This paper deals with the modeling of multi-winding transformers. Based on the real example of a railway DC - supply, a modeling methodology is presented as the original part of the paper. Afterwards two application examples of multi-winding phase shifting transformers (18 and 24 pulse) are described, these applications have been simulated using the SIMSEN simulation software package [1,2]. Harmonics analysis of simulation results (elimination of low order harmonics 5, 7, 11, 13) has validated the proposed methodology

Allenbach, P.

Sapin, A.

Simond, J.-J.

This paper deals with the modeling of multi-winding transformers. Based on the real example of a railway DC - supply, a modeling methodology is presented as the original part of the paper. Afterwards two application examples of multi-winding phase shifting transformers (18 and 24 pulse) are described, these applications have been simulated using the SIMSEN simulation software package [1,2]. Harmonics analysis of simulation results (elimination of low order harmonics 5, 7, 11, 13) has validated the proposed methodology.LM

Infoscience - École polytechnique fédérale de Lausanne

   "#D%LI() #* "+LTI-WI(DI() /HA2% 2HI*TI() TRA(2*#R"%R2 A//LICATI#(2 T# DC A(D "+LTI-L%5%L 52I 2+//LI%2  On request of the PCIM organizers, this paper is a reviewed issue of an ICEM 2000 contribution [6]  A. Sapin, P. Allenbach, J.-J. Simond Swiss Federal Institute of Technology Electrical Machines Laboratory (LME) CH-1015 Lausanne, Switzerland Tel: +41 21 693 56 09, Fax: +41 21 693 26 87, E-mail: alain.sapin@epfl.ch    A62TRACT  This paper deals with the modeling of multi-winding transformers. Based on the real example of a railway DC - supply, a modeling methodology is presented as the original part of the paper. Afterwards two application examples of multi-winding phase shifting transformers (18 and 24 pulse) are described, these applications have been simulated using the SIMSEN simulation software package [1,2]. Harmonics analysis of simulation results (elimination of low order harmonics 5, 7, 11, 13) has validated the proposed methodology.  Key:ords? Multi-winding transformer, phase shifting, harmonics reduction, DC supply, simulation, harmonics analysis.   1 I(TR#D+CTI#(  The DC power supplies are made of rectifiers [3]. Those are providing harmonics leading to a great inconvenience for the power networks. Even if additional L-C filters are able to absorb the most important part of the harmonics content, they may introduce resonance problems. A more suitable solution is to increase the pulse number of the rectifiers. This requires phase shifting transformers having multi-windings [4]. The most complex part to modelize such special transformers is the determination of all the needed parameters: inductance and resistance values for each winding, coupling coefficients between all the considered windings. These parameters must be determinated using the measurements obtained from the typical tests under no-load and short circuit conditions.     1V 1W1U 2W2V2U 3U3V 3W1U1V1W2U2V2W-7.5°3U3V3W22.5°Primary Screen Secondary 1 Secondary 2Vector group T1 = Dd11.75d0.75   LA1D LB1D LC1D1V 1W1ULA2Y2V 2W2ULA2DLB2YLB2DLC2YLC2DLB3Y3W 3U3VLB3DLC3YLC3DLA3YLA3D2V' 2W'2U' *igure 1 - "ulti-:inding phase shiJting transJormer example   2 "%TH#D#L#)N  2.1 Systematic names definition  The developed methodology is explained using the multi-winding phase shifting transformer example given in the left part of Fig. 1. The initial stage is to define the different names and indexes for inductances and to use for the development, as shown in the right part of Fig. 1:  L     Inductance  X    phase x (a, b, c)   I   winding I (1, 2, 3…)    Y/D  star/delta connection   Example: LB2D inductance, phase b, winding 2, delta-connected.  