Abstract - To reduce the system imbalance, various types of specially-connected transformers have been adopted in Taiwan and abroad. Because three-phase.
LEEE Transactions on Power Delivery, Vol. 11, No. 1, January 1996
323
THREE PHASE MODELS OF SPECIALLY CONNECTED TRANSFORMERS Bin-Kwie Chen Member, IEEE
Bing-Song Guo
Tatung Institute of Technology Taipei, Taiwan Abstract - To reduce the system imbalance, various types of specially-connected transformers have been adopted in Taiwan and abroad. Because three-phase transformer models are necessary for three-phase load flow simulation, specially-connected transformers which have been adopted so far are selected to derive three-phase models in this paper. These transformers include the single-phase connection, the V-V connection, the Wye-Delta connection, the Scott connection, the Le Blanc connection, and the Modified-Woodbridge connection. The three-phase models with the specially-connected transformers offered in this paper distinctly represent the HSR feeding system by means of the physical three-phase circuit and the symmetrical-component equivalent circuit. The three-phase model could be joined to the three-phase load flow program to exactly analyze the system imbalance in the future. The symmetrical-component equivalent circuit may also be used to derive the approximate unbalance-estimating formula.
The imbalance in the three-phase power system, which comes from unbalanced loads, untransposed transmission lines, blown fuses on power factor correction capacitors, and so on, has been existing for a long time. However, the problem has not received the deserved attention in Taiwan. Now with the construction of the high speed railway (HSR) impending, the use of single phase loads may further unbalance the Taipower system. There are various types of specially-connected transformers employed in the railway substation. Different connections lead to different influences on imbalance. TO make realistic comparison of the imbalances for the various specially-connected transformers, computer simulation by the three-phase load flow program is the most effective and reasonable method [ 1,2,3I . In order to use the three-phase load flow program to further analyze the imbalance level of the Taipower system which comes from the planned HSR loads, the three-phase models with the specially-connected transformers will be derived for the future program analysis.
95 SM 351-7 PWRD A paper recommended and approved by the IEEE Transformers Committee of the IEEE Power Engineering Society for presentation at the 1995 IEEE/ PES Summer Meeting, July 23-27, 1995, Portland, OR. Manuscript submitted July 29, 1994; made available for printing April 27, 1995.
The specially-connected transformer is employed in electric railway substations in many countries. The single-phase connection is used in Italian railways [21, French TGV [ 4 1 and New Zealand Railways [51. The V-V connection is used in French TGV 161, British Rail ECML 171 and Finnish state Railways [ 8 ] . The Wye-Delta connection is used in China railway [ 9 ] . The Scott connection is used in Japanese Tokaido-Shinkansen[lO]. The Le Blanc connection is used in Taiwan railway[lll. The Modified-Woodbridge connection is used in Japanese Sanyo-Shinkkansen 1121. In this paper, these specially-connected transformers are chosen to derive the three-phase models to represent the unbalanced traction loads on electric railway systems. In order to make a comparison of the effects on the imbalance for the various types of specially-connected transformers, in this paper the unbalance-estimating formulas for the specially-connected transformers will be derived in association with the symmetrical-component equivalent circuits.
-PHASE TRANSFORMERS
WITH SPECIAL CONNsCTIOHS As shown in Figure 1, the specially-connected transformers in HSR substations only feed power to traction loads. The three-phase transformer model derived in this paper will be used to form a three-phase equivalent network viewed from the source terminal of the transformer. The unbalance level on the duty point which the utility is concerned about can be reflected on the source terminal of the feeding transformer by the three-phase transformer model. The following notations will be used all through this paper : Y, :denotes the total admittance and equals the reciprocal of all the impedances on the secondary of the dedicated transformer, such as catenary impedance, rail impedance, equivalent impedance of the traction load, and the transformer impedance referred to its secondary. Q :indicates the turn ratio NI/N2 :indicates the turn ratio NJN,
a=eJ%n utility
source HSR substation
traction loads Figure 1 substation used to exclusively supply HSR loads
0885-8977/96/$05.00 0 1995 IEEE
324 Three-phase Transformer
Model
for
Sinqle-Phase
Connected
Kirchhoff's Voltage Law following relations :
The feeding transformer with single-phase connection in HSR substations is shown in Figure 2. The following relations exist :
v,, = a I, = I,
I, =
P
1,
=
I C
=
a v, a v,
B P
1, I 2
= YL, V I
I,
1,
I, = YL,
v,
= Y,
= =
I, + I, + I, = 0
V,
-
v,, v,,
(KVL), we will have the
v2
The primary currents I,, expressed as :
and
I,
I,
can
be
Similarly,
-
I C
=
P
Figure 2 The feeding transformer with single-phase connection
I,
I2
p2 ZL'
=
B'
=
-
(I,+I,)
=
-
B2
y,,
f
v,
p2 +
v,
yL2
p'
(YL, + Y,,
)
v, -
Y,,
vc
The primary currents I,, I, and I, can be expressed as I, = 0 I, =
p
=
p
I, = =
I,
=
Y,V,
p
Y,V,
-
P'
= Y,
p
Y,
P
(V.
