Calculation of Voltage Unbalance Factor in Power System Supplying Traction Transformers

Transcription

1 Calculation of Voltage Unbalance Factor in Power System Supplying Traction Transformers Lesław Ładniak Wroclaw University of Technology Wroclaw, Poland 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 ε u depending on the transformer power S T and the shortcircuit power of the supplying system 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 threephase 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 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 phaseshift 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 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 and positive-sequence voltage at the supply site (expressed in percentage): ε u = 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

2 When formulating equations presenting the relationships between voltages and currents at the input and output terminals of traction transformer we use mainly the balance of power, assuming that the output power is equal to the input power: u AN i A + u BN i B + u CN i C = u x i x + u y i y (2) u A, u B, u C voltages on the primary side of the i A, i B, i C currents on the primary side of the u x, u y voltages on the secondary side of the i x, i y currents on the secondary side of the transformer. The relationship between the voltages and currents on the primary and secondary sides depend mainly on the connections of transformer windings and turns ratio n = N 1 /N 2. The relationship between the voltages on the secondary side of the transformer and the supply voltages is presented in the For a transformer whose windings are connected in an open delta Vv (Fig. 2), equation (3) presenting voltage at the output terminals of the transformer depends on the values of input voltages is as follows: u x = n u y For the transformer in question equation (4) describing phase currents on the primary side of the transformer depending on the currents values on the secondary side of the transformer takes the following form: i A 1 0 i B = 1 n -1 1 i C 0-1 i x i y u A u B u C (5) (6) n U L = N u U H (3) U L column matrix of the voltages on the secondary side of the U H column matrix of the voltages on the primary side of the N u the matrix of coefficients of turns ratio of the transformer. At non-zero values of the ground current the relationship between the primary side currents of the transformer and the values of currents on the secondary side is described in the n I H = N i I L + I N (4) I H column matrix of currents on the primary side of the I L column matrix of currents on the secondary side of the I N column matrix of earth currents of the N i matrix of coefficients of the transformer s current ratio. Fig. 3. Phasor diagram of volatges and currents in relative units Using equation (5) and the relationship between sinusoidally changing voltages in a symmetrical 3-phase system we can state that in this case output voltages of the transformer have the phase shift of 120 o (Fig. 3). A B C U x = 1 n U AB = 1 n 3 U A e jπ/6 (7) i A i B i C N1 N1 U y = 1 n U BC = 1 n 3 U A e -jπ/2 (8) x ix N2 N2 iy y If the voltage shape is sinusoidal the values and character of the transformer s load can be expressed by the value of admittance Y xl and Y yl connected to secondary windings of the transformer: Fig. 2. Open delta transformer connection (Vv) o I L = Y L U L (9) where Y L is the admittance matrix of the transformer load.

3 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: I H = 1 n 2 N i Y L N u U H + I N (10) As is shown in equation (10) the circuit composed of a transformer and load is described by the admittance matrix of the circuit Y T, which is the product of the matrix of transformer turns ratio and the load matrix: Y T = 1 n 2 N i Y L N u = N i Y H N u (11) In equation (11) Y H is the admittance of the transformer load calculated for the primary side according to the following relationship: Y H = 1 n 2 Y L (12) The admittance matrix of the circuit Y T can be considered the sum of the admittance matrix Y Tx and Y Ty : Y T = Y Tx + Y Ty = Y x N i 1 x N u + Y y N i 1 y N u (13) where Y x and Y y 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: where I H = (Y Tx + Y Ty ) U H + I N = = 1 n 2 (Y xl N i 1 x N u + Y yl N i 1 y N u ) U H + I N = = (Y x M x + Y y M i ) U H + I N (14) M y = N i 1 y N u (15) For a transformer which windings are connected in an open delta Vv, equation (9) is as follows: I A 0 1 Y xl U A I B = 1 n Y yl U B (16) I C 1 0 U C The admittance matrix of the circuit for a transformer with Vv wiring is as follows: Y y -Y y Y y Y x + Y y -Y x =Y x Y y Y x Y x (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 ε u depending on the transformer power S T and the shortcircuit power of the supplying system at the place of connection of the transformer to the supplying grid. The general formula of transformer equations for symmetrical components is as follows: I s = Y Ts U s (18) I s the column matrix of symmetrical components of transformer currents, U s the column matrix of symmetrical components of transformer voltages, Y Ts 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 M x and M y according to the following equations: M xs = S N i 1 x N u S -1 (19) M ys = S N i 1 y N u 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: S = a a 2 S -1 = 1 a 2 a 1 a 2 a 1 a a 2 (21) The admittance matrix of the loaded transformer for symmetrical components is expressed in the following equation: Y Ts = Y x M xs + Y y M ys (22)

4 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: I o U o I 1 = 0 Y x + Y y - Y x - a Y y I Y x - a 2 Y y Y x + Y y (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. 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: = - Y x 1 + a 2 2 Y y 2 = - S x + a 2 S y (Y s + Y x + Y y ) + S x + S y (27) S x, S y the load of particular windings on the secondary side of the short-circuit power of the system at the place of transformer connection. Because S x + S y, the relationship between voltages for symmetrical components can be presented as: = - S x + a 2 S y (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 S T the above equation takes the following form: = - k S T + a 2 (1 - k)s T = = - (k + a 2 - a 2 k) S T - 3k 2-3k + 1 S T (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 ε u is the product of the factor k f describing the transfer energy by the transformer and the ratio of rms power of the transformer S T and short-circuit power of the system at the supplying place: 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 A 2 node it can be stated that: (Y 2 + Y x + Y y ) Y 2 E 2 = (Y x + a 2 Y y ) (24) In the case when the source of supplying voltages is symmetrical, i.e. E 2 = 0 and Y 2 = Y s, we obtain: (Y s + Y x + Y y ) = (Y x + a 2 Y x ) (25) As can be seen in the above equation, the relationship of the negative sequence voltage to positive sequence voltage, is described in the = - Y x + a 2 Y y Y s + Y x + Y y (26) ε u = k f S T 100% (30) Generally, the value of coefficient k f 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 120 o, k f coefficient is calculated from the k f = 3k 2-3k + 1 (31) In the case when the voltages on the secondary side of the traction transformer have the phase shift of 90 o, the value of k f coefficient is calculated according to the equation below: k f = 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.

5 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 , [2] S. L. Chen, R. J. Li, P. H. Hsi: Traction system unbalance problem - analysis methodologies, IEEE Tran on Power Delivery, Vol. 19, pp , 2004 [3] L. Ładniak, W. Rojewski, M. Sobierajski: Opracowanie koncepcji budowy układów zasilania kolei dużych prędkości, Raport Serii Sprawozdania, Politechnika Wrocławska [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 [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, [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, [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 , 2008.