Download estimation of parameters of multi-port equivalent scheme for

Zeszyty Problemowe – Maszyny Elektryczne Nr 3/2014 (103)
19
Joseph El Hayek *, Tadeusz Sobczyk **
* University of Applied Sciences and Arts Western Switzerland, School of Engineering, Sion
** Politechnika Krakowska, Instytut Elektromechanicznych Przemian Energii
ESTIMATION OF PARAMETERS OF MULTI-PORT EQUIVALENT
SCHEME FOR MULTI-WINDING TRACTION TRANSFORMERS
Abstract: This paper aims to present a new equivalent scheme of multi-windings traction transformers, based
on multiport purely inductive circuit. The mathematical background of this equivalent scheme is described.
The determination of the different scheme elements is made through a finite-elements calculation of both main
and leakage inductances, for the case of a four-winding transformer. A procedure is defined, which allows to
estimate the values of these elements from some measurements on the transformer at no-load and short-circuit
operations. A specific strategy of short-circuit tests is described, allowing to determine all parameters in a
rather simple way.
Keywords: multi-winding transformer, equivalent scheme, multi-port circuit, traction transformers
1. Introduction
The equivalent scheme of a transformer is a
basic tool in electrical engineering of alternating currents. Its “classical” form is used
everywhere when magnetic coupling exists.
Commonly an equivalent scheme of T-type
with one vertical magnetizing branch is used.
It is well known that a transformer having
more than three magnetically coupled
windings cannot be represented uniquely by a
T-type equivalent scheme. In order to be described correctly, three coupled windings need
three self- and three mutualinductances.
Anequivalent scheme has to include the same
number of independent parameters. The Ttype equivalent scheme for three windings
needs accordingly six parameters: three leakage inductances, one common magnetizing inductance and two winding ratios, recalculating
the differentparameters to a reference winding.
So, six independent inductances can be
uniquely represented by six parameters of the
equivalent scheme.In case of four coupled
windings there are ten independent quantities:
four self-inductances and six mutualones, but
the T-type equivalent scheme with one magnetizing branch presents only eight
parameters: four leakage inductances, the
common magnetizing inductance and three
winding ratios. It is thereforeimpossible to
represent in auniquewaymore than three
magnetically coupled windings by the
equivalent scheme of T-type.
In [2]the multi-port circuit has been proposed
as an equivalent circuit of an N
magneticallycoupled
coilsset.
In
that
representationthe number of inductive
elements is always equal to the number of
independent self and mutual inductances for
an arbitrary number of coils. In [3][4][5] this
approachhas been applied as an equivalent
scheme ofamulti-winding traction transformer.
Traction transformers have many windings
with very different tasks within the
locomotive. There are windingsconnecting the
locomotive to the traction supply system,
some ones supplying power electronics drives
and others feeding the auxiliary systems of the
train. A proper representation of traction
transformers is of a great interest for designers
as well as for users.
In [5] an equivalent scheme of a laboratory
model traction transformer with four
windingshas been developedand a procedure
of determining its parameters by field
computation has been described, including
magnetic non-linearity of the transformer core.
This paper aims to present a procedure, which
allows estimating these parameters from some
measurements on the transformer by no-load
and short-circuit operations.
2. Background for themultiport equivalent
scheme
A set of magnetically coupled coils is
modelled by relations between coils flux
linkages and coils currents. Assuming
magnetic linearity, a set of coils is described
by relation (1)
Ψ =Li
(1)
Zeszyty Problemowe – Maszyny Elektryczne Nr 3/2014 (103)
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where:
(5)
i=WΨ
Ψ - is the vector of flux linkages
where:
Ψ T = [ψ 1 ψ 2 L ψ N ]
 W1,1 W1,2 L W1, N 

W2,2 L W2, N 
−1

W=L =

O
M 


WN, N 
( sym)
i - is the vector of winding currents
iT = [i1 i2 L iN ]
Taken into account (4) the matrixWcan be
written in the following form
L - is the inductance matrix
L12 L L1N 
 L11

