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Finite Element Analysis of the 500 MVAHydro-Generators at the Bieudron Power PlantErich Schmidt, Member, IEEE

Vienna University of Technology, Institute of Electrical Drives and MachinesGusshausstrasse 25-29, A-1040 Vienna, Austria,Email: eri ch. schmdtQtuw en. ac. at-Christian Grabner

Graz University of Technology, Institute of Electrical Machines and Drives+Kopernikusgasse24, A-8010Graz, AustriaEmail: grabner@ema. u-grazac. at

Georg Traxler-SamekALSTOM Power Ltd.,Hydro Generator Technology CenterCH-5242 Birr, SwitzerlandEmail: georg. t raxl er@power. al st omcom

Abstract- For an implementationof the finite el-ement analysis in the design optimization of largesynchronous machines, the efficiency of modelingand analysis methods is of great importance. Thepresented modeling and analysis methods utilizeonly one finite element model for all angular rotorpositions without any remeshing. These methodsare applied to the nonlinear finite element analysisof the salient-pole hydro-generatorsat the Bieudronpower plant in Switzerland. The comparison withmeasurementsof the steady-state,parametersshowsan excellent agreement and validates the proposedmethods.Keywords-Space vector calculus, Reactancecal-culation, Synchronous machine, Domain decamp+sition, Finiteelement analysis

1. INTRODUCTIONThe finite element analysis of large synchronous

generators provides a significant improvement in theaccuracy of the predicted performance. Therefore, thefinite element method is increasingly utilized not onlyfor the verification of contractual values of existingmachines, but also for the design optimization of newmachines [l],2]. In the caseof fractional slot statorwindings, the inclusion of more than one or two polepitches is necessary for the consideration of a precisemodel of the synchronous machine. These large finiteel

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11. FINITELEMENTODELINGDue to the fractional slot stator winding with 23/7

slots per pole and phase, the 2D finite element modelhas to consider one half of the synchronous genera-tor. The periodicity is obtained by using anti-periodicboundary conditions along the two boundaries beingseven pole pitches apart. Fig. 1depicts the mesh ofone pole pitch in the iron and the winding regions.The data of the complete model are summarized inTable11

Fig. 1: Finite element model of one pole pitch

For the purpose of the intended domain decom-position, the complete 2D model is divided into twodistinct parts. The stator part consists of 69 statorslots with 138stator bars representing the fractionalslot stator winding according to the winding diagram.The rotor part holds seven poles including the damperwindings, the field windings and the rotor yoke. Asshown in Fig. 1, additional 1D elements consider theairgap between the poles and the rotor yoke due tothe construction.

Both model parts have an equidistant discretizationin moving direction on the cylindrical sliding surfaceinside the airgap. This sliding surface between the in-dependent stator and rotor parts facilitates the analy-sis of different rotor positions without any remeshingof the airgap regions [3],[6].

The complete finite element model is solved withthe nonlinear magnetostatic analysis of the EMASsolver [7]. Within the solver, a domain decompositionalgorithm based on 181and fully described in [3]is uti-lized for the analysis of different rotor positions. Thisalgorithm reduces the number of unknown potentialsin the analysis of each rotor position as listed in Ta-ble 11.Thus, a significant reduction of the calculationtime for successive rotor positions is established.

TABLE I 1FINITE EL EM ENTODE L DATANumber of elements 102296Number of nodes 89935 'Unknown potentials 87806Residual potentials 2268

111. ANALYSISESULTSA . Fi el d Results

Fig. 2 shows the distribution of the magnetic fluxdensity in the airgap and the fundamental harmonicwith a field current excitation of I f =1.5kA. As ex-pected, the stator slots cause ripple componentsof approximately 30% of the fundamental magnitude. Dueto the fractional number of slots per pole and phase,these higher harmonics do not cause any equivalenthigher harmonics in the sinusoidal induced generatorvoltage.

Is 2r 3r I n 5s 6r 7s1.51

1p (WFig. 2: Magnetic fl ux density Br(v) n the airgap versusangular position, field current excitation o f Zf =1.5kAFig. 3and Fig. 4 show the distribution of the mag-netic fl ux density in the airgap with rated stator cur-rent excitation. In both pictures, the fundamentalharmonic and the third harmonic are drawn,addition-ally. This harmonic component in theflukdensity di stribution arises from the saturation and the saliency

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of the synchronous generator, in particular in case ofa stator current space vector in quadrature axis.E.(+ (T)

I

I-0.75 Ic(4

Fig. 5: Magnetic flux density Br ( v) n the airgap versusangular position, idealized short circuit condition,field current excitation of 1, =1.5kAwhich is obtained directly from theEMAS solver. Re-ferring to the base quantities (7) 191 the magneticenergy (1) can be written in a normalized form i t s

w , = i a ~ a + i b $ b + i c $ c . (2)As the stator is star-connected, zero-sequence statorcurrents are impossible. Thus, the transformation tothe dq rotor fixed reference frameas described in [9],[lo)yields

