18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

Calculation of Voltage Unbalance Factor in Power

System Supplying Traction Transformers

Lesław Ładniak Wroclaw University of Technology

Wroclaw, Poland l.ladniak@pwr.edu.pl

Abstract - The article presents the models of transformers with specially connected windings for supplying catenary systems of high-speed rail. Using the symmetrical components method and equations describing the relationship between voltages and currents on the primary and secondary sides of the transformer one can derive equations describing the value of the voltage unbalance factor εu depending on the transformer power ST and the short-circuit power of the supplying system Ss at the place of connection of the transformer to the supplying grid. The applied procedure allows for estimating voltage unbalance for single-phase, open delta Vv, Way-Delta and Scott as well as Le Blanc, Woodbridge and Roof-Delta transformer connections. The article proves that influence caused by catenary systems of HSR depend on transformer connections, the load and the short-circuit power of the supply network. The voltage unbalance factor calculated for various transformer connections can be very useful when choosing an appropriate supplying method and will help create an optimal train timetable to minimise the unbalance in supplying power system.

I. INTRODUCTION The catenary of high-speed rail are supplied from three-

phase energy system via a specially designed transformer. The main task of this transformer is to convert the three-phase voltage into a two-phase one. A well-known and widely applied transformer used for converting the three-phase voltage system into two-phase one is the Scott’s transformer, whose secondary voltages have the 90° phase shift. By using the solutions which apply the Way-Delta connection or the open delta Vv connection we obtain the output voltage phase shift of 120°. In the extreme case the catenary can be powered by two single-phase transformers or a transformer with push-pull connection connected by secondary windings, whose phase-shift is 180°. For transforming voltage for the purposes of rail powering systems more complex winding connections are used, such as Le Blanc or Woodbridge ones. Among the modern solutions the Roof-Delta connection [5] and so-called balanced connection Yd11 [5] are worth noting. Each transformer has its advantages and disadvantages. These solutions also differ in properties and characteristics. As a result, all kinds of disturbances and load unbalances are transferred to the energy network to varying extent. Regardless of the used connections of primary and secondary windings of the supply transformer, unbalance of voltage and currents is introduced into the supplying system [1], [2], [6]. This results from the fact that the loads connected to two phases of the

secondary side of the transformer change regardless of each other in a very broad range.

The measure of the unbalance introduced to the supply system is the ratio of rms of negative-sequence voltage U2 and positive-sequence voltage U1 at the supply site (expressed in percentage):

εu = U2U1

100% (1)

The article describes the way of determining the values of voltage unbalance factor depending on the connections of the traction transformer windings and the value and relation of the power of receivers connected to the secondary side of this transformer. The voltage unbalance factor calculated for various traction transformer connections can be very useful for choosing an appropriate supplying method to minimise the unbalance in supplying power system.

II. TRACTION TRANSFORMER MODEL Traction transformer is a multi-port with three input terminals, reference terminal and two outputs [4]. In Fig. 1 particular voltages and currents at the input and output terminals of the transformer are marked.

Fig. 1. Traction transformer model

18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

When formulating equations presentingbetween voltages and currents at the input anof traction transformer we use mainly the bassuming that the output power is equal to the

uAN iA+ uBN iB + uCN iC = ux ix + uy iy

where:

uA, uB, uC – voltages on the primary side of iA, iB, iC – currents on the primary side of th ux, uy – voltages on the secondary side of th ix, iy – currents on the secondary side of the

The relationship between the voltages anprimary and secondary sides depend mainly oof transformer windings and turns ratio n = N

The relationship between the voltages on of the transformer and the supply voltages ifollowing equation:

n UL = Nu UH

where: UL − column matrix of the voltages on thof the transformer, UH − column matrix of the voltages on of the transformer, Nu − the matrix of coefficients of turtransformer.

At non-zero values of the ground currenbetween the primary side currents of the travalues of currents on the secondary side isfollowing equation:

n IH = Ni IL + IN

where: IH − column matrix of currents on the prtransformer, IL − column matrix of currents on the secotransformer, IN − column matrix of earth currents of the Ni − matrix of coefficients of the transform

x

BA C

o

yN2 N2

N1 N1

ix iy

iA iB iC

Fig. 2. Open delta transformer connecti

g the relationships nd output terminals balance of power, e input power:

(2)

f the transformer, he transformer, he transformer, e transformer.

nd currents on the on the connections

N1/N2.

the secondary side is presented in the

(3)

he secondary side

the primary side

rns ratio of the

nt the relationship ansformer and the s described in the

(4)

rimary side of the

ondary side of the

e transformer, mer’s current ratio.

y

ion (Vv)

For a transformer whose widelta Vv (Fig. 2), equation (3) terminals of the transformer dvoltages is as follows:

ux= 1

n 1 -1 0

uy 0 1 -1

For the transformer in quphase currents on the primdepending on the currents valutransformer takes the following

iA 1 0 iB = 1

n -1 1

iC 0 -1

Fig. 3. Phasor diagram of volat

Using equation (5) ansinusoidally changing voltagesystem we can state that in thtransformer have the phase shif

Ux = 1n UAB =

1n 3 UA e jπ/6

Uy = 1n UBC =

1n 3 UA e -jπ/2

If the voltage shape is sinus

the transformer’s load can badmittance YxL and YyL connecttransformer:

IL = YL UL

where YL is the admittance mat

indings are connected in an open presenting voltage at the output

depends on the values of input

uA uB

uC (5)

uestion equation (4) describing mary side of the transformer ues on the secondary side of the g form:

ix

iy

(6)

tges and currents in relative units

nd the relationship between es in a symmetrical 3-phase his case output voltages of the ft of 120o (Fig. 3).

