
November
30.
1990.
361
b)
Modeling based on leakage inductances.
These models frequently
use an inverse inductance matrix, often referred to
as
r
[7].[11].
r
models reflect accurately the transfer characteristics of the
transformer (short circuit response) because their parameters are
obtained from short circuit tests. As we will show later in this
paper, there is no need
to
use the
r
matrix to construct a model
from leakage inductance information. The
r
models present a
difficulty in relation with the iron-core which has to be attached in
a heuristic way since the magnetizing effects are lost in the short
circuit tests.
Modeling based on self and mutual inductances.
There exist accu-
rate formulae to compute the self and mutual inductances for sec-
tions or windings of a transformer [12,13]. However, because of
the presence of the iron-core, the self and mutual inductances have
values that are very close in magnitude,
so
that this approach leads
to an ill-conditioned set of equations.
We use a combination of the first two approaches: leakage induc-
tances (turn-to-turn) are used for the modeling of the windings and the
principle of duality
is
used for
the
iron-core.
A model considering each turn as an independent entity would be
impractical due to its large size. To overcome
this
problem, we lump, by
matrix manipulations, as many turns
as
the frequency range we are
interested in permits. We assume for the calculation of the winding
inductance matrix that the geometry is axisymmetrical and that the iron-
core has infinite permeability. The magnetizing effects are then included
into the model by the use of test turns and the principle of duality.
>
c)
HIGH-ORDER WINDING MODEL
In reference [l] we have calculated the parameters taking the turns
as
basic elements for the capacitances and the leakage inductances. For
the calculation of the capacitances we have used the charge simulation
method with axisymmetric geometry. We have used the image method
for the calculation of the turn leakage inductances, based
on
axisym-
metric geometry. As a consequence, our model for the windings is not
able to represent the mutual leakage effects between different legs of a
transformer, it accounts only for the mutual effects between the turns
wound on the same leg. The inductive coupling through the leakage
paths (between two different legs)
is
believed to be negligible, especially
when it is compared with the tight coupling from the yokes. However,
the capacitive coupling between the windings on different legs and to the
tank may not be negligible. We can add external capacitance
to
represent
this
effect after the leg-by-leg model is constructed.
Inductive Model
For an
N
turn transformer the voltage-current governing equation
for the inductive part of the transformer (using leakage inductances,
equation (17) from
[I]).
is
The order of the matrix
L
'
is
N-1
because one of the turns serves as
.reference.
Relation (1) constitutes the backbone of the inductive model for
windings. A similar process for obtaining a model was described in [IO]
for complete. coils and it involves the inversion of the matrix L' to get an
inverse inductance matrix
r
'.
As we will
see,
obtaining the
r
'
matrix is
not necessary since we can integrate equation
(1)
and get a Norton
equivalent directly. We will lump several turns in series before integrat-
ing in order to reduce the model to a manageable size.
0885-8977/91rS3.ooQ1992
IEEE
362
Capacitive Model
The turn capacitance matrix
C'"",
obtained
in
reference
[I]
from
the charge simulation method, relates currents and voltages through the
turn-to-turn capacitances. We can use this information to produce a
nodal model. Shifting half of each capacitance to the two ends of the
turn, we can write the nodal equation
(2)
icnodc=C-
-
vdc
where the nodal capacitance matrix
Cde
can
be
obtained fmm the turn
capacitance matrix
c'"'"
by recognizing which elements are connected to
each node. The resulting matrix
Cnoh
is of order
2N,
with the general
terms given by
d
dt
C2i42j-1
=
CZ,2j
=
$5
cy (3a)
C2i42j
=
CZJj-1
=
0
(3b)
Figure
1
shows an equivalent circuit for the winding model. The
inductances are not explicitly shown. This model is of a very high order
since it is based on turn-to-turn variables.
TURN
3
I
I
C5.G:'
-
6*c6.G
-
Figure 2a. Turn-to-turn model (turns disconnected)
inductive
couuline
Figure
1.
Winding model
Figure
2b.
First lumping step
mdsnrmbo
d
mhr
formum3
fa.acuan
1
\
/
indueuvc
coupling
MODEL REDUCTION
To
reduce the model, we lump several series connected turns
to
form sections. The essence of the process is the shifting of some (inter-
nal) capacitances towards pre-established section nodes. Once the capa-
citances are moved to sectiori nodes, the loop inductances are easily
lumped.
In
order to show how the reduction is done, a 3-turn section
will be formed step by step; see Figures 2a to 2e.
