Performance and mathematical model of three-

phase three-winding transformer used in 2
25 kV

electric railway
W. Mingli, X. Chengshan, Y. Fan and T.Q. Zheng

Abstract: A three-phase three-winding transformer, the so-called cross-connected transformer, has

been used in a 2
25 kV electric railway in China. Based on the magnetomotive force balance
equations, winding connection equations, output-port equations and voltage transfer equations,
the performance and mathematical model of the cross-connected transformer are analysed. The
input current and output voltage equations and nodal admittance matrix of the transformer
are obtained, which can be used in future analyses. The problem of omitting substation
autotransformers is addressed by virtue of two equivalent circuits put forward. The applicability of
the proposed formulas and mathematical model is validated by comparison of measured and
calculated busbar voltages and short-circuit simulations. Finally, the reasons that led to the low
catenary voltage of one section of the Datong–Qinghuangdao railway, supplied by a cross-
connected transformer, are investigated.

1 Introduction the cross-connected transformer and in the upgrading of

the electric feeding systems of the Datong–Qinhuangdao
Various connection transformers, including the single- railway.
phase, Scott, modified Woodbridge and cross-connected
transformers, have been used in autotransformer (AT) 2 Cross-connected transformer
power supply systems for electric railways throughout the
world [1–5]. The cross-connected transformer was proposed 2.1 Connection principle
by M. Yaoshan in 1980 when China decided to adopt A cross-connected transformer, as shown in Fig. 1a, is in
2
25 kV AT feeding systems in railway electrification [6]. fact a three-phase three-winding transformer and can be
In 1988, this specially-connected transformer was installed visualised as a combination and simplification of two wye–
in two substations, Yanqing and Zhuolu, of the Datong– delta transformers. The capacity of the two delta windings
Qinhuangdao railway of North China. In recent years, the should be designed equal to 50% of rated capacity of
electric power supply systems of the Datong–Qinhuangdao the transformer. In the voltage phasor diagram of Fig. 1b,
line have frequently suffered from overloads owing to the the voltages of two output-ports b1c2 (a phase) and b2a1
continuous increase of the heavy-haul freight services. The (b phase) form a cross at the common vertex of the two
catenary voltages in one section supplied by Yanqing deltas, hence the name cross-connected transformer. Com-
substation are even below 19 kV under unfavourable pared with the single-phase (two-winding), Scott and modified
conditions. The low voltage often results in slow progress Woodbridge transformers, this transformer has several
of trains and a bottleneck can build up in this area. The properties: (a) Two autotransformers (ATs) for single-track
railway authority is planning to improve the electrical railways, four for double-track railways, which are conven-
capacity of this railway. As a result, the performance of tionally installed in traction substations, can be omitted; (b) If
the cross-connected transformer needs the attention of necessary, the primary neutral can be connected to ground;
engineers. and (c) Three-phase AC source can be conveniently obtained
Although the cross-connected transformer has been in on the low-voltage side of the substation.
operation for about 17 years, analyses of its performance
and a mathematical model were never conducted in a deep
2.2 Substation installation
way in publications. In an early study, a simplified method
The main connection of Yanqing substation is shown in
was used in electrical calculations, which cannot account for
Fig. 2 [3], and the feeding arrangement of the traction
the inherent characteristics of a three-winding transformer
network is shown in Fig. 3. Under normal conditions, one
[7]. It is hoped that this paper be helpful in understanding
line-transformer combine is put into operation, the other
reserved. These two combines alternate according to a
r IEE, 2006 maintenance schedule. A parallel capacitor compensator
IEE Proceedings online no. 20045281 (PCC) is installed on each 55 kV busbar for power factor
doi:10.1049/ip-epa:20045281 correction, which consists of a capacitor bank and an air-
Paper first received 26th December 2004 and in final revised form 16th October core reactor connected in series. Related parameters are as
2005 follows:
The authors are with School of Electrical Engineering, Beijing Jiaotong
University, Beijing 100044, People’s Republic of China (a) Two 110 kV transmission lines are supplied from
E-mail: mlwu@center.njtu.edu.cn different busbars of the same substation of the electric

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 271
 A B C The second group contains winding connection equations:
∗
8
< IA þ IB þ IC ¼ IO ¼ 3I0
Ua c þ Ub1 a1 þ Uc1 b1 ¼ 0 ð2Þ
: 11
Ua2 b2 þ Ub2 c2 þ Uc2 a2 ¼ 0
O

T: trolley wire The third group includes output-port equations:

R: rail
8
F: positive feeder
>
> Ua ¼ ðUc1 b1 þ Uc2 a2 Þ
∗ ∗
>
> Ub ¼ ðUa1 c1 þ Ua2 b2 Þ
>
>
a1 b1 c1 a2 b2 c2 >
> ITa ¼ Ic1 b1  Ib1 a1
T N T >
>
<I ¼ I
Fa c2 a2  Ib2 c2
ð3Þ
R R >
> I T b
¼ Ia2 b2  Ib2 c2
>
>
>
> I F ¼ Ia1 c1  Ib1 a1
>
>
b