(7)

2 (8)

soidal the values and character of be expressed by the value of ted to secondary windings of the

(9) trix of the transformer load.

18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

III. RESPONSE EQUATION OF ANALYSED CIRCUIT In order to describe the relationship between the current

values on the primary side of the transformer and the values of supply voltages and the relationship between the current and voltage at the output terminals of the transformer we need to use equation (4). If in the equation we take into account the relationship (8) and the relationship (3), we obtain:

IH = 1n2 Ni YL Nu UH + IN (10)

As is shown in equation (10) the circuit composed of

a transformer and load is described by the admittance matrix of the circuit YT, which is the product of the matrix of transformer turns ratio and the load matrix:

YT = 1n2 Ni YL Nu = Ni YH Nu (11)

In equation (11) YH is the admittance of the transformer

load calculated for the primary side according to the following relationship:

YH = 1n2 YL (12)

The admittance matrix of the circuit YT can be considered

the sum of the admittance matrix YTx and YTy:

YT = YTx + YTy = Yx Ni 1x Nu + Yy Ni 1y Nu (13) where Yx and Yy are the admittances of load calculated for the primary side of the transformer including the transformer turns ratio.

The response of the circuit composed of a transformer and load can be divided into a sum of responses dependent on particular transformer loads and the matrix of the turns ratio:

IH = (YTx + YTy) UH + IN =

= 1n2 (YxL Ni 1x Nu + YyL Ni 1y Nu) UH + IN =

= (Yx Mx + Yy Mi) UH + IN

(14)

where

My = Ni 1y Nu (15)

For a transformer which windings are connected in an open delta Vv, equation (9) is as follows:

IA 0 1 YxL 0 0 1 -1 UA IB = 1n2 -1 -1 0 YyL

1 -1 0 UB

IC 1 0 UC

(16)

The admittance matrix of the circuit for a transformer with Vv wiring is as follows:

Yy -Yy 0 0 0 0 1 -1 0 -Yy Yx + Yy -Yx = Yx 0 1 -1 + Yy -1 1 0 0 -Yx Yx 0 -1 1 0 0 0

(17)

IV. TRANSFORMER EQUATIONS FOR SYMMETRICAL COMPONENTS

Using the symmetrical components method and equations describing the relationship between voltages and currents on the primary and secondary sides of the transformer one can derive equations describing the value of the voltage unbalance factor εu depending on the transformer power ST and the short-circuit power of the supplying system Ss at the place of connection of the transformer to the supplying grid.

The general formula of transformer equations for symmetrical components is as follows:

Is = YTs Us (18) where: Is − the column matrix of symmetrical components of transformer currents, Us − the column matrix of symmetrical components of transformer voltages, YTs − the admittance matrix of symmetrical components for the transformer.

In order to go from the phase values described in equation (14) to the values for symmetrical components described in equation (18) one only needs to convert the matrixes Mx and My according to the following equations:

Mxs = S Ni 1x Nu S -1 (19)

Mys = S Ni 1y Nu S -1 (20) In equations (19) and (20), the matrix S is the symmetrical

components transformation matrix and S-1 is the inverse matrix of S:

1 1 1 1 1 1

S = 13 1 a a2 S -1 = 1 a2 a

1 a2 a 1 a a2

(21)

The admittance matrix of the loaded transformer for

symmetrical components is expressed in the following equation:

YTs = Yx Mxs + Yy Mys (22)

18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

For traction transformer with windings connected in an open delta Vv, which is presented in equations (16), using the conversion of the phase values into symmetrical components we obtain:

Io 0 0 0 Uo I1 = 0 Yx + Yy - Yx - a Yy U1 I2 0 - Yx - a2 Yy Yx + Yy U2

(23)

Analysing the above equation one can note that the circuits

for the positive and negative sequences are connected parallel and the circuit of the zero sequence is disconnected. Fig. 4 presents the connections of symmetrical components circuits for a Vv connected transformer.