Reduction
of
Capacitances
The first step is to move the capacitances connected to internal
nodes to the external ones (section nodes). For convenience, an odd
number of turns per section is chosen. The reduction of the capacitance
network can
be
done by operating on the nodal capacitance matrix
Cnodr.
Consider first that the
three
turns are disconnected (Figure 2a). The
nodal capacitance matrix for
this
arrangement is
Figure 2c. Capacitances moved to the section nodes
TURNS
NODES
123456
SECTION
NODES
12
6
Lower case
c
is used for capacitances in the circuit; upper case
C
for the
elements
in
the matrix. For example, Cll =clc +c13 +c15 and
C
13
=
-
c
13.
Connecting the end of one turn
to
the beginning of the next
one, gives the circuit shown in Figure 2b. By moving the capacitances
connected to the internal nodes (2-3 and
4-5)
towards the nearest external
node
(1
or
6).
we obtain the circuit shown
in
Figure 2c.
This
circuit has
the section capacitance matrix
CZ
=
Figure 2d. Reduction process
Note that we can obtain this matrix by operating on the
Cnde
matrix
(equation
(4)):
partitioning
Cnde
in four square matrices of order 3 and
adding all the elements in each submatrix, we obtain the capacitance
matrix
(5)
of the reduced model
(see
Figure
2d).
In
terms of the ele-
ments of the matrix
Cde,
the last equation can
be
written as
1
11
c22K33+2c13
c15K24+c26K35
C15+C24+C26fC35 c44c55+c66+2c46
(6)
r+
c;y&
=
363
There is no need to write explicitly the matrix
Cnde
of order
2N
since
all
the needed information is contained in the matrix
Ct"
which is
of order
N.
For our example, using equation
(3a)
and the definitions of
self
(CJ
and mutual (C,
,
i#J
capacitances in equation
(6).
we have
cy;
=
ciy+
!L?
(ciy
+
Cyy)
+
!L?
ciy
czg
=
c;y+
%
(Cg
+
Ciy)
+
!L?
ciy
cyg
=
CTf
=
c;y+
%
(
Cyy
+
Cyy)
The procedure described above can be generalized for any (odd)
number of
turns
per section and any number of sections per winding. The
general equatlons to form a winding are
1,
1.
1.
1,
Cfyde=x
CC!?
+
%xC:y
+
%xCEy
+
YZCE~
(7a)
1=1,
J
=4
where
Ii
=
ii
+
(wk-1)/2
-
1
ii
=
first turn in section
i
rn
=
turn
in the center of this section
wk
=
number of lumped
turns
in section
k
From equations (7a) and (7b) we can see that the reduction process con-
sists simply of an addition of elements in the turn capacitance matrix
by blocks. The boundaries for forming the blocks are the
turns
located at the beginning, at the center, and at the end of the section to be
formed.
Each section produces four entries
in
the node capacitance
matrix for sections
CFA,.
The procedure is presented schematically in
Figure 2e.
ctwn
1
2
. .
NS
-SECTIONS
\
1
2
3
4
. . .
2Ns-
SECITONNODES
12...2w,
...
2N
-
TURN
NODES
a
12
3
4
...
2Ns
.I
1
I
Figure 2e. General reduction procedure for the capacitive network
1
2
...
Ns
-
SECTlONS
,
L'
Figure
3.
General reduction procedure for the loop inductance matrix
Reduction
of
Inductances
Once the capacitances have been moved to the ends of the section,
the
turns
inside each section become connected in series (see Figure 2c).
Then, for a section containing
k+l
turns,
we have
(8)
j.
-
i.
.
1
-
=
Ii+Z
=
.
. .
=
Ii+k
Applying this condition to equation
(1).
we see that the equivalent induc-
tance is the sum of
all
elements (self and mutual). In our example
i
1
=
i2
=
i3
and equation (1) with turn
N
as reference becomes
v1
-vN
L'11 L'12 L'13
!v2
-
=
[
L'21 L'ZZ L'23]
5
1::
v3
-vN
L'31 L'32 L'33
Adding the three rows, yields
or, in compact form,
This process can
be
generalized to any number of
turns
per section
as illustrated in Figure
3.
Note that the reduction process is simply the
addition of elements in the matrix
L'
by blocks. Equation
(1)
would
become
where
M
=
number of sections
wi
=
number of lumped
turns
in section
i
N
=W1+WZ+
...
+wM+~
The matrix
L"
of equation
(9)
is a modified (reduced) version of
the loop inductance matrix
L'.