F F >
> I N ¼ I c1 b1 þ Ia2 b2  Ia1 c1  Ic2 a2
:
a ¼ ITa þ ITb  IFa  IFb
UB The fourth group consists of voltage transfer equations:
8
>
> UAO  nUa1 c1 ¼ Z1 IA  nZ2 Ia1 c1
>
>
>
> U AO  nUa2 b2 ¼ Z1 IA  nZ3 Ia2 b2
<
UBO  nUb1 a1 ¼ Z1 IB  nZ2 Ib1 a1
ð4Þ
> UBO  nUb2 c2 ¼ Z1 IB  nZ3 Ib2 c2
>
>
>
>
> U  nUc1 b1 ¼ Z1 IC  nZ2 Ic1 b1
: _ CO
UCO  nUc2 a2 ¼ Z1 IC  nZ3 Ic2 a2
UA UC
In these equations, n ¼ N1/N2, where N1 is the number of
turns of the primary winding and N2 is the number of turns
of the secondary winding. The direction of Ia1 c1 is defined as
b1 b2
from a1 to c1, and the voltage Ua1 c1 is equal to Ua1  Uc1 ,
c1 and so on. The directions of IA, IB, IC, IFa, IFb and IN are
all defined as into the transformer, but ITa and ITb are out
a2 of the transformer, as shown in Fig. 4. The voltage transfer
a1 c2 equations in (4) are based on the star equivalent circuit of
three-winding transformers, as shown in Fig. 5 for phase A,
b
and the interphase couplings are ignored.
Fig. 1 Cross-connected transformer
a Connection principle
b Voltage phasor diagram
3.2 Current distribution
In China, the 110 kV grids are effectively grounded
networks. Traction supply substations, owned and managed
by the railway department instead of by the electric power
utility network. At their sending end, the maximum system company, are, generally, of the terminal substations in the
impedance is j 3.4396 O. The length of the transmission lines power system. If the supply transformer has a neutral on its
is 73.5 km, and the impedance is 0.131+j 0.386 O/km at primary side, whether it should be connected to ground or
50 Hz. not is determined by the authorities of the electric power
(b) The short-circuit tests of one cross-connected transfor- company according to the assessment of zero-sequence
mer were performed in factory before field installation. The protection. The cross-connected transformer can serve both
results are shown in Table 1. in the mode of neutral grounded and isolated. This merit
extends its adaptability to electric utility networks.
The distribution of load currents in the windings of
3 Performance and mathematical model a cross-connected transformer can be derived from (1)–(4).
The formulas are given in Appendix 10.1, and the
3.1 Fundamental equations distribution in the secondary windings is shown in Fig. 6.
According to the voltage phasor diagram of Fig. 1b, it There would be no zero-sequence current if the primary
is convenient for the later analyses to give the winding neutral were isolated. However, when operating with the
expanded diagram of Fig. 4. Because of the special neutral earthed, there is probably a zero-sequence current
connection, the cross-connected transformer must be caused by the zero-sequence component of source voltages,
analysed based on elementary principles of transformers and the distribution of this current in the two delta windings
and circuits. The following four groups of equations can is inversely proportional to their equivalent impedance. It
describe the operating characteristics, except magnetisation, should be noted that traction loads cause no zero-sequence
of a cross-connected transformer under steady state. currents under both cases. Generally speaking, a cross-
The first group consists of magnetomotive force balance connected transformer is similar to a wye–delta transformer.
equations: The latter was once widely used in TR (trolley–rail),
8 TR+NF (trolley–rail with negative feeder) and BT (booster
< nIA ¼ ðIa1 c1 þ Ia2 b2 Þ transformer) traction power feeding systems of the electric
nI ¼ ðIb1 a1 þ Ib2 c2 Þ ð1Þ
: B railways in China. Therefore, the capacity availability of a
nIC ¼ ðIc1 b1 þ Ic2 a2 Þ cross-connected transformer is also 75.6% [8].
272 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
 110 kV 110 kV

2PT 1PT

75 MVA 75 MVA
110 kV/27.5 kV/27.5 kV 110 kV/27.5 kV/27.5 kV

2T 1T
PCC PCC

5PT 4PT
4CT 3CT

N (c1,a2)
T
(b1)
F
(c2)
T (b2)
F (a1)

2SG 1SG

3PT
6PT
3T

10.5 kV

FRT F T F T FRT
2 × 27.5 kV 2 × 27.5 kV 2 × 27.5 kV 2 × 27.5 kV

to Xiazhuang to Shacheng

Fig. 2 Yanqing substation

If Z2 ¼ Z3, we can obtain (see Appendix 10.2 for derivation):

110 kV 2 3 2 32 3 2 3
IA 1 1 1 UA 1 2

AT AT AT AT
4 IB 5 ¼ 1 4 1 1 1 54 UB 5 þ 1 4 1 1 5 a
I
2 × 27.5kV Ib
3Z0 3n
22.2 km 22.7 km 25.5 km 19.1 km IC 1 1 1 UC 2 1
ð5Þ
Xiazhuang Tielu Yanqing Beixinpu Shacheng





Fig. 3 Feeding scheme Ua Ua0 Z 2 1 Ia
¼  ð6Þ
Ub Ub0 3 1 2 Ib
where
Table 1: Short-circuit tests


Ia ITa þ IFa
¼ ð7Þ
Number Test Short-circuit Base of Percent Ib ITb þ IFb
side side capacity, MVA impedance
2 3


 U
1 ABC a1b1c1, a2b2c2 75 9.81 Ua0 2 1 1 2 4 A 5
¼ UB ð8Þ
2 ABC a2b2c2 37.5 5.34 Ub0 3n 2 1 1
UC
3 a1b1c1 a2b2c2 37.5 1.82
n2 Z2
Z0 ¼ Z1 þ ð9Þ
2
3.3 Input current and output voltage
2
equations Z ¼
Z1 þ Z2 ð10Þ
For normal three-phase power transformers, the generalised n2
matrices, or ABCD parameters, can be developed to If there is no zero-sequence component in the source
describe their behaviour as shown in [9]. Owing to the voltages, the no-load output voltages will be Ua0 ¼ 2UC/n
special supply scheme, the transfer characteristics of a cross- and Ub0 ¼ 2UA/n. Relatively speaking, the a phase can be
connected transformer can be represented preferably by nominated as the leading phase, and b phase the lagging
the following input current and output voltage equations. phase. Actually, if the contributions from the PCCs and the

