Predcton of the o-load Voltage Waveform of Lamnated Salent-Pole Synchronous Generators S. Keller M. Tu Xuan J.-J Smond Member IEEE Ecole Polytechnque Fédérale de Lausanne (EPFL) Laboratore de Machnes Electrques CH-05 Swtzerland Tel.: 4 2 693 46 90 Fax: 4 2 693 26 87 Emal: stefan.keller@epfl.ch Abstract - Ths paper presents a combned analytcal and FEM method for predcton of the no-load voltage waveform of lamnated salent-pole synchronous generators. The descrbed method takes nto account saturaton effects as well as the damper bar currents due to the slot pulsaton feld. The method calculates frst the damper bar currents and then ncludes them n the calculaton of the no-load voltage. The combnaton of magnetostatc 2D fnte element smulatons for calculatng the magnetc couplng between the machne wndngs and of an analytcal resoluton results n a very precse predcton of the no-load voltage. At the same tme smulaton tme s drastcally reduced compared to transent magnetc 2D fnte element smulatons. The method was verfed on several examples comparng the obtaned results (damper bar currents and no-load voltage) to results obtaned from transent magnetc fnte element smulatons and n one case to the measured no-load voltage. A tool mplementng ths method s currently used by one of our maor ndustral partners. It s planned to use ths method for analyss of the effects of varous rotor eccentrcty condtons n salent-pole synchronous generators. Index terms - Modelng no-load voltage waveform damper wndng damper currents synchronous machnes I. ITRODUCTIO It s very mportant for the desgner of salent-pole synchronous generators to be able to predct the no-load voltage waveform n a fast and relable way. Ths s above all true n the case of a desgn where a low number of stator slots per pole and per phase have been chosen to reduce costs. Knowng n advance the harmoncs content of the no-load voltage s mportant for satsfyng standards requrements (Telephone Harmonc Factor etc.). Also the predcton of the losses due to currents n the damper bars nduced by the slot pulsaton feld s mportant for the desgn. Furthermore systematc ndustral applcaton calls for a fast method to be appled n a user-frendly way. Also the analyss of the nfluence of the damper bar currents as well as of dfferent stator wndng schemes on unbalanced electromagnetc pull n the case of varous rotor eccentrcty condtons calls for a fast and n the same tme precse calculaton method. These problems have been addressed by several authors applyng dfferent analytcal numercal and combned methods. Traxler-Samek Schwery and Schm [] apply a fast analytcal method whose output can be used as a crtera for the selecton of the number of stator slots. On the other extreme transent fnte element smulatons allow very precse predcton not only of the no-load voltage waveform but also of the damper bar currents or of the magnetzaton of the machne. In [2] transent fnte element analyss s used for predcton of the no-load voltage shape and n [3] ths same method s used for the desgn of the damper wndng of a sngle-phase generator. As ths method s very tmeconsumng especally n the case of a fractonal slot stator wndng t s not very sutable for comparng several dfferent machne geometres. Also a certan number of combned methods were presented. In [4] a combned analytcal and fnte element modelng method s used for calculaton of the currents nduced n the damper wndng and for calculaton of the force-densty harmoncs ncludng the effects of these currents. Fnally n [5] and [6] the modfed wndng functon approach and the magnetcs crcuts approach have been used for modelng the synchronous machne performance under dynamc ar-gap eccentrcty. In ths artcle the authors wll present a combned analytcal and fnte element method for predcton of the damper bar currents and the voltage waveform n no-load condtons. Ths method takes nto account saturaton effects as well as all geometrcal data of the machne (except effects of the end regons). The results obtaned were compared to the results obtaned from transent fnte element smulatons and n one case to the measured no-load voltage. These comparsons were done on several synchronous generators n the range
from 0MVA to 30MVA ncludng nteger and fractonal slot stator wndngs and damper wndngs centered or shfted on the pole shoes. The descrbed method was mplemented n a tool whch s currently used by one of our maor ndustral partners. A generalzaton of the method for analyzng varous rotor eccentrcty and stator deformaton condtons n synchronous machnes s n development. Summarzed the method conssts n calculatng usng magnetostatc 2D fnte element smulatons the magnetc couplng of the machne electrcal conductors (damper bars feld and stator wndngs) for a certan number of postons of the rotor consderng the machne rotatonal perodcty. In a second step the damper bar currents and the no-load voltage can be calculated by solvng the dfferental equaton system formed of the nductances calculated n the frst step. The hypotheses assumed are the followng: Constant feld current o eddy currents eglect effects of end regons II. MAGETIC COUPLIG OF THE MACHIE CODUCTORS Due to magnetc couplng a voltage s nduced n each conductor of the machne. The voltage nduced n conductor s gven by the dervatve of the flux seen by conductor : dϕ v = () The flux seen by the conductor (damper bar or conductor on the stator) can be expressed as the sum of the flux contrbutons of the currents n the feld wndngs and n the damper bars. As the MMF caused by the damper bar currents s sgnfcantly lower than the MMF caused by the current n the feld