The new loops involve several simple
loops
in
series; the definition of a simple loop was given in
[
11.
COMPLETE MAGNETIC MODEL
The model for the windings of a transformer is given by equation
(9).
This model is calculated from leakage inductances assuming that the
iron-core is perfect. The model can predict accurately the transfer
characteristics of the transformer but it assumes that
p
=
-,
so that the
iron-core (i.e. the magnetizing current) is not properly represented. On
the other hand, iron-cored devices have traditionally been modeled using
the principle
of
duality with the disadvantages noted in the Introduction.
In
this section we show how the two models can be used together so that
their drawbacks are eliminated.
Test Turns
For the purpose of interconnecting the two models we require test
turns
that measure the leakage inductance and simultaneously the leak-
age flux.
This
can be achieved if the test turns are very thin. To measure
the total leakage flux per transformer leg, we use two test turns
as shown
in Figure
4.
Turn
CY
Figure 4. Test
turns
to measure the leakage flux
364
In Figure
4
turn
a
links the leg flux while turn
p
links only the
yoke flux. Thus the flux difference
$,
-
$p
is the leakage flux. Test turns
such
as
a
and
p
can be used to define sections of the iron-core for lump-
ing leakage fluxes into a single node. For example, the leakage flux
between our two test
turns
is lumped to node A (see Figure
4).
We cannot simulate turn
p
around the yoke with our image method
as
the required geometry is axisymmetric. We can, however, choose
another location for this test
turn
so that it links the same flux; see Figure
5.
If we consider that the leakage flux is axial, or almost axial, we can
estimate the radius of turn
p
as half the distance between the centerlines
of the legs of
the
transformer.
Figure
5.
Alternative test turn
The Augmented Matrix
L’,
We can compute an augmented matrix
L’,
including the
p
test turn,
using the test turn
a
as
the reference and following the process esta-
blished in reference [l]. Then, we have
In compact form, (1 1) becomes
(12)
d
dt
v,-~,e,=L”~--i,
where
V,=[V~,~,I~=[V~.Y~.
...
,vM,eyI
lT
i,
=
[iT
,
ip
lT
=
[i1
,
i2
,
. . .
,
iM
,
ig
1’
w,
=[WT.
1
]T=[w1
,
w2,
‘.’
,
WM,
1
IT
We have substituted
e,
for
vu
and
er,
for
vp
in equation (12) to make
clearer that they represent voltages due to the iron-core. There
are
two
reasons for taking turn
a
as the reference. First, it is a fictitious turn,
and thus equation (12) includes the voltages and currents of all actual
turns as variables. Second, its flux is the common flux that all turns link:
therefore, the voltage of turn
a
represents the common voltage
e,
of all
turns. Thus equation (12), rewritten as
(13)
d.
Z’,
v,
=
w,
e,
+
L”,
shows that the voltage of each section (or turn) has a common com-
ponent
e,
due to the flux through the leg and a component due to the flux
in the air.
We can include in equation (13) the resistance of the conductors, as
follows
v,
=
w,
e,
+
R,
i,
+
L”,
Ai,
(14)
dt
Here
R,
is a diagonal matrix whose elements are the sum of the resis-
tances of the turns in each section.
Note that
all
inductance matrices are calculated with the use of the
image method based on axisymmetric geometry.
As
a consequence, the
magnetic effects represented by these matrices have mutual coupling
only with the turns (or windings) that
are
wound on the same leg. The
image method does not permit to include the (small) mutual inductances
due to leakage fluxes between coils wound on different legs.
Model
from
Duality
Applying the principle of duality to a
three
phase (three-legged)
transformer, we obtain the electric equivalent circuit shown in Figure 6,
described in references [2]-[6]. This equivalent circuit consists of five
nonlinear inductors,
La, Lb, Lc, Ly,
and
b2,
that represent the flux in the
iron-core. It has three liiear (positive) inductors that represent the leak-
age flux,
La,, Lb,
and
L,.
The circuit also has a number of negative
(linear) inductors in series with the terminals (only two terminals per leg
are shown),
La,
. .
.
Lc2.
For the nonlinear inductor on leg
a
(if the current through it is
i,),
we have
4
dt
e,
=
-
$=$(i,)
from where we get
(16)
d
dt
e,
=
La(ia)
-i,
where
A similar equation can be obtained for each nonlinear inductor.
I
I I
0
Figure 6. Application of the principle of duality
Complete Model
The interfaces between the model based on leakage inductances
and the model from duality are the test turns
a
and
p
on each leg. Turn
a
can be considered
to
be the connection to the leg inductor
(La,
for exam-
ple) while turn
p
is used
to
connect the model to the yoke inductor (such
as
Lyl
).