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 273
 B N
b1 b2
2 T
1I I
3 F 3 T

2 1I
I
3 T
c1 3 F

1 a2 2
1 I 1
I 3 F
I
3 Tα 3 T

2
I
O a1 3 F
F
c2

A C

N T T
F

b1 +
b2 Fig. 6 Distribution of load currents in secondary windings
c1
The input current and output voltage equation can be
2 3 U  phase used in the calculation of operating conditions of the
a2
transformer for a specified load condition, including short-
circuit of traction networks. If the source impedance of the
a1 c2 power system needs to be taken into account, just substitute
−
F Z1 by ZS+Z1 in (10), where ZS is the system impedance
of the utility network (assuming that the positive- and
negative-sequence impedance are equal).
+ U
−
T
F
3.4 Superimposition effect of voltage loss

phase Let Z ¼ R+jX; by virtue of (6), the voltage loss on 55 kV
busbars can be approximately calculated by
Fig. 4 Winding expanded diagram " #" #


DUa R 2 cos ja cosð60  jb Þ jIa j
¼  
Z2
DUb 3 cosð60 þ ja Þ 2 cos jb Ib 
Ua c
1 1 " #" #
Z1/n2 X 2 sin ja  sinð60  jb Þ jIa j
Ia c
þ  
UAO / n 1 1
3 sinð60 þ ja Þ 2 sin jb Ib 
nIA Z3
Ua b
ð14Þ
2 2

Ia b
where ja and jb are power factor angles. It is clear that the
2 2 voltage loss of one phase is not only determined by its own
load, but it also has a relation to the load of the other phase.
For common thyristor-controlled AC–DC locomotives, the
Fig. 5 Star equivalent circuit for phase A
load of the leading phase leads to a voltage drop in the
lagging phase, but the load of the lagging phase results in a
distributive capacitance of the traction networks are voltage rise in the leading phase, as indicated in Fig. 7.
deducted, Ia and Ib are, respectively, equal to the sum of Consequently, the voltage of the lagging phase is, in general,
the train currents of the corresponding phase. lower than that of the leading phase due to this super-
Furthermore, if the magnetising current of the transfor- imposition effect of voltage loss.
mer needs to be taken into account, the magnetising
admittance matrix YM should be added to the zero-
sequence admittance matrix in (5): U
0
2 3 2 1 

I − I 
ZMs ZMm ZMm 3  3
U

120°
Z M ¼ 4 ZMm ZMs ZMm 5 ð11Þ 2I − 1I
ZMm ZMm ZMs I 3
3 

I

where U
1
ZMs ¼ ðZz þ 2Zp Þ ð12Þ
3
1
ZMm ¼ ðZz  Zp Þ ð13Þ U0
3
Zp and Zz are, respectively, the positive- and zero-sequence Fig. 7 Superimposition effect of voltage loss
magnetising impedances of the transformer, and they can be
obtained from no-load tests. If the primary neutral is 3.5 Node admittance matrix
ungrounded, the first item on the right-hand side of (5) In three-phase power systems, the topology of traction
should be omitted. substations is generally unsymmetrical. So, if problems

274 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
involving traction substations need to be analysed, it is can be obtained. This can be used in analysing relevant
convenient to apply the node admittance matrix to the problems, such as the calculation of three-phase power flow
phase co-ordinates. To avoid undue complexity of the or three-phase short-circuit, or the distribution of traction
expressions, we still assume Z2 ¼ Z3. If the primary neutral harmonics in power systems.
is grounded, the node admittance equation of a cross- Similarly, if it is necessary to consider the magnetising
connected transformer can be derived as current of the transformer, the magnetising admittance