Y,, = Y,, the relation between P' Ybl = Y , and I,, and I, can be expressed as
Glven
- V,)
I,,
v,
- I, - p z Y, v,
+ p'
Y,
Yl
-Y1
0
-Y2
v,
Given p' Y , = Y, the relation between V, be expressed as
and I,
can
(3)
Yz
BY transforming equation 3 into symmetrlcal components, the relation between V,,, and I,,, can be obtained as 0
(4) -a2Y1-Yz YI+Yz
0
By transforming equation 1 into symmetrical components, the relation between V,,, and I,,, can be obtained as
Three-phase
Model
for
the
Scott-Connected
Transformer
11
The feeding transformer with the Scott connection in HSR substatlons is shown in Figure 4(a). Both the voltage-phasor and current-flow diagrams are shown in Figure 4(b). YLz and Y,, denote separately the, loads in the two sides of the center-feed system. By flux coupling, KCL and KVL, the following relations will be found . 2
I, = Y,,V,=
Qc
fi
=
I C
Figure 3 Feeding transformer with V-V connection Three-phase Model for V-V Connected Transformer The feeding transformer with V-V connection in HSR substations is shown in Figure 3 . Y,, and Y,, denote separately the equivalent admittance in the two sides of the center-feed system. By flux coupling, Kirchhoff's Current Law (KCL), and
I, = -Ic,
I, =
-
IC, -
I,, 'A,
p
(v,-~va-~Vc)YL1
(sa)
325
By transforming (5) into symmetrical components, the relation between V,,, and I,,, can be obtained as
'3
lh/rN1/2
-
0
0
0
0
Y1-Y2
Yl+Y2
(6)
Three-vhase Model for Le Blanc-Connected Transformer
. d
Figure 4(a) The feeding transformer connection
with
The feeding transformer with Le Blanc connection in HSR substations is shown in Figure 5(a). Both the voltage-phasor and current-flow diagrams are shown in Figure 5(b). By flux Coupling, KCL, KVL, and Ohm's law,
Scott
.voltage phasorin primary
voltage phasor in secondary
v,
I,
+m
=
-p1
=
31p
=
P
I, = I, = I, = I, =
secondarycurrentflow
primary current flow
Figure 4 ( b ) Voltage phasor and current flow diagram
I, =
(V,+V,-2V,)
&-
~~~,~-~,,-~,,~
(V,-V,) YLl Vl YL2 v2
-
IC,
+
I,,
+ I,,
I,,
+
1 I,, = p (-1, I,
:.
$a
=
I, = $3 I,
-
=
- a - a =
:.
I,
I,, = - a I,
3
a I,, +
+
I,,
-
a I,, -
*
-
P I 2
a I,,
4a
I, -
7 a I, =
n
7P
'BC
=
I,,
= -
.