In Figure 7 we show the complete model for one leg.
We can see that the sum of currents entering block
L”,
has to
be
zero:
N
k=l
cwk in,
-
i,
-
ip
=
0
(17)
The sum of
i,
and
ig
is the magnetizing current. Thus, from (17), we
have
N
k=l
cwk i,,
=
i,
+
ip
=
i,,
(
17a)
Figure 7. Complete model for one leg
of a transformer
We have a total of N
+
4
unknowns:
N
voltages
or
currents for the
sections
or
tums,
plus
e,,
eb, i,,
and
ip.
The equations for the complete
model are the N+l voltage equations given by equation (14), and two
nonlinear equations, as (16). for the iron-core elements, and the KCL
equation (17). Thus, we have a proper set of equations, some of which
are
nonlinear.
For a three-legged transformer we would have a model as shown in
Figure
8.
This
model can be obtained by substituting an L" block (see
Figure 7) for the inductance network in Figure
6.
It is important
to
note
that all the equivalent circuits derived from the principle of duality are
only electric representations of magnetic circuits. These circuits
are
nor-
mally referred to a common number of turns
(in
our models
this
number
is one).
In
all
cases (Figures 6, 7 and
8),
the actual electric connections
have
to
be done externally. Also,
the
capacitances
are
to
be connected to
the terminals of the
turn
(or
sections) externally to the magnetic model.
The equations describing
the
three-legged transformer are
Figure
8.
Complete model for a three-legged transformer
N
k=l
xwk
i,
-i,
-ig,
=O
N
k=l
N
k=l
xwk ib,
-
i,
-
is,
+
iy,
= 0
(21a)
cwk ib,
-
i,
-
ig,
+
iy,
=
0
(21b)
The last two equations (equations (21))
are
very important conceptually
since they represent the fact that the sum of the currents inside a window
is zero. These two equations correspond to the KCL in the nodes 1 and 2
of Figure 8 and represent, in the electric equivalent circuit, the mmfs (of
the magnetic circuit) around the windows.
Equations (18)
to
(21) form a fully determined set of equations for
a three phase (three-legged) transformer (magnetic model).
CALCULATION
OF
TRANSIENTS
Inductive Model
For the single legged transformer described above (by equations
(14). (16) and (17)) we obtain the transient model by applying the
trapezoidal rule of integration. From equation (14) we have
(22)
RTH
i,
+
w,
e,
=
v,
+
v!$
and integrating equation (16) we get
i,
=
G,
e,
+
i
p
The superscript
hisf
stands for history and accounts for the previous
values of current and voltage in the circuit. Equation (22) represents a
Thevenin equivalent for the windings, and equation (23) models the
iron-core with a Norton equivalent. Note that in equation (23) the con-
ductance
G
is a function of the current through the nonlinear inductor.
The derivation
of
these equations can be found in Appendix 1.
These two equations form together with
the
KCL
equation (equa-
tion
(17))
the transient model for the magnetic part of one leg of a
transformer. The equations have
to
be
solved by an iterative method
since they contain some nonlinearities.
When the core is not saturated
(or lightly saturated), the currents
i,
,
ig
through the inductors represent-
ing the iron are small compared with the currents in the actual turns.
This
allows to transfer them to the history vector
as
a first approximation, and
then solve the equations iteratively. This gives fast convergence and,
more importantly, permits to decouple the magnetizing (nonlinear) equa-
tions from the leakage equations. A more detailed discussion on conver-
gence is presented in Appendix 2. We can obtain a first approximation if
we estimate the magnetizin current
img
=
i,
+
ig
in equation (17a) and
transfer it to the history as
i&
.
men we get
k=l
After one iteration we can update this value. Combining equations (22)
with (24) we form
LJ
In compact form, equation (25) is
R,,
i,,
=
vOug
+
V&
(26)
For the iron-core we have the following relations: from equation (23),
i,
=
G
,
e,
+
ip
(27a)
and, similarly, for the inductor representing the yoke,
ig
=
~p
eyl
+
ir
(27b)
Equation (26) is
a
hybrid system of linear equations that can be solved
(after estimating an initial value for
i,
and
ip)
in combination with the
external circuit (including the capacitances) to get the first approximation
foril,
i2;.
.
,iN, v1,vZ;.~ .VN, i,, ip,
e,,ande,,.Later, weuseequa-