 matrix YM should be added to the 3
3 sub-matrix YPP:
IP Y PP Y PS V P 2 3
¼ ð15Þ
IS Y TPS Y SS VS y
1 4 Ms
yMm yMm
YM ¼ yMm yMs yMm 5 ð22Þ
where 3
yMm yMm yMs
I P ¼ ½ IA IB IC T ; V P ¼ ½ UA UB U C T where
I S ¼ b I Ta I Fa I Tb IFb IN c ;T 1 2
yMs ¼ þ ð23Þ
Zz Zp
VS ¼ b U Ta U Fa UTb U Fb UN cT
2 1 1
4ðym  yn Þ þ n2 y0 2ðym  yn Þ þ n2 y0 yMm ¼  ð24Þ
1 6 Zz Zp
Y PP ¼ 4 2ðym  yn Þ þ n2 y0 4ðym  yn Þ þ n2 y0
3n2
2ðym  yn Þ þ n2 y0 2ðym  yn Þ þ n2 y0 4 Omitting substation ATs
2
3
2ðym  yn Þ þ n y0 4.1 Equivalent circuit A
7
2ðym  yn Þ þ n2 y0 5 Because the common vertex of the two delta windings can be
4ðym  yn Þ þ n2 y0 connected to rails, a cross-connected transformer can directly
feed 2
25 kV traction networks and the ATs that are
ð16Þ conventionally installed on the traction side of substations
2 can be omitted. This can simplify the substation connection,
0 0 reduce the area occupation and decrease operating energy
16
Y PS ¼ 4 ðym  yn Þ ym  yn losses. If substation ATs are omitted, there will be some
n requirements on the impedance of the supply transformer.
ym  yn ðym  yn Þ
3 The problem of omitting substation AT has been discussed
ym  yn ðym  yn Þ 0 for single-phase three-winding transformers in the literature,
7 e.g. in [10] and [11]. For convenience, Appendix 10.3 gives a
ðym  yn Þ ym  yn 05 ð17Þ
concise description, where the main idea is based on [11].
0 0 0
Using the node admittance matrix in (15), this issue can be
2 addressed for a cross-connected transformer.
2ym 2yn yn
6 The self-admittance sub-matrix of the secondary side can
6 2yn 2ym ym be written as
6
Y SS ¼6
6 yn ym 2ym Y SS ¼ Y 1 þ Y 2 þ Y 3 ð25Þ
6
4 ym yn 2yn where
2
ðym þ yn Þ ðym þ yn Þ ðym þ yn Þ 7ym  yn 7yn  ym ð5yn þ ym Þ
3 6
ym ðym þ yn Þ 6 7yn  ym 7ym  yn ð5ym þ yn Þ
yn
7
ðym þ yn Þ 7 16
6
7 Y 1 ¼ 6 ð5yn þ ym Þ ð5ym þ yn Þ 7ym  yn
46
2yn ðym þ yn Þ 7
7 ð18Þ 4 ð5ym þ yn Þ ð5yn þ ym Þ 7yn  ym
7
2ym ðym þ yn Þ 5 0 0 0
3
ðym þ yn Þ 4ðym þ yn Þ ð5ym þ yn Þ 0
7
Z1 þ n2 Z2 ð5yn þ ym Þ 0 7
ym ¼ ð19Þ 7
Z2 ð2Z1 þ n2 Z2 Þ 7yn  ym 077
7
7ym  yn 05
Z1
yn ¼ ð20Þ 0 0
Z2 ð2Z1 þ n2 Z2 Þ
ð26Þ
1 2 2 3
y0 ¼ ¼ ð21Þ 1 1 1 1 0
Z0 2Z1 þ n2 Z2
6 1 1 1 1 0 7
When operating with its neutral isolation, the nodal ym þ yn 6
6
7
Y2 ¼  6 1 1 1 1 077 ð27Þ
admittance matrix keeps unchanged except for dropping 4 4 1 1 1 1 05
the y0 items in YPP. It should be pointed out that the
currents in (15) are all defined as from the node into the 0 0 0 0 0
transformer and the directions of ITa and ITb are different 2 3
1 1 0 0 2
form those shown in Fig. 4. 6 1 1 0 0 2 7
Superimposing the node admittance matrices of the ym þ yn 6
6
7
Y3 ¼ 6 0 0 1 1 2 7
7 ð28Þ
supply transformer, PCCs, traction networks and the three- 2 4 0
phase utility grid together according to the network 0 1 1 2 5
connection, the node admittance matrix of the total system 2 2 2 2 8

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 275
 B Z10 þ n2 Z20
ym0 ¼ ð31Þ
Z1 Z20 ð2Z10 þ n2 Z20 Þ
Z10
yn0 ¼ ð32Þ
Z20 ð2Z10 þ n2 Z20 Þ
n2
Z10 ¼ Z1  Z2 ð33Þ
4
3
Z1 Z1
Z20 ¼ Z2 ð34Þ
2
A C
AT
b1 b2 T and Y3 is the same as in (28). If the relation ym  yn ¼
ym0  yn0 is noted, the equivalent circuit B of Fig. 9 for a
Z2
Z2
Z2 Z2 Z2
cross-connected transformer can be worked out. So we can
c1 2 say that a cross-connected transformer, when operating
 phase R
a2
with its node N connected to rails, can equivalently
Z2 Z2 substitute a cross-connected transformer, with N isolated,
a1 c2 and two ATs. This conclusion is useful in the determination
F of the impedance parameters when designing a cross-
connected transformer.
∗
B
∗
− 4Z 2
n2
Z1 − Z2

phase 4
AT
Z2
2

T
R F
n2 n2
Z1− Z2 Z1− Z2
4 4
Fig. 8 Equivalent circuit A
A C
Y1 can be considered as the result that the node N has been AT
eliminated from the original admittance matrix, and this b1 b2 T
3 3
can be verified by a matrix reduction process with (15) by Z Z Z2
2 2 2 2 3
setting IN ¼ 0. Noting that ym+yn ¼ 1/Z2, Y2 can be viewed 3
Z
Z
2 2
2 2 c1 2
as a 1 : 1 single-phase transformer connected to the four  phase R
3
terminals Ta, Tb, Fa and Fb, with leakage impedance 4Z2 3
Z
a2 Z
2 2
[12]. Y3 represents two ATs that are connected to the output 2 2
ports of the a and b phases, with leakage impedance Z2/2. a1 c2
Therefore, the circuit of Fig. 8, which consists of ideal F
transformers and impedances, can equivalently represent a

phase
cross-connected transformer.
AT

4.2 Equivalent circuit B Z2

The function of the 1 :1 single-phase transformer in the 2
equivalent circuit A is to correct the circulation current that
flows through the mid-point of ATs, as illustrated in Appendix
T
R F