The different coupling coefficients also have to be defined carefully:  k     Coefficient  i    winding i (1, 2, 3…)   y/d   start/delta connection    j  winding j (1, 2, 3…)     y/d star/delta connection   Example: k1d2y coupling coefficient between delta-connected winding 1 and star-connected winding 2.  2.2 Necessary input data  The necessary input data correspond to the data plate of the transformer as well as some additional measurements coming from the typical tests under no-load and short circuit conditions. These input data are:  SN1 = 3146 kVA apparent power. UN1 = 22 kV primary side line voltage. FN = 60 Hz frequency.     UN20 = 569 V  secondary 1 line voltage     (no-load) UN30 = 569 V secondary 2 line voltage     (no-load) i0 = 0.00569 p.u. no-load primary side current. xcc12 = 0.0878 p.u. short circuit reactance 1-2,    3 open. xcc13 = 0.0878 p.u. short circuit reactance 1-3,    2 open. xcc23 = 0.0112 p.u. short circuit reactance 2-3,    1 open R1 = 0.611 Ohm primary terminal resistance. R2 = 0.00088 Ohm secondary 1 terminal resist. R3 = 0.00104 Ohm secondary 2 terminal resist.  2.3 Stray reactances in per unit  In a first stage, the stray reactances (xs1, xs2 and xs3) of each terminal are calculated in [p.u.]:  p.u. 0822.022313121 =−+= xccxccxccxs  (1) p.u. 0056.021323122 =−+= xccxccxccxs  (2) p.u. 0056.021223133 =−+= xccxccxccxs  (3)   2.4 Primary winding inductances  For the primary winding, the rated impedance ZN1Y is first calculated according to equation (4):  Ohm 8.1531112==SNUNYZN  (4)  Using equations (1) and (4), the equivalent star-connected stray reactance XS1 is:  Ohm 64.12111 =⋅= YZNxsXS  (5)  The star-connected equivalent no-load impedance Z1Y is calculated taking into account the no-load current i0:  kOhm 04.271011 =⋅= YZNiYZ  (6)  Using the angular frequency, the equivalent star-connected inductance L1Y is calculated:  H 72.71211 =⋅⋅=FNYZYLpi (7)  To obtain the delta-connected inductance L1D, the previous value has to be multiplied by 3:  H 2.215131 =⋅= YLDL  (8)  Doing the same operation for the delta-connected stray inductance LS1D, we obtain:  H 1006.02131 =⋅⋅⋅=FNXSDLSpi (9)  Finally, the main delta-connected inductance LH1D is:  H 1.215111 =−= DLSDLDLH  (10)   2.5 Secondary 1 winding inductances  The mathematical development is based on the elements represented in Fig. 2. For the secondary 1 winding (index 2), the desired phase shifting angle is equal to –7.5°. This means that this secondary winding has to be a combination of Dy11 (-30°) and Dd0 (0°). The first step is to define the ratios n2y/n1d   and n2d/n1d between the turns numbers of the windings 2Y, 2D and 1D according to the equations (11) and (12):.  adnyn=12   (11) bdndn=12             (12)  The phase angle between the line voltages U1U-1V and U2U-2V must be 7.5 o. The line voltage U2U-2V is the sum of the three partial voltages: U2U-2U’, U2U’-2V’, U2V’-2V, as represented in Fig. 2.  WVVUVUVU UaUbUaU 11111122 ... −−−− −+=      (13)  2U2V2W52.5°60°7.5°2V'2U'b U1U-1V_a U1U-1V_-a U1U-1V_ *igure 2 O 5ector diagram Jor the secondary 1 :inding Under the assumption that primary and secondary line voltages are equal (a and b become a’ and b’ in equations 11 and 12) and by taking into account the angles at points 2V’ and 2V equal to 60o and 60 – 7,5 = 52,5o, the projections of equation (13) on both horizontal and vertical axis lead to:  cos 60o. a’ + cos 60o. b’ + a’ =cos 52.5o                                                                                           (14)  sin 60o. a’ + sin 60o. b’ = sin 52,5o  and finally:  00389819,0120'.12===UNUNadnyna               (15) 019795,0120'.12===UNUNbdndnb                      (16)                                            The delta-connected inductance L2D is:  mH 31.8411222=⋅!"#$%&= DLdndnDL  (17)  Doing the same for the star-connected inductance L2Y, we obtain:  mH 269.322222=⋅!"#$%&= DLdnynYL  (18)  The equivalent star-connected stray reactance is calculated using equations (2):  mOhm  5763.01201222=!"