I1 2r 3n 4 r 5~ e rr1.5 1

p (rad)Fig. 3: Magnetic flux density Br(q) n the airgap versusangular position, rated stator current excitation indirect axis

Finally, Fig. 5 shows the distribution of the mag-netic flux density in the airgap in the idealized shortcircuit condition with a field current excitation of1 = 1.5kA . With this condition, the fundamentalharmonic vanishes. Due to the fractional slot statorwinding, the local compensation of the field and sta-tor current excitations differs among the several poles.This can be seen with the additionally drawn thirdharmonic. Fig. 6 shows the magnetic flux distribu-tion in the stator slots and teeth, the airgap and therotor poles according to this short circuit condition.B. Steady-state Reactances

The reactance calculation of the synchronous ma-chine utilizes the total magnetic energy of the threestator phases in steady state,

Due to the fractional slot stator winding, the cztlcu-lated total magnetic energy has nearly no dependencyon the rotor position. Therefore, the evaluation ofboth reactances X d, xq from the magnetic energy (3)isstraightforward for the current injection accordingto i d = 0.. .1.4 and iq = 0.. .1.4. As depicted inFig. 7, the saturation gains influence with stator (cur-rent magnitudes of is >0.6.

The splitting of the synchronous reactances asZ d =XU +ZdhZ q =2 s +Zqh

, (44(4b)

allows for the evaluation of the stator leakage mac-tance zo from the distribution of the magnetic ihxdensity within the airgap. The magnitudes Bid, .B1,obtained from a Fourier analysis of the magnetic h xdensity distributions can be used as

(5)- qhli,=l xqhB l d $dh l i d=l xdh- - .Therefore, the stator leakage reactancex u is evaluatedby using (4) as

xq x qh---

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Fig. 6: Magnetic flux distribution n the idealized short circuit condition, field current excitationof I f =1.5kA

The calculated valueas listed in Table I11shows al-most no dependency on the current excitation in therangeof i s

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ered. The nonlinear analysis uses a domain decom-position algorithm with fully independent stator andrcitor model parts. This analysis method facilitatesa finite element mesh which remains completely un-changed forall rotor positions without .any remeshing.Moreover, the domain decomposition within the non-linear analysis provides a significant reduction of thecalculation time for successive rotor positions. Thecomparison with measurements shows the high suit-ability of the proposed methods for an applicationin the integrated design optimization of large syn-chronous generators.

APPENDIXAs proposed in [9],base quantities using the effec-

tive rated phase current I N , the effective rated phasevoltageUN and the rated frequency f N are definedasbase current I b = f i I N 1

4 N2 r f N 'ase fl ux linkage $b =-

REFERENCESZhou P., McDermott T.E., Cendes Z.J ., Rahman MA.:"Steady State Analysis of Synchronous Generators by aCoupled Field-Circuit Method". Proceedings of the IEEEInternational Electric &fachines and Drives Conference,IEMDC, Milwaukee(WI,USA), 1997.Ramirez C., 'h Xuan M., Simond J .J ., Schafer D.,Stephan C.E.: "Synchronous Machines Parameter:, De-termination using Finite.Elements Method". Proceedingsof the International Conference on Electrical Macliines,ICEM, Espoo (Finland), 2000.Schmidt E.: "Electromagnetic Finite Element Aniilysisof Electrical Machines using Domain Decomposition andFloating Potentials". Proceedings of the 13th Conferenceon the Computation of Electromagnetic Fields, COM-PUMAG, Evian (France), 2001.HowaldW., Stijckli F.: "Generatoren fur daa weltgrhssteHochdruck-Wasserkraftwerk" . ABB Technik, 10/1994.HowaldW.: "Die Generatorenfur das Hochdruck-Wasser-kraftwerk Bieudron". ABB Technik,02/1998.Zhou P., StantonS., CendesZ.J.: "Dynamic ModelingofElectric Machines". Proceedingsof the2nd Naval Syrnpesium on Electric Machines, Annapolis (MD, USA), 1998.Brauer J.R., MacNeal B.E.: MSC/EMAS User Mariual.MacNeal-Schwendler Corporation, Los Angeles, 1994.N.N.: MSC/NASTRAN Superelement Analysis, Sem-nar Notes. MacNeal-Schwendler Corporation, Los Ange-l es, 1997.Kovacs, P.K.: Tbansient Phenomena in Electrical Ma-chines. Elsevier, Amsterdam, 1984.Nabeta SI.: &tudedes Regimes Ttansitoires des ,Ma-chines Synchrones par la Mkthode des filkrnents Finis.Dissertation, I nstitut National Polytechnique de Greno-ble, 1994.

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