10.4. The presence of this single-phase transformer is somewhat
awkward. In fact, another decomposition method for YSS exists: Fig. 9 Equivalent circuit B
Y SS ¼ Y 01 þ Y3 ð29Þ
4.3 Impedance determination
where AT traction power feeding systems, for the purposes of
2
7ym0  yn0 7yn0  ym0 ð5yn0 þ ym0 Þ alleviating interference on adjacent communication lines
6 0 and improving the quality of power supply, generally have
6 7yn  ym0 7ym0  yn0 ð5ym0 þ yn0 Þ
16 a requirement on the leakage impedance of ATs [5, 13–16].
Y1 ¼ 6
0
ð5yn0 þ ym0 Þ ð5ym0 þ yn0 Þ 7ym0  yn0
46
6
A quantity of j 0.45 O was specified for the Datong–
4 ð5ym0 þ yn0 Þ ð5yn0 þ ym0 Þ 7yn0  ym0 Qinhuangdao railway. This corresponds to Z2 ¼ j 0.9 O by
0 0 0 virtue of the equivalent circuit of Figs. 8 and 9. Since
3
0 0
ð5ym þ yn Þ 0 1
7 ðZ21 þ Z23  Z31 Þ
Z2 ¼ ð35Þ
ð5yn0 þ ym0 Þ 0 7 2
7 ð30Þ
7yn0  ym0 077
where Z21, Z23 and Z31 are the short-circuit impedances
7 between windings 2 and 1, 2 and 3, and 3 and 1,
7ym0  yn0 05
respectively, and Z21 ¼ Z31 because of the symmetry of
0 0 the winding arrangement, the requirement becomes
276 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
Z23 ¼ j1.8 O. For a double-track railway, it is in fact four, The simulation results are given in Table 2. The errors are
not two ATs that are omitted, so the condition should be caused mainly by the adoption of an average value for Z2
Z23 ¼ j0.9 O. As long as this condition be satisfied, it can be (see Appendix 10.5).
ensured that the performance of the AT feeding system will
not deteriorate because of the application of a cross-
connected transformer and omitting substation ATs. This 60 U
3.0
condition can be realised by the appropriate arrangement of
the windings to enhance magnetic couplings between the 50 U 2.5

two secondary deltas. After Z2 is determined, Z1 can be 40 2.0

voltage, kV

current, kA
selected by considering the requirement of voltage regula-
tion and the restriction of short-circuit current. 30 I 1.5

20 1.0
5 Verification of formulas and mathematical I

model 10 0.5

5.1 Field measurements 0 0.0

To evaluate the validity of the foregoing formulas and 14:00 14:05 14:10 14:15 14:20 14:25 14:30
time
mathematical model, field measurements of the busbar
a
voltages and load currents were performed in Yanqing
substation. During the tests, transformer 2T was in service
phase
1.0
with its primary neutral isolation. Voltage signals of 55 kV

inductive
busbars were obtained from the secondary circuit of 5PT 0.8  phase
and 6PT. Current signals of Ia and Ib were taken from a 0.6
secondary winding of 4CT, which was used for overcurrent 0.4
protection of the transformer, as shown in Fig. 10.
power factor
0.2
0
−0.2
to measuring −0.4
4 CT instrument
capacitive

I> kA
T −0.6
−0.8
link −1.0
14:00 14:05 14:10 14:15 14:20 14:25 14:30
F time
b

Fig. 11 Measured results

Fig. 10 Current measuring loop a Voltages and currents
b Power factors
Figure 11 shows the measured results during one half-
hour. The measured and calculated busbar voltages are
compared in Fig. 12. In calculation, (14) was used to obtain 65
voltage losses DUa and DUb, the currents and their power U
calculated
factor angles use measured values, then the busbar voltages
were computed by Ua ¼ Ua0DUa and Ub ¼ Ub0DUb, 60

where Ua0 ¼ Ub0 ¼ 56.5 kV. The impedance parameters are

computed in Appendix 10.5. The calculated results conform
voltage, kV

55
U
measured
closely to those measured in Fig. 12. U measured

50
5.2 Short-circuit simulations
Using the node admittance (15), short-circuit simulations 45 U calculated
were conducted to verify the measured data of the short-
circuit impedances of the transformer shown in Table 1.
40
The impedances were first calculated with a matrix
14:00 14:05 14:10 14:15 14:20 14:25 14:30
reduction program by setting appropriate voltage and time
current conditions on the primary and secondary side. Then
the actual quantities were converted into percent values. Fig. 12 Comparison between measured and calculated results