#$%&⋅⋅=UNUNYZNxsXS  (19)  The corresponding inductance is:  H 528.1222 µpi=⋅⋅=FNXSLS  (20)  Now, the global stray inductance LS2 has to be distributed on both star/delta-connected part of the secondary 1 winding. Therefore, a total equivalent star-connected inductance L2YTOT has to be defined:  mH 37.3123122 =⋅+= DLYLYTOTL  (21)  Then, the star-connected stray inductance is:  H 159.02222 µ=⋅= LSYTOTLYLYLS  (22)  The remaining delta-connected stray inductance LS2D is:  H 108.4)22(32 µ=−⋅= YLSLSDLS  (23)  Finally, the two main inductances of the secondary 1 winding are deduced:  mH 31.84222 =−= DLSDLDLH  (24)  mH 269.3222 =−= YLSYLYLH  (25)  For all the secondary 1 winding, the total, main and stray inductances have been defined for both star/delta-connected windings. The same operations can be applied to the secondary 2 winding.   2.6 Coupling coefficients  The last step is to define the coupling coefficients between all the windings. The coefficients are calculated with the assumption that the 3 magnetic columns of the transformer are symmetrical. This leads to a magnetic coupling coefficient km equal to –0.5 between the columns. Two examples of coupling coefficients are calculated in equations (26,27):  9997.0212121 =⋅⋅=DLDLDLHDLHddk  (26)  9997.0212121 =⋅⋅=YLDLYLHDLHydk  (27)     2.7 Winding resistances  Finally, the resistance of each winding is calculated according to the measured terminal resistance and the related connections. For the primary winding, this is easy, because the terminal resistance is equal to the winding resistance (delta-connection).  Ohm 611.011 == RDR  (28)  With the assumption that the resistance is proportional to the square root of the inductance, we can write:  mOhm  1243.022222 =+=YLDLRYR (29)  mOhm  6313.02222 =⋅= YRYLDLDR  (30)   3 "#D%LI() A(D I"/L%"%(TATI#( I( 2I"2%(  The SIMSEN simulation software has been developed to simulate electrical power and adjustable speed drive systems having an arbitrary topology [1,2]. A given system topology (i.e. fig. 3 and 6) is built by choosing and linking the different necessary elements or modules from a module’s list. The actual module’s list of SIMSEN offers about one hundred modules, one of them is the so-called “linked inductor”, this module is suitable for modeling multi-winding transformers. Table 1. shows the input data for one linked inductor. In the section - LINKED INDUCTORS - all the coupled linked inductors as well as their related coupling coefficients are mentioned.    *igure 3 - 24-pulse DC supply Jor rail:ay suSstation.   - GENERAL DATA :Name = T1LA1DComment =Writing = SI- LINKED INDUCTORS :;T1LA1DT1LB1D -k1d1d*kmT1LC1D -k1d1d*km;---------------T1LA2Y k1d2yT1LB2Y -k1d2y*kmT1LC2Y -k1d2y*km;---------------T1LA2D k1d2dT1LB2D -k1d2d*kmT1LC2D -k1d2d*km;---------------T1LA3Y -k1d3y*kmT1LB3Y k1d3yT1LC3Y -k1d3y*km;---------------T1LA3D -k1d3d*kmT1LB3D k1d3dT1LC3D -k1d3d*km- RATED VALUES:  Sn   [VA] =  0.00000000000E+0000 Un    [V] =  0.00000000000E+0000 Fn   [Hz] =  0.00000000000E+0000  - PARAMETERS :  R   [Ohm] =  6.11000000000E-0001 L     [H] =  2.15161475046E+0002  - INITIAL CONDITIONS:  I     [A] =  1.37001402709E+0001  - CALCULATED VALUES:  Leff  [H] =  2.15161475046E+0002 TaSle 1. Input data Jor linUed inductor T1LA1D  The SIMSEN simulation software automatically takes into account all the differential equations of a multi-winding transformer defined with linked inductors. The fact that a linked inductor can be coupled with an infinite number of other linked inductors is one of the SIMSEN’s features, it makes possible combining any kind of magnetic coupled circuits.  For further information: http://simsen.epfl.ch     4 A//LICATI#( %VA"/L%2  4.1 24-pulse transformer for a railway DC supply  The first application example deals with the 24-pulse DC supply of a railway substation. The corresponding circuit is represented in Fig. 3.  