'CA
1 1 - -I,
3
'2
p
(
1 3 I, +
1
(5c)
I,
=
=
q
n
+ -a 2
I,,
2
IAB
-
+a
+a
I,,
I,,
- 1 2 a IAB
I,
*I1 P I 1 I2
Because the system primary is ungrounded, I, =
-
I,-
substituting I,
=
f
p 2
=
I,
p
I,
- 1 JT
(5a) into
Y,,
v,
-
I,
Nz/3
(5c) yields
$ P'
Y,,
v, -
5 P2 YL,
Figure 5(a) The feeding transformer with Le Blanc connection
vc
Substituting (5a) and (5b) into (5e) to yield
-
p2 Y L 2 ) vc
I, = +
P2 Y L 2 )
vc
Given pY,,=Y, and vYL2=Y,, the relation between , V and , ,I can be expressed as
-3Y1
+Yl+Y2
fY1-Y2
-4Y1
fY1-Y2
4Yl+Y2
(5)
voltage phasor in primary
voltage phasor in secondary
Figure 5(b) Voltage-phasor diagram.
326 We will find out that
--
I
J7 p
I,
2
A-
I,=
2-
p
I,
+ p
I,
IC=
2
p
I,
-
p
I,
J7
fi
voltage phasor in primary
voltage phasor in secondary
By appropriate substitution, we have
I, =
v,
Y,,
$32
I, =-2p2y,,v,+
v,
Y,,
-$Y
3
-7 2 p2Y,,v,+
Y,,
v,
($32YLl1 PZY,,)
(+Y,,+p'Y,,)VB+
3
I, =
- $32 2
(~p2Y,,-p~Y,,)V,+
v,
(~p~Y,,+p2Y,,)v,
3
Given PY,, = Y, and $'Y,, = Y,, the relation between ,V and , I can be expressed as
[ i: I=[
+Yl
-4Y1
-3Y1
fYl+Y2 )Y1-Y2
-)Y1
][ ]
-)Y1 )Y1-Y2 fYI+Y2
t vi current flow in secondary
current flow in primary
Figure 6(b) Voltage phasor and current flow diagrams (7)
Adding
(9f) to (99) yields
By transforming ( 7 ) into symmetrical component, the relation between V,,, and IO1,can be obtained as
[ ;:]=[ :
0 Y1-Y2 Y1+Y2
Y1!Y2 Y1-Y2
0
[I 5]
na
(8)
- -
J3
Three-phase Model
fi
v,= p +
v,
I,
+
I,= I,, -
I,=
nll
=
I,, +
I A l
fi
=
vP3P4
(
-
v,
P
vA
)
I, = I,
I61
-
(9f)
I C 2
n 1 1 = I,, + I,, - I,, - I,, 'A = p 'A1 + P 'A2 I, = p ' B l + P I62 I C
=
P
I C 1
"*
+
P
I.
(9g)
'a
r'
&-
3
(
Is1
=
+(IB-
I,
-
+ I,,
)
-
1 2
I,,
(
+ I,,
)
I,)
I, = 2 p 1 ,
611 + I , = I,, =
+ I,, a ( I,-I= )
I,,
- I,,
Subtracting (9g) from (9d) yields
fir, -
I, = =
'A1
+
I A 2
-
'61
-
'62
a
(1,-1,) Adding (9j) to (9k) yields
P
I, = 21, - (Ia + I C )
which can be substituted by (9h) to yields 2 I, + I, = TI, - p, 2 I, 3 Jj Adding (9i) to (91) yields
I.
+ Y
;
I, = -
2 f i
I C ,
I,
Adding (9d) to (9e) yields
:.
(9e)
-
=
Adding (se) to (9g) yields
(9d)
+
1 + -I,
2 -PI,
(9b) (9C)
I C 2
I,
1 2 J-7
a ~ -,
(9a)
I C ,
I A 2
+
I,
2 f i
aln
:. I,
The feeding transformer with Modified-Woodbridge connection in the HSR substation is shown in Fig 6(a) . The voltage-phasor and current-flow diagrams are both shown in Fig 6(b) . BY flux coupling, KCL and KVL, we have the following relations.
I,
...-
for Modified Woodbriae-Connected
transfor=
vl=
~, l-ac
-
'I,
=
- +311
n
-
t V,
3
n
3
The delta connection on the secondary assures that
Nz NI
-V
Figure 6(a) The feeding transformer with Modified-Woodbridge connection
11,
Subtracting (91) from (9i) yields I, = - -PI11 P I , + 11,
-
Note:
+ PI2 +
N2
AN
+-V
NI
N2 NI
+-Vm
= 0
(KVL)
.
In addition, if the primary side is solidly grounded, the relation V,+V,+ V,
= 0
will be held .