L 22 L L 2N 

L=

O
M 


L NN 
( sym)
In order to obtain the multi-port equivalent
scheme,one has to express the coil currents in
function of the flux linkages. This means that,
instead of the classical relation(2),
ψn = Ln,1 i1 + L n,2 i2 + L + L n, N iN
(2)
weshould apply the expression(3)
in = Wn,1 ψ1 + Wn,2 ψ2 + L + Wn, N ψ N
(3)
for n = 1,2,..., N , in which the current of the
winding ‘n’ depends on each flux linked to
each winding.Because the windings magnetic
coupling is not ideal, the matrix of inductances
L is not singular and the relations (2)
exist.Using an analogy to a description of
purely resistive circuit by the node potential
method, relations (3) can be writtenas (4)
in =
1
1
1
⋅ (ψ n −ψ 1 ) + L + e ⋅ψ n + L + e ⋅ (ψ n −ψ N )
Len,1
Ln
Ln, N
(4)
It this expression the flux linkages replace the
node potentials and the inductances Len ,k and
Len replace therespective resistances. Actually,
this substitution is possible, since the relations
between flux linkages and winding currents
are algebraic, like the relations between
potentials and currents in a resistive
network.Therefore one can write the relation
N
1
1
 e +∑ e
L
L
 1 kk ≠=11 1,k


W=



 ( sym)


−
1
Le1,2
N
1
1
+∑ e
e
L 2 k =1 L 2,k
k ≠2




1

L
− e
L 2, N 


O
M

N
1
1 
+
∑
LeN k =1 LeN,k 
k ≠N

L
−
1
Le1, N
(6)
The inductances in this matrix with the
superscript'e'are related to the elements of the
matrix W according to the following
expressions:
1
Len ,k
= − Wn ,k , for k ≠ n
N
1
∑ Wn ,k
e =
L n k =1
(7)
(8)
Treating the matrix W in the form (6) as a
conductance
matrix
of
purely
G
resistivecircuit described by the relations
i = G v (where v is a vector of node
potentials), the relations i = W Ψ can be
interpreted as an N-port, purely inductive
circuit with ‘N’ nodes, providing the flux
linkages ψ1 , ψ 2 ,..., ψ N . The nodes are
connected to each other by the inductances
Len ,k and to a reference node by the
inductances Len . The winding currents
i1 , i2 ,..., iN supply respective nodes. To fulfil
the voltage equations
un = R n in +
dψn
, for n = 1,2,..., N
dt
(9)
Zeszyty Problemowe – Maszyny Elektryczne Nr 3/2014 (103)
the winding resistance R n has to be added to
each port.
A typical multi-port equivalent scheme for a
four-winding transformer is shown in Fig.1, in
which the inductances Len lay in vertical
branches connecting all nodes to the reference
e
one and the inductances Ln , k connect all
nodes constituting the N-polygon.
Le3,4
i4
u4
L 2,4
R4 Le1,4
Le1,2
R1
u1
i1
ψ1
Le4
Le1
i3
e
Le1,3
i2
ψ4
ψ2
R3
Le3,4
Le3
ψ3
u3
R2
Le2
u2
Fig. 1. Equivalent scheme of a four-winding
transformer
In the new multi-port equivalent scheme there
are many ‘vertical inductances’ instead of a
single common ‘magnetizing inductance’ in
the “classical” T-type equivalent scheme. The
total number of equivalent inductances in that
multi-port equivalent scheme is exactly equal
to the number of independent elements of the
inductance matrix. So, the multi-port
equivalent scheme can represent precisely any
inductance matrix, i.e. any multi-winding
transformer, even without recalculating the
parameters to one reference side. Inductances
appearing in the multiport equivalent scheme
can be collected into the matrix (10).
 Le1
Le1,2 L Le1, N 


Le2 L Le2, N 

Le =

O
M 


LeN 
( sym)
(10)
The elements of that matrix characterize all
inductances of the multi-port equivalent
scheme. The inductances Len are located in the
vertical branches and the inductances Len,k
connect the respective nodes of the upper
polygon.
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3. Equivalent scheme of traction transformers
A traction transformer is located in the train.
In this sense, it does not belong to a static
supply station. Its main usage is to bring the
single-phase catenary’s voltage level to
values, which are appropriate to the power
electronics and the traction motors. Static
converters are fed by the means of low voltage
sources, which are the secondary windings of
traction transformers. Whenever more than
two single-phase power sources are needed,
we have recourse to a transformer with several
secondary coils in order to save place and
weight. Generally such transformers have also
other low voltage windings used for supplying
different auxiliary devices within the train,
such as lights, heaters or air conditioners for
example. Moreover, especially in Europe
where the railways supply networks operate at
different
voltages
and
frequencies,
transformers should be able to work under
diverse systems; this leads to use even more
coils in order to achieve a wide range of
operations. Typically, a transformer can
involve up to 24 different winding parts.
In [4] the multiport equivalent scheme for a
model transformer has been created and
investigated. This transformer operates at two
network’s frequencies: 16.7 Hz and 50 Hz. It
consists of a two-limb magnetic core, and
several windings dispatched as in Fig.2:
- on the primary side:
4 x High-voltage windings (HT) connected in
parallel in normal configuration.
- on the secondary side:
4 x Traction windings (Tr) which supply the
locomotive motors through static converters.
4 x Filter windings (Fi) designed for filtering
harmonic currents on the transformer primary
side and further to the supply network.
In order to insure the dual frequency
operation, one winding per block had to be
divided into two parts. This leads to the
representation of each windings group by four
windings, instead of three like in Fig.2. In [4]
a group of four windings has been modelled
and the equivalent scheme has been
determined. Such windings are magnetically
coupled through a common flux in the iron
core. However, windings are magnetically
coupled in the air also, which varies for each
pair of windings. So, the inductance matrix
Zeszyty Problemowe – Maszyny Elektryczne Nr 3/2014 (103)
22
can be divided into two parts: the matrix of
main magnetizing inductances, due to the
coupling through the iron, and the matrix of
leakage inductances representing the coupling
in the air.
- the matrix of leakages inductances
53.1
 1.1
Lσ = 
 49.7