Table 2: Short-circuit simulations

Number Test Short-circuit Impedance, O Base of Calculated percent Error

side side capacity, MVA impedance

1 ABC a1b1c1, a2b2c2 15.83 75 9.812 0.02%

2 ABC a2b2c2 17.30 37.5 5.362 0.41%
3 a1b1c1 a2b2c2 1.102 37.5 1.821 0.05%

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 277
6 Discussion Institute for their helpful discussions, and Prof. S. William-
son of UMIST, UK, for his encouragement and advice
According to the analysis in Section 3 and the result of on revision of the manuscript. Thanks are also due to
practical measurement, the reasons leading to low catenary Z. Yinbao and C. Guang for their assistance with the
voltage in the section from Yanqing to Xiazhuang of the measurement work. The first author acknowledges the part
Datong–Qinhuangdao railway were investigated. The financial support from BJTU Foundation of China, under
reasons can be summarised as the following four aspects: grant PD265.
(a) The electric source of Yanqing substation is too weak.
The short-circuit level on the 110 kV side is only 364 MVA 9 References
under the minimum mode of the supply network.
1 Kan, H., Tsuruta, K., Takechi, M., and Kodama, T.: ‘Transformers
(b) The practical daily load has observably exceeded the for the new Sanyo trunk line’, Tech. Rev. Mitsubishi Electr., 1971, 45,
original design capability. (4), pp. 450–457 (in Japanese)
2 Glover, J.D., Kusko, A., and Peeran, S.M.: ‘Train voltage analysis for
(c) The superimposition effect of voltage loss of the cross- ac railroad electrification’, IEEE Trans. Ind. Appl., 1984, 20, (4),
connected transformer makes the b phase present a very pp. 925–934
low voltage when both phases have heavy loads. 3 Yaoshan, M., and Dongbai, C.: ‘Traction power supply systems’
(China Railway Press, Beijing, China, 1988), (in Chinese)
(d) The AT intervals in the traction network are too long, 4 Roussel, H.: ‘Power supply for the Atlantic TGV high speed line’.
which cannot adapt to the increased loads. Proc. IEE Int. Conf. on Mainline Railway Electrification, York, UK,
September 1989, pp. 388–392
To improve the voltage level of the b phase, a desirable 5 CCITT: Directives concerning the protection of telecommunication
lines against harmful effects from electric power and electrified railway
scheme is to install a voltage regulator, which can be lines, Volume I and IV, Geneva, Switzerland, 1989
automatically taped by a controller according to the real-time 6 Yaoshan, M.: ‘Different connection transformers applicable to
substations of AT traction power supply systems’, Electr. Railw.
busbar voltage and load current. In fact, the insufficiency of Dyn., 1980, (6), pp. 1–5 (in Chinese)
supply capability arises not only at Yanqing substation; the 7 Yaoshan, M.: ‘The electrical calculation of the transformer with two
reconstruction of the traction power supply systems of the secondary windings connected in a common vertex for 2
25 kV
traction systems’, J. Southwest Jiaotong University, 1986, 21,
Datong–Qinhuangdao railway has been planned as a whole. (Supplement), pp. 36–44 (in Chinese)
Various countermeasures, including the installation of addi- 8 Jianyou, C.: ‘Power supply systems of AC electrified railways’ (China
tional feeder substations and AT posts, are under considera- Railway Press, Beijing, China, 1981), (in Chinese)
9 Kersting, W.H.: ‘Distribution system modeling and analysis’ (CRC
tion. In view of the actual instance of Yanqing substation, it is Press, 2002)
recommended to parallel the two 110 kV incoming lines as a 10 Iwashita, T.: ‘Economical formation method of AC feeding circuit’,
Electr. Railw., 1981, 35, (10), pp. 6–10 (in Chinese)
temporary remedy measure; this can decrease the voltage loss 11 Chengshan, X.: ‘Discussion on omitting substation AT’, Electr.
by about 1/3 and needs almost no investment. Railw., 1998, 52, (1), pp. 1–4 (in Japanese)
12 Laughton, M.A.: ‘Analysis of unbalanced polyphase networks by the
method of phase co-ordinates Part 1. System representation in phase
7 Conclusions frame of reference’, Proc. IEE, 1968, 115, (8), pp. 1163–1172
13 Hayashi, M., and Iwashita, T.: ‘Theoretical analysis of feeding system
The performance and mathematical model of the cross- with autotransformer – formulae for calculating power characteristics’.
Research report (NO.735) of the Railway Technical Research Institute
connected transformer have been analysed in a relatively of Japanese National Railways, November 1970, (in Japanese)
strict way in this paper. The input current and output voltage 14 Chengshan, X.: ‘A research on autotransformer power supply systems
equations that can be applied to compute voltage losses and – selection of feeding circuit parameters’. Research report of China
Academy of Railway Sciences, February 1984, (in Chinese)
to perform short-circuit analysis of traction networks have 15 Mellitt, B., Allan, J., Shao, Z.Y., Johnston, W.B., and Goodman, C.J.:
been derived. The nodal admittance matrix of the transfor- ‘Computer-based methods for induced-voltage calculations in AC
mer has been obtained, which can be conveniently used to railways’, IEE Proc. B, Electr. Power Appl., 1990, 137, (1), pp. 59–72
16 Menuet, J.P.: ‘The autotransformer and other equipment’. IEE Coll.
interface with power systems and traction networks in on 50 kV autotransformer traction supply systems – the French
related analyses. The problem of omitting substation ATs experience, November 1993
has been discussed by virtue of two equivalent circuits put
forward. A comparison of measured and calculated busbar 10 Appendixes
voltages and short-circuit simulations validated the applic-
ability of the proposed formulas and mathematical model. 10.1 Current distribution
Finally, the reasons that led to the low catenary voltage of If a cross-connected transformer operates with its primary
one section of the Datong–Qinghuangdao railway, supplied neutral grounded, the current distribution in windings can
by Yanqing substation, have been investigated. be derived from (1)–(4) as
The practical application of the cross-connected trans- 2 3 2 3
former is instructive for the design of AT power supply Ia1 c1 1
4 Ib1 a1 5 ¼  nZ 3
systems of electrical railways. This supply transformer has I0 4 1 5
the merits of (a) omitting substation ATs, (b) capability of Z2 þ Z3
Ic1 b1 1
neutral grounding on the primary side and (c) easy 2 3
2 3 ITa
obtaining of three-phase AC source on the secondary side. 1 0 0 2 6 7
If appropriate selection is given to the transformer 1 IFa 7
þ 4 1 0 0 1 5 6 4 I Tb 5 ð36Þ
impedances, i.e. considering the requirement of AT feeding 3
2 0 0 1
networks and the co-operation with the system impedance, I Fb
the cross-connected transformer is valuable in electric 2 3 2 3
Ia2 b2 1
railway engineering. Owing to the superimposition effect 4 Ib2 c2 5 ¼  nZ2 I0 4 1 5
of voltage loss, it is advisable to arrange the heavy load Z2 þ Z3
section to the leading phase of the transformer. Ic2 a2 1
2 3
2 3 IT
8 Acknowledgments 0 1 2 0 6 a 7
1 IFa 7
þ 4 0 1 1 0 5 6 4 ð37Þ
The authors wish to thank M. Yaoshan and X. Dunqing of 3 I Tb 5
0 2 1 0
China Railway Electrification Survey Design & Research I Fb

