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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) 20 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. 21 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) 23 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]