 52.2
1.1 49.7 52.2
53.1 52.3 49.8 
[H]
52.3 53.1 1.1 

49.8 1.1 53.1 
(13)
The equivalent scheme of themodelled group
of windings is exactly the same as in Fig.1.
The inductances appearing in this equivalent
scheme, obtained by the procedure described
in the previous sections and arranged as in
(10), are summarized in (14):
Fig. 2.Layout of a traction transformer
After recalculating all inductances to a
reference number of turns, exemplary of the
winding denoted as '1', the inductance matrix
takes the form
1
1
L = Lµ 
M

1
1
1
M
1
σ
 Lσ1
L'1,2

L'σ2
+


(sym)
L 1
L 1
+
O 1

L 1
L L'1,σ N 

L L'σ2, N 
(11)
O
M 

L' σN, N 
In [5] calculation results of inductances in
those matrices are presented,using FLUX2Das
finite elements software.It has been shown that
whereas the Lµ value changed with the
saturation of the iron core, the leakage
inductances in the second matrixremain
constant.For unsaturated state of the
transformer the following data have been
obtained:
- the matrix of main inductances
1
1
L µ = 4600
1

1
1 1 1
1 1 1
[ H]
1 1 1

1 1 1
 18521 67.3 - 176 - 210 
 67.3 18592 - 209 - 175 
[H] (14)
Le = 
 - 176 - 209 18555 67.3 