327 I, = YLIV,
Substituting yields:
2 --p n
I, =
--
eq (9a) into eq (9h)
With the following relations in existence,
v, + v, + v,
1 + -I, 3
YL,V,
2 -
p ,YL1[2vA-
v, I,
+ v, +
(V,
3
3
Similarly, 1 I, = - -PI,
+
JT
V,)
]
-2 p‘ Y L1 v +
q32YL1VII -+YL1vB
3
PI,
=
Vl
+ L3I ,
= 2 p2y,,v, =
and
=
+
=
-p ‘
I,
fi
=
-
-
=
--p2YL1vA+ 2
-+YLIV,
JT (
3
‘BN
+
I, = 0
$IN
I,
= UI,
- CLI,
I, = =
YLlVl
I,
YL2V2
We will find out that
3
--payLl1
p
- UI,
11,
PYL2V,
I,
= 3VN
I, = U I ,
1 + -I, 3
- PI2 +
VAN
+ 131 ,
+ 11, 3 p a Y L 2 ) v B +(
31P 2 Y L 1 +
p
I,
+ p
- p -
I,
-
I,
+ p
31,
= 2
31,
=
31, = VAN =
I,
B
v,,
=
v,,
=
I,
2
p
I,
I,
v, - v,
+
-v, 1
-
LV,
3
3
$v, -
$7,
-
IV,
3
+ L3V ,
By appropriate substitution and arrangement and p2 Y,, = Y, and p2YL, = Y,, the relation assuming between V, and , , I can be expressed as
31
-4Y1 - f Y 2
-$Y1 +;y2
gYl-gY2
+Yl+)Y,
-+y1+gy2
Assuming p2YL,= Y, and p2YL, = Y2, the relation between and ,I can be expressed as
V,
[ ;:
$Yl
]=[ -4Y1 -4Y1
-$Y1
-fY1
fY,+Y2 fY1-Yz
?Y,-Y2 4YI+Y2
][
;]+$I{
;]
BY (91
(11)
transforming
components, the
into
symmetrical
relation between V,,,
and
I,,,
can be obtained as 0
0
0
0
Yl+dY2
Y1+Y2
(12)
By transforming (9) into symmetrical components, the relation between Vola and I,,, can be obtained as
[ I: ]=[
0
0
0
Y1-Y2
(10) Y1+Y2
Three-phase Model for We-Delta Connected Transformer The feeding transformer with Wye-Delta connection in the railway substation is shown in Figure 7. I’
EQUIVALENT CIRCUITS AND UNBALANCE ESTIMATING FORMULAS The three-phase equivalent circuit represents the traction load in the transformer secondary and can be referred to the primary. The circuit is directly connected with the utility. The symmetrical component equivalent circuit can be used to derive the simplified formula to rapidly estimate the voltage imbalance.
Railway
vc
IC
1
power SllPPlY
system
Figure 7 The feeding transfoemer with Wye-Delta connection Figure 8 The Equivalent three-phase circuit for a utility system.
328 When the utility system, looked at from any one of the railway substations, is assumed to be well-balanced and can be replaced by the Thevenin equivalent circuit as shown in Figure 8 , where V,, V, and V, are the three-phase line-to-ground voltage on the duty point.
[ ; ;j=[ ::f: [I: ;; 51 J+[
ZM
J+[
ZP
ZW
By symmetrical transformation arrangement ,
[ is
]=[
zo+;zN 0
where Z, = Z,
z", 0
+ 2
Z,,
and some
41
(13)
= Z, = ZS= Z,-
Z,,, ES = E, = EAs
: ][ ;:
]+[
zs
Z,
where S, and S,, corresponding to Y, and Y, respectively, are expressed in MVA, and S,+ is the three phase short circuit capacity. Suppose is = S, + the total load in the center-feed system. S,. Then,
Transformer with V-V Connection
The utility system can be expressed in terms of the symmetrical-component equivalent circuit as shown in Figure 9. And the railway power supply system on the right-hand side of Figure 8 is represented in terms of any one type of specially-connected transformer stated previously .