278 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
 2 3 Because of the symmetry of winding arrangement, it can be
2 3 2 3 2 3 ITa
IA 1 1 1 2 2 6 7 assumed that Z2 ¼ Z3 in practical applications, hence (42)
6 7 6 7 1 6 7 6 IFa 7 can be written as (6).
4 IB 5 ¼ I0 4 1 5 þ 4 1 1 1 1 56 7
3n 4 I Tb 5 If the cross-connected transformer operates with its
IC 1 2 2 1 1 primary neutral isolated, the neutral potential is
I Fb
2 3 2 3
1 1 2 " #
6 7 1 6 7 I Ta þ I Fa 1
¼ I0 4 1 5 þ 4 1 1 5 ð38Þ UO ¼ ðUA þ UB þ UC Þ ð43Þ
3n I Tb þ I Fb 3
1 2 1
and (41), (42) and (6) still stand.
where U0
I0 ¼ ð39Þ
n2 Z2 Z3
Z1 þ 10.3 Omitting substation AT for single-
Z2 þ Z3 phase three-winding transformers
1
U0 ¼ ðUAO þ UBO þ UCO Þ ð40Þ Figure 14 shows a combination of an n : 2 single-phase two-
3 winding transformer and a 1 : 1 AT. The node admittance
n2 Z2 Z3 equation can be derived as:
and Z1 þ is, in fact, the zero-sequence impedance of
Z2 þ Z3 2 3
1 1 n n
the transformer measured on the primary side. Figure 13 0
shows the equivalent circuit for zero sequence current. 6 Zt Zt 2Zt 2Zt 7
6 7
6 7
2 3 6 1 1 n n
0 7 2 3
IA 6 Z Zt 2Zt 2Zt 7 UA
n2Z2 6 t 7
6 IB 7 6 76 7
Z1 6 7 6 n n n2 1 n2 1 1 76 UB 7
6 IT 7¼6 þ þ 76 UT 7
6 7 6 2Zt 2Zt 4Zt 4Zg 4Zt 4Zg 2Zg 76 7
n2Z3 4 IF 5 6 74 UF 5
6 7
IR 6 n n n2 1 n2 1 1 7 UR
6 þ þ 7
6 2Zt 2Zt 4Zt 4Zg 4Zt 4Zg 2Zg 7
6 7
4 1 1 1 5
Fig. 13 Equivalent circuit for zero-sequence current 0 0
2Zg 2Zg Zg
ð44Þ
10.2 Derivation of input current and output
voltage equations In (44), the bottom-right 3
3 sub matrix contains
Substituting (39) into (38) yields the input current equation 2 3
(5). Substituting (36), (37) and (38) into (4) gives: 1 1 2
1 4
2
Uc1 b1
3 2
1 1 2 2
3
3 Y AT ¼ 1 1 2 5 ð45Þ
UA 4Zg
6 U 7 6 7 2 2 4
6 c2 a2 7 1 6 1 1 2 76 7
6 7 ¼ 6 7 4 UB 5 which is the admittance matrix of the AT [5, 11].
4 Ua2 b2 5 3n 4 2 1 1 5
UC Figure 15 shows an n : 1 : 1 single-phase three-winding
Ua1 c1 2 1 1
2 transformer, where Z1 and Z2 are the equivalent impedance
2ðZ1 þ n2 Z2 Þ 2Z1 of the primary and secondary winding. The corresponding
6
1 6 2Z1 2ðZ1 þ n2 Z3 Þ node admittance equation is
 26
3n 4 Z1 ðZ1 þ n2 Z3 Þ 2
2 2 2 n
ðZ1 þ n Z2 Þ Z1 6 2Z1 þ n2 Z2
32 3 6 2Z1 þ n2 Z2 2Z1 þ n2 Z2
Z1 ðZ1 þ n2 Z2 Þ ITa 2 3 6
7 6 7 IA 6 2 2 n
2 6
ðZ1 þ n Z3 Þ Z1 7 6 IFa 7 6 7 6 1 þ n2 Z2
2Z 2Z1 þ n2 Z2 2Z1 þ n2 Z2
7 6 7 6 IB 7 6
2ðZ1 þ n2 Z3 Þ 2Z1 5 4 ITb 5 6 7 6
6 IT 7¼6 n n Z1 þ n2 Z2
2Z1 2ðZ1 þ n2 Z2 Þ IFb 6 7 6 2Z1 þ n2 Z2 2Z1 þ n2 Z2 Z2 ð2Z1 þ n2 Z2 Þ
6 7 6
ð41Þ 4 IF 5 6 n n Z1
6
IR 6 6 2Z1 þ n2 Z2 2Z1 þ n2 Z2 Z2 ð2Z1 þ n2 Z2 Þ
thus 6
2 3 4 1