Fig. 4. Connections of symmetrical components circuits for Vv transformer

V. VOLTAGE UNBALANCE FACTOR Taking into account the connections in particular circuits

for symmetrical components (Fig. 4) and Kirchhoff’s Current Law for A2 node it can be stated that:

(Y2 + Yx + Yy) U2 – Y2 E2 = (Yx + a2 Yy) U1 (24)

In the case when the source of supplying voltages is symmetrical, i.e. E2 = 0 and Y2 = Ys, we obtain:

U2 (Ys + Yx + Yy) = U1 (Yx + a2 Yx) (25)

As can be seen in the above equation, the relationship of the negative sequence voltage U2 to positive sequence voltage U1, is described in the following equation:

U2U1

= - Yx + a2 Yy

Ys + Yx + Yy (26)

If we take into account the relationship between the rms power S, and the admittance element of the circuit Y, it can be stated that:

U2U1

= - Yx U1

2 + a2 Yy U12

(Ys + Yx + Yy) U12 = -

Sx + a2 SySs + Sx + Sy

(27)

where: Sx, Sy − the load of particular windings on the secondary side of the transformer, Ss − short-circuit power of the system at the place of transformer connection.

Because Ss ≫ Sx + Sy, the relationship between voltages for symmetrical components can be presented as:

U2U1

= - Sx + a2 Sy

Ss (28)

Due to the fact that the sum of loads on the secondary side

of the transformer should be lower than the nominal power of the transformer ST the above equation takes the following form:

U2U1

= - k ST + a2 (1 - k) ST

Ss =

= - (k + a2 - a2 k) STSs

≅ - 3k2 - 3k + 1 STSs

(29)

where k is the factor of load distribution on the secondary side of the transformer.

From equation (29) we can see that the voltage unbalance factor εu is the product of the factor kf describing the transfer energy by the transformer and the ratio of rms power of the transformer ST and short-circuit power Ss of the system at the supplying place:

εu = kf STSs

100% (30)

Generally, the value of coefficient kf depends on the phase shift between voltages on the secondary side of the transformer [4]. For transformer in which voltages on the secondary side have the phase shift of 120o, kf coefficient is calculated from the following equation:

kf = 3k2 - 3k + 1 (31) In the case when the voltages on the secondary side of the traction transformer have the phase shift of 90o, the value of kf coefficient is calculated according to the equation below:

kf = |2k - 1| (32)

Equation (30) shows that the value of voltage unbalance factor is mainly dependent on the value of short-circuit power in a traction substation. The value of voltage is much less dependent on the power distribution on the secondary side of the transformer and the value of the phase shift between voltages supplying the rail traction.

18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

VI. CONCLUSIONS The analysis for supplying of High Speed Rail catenary in

Poland has been conducted [3]. All nine substations have two 60 MVA transformers with windings connected in open delta (Vv). In normal supplying condition when all substation are fully operational then value of voltage balance is small and all requirements are fulfilled from Operation and Maintenance Manual of the National Energy System.

Unfortunately, in an emergency situation, when the supplying system needs to function on one transformer only, or in the case of lengthening the supplied section of catenary due to a fault in the neighbouring substation, the voltage unbalance factor reaches the value of about 1,5 %. This problem concerns 2 out of 9 traction substations which are supposed to supply traction network of HSR. For the two substations both the instantaneous and the mean value of voltage unbalance calculated for a 10 minute period exceed the boundary values set out in the regulations.

If we assume that the power of a typical HSR traction substation is 60 MVA, and the windings of transformer is Vv connection, and when the admissible value of the voltage unbalance factor need to be less than 1 %, the short-circuit power of the system should be higher than 6 GVA. The choice of the level of supply voltage and the type of winding connection have significant influence on the decision whether or not to use additional compensators for instantaneous voltage changes in the supplying grid. Of course, an important aspect when selecting the way of supplying traction systems is the

cost of production, installation and maintenance of particular transformers.

REFERENCES [1] G. Burchi, C. Lazaroiu, N. Golovanov, M. Roscia: “Estimation of

Voltage Unbalance in Power Systems Supplying High Speed Railway”, Electrical Power Quality and Utilisation, Vol. XI, No. 2, pp. 113-119, 2005.

[2] S. L. Chen, R. J. Li, P. H. Hsi: “Traction system unbalance problem - analysis methodologies”, IEEE Tran on Power Delivery, Vol. 19, pp. 1877-1883, 2004

[3] L. Ładniak, W. Rojewski, M. Sobierajski: “Opracowanie koncepcji budowy układów zasilania kolei dużych prędkości”, Raport Serii Sprawozdania, Politechnika Wrocławska 2011.

[4] L. Ładniak: “Transformacja napięć i prądów w układach zasilania trakcji kolei dużych prędkości”, Transcomp XIV International Conference Computer Systems Aided Science, Industrial and Transport, Zakopane 2010.

[5] H. Morimoto: “New-Type Transformer for AC Feeding”, Railway Technology Avalanche, Newsletter on the Latest Technologies Developed by RTRI, RTA-20-3, Dec. 18, 2007.

[6] P. E. Sutherland, M. Waclawiak, M. F. McGranaghan; “Analysis of Harmonics, Flicker and Unbalance of Time-Varying Single-Phase Traction Loads on a Three-Phase System”, International Conference on Power Systems Transients, Canada, June 19-23, 2005.

[7] Wang Guo, Ren Enen, Tian Mingxing: “A hybrid Active Compensation Method for Current Balance Based on Yd11 connection traction transformer”, Workshop on Power Electronics and Intelligent Transportation System, IEEE PEITS, pp. 582-586, 2008.