By combining with eq.13, the symmetricalcomponent equivalent circuit with the V-V connected transformer is obtained as shown in Figure 11. Picking node x to apply XCL yields 1,
= - ys
_ -
-
a
(
v2
a2 Y, V, + Y , V,
a' Y, + Y,
)
v,
=
+
Y,
(v, - v,)
( Y , + Y, + Y,
)
v,
usually, Y,>> Y,, Y,. Then,
&,&A- cl
v2
4T
Figure 9 The symmetrical-component equivalent circuit for a utility system. The symmetrical component equivalent circuit will be drawn by combining eq.13 with eqs. 2,4,6,8, 10 and 12 separately. The results will be exploited to derive the formula to estimate voltage imbalance on the duty point.
ES
@
v1
y1
v2
Ys
aY v 2 -
Y1
2 Y1Vl
-
Transformer with Sinqle-phase Connection By combining with eq.13, the symmetrical-component equivalent circuit for the single-phase connected transformer is obtained as shown in Figure 10. The simplified formula to estimate imbalance can be obtained. From Figure 10
is the total load in the center-feed Suppose SI, .where S, indicates the MVA load on system. S,( =S,+S, the left-hand side of the center-feed system and s, the MVA load on the right-hand side. Let SI= k S l ~ and S, = (1-k) 5'10, where k depends on the train dispatch schedule. Then
Usually, Y,>> Y [l]. Thus
Let Y = Y, + Y,. Y is the total equivalent admittance in the center-feed system where Y, indicates the admittance on the left-hand side of the center-feed system and Y, on the right-hand side. Then,
I
rL VI
Figure 10 The symmetrical-component equivalent circuit for single-phase connection
(17)
Transformer with Scott, Modified-Woodbridqe Connection
Le
Blanc,
and
The relation between V, and I,, in the primary of the feeding transformer with Scott, Le Blanc and Modified Woodbridge connection is the same. By combining of eq.13 with the symmetrical -component circuit of the Scott or Le Blanc connected, or Woodbridge connected transformer, the equivalent circuit is obtained as shown in Figure 12.
329
m@*y, (18)
YS
Es
Choose the transformers used in the center-feed system for comparison. Assume the total load on the two sides of the center-feed system is s l 0 and the three-phase fault level is S 3 0 . Here, and S3a, are two constants. Set Sr = ’1’. sr is also a
s10
s30
constant. From the unbalance-estimating formulas in eqs. (15), (17),(19) and (21), the performance of the imbalance reduction due to different transformers can be divided into three category:
%
-
-
(i) Single-phase connection:
Figure 12 The Symmetrical-Component Equivalent Circuit for Scott, Le Blanc and Woodbridge connection A s before, let S, = k S10 and s, = ( l - k ) S l @ , where k depends on the train disDatch schedule. Then
I$I=
(
3k
3k2-
+ 1
)l’,
Sr
(17,21)
(iii) Scottconnection,Leblanc connection and Modified-woodbridge connection:
Transformer with We-Delta Connection By combining eq.12 with eq.13, the symmetrical-component equivalent circuit with the Wye-Delta connected transformer is obtained as shown in Figure 13.
where 05 k 5 1 . k is relative to the train dispatch schedule. The results are plotted in Figure 14 [4,131. Figure 14 shows the comparison of imbalance for different A types of specially-connected transformers. comparison reveals that the V-V and Wye-Delta connections are better in the reduction of the imbalance than the single-phase connection, and the Scott, Le Blanc or Modified-Woodbridge connections are better than the V-V and Wye-Delta connections. With the exception of the single-phase connection, the other special connections can surely reduce the imbalance.
V Z- -Y1-aY2
*
(ii) V-V connection and Wye-Delta connection:
E-YS+Yl+Y*
Usually, Y,>> Y,, Y,. Then
Smgle-Phase Connectlon V-V and Wye-Delta Connectlon (111) Scott, Le Blanc ModifiedWoodbndge Connectlon (1)
(11)
I
I
I
I
,
Figure 13 The Symmetrical-Component Equivalent Circuit for Wye-Delta COnneCtiOn where S , , corresponding to Y,, and S,, corresponding to Y2, are expressed in MVA. Suppose S l 0 is the total load in the center-feed system. 5’10 = S, + S, ,where S, indicates the MVA load on the left-hand side and S, the MVA load on the right-hand side. Let S, = kS10 and S, = (l-k)S1a,, where k depends on the train dispatch schedule. Then,
0
I
1 2
1
Figure 14 A comparison of imbalance for various types of specially-connected transformers.