The DC supply is made of two 12-pulse transformers parallel connected. Each of them is modeled by using 15 linked inductors. The first transformer has the connection Dd11.75d0.75 and the second one has the connection Dd0.25d11.25. The parallel connection allows supplying the 4 diode rectifiers with shifted angles –22.5°, -7.5°, +7.5° and +22.5°. This leads to a 24-pulse rectifier system. Note that the rectifiers are parallel connected. To equalize the load of both 12-pulse transformers, additional chokes are connected in the DC-link. The next figures present simulation results in steady state at 100% load with symmetrical transformers, as calculated in point 2.  As expected, the first dominant AC supply phase current harmonics are the 23rd and the 25th. The harmonics analysis results confirm the proposed 24-pulse modeling.    *igure 4 - AC supply phase current.    *igure W - 2pectrum oJ AC supply phase current.    4.2 Multi-level Voltage Source Inverter 18-pulse DC supply  The studied circuit (see Fig. 6) corresponds to a multi-level Voltage Source Inverter (VSI) feeding an AC motor. This topology has been presented in [5].  *igure X - 1Y-pulse DC supply Jor multi-level 52I.  The VSI is made of several DC-cells, each supplied through a 6-pulse diode rectifier. Regarding the present paper, the most interesting part of the above topology is the 18-pulse multi-winding transformer. The VSI requires many floating voltage potentials. A special transformer having many output or secondary windings fulfills this specification.   *igure [ - AC supply phase current.     *igure Y - 2pectrum oJ AC supply phase current.   *igure \ - "otor phase voltage, current and air-gap tor^ue  The transformer is modeled using 48 linked inductors. Each linked inductor is coupled with the 47 other linked inductors. The calculation of all the needed parameters is based on the same principle as the one developed in point 2. As expected, the first dominant AC supply phase current harmonics are the 17th and the 19th. The simulation results have been compared successfully with measurements coming from [4].  W C#(CL+2I#(2  Multi-winding phase shifting transformers are a suitable solution to supply DC systems with low harmonics content in the AC supply phase current. Based on a real example, a modeling methodology has been developed. This method requires the main data of the transformer, the phase shifting angles as well as the windings connections. Multi-winding phase shifting transformers have been simulated using the SIMSEN simulation software, which proved to be a powerful tool to simulate such complex transformers. Harmonics analysis of simulation results (elimination of low order harmonics 5, 7, 11, 13) has validated the proposed methodology.   R%*%R%(C%2  [1] A. Sapin, J.-J. Simond: SIMSEN : A modular software package for the analysis of power networks and electrical machines, CICEM’95, Hangzhou, China. [2] J.-J. Simond, A. Sapin, B. Kawkabani, D. Schafer, M. Tu Xuan, B. Willy: Optimized Design of Variable Speed Drives Based on Numerical Simulation, EPE 1997. [3] Paice D.: Power Electronic Converter Harmonics, Multipulse methods for clean power, IEEE Press. [4] Enjeti P.N, Choi, S., I.J. Pitel : Polyphase Transformer Arrangements with Reduced kVA Capacities for Harmonic Current Reduction in Rectifier-Type Utility Interface, IEEE Trans. on Power Electronics, Vol. 11, No. 5, Sept. 1996. [5] Peter W. Hammond: A New Approach to Enhance Power Quality for Medium Voltage AC Drives, IEEE Transaction on Industry Applications, Vol. 33 No 1, 1997.  [6] A. Sapin, P. Allenbach, J.-J. Simond: Modeling of        Multi-Winding Phase Shifting Transformers,  Applications to DC Supplies. ICEM 2000, Helsinki    

Modeling of Multi-winding Phase Shifting Transformers : Application to DC and Multi-level VSI Supplies.

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Modeling of Multi-winding Phase Shifting Transformers : Application to DC and Multi-level VSI Supplies.

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