67.3 18558
 - 210 - 175
These results show consequently, that the
procedure based on field computation, allow
the determination of the multi-port equivalent
scheme.
4. Estimation of
equivalent scheme
measurements
the multiport
parameters by
Usually the equivalent scheme parametersof
transformers are determined by no-load and
short-circuit measurements. The elements of
the multi-port equivalent scheme can be also
found from such testscarried out in a different
manner.
4.1. The no-load test
For the no-load operation, when only the
winding '1' is supplied by asinusoidal voltage
with apulsation Ω , and all the other windings
are open, the transformer is described by the
equations
(
)
U0 = R + jΩ(Lµ + Lσ I 0
in which:
I 0T = [I1 0 0 0]
(12)
U 0T = [U1 U'2
U'3 U'4 ]
(15)
Zeszyty Problemowe – Maszyny Elektryczne Nr 3/2014 (103)
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As a first estimation, if we neglect resistances
and leakage inductances of all windings, (15)
leads to U1 = jXµ I1 . On the other hand the
Assuming that X e µ is sufficiently high, the
currents in the short-circuited windings are
multi-port equivalent scheme under such
assumption becomes as in Fig. 3.
I'2,sc =
I'3 = 0
I' 4 = 0
U '3
I' 2 = 0
U'4
I1
jX e 4
U'2
e
e
e
fulfils therelation U1 = j (Xeµ /4) I1 . So, in
general case the vertical inductances in the
multiport equivalent scheme can be estimated
as
Le1 = Le 2 = L = Le N = N Lµ
(16)
4.2. The short-circuit tests
The classical short-circuit test for the T-type
equivalent circuit is not useful for the
multiport one. In [3] the following strategy has
been shown to be very effective:one winding
is supplied by a limited voltage having all
other windings short-circuited; and this
issuccessively repeated for all windings. The
equations, when the winding '1' is
supplied,take the form
)
Usc = R + j ( Xµ + Xσ ) Isc
(17)
where:
T
[
I'2,sc
I'3,sc
I'4,sc
]
UscT = [U1 0 0 0]
As a first estimationall windings’ resistances
can be omitted, which leads to equations (18).
I sc =
1
W U sc
jΩ
U1,sc
jX e1, k
(18)
; I'4,sc =
U1,sc
jX e1, 4
(19)
= I'2,sc + I'3,sc + I'4,sc
jX e 3,4
I'4
= X 2 = X 3 = X 3 = X µ , the current I1
I sc = I1,sc
jX e1,3
(20)
i.e. it is just the sum of currents in theshortcircuited windings. These results are evident
from the equivalent scheme at aconsidered
short-circuit condition, shown in Fig.4.
e
(
U1,sc
and the current in the supplied winding is
k ≠1
jX e 3
jX e 2
; I'3,sc =
I1,sc = ∑
Fig. 3. Equivalent scheme of a four-windings
transformer for the no-load condition
As Xe1
jX e1, 2
N
U1
jX e1
U1,sc
jX e1,4
U'4 = 0
I1
U1
jX e1,2
jX
I'3
e
2,4
jX e1,3
jX e 3,4
U'3 = 0
I'2
U'2 = 0
Fig. 4. Equivalent scheme of a four-winding
transformer
for
the
short-circuit
conditionwhen the winding '1' is supplied
This test allows to determine the inductances
Le1, 2 Le1,3 and Le1, 4 . When the winding '2' is
supplied and the other ones are short-circuited,
the inductances Le 2,1 = Le1, 2 , Le 2,3 and Le 2, 4
can be found, and so on. Then, from four
short-circuit
tests
all
theinductances
constitutingthe upper polygonal of the
equivalent circuit can be determined.
However, it should be noticed that some
values of inductances Le n,k for the analysed
traction transformer are negative, as it can be
seen in the matrix (14). So, whenmeasuring
the short circuit currents, one has totake, not
only their rms values, but also their phases
with respect to the phase of the supplied
winding current.
24
Zeszyty Problemowe – Maszyny Elektryczne Nr 3/2014 (103)
5. Conclusions
6. References
In this paper, a newmulti-port equivalent
scheme of the multi-winding traction
transformers is presented. It is universal
because the number of its inductive elements
is exactly equal to the number of self and
mutual inductances of transformer’s windings.
Thanks to that, any traction transformer with
an arbitrary high number of windings can be
correctly represented by such an equivalent
scheme. The centrally located magnetizing
inductance in the T-type equivalent scheme is
decentralized in the multi-port equivalent
scheme to many magnetizing inductances
distributedat each portrespectively.
Determining theinductances in that equivalent
schemerequires that the finite elements
simulation has to calculate precisely the self
and mutual inductances of transformer’s
windings, especially their coupling by leakage
fluxes in the air.
In this paper the procedure of estimation of
themulti-port equivalent scheme elementsis
presented, basing on no-load and short-circuit
tests, rather similar as for the classical T-type
scheme, which should be well accepted by a
majority of engineers. However, more precise
determination of those elements is much more
complicated.
[1]. Erickson R.W., Maksimovič D.: A multiplewinding magnetic model having directly
measurable parameters, IEEE PESC, Vol.2, 1998,
pp. 1472-1478.
[2]. Sobczyk T.J.: On a circuital representation of
magnetically coupled coils, (in Polish), Proc. of Int.
Conf. on Fundamentals of Electrical Engineering
and Circuit Theory (IC-SPETO), Poland, Vol. 2,
2003, pp. 493-496.
[3]. Sobczyk T.J.: Equivalent schemes of multiwinding one-phase transformers, (in Polish), Proc.
of Int. Symp. on Electrical Machines (IS-SME),
Poland, 2004, pp. 452-455.
[4]. El Hayek J., Sobczyk T.J.: Equivalent circuit
of multi-windings traction transformers including
magnetizing currents, Proc. of ICEMS, China,
Vol.3, 2005, pp. 1740-1745.
[5]. El Hayek J., Sobczyk T.J.: Multi-port
equivalent scheme for multi-winding traction
transformers, COMPEL, Vol. 31, Issue 2, 2012, pp
726-737.
Authors
Joseph El Hayek, Ph.D., Eng.
University of Applied Sciences and Arts Western
Switzerland, School of Engineering,
Route du Rawyl 47, CP, 1950 Sion 2, Switzerland
[email protected]
Tadeusz J. Sobczyk, Prof. dr hab. inż.
Politechnika Krakowska,
Instytut Elektromechanicznych Przemian Energii,
Kraków, 31-155, ul. Warszawska 24,
[email protected]