CO” (21)
The Comparison of Imbalance for Different Tvpes of Specially-Connected Transformers The unbalance-estimating formulas derived in this paper are based on two assumptions. First, the utility system is assumed to be three-phase balanced system. Second, the equivalent traction load impedance is much larger than the utility source impedance.
The specially-connected three-phase transformer model to be used in.railway electrification is built up as described in this paper. The equivalent circuits of traction systems are combined with these three-phase transformers such that the electric railway system is completely joined to the utility system in terms of the three-phase equivalent circuits; that is to say, the single-phase traction loads on the secondary side of a specially-connected transformer can be equivalently transformed on the primary side and be represented by the derived three-phase model.
The various types of specially connected transformers have been employed in the railway substation. Different connections lead to different influences on imbalance. The unbalance-estimating formulas for different types of specially-connected
330 transformers derived from the symmetrical- component equivalent circuits have been presented in this paper. The V-V and Wye-Delta connections result in less of imbalance than the single-phase connection. In association with the reasonable train dispatch schedule, the use of Scott, Le-Blanc and Modified-Woodbridge connections can more effectively reduce the imbalance level than can the V-V and Wye-Delta connections. The EPRI report [l] mentioned that 'the rule of thumb" for the voltage imbalance actually underestimates the voltage imbalance. The elaborative analysis only depends on the three phase load flow analysis [1,2,31. The three phase load flow program will be the most necessary and important tool, and the three-phase transformer model offered in this paper can be joined to the three-phase load flow program to handle the exact analysis of unbalanced system in the presence of the traction loads. Thus, the imbalance reduction by using the different specially-connected transformers can be actually compared.
Ac know1edaement The authors would like to thank Mr. Chang S. Chou, General Manager of the Heavy Electric Apparatus Plant of Tatung Company, for providing the helpful information that makes the model derivation smooth.
REFERENCES J. J. Burke and J. W. Feltes, Railroad Electrification on Utility Systems, EPRI Report E L - 3 0 0 1 , March 1983. G. Astengo, C. oss si, and G. sciutto, "Single phase 25kv electrification on Sardinia railway: some particular problems related to an electrified rallway located in an island," IEE Conference Publication NO. 312, International Conference on Main Line Railway Electrification, pp. 178-181, 1989. C. Howroyd et al., 'Discussion on D. electrification of Taiwan maln line railway IEE Proceedings, from Keelung to Kaohsiung, vol. 131, Pt. B, no. 5, pp. 226-230, September 1984. M. Methinier, High Speed Railway, Electrical Power Supply Preliminary Report, May 1992. M. R. Denley, A . W. bregor, and W. B. Johnston, "The electrification of New Zealand railways north island main trunk route, IEE Conference Publication NO. 279, International Conference on Electric Railway systems for a new Century, pp. 189-193, 1987. H. Roussel, "Power supply for the atlantic TGV high speed llne,'I IEE Conference Publication No. 312, International Conference on Main Line Railway Electrification, PP. 388-392, 1989. P. T. Williment, "East coast main line electrification 25kv AC system design," IEE
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Bioaraghiea Bin-Kwie Chen was born in Kaohsiung, Taiwan on June 3, 1953. He received his B.S. degree from the National Cheng Kung University in Taiwan in 1976, and his MSEE and Ph.D. degrees from the University of Texas at Alington in 1982 and 1986, respectively. After finishing his compulsory military service in 1978, he worked for two years as a design engineer in the Design and Construction Department of Taiwan Power Company, where his duties and responsibility included the design of power systems, lighting systems, MCC, substations, and control room arrangements for power plants. Dr. Chen 1s currently teaching at the Tatung Institute of Technology. His main interest lies in exploiting computer methods for power system analysis in the area of planning, operation and control.
Bing-Song Guo was born in Hua-lien, Taiwan on December 7, 1969. He received his B.S. and M.S. degrees in Electric Engineering from Tatung Institute of Technology, Taipei, Taiwan in 1992 and 1994, respectively. Currently he is taking the military service in Army as an officer.