 UA 0 0
Ua 2 1 1 2 6 7 Z2
¼ 4 UB 5 n 3
Ub 3n 2 1 1 0
UC 2Z1 þ n2 Z2 7

72
1 2ð2Z1 þ n2 Z2 Þ 2ð2Z1 þ n2 Z3 Þ n
0 7 U 3
 2 2 7 A
2Z1 þ n Z2 76
3n ð2Z1 þ n2 Z2 Þ ð2Z1 þ n2 Z3 Þ 7 6 UB 77
2 3 Z1 1 7 6 7
ITa 76 7 ð46Þ
 Z2 ð2Z1 þ n Z2 Þ Z2 7
2
766 U T 7
ð2Z1 þ n2 Z3 Þ ð2Z1 þ n2 Z2 Þ 6 IFa 7
6
7 7 7
6 7 Z1 þ n2 Z2 1 7 4 UF 5
2ð2Z1 þ n2 Z3 Þ 2ð2Z1 þ n2 Z2 Þ 4 ITb 5 7
Z2 ð2Z1 þ n2 Z2 Þ Z2 7
7
UR
IFb 1 2 5
ð42Þ Z2 Z2

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 279
 I
Zt R
1:1
n:2
A T
b1 b2
∗ ∗ T
3
Zg 2I /
N
R c1

a2 I/3

B F a1 c2 F

Fig. 14 Combination of single-phase two-winding transformer and AT

T
F

Z1 Z2
n:1:1 a
A T
3I/4 2I/3 AT
b1 b2 T
∗ ∗
I /4 12
5I / Z2 I
I/3
c1 2
Z2
I /1 Z 2 I/12 R
R I/4 I /4 a2 2 I/3
2I/3
∗ I/3
a1 I/3 c2
I/4
F

I/12 I/4
Z2 ∗
B F
∗
−4 Z 2 I/12 I/3
Fig. 15 Single-phase three-winding transformer
I/6 I/6
AT
Note that the bottom-right 3
3 sub matrix in (46) can be Z2
separated into two matrices: 2
2 3
Z1 þ n2 Z2 Z1 1
6 Z2 ð2Z1 þ n2 Z2 Þ Z2 ð2Z1 þ n2 Z2 Þ Z2 7 T
R F

6 7
6 7 b
6 Z1 Z1 þ n2 Z2 1 7
6 7
6 Z2 ð2Z1 þ n2 Z2 Þ Z2 ð2Z1 þ n2 Z2 Þ Z2 7 2I/3 AT
6 7
4 1 1 2 5 b1 b2 T
2I / 18
Z2 Z2 Z2 9 7I / Z2 I
2 3 I/3
2 2 3 3
n n I/18 Z
c1
Z2 2
6 2ð2Z þ n2 Z Þ 2ð2Z þ n2 Z Þ 0 7 2 2 I/9 2 R
6 7 8 5I/18
1 2 1 2 I/1 a2
6 7 5
I/3
2I/3
¼ 6 n2 n2 7
6 0 7
4 2ð2Z1 þ n2 Z2 Þ 2ð2Z1 þ n2 Z2 Þ 5 a1
I/3
c2
F
0 0 0 I /6
2 3 I /6
1 1 2 I /6 I /6 I/3
1 6 7 ð47Þ AT
þ 4 1 1 2 5 Z2
2Z2
2 2 4 2

The latter is just the admittance matrix of an AT. It is

clear that, in designing the single-phase three-winding T
R F

transformer, if
c
Z1 ¼ Zt  n2 Zg ð48Þ
Fig. 16 Current distribution for TR load
Z2 ¼ 2Zg ð49Þ a Original circuit
Eq. (46) will become identical to (44). Hence, a single-phase b Equivalent circuit A
c Equivalent circuit B
three-winding transformer can equivalently substitute the combi-
nation of a single-phase two-winding transformer and an AT.

10.4 Illustration of equivalent circuits verified that the terminal voltages, such as U Ta Fa , U Tb Fb ,
For TF loads, the equivalence is relatively easy to under- U Tb R  U RFb , U Ta Tb  U Fb Fa etc., are equal for these three
stand. For TR loads, Fig. 16 can be taken as a reference. In circuits. If the 1 : 1 single-phase transformer in equivalent
these circuits, only the secondary side is shown. It can be circuit A were left out, the circulation current that flows

280 IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
through the mid-point of ATs should be 3I/8, instead of I/3. where Z1%, Z2% and Z3% are the percent impedance of
Then this will not be equivalent to the original circuit (the windings with a base capacity of 75 MVA. Thus Z1% ¼ 8.90,
node admittance matrix has been changed since dropping of Z2% ¼ 1.86 and Z3% ¼ 1.78. Neglecting the difference
the 1 : 1 single-phase transformer). between Z2% and Z3%, and adopting an average value
1.82, the transformer actual impedances can be calculated as:
10.5 Impedance parameters 8
On the 110 kV side of Yanqing substation, the system >
> 8:90 1102
< Z1 ¼ j
¼ j14:36 ðOÞ
impedance can be calculated as 100 75
ð52Þ
>
> 2
ZS ¼ j3:4396 þ 73:5
ð0:131 þ j0:386Þ
ð50Þ : Z2 ¼ j3
1:82
27:5 ¼ j0:551 ðOÞ
¼ 9:629 þ j31:81 O 100 75
According to the measured data of the short-circuit tests Therefore, the impedance Z in (6) is
of the transformer and the star equivalent circuit of three- 2
winding transformers, the following equations can be Z ¼ ðZS þ Z1 Þ þ Z2 ¼ R þ jX
obtained: n2
8 ¼ 3:611 þ j17:86 ðOÞ ð53Þ
> Z2 %
Z3 %
>
< Z1 % þ Z2 % þ Z3 % ¼ 9:81
>
where
ð51Þ
>
> Z1 % þ Z3 % ¼ 2
5:34 110
>
: n ¼ pffiffiffi ð54Þ
Z2 % þ Z3 % ¼ 2
1:82 27:5 3

IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006 281