wndngs the flux created by the damper bar currents s supposed not to nfluence the level of saturaton of the generator. Therefore the contrbuton of the damper bar k can be expressed as the multplcaton of the current n the damper bar k wth a mutual dfferental nductance descrbng the change n flux seen by conductor when the current n the damper bar k changes: ϕ = = ϕ exc ϕ exc ϕ k These dfferental nductances depend on the rotor poston and on the saturaton of the machne. Thus they have to be determned for a gven man flux (gven feld current) and for dfferent rotor postons. As the stator has as seen from the rotor a rotatonal perodcty of one stator slot ptch the L dffk k (2) nductances have to be determned only for some postons of the rotor wthn one stator slot ptch. The nductances are obtaned usng magnetostatc 2D fnte element smulatons. For the determnaton of the results of two fnte element smulatons are necessary: one wth the feld wndngs suppled wth the current correspondng to the chosen man flux and one wth the feld wndngs suppled wth the same current and wth damper bar k suppled wth a test current. follows: Where: ϕ exck ϕ exc L dffk ϕ exck ϕ exc L dffk = ------------------------------------- k L dffk s then calculated as : flux created by the feld wndngs and the current n conductor k seen by conductor : flux created only by the feld wndngs seen by conductor As each smulaton can be used for the calculaton of several nductances the necessary number of magnetostatc fnte element smulatons s: ()W where s the number of damper bars and W s the number of rotor postons consdered wthn one stator slot ptch. The smulatons wth only the feld wndngs suppled provde also the values of flux caused by the current n the feld wndngs (as used n equaton 2). As n the case of transent fnte element smulatons only a part of the machne has to be consdered therefore a typcal number of magnetostatc fnte element smulatons could be: (20)20 = 420 (one pole par 0 damper bars per pole 20 postons of the rotor). Usng magnetostatc fnte element smulatons to determne the magnetc couplng of the machne electrcal conductors allows to take nto account precsely saturaton effects as well as all geometrcal data of the machne (pole shoe shape rotor and stator slottng damper bar dstrbuton etc.) except end regon effects. All these nfluencng factors are contaned n the values of flux and n the dfferental nductances whch can be calculated usng a standardzed scheme of magnetostatc fnte element calculatons for any type of salent-pole synchronous generator (nteger and fractonal slot stator wndngs varous damper bar dstrbutons varous pole shoe shapes etc.). III. ELECTRICAL CIRCUIT OF THE DAMPER CAGE AD CALCULATIO OF THE DAMPER BAR CURRETS The machne conductors form the followng 3 galvancally separated crcuts: Feld wndngs Damper cage Stator wndngs (3)
As the current n the feld wndngs s consdered constant and the machne s consdered n no-load condtons the feld wndngs and the stator wndngs do not have to be modeled. The electrcal crcut of fgure s assocated wth the damper cage. Therefore and expressng nductances: ϕ also usng the dfferental L dff dϕ d = L (7) dff Ω The followng equaton s obtaned for each loop of fgure : R bar bar R bar bar dϕ exc (8) Fgure. Electrcal crcut assocated wth the damper cage In fgure only a part of the damper crcut s shown the whole crcut conssts of one branch for each damper bar present n the part of the machne consdered for calculaton of the nductances as descrbed n paragraph II. As can be seen n fgure each damper bar s modeled as one branch composed of a resstance and a voltage source. The resstance models obvously the resstance of the damper bar and the voltage source models the voltage nduced whch s the dervatve of the flux seen by the damper bar (as n equaton ). The resstance of the short crcut rngs s neglected but could also be consdered changng slghtly the equatons descrbed below. For each loop n the electrcal crcut the followng equaton can be wrtten: dϕ dϕ R bar bar R bar bar = 0 (4) The flux seen by each one of the two bars ϕ and ϕ can be replaced wth the expresson of equaton 2 and the dervatve n tme of the flux terms n equaton 2 can be replaced wth ts partal dervatves as follows: dϕ ϕ = d ϕ Ω Where α s the poston of the rotor and ts dervatve n tme the rotatng speed Ω. The dervaton of the flux wth respect to the current can be replaced wth a dfferental nductance as descrbed n paragraph II: ϕ = L dff (5) (6) d k L dffk L dffk k Ω dϕ exc L dffk d k L dffk k Ω Fnally the current n the last bar can be expressed as the negatve sum of the currents n all other damper bars (Krchhoffs law): R bar bar R bar d k Ldffk bar ( L dff L dffk L dff ) L dff L dffk L dffk Ω k α dϕ dϕ exc exc = 0 As all the L dffk and ϕ exc have been determned as descrbed n paragraph II and the only remanng unknowns beng the damper bar currents the system of dfferental equatons formed of - equatons (where s the number of damper bars the current n the last bar beng calculated as mentoned above) of the type of equaton 9 can be solved usng a numercal method. In the case of ths artcle the 2nd order Runge-Kutta method (also called Heuns method) was appled for solvng the system of dfferental equatons. The descrbed method could also be used n the case of several galvancally separated damper cages (e.g. one on each pole shoe) or n the case of parallel crcuts on the stator ncludng the effects of currents crculatng n the parallel crcuts n noload condtons (especally n the case of an eccentrc rotor). In these cases the equatons descrbed above have to be modfed. = 0 L dff (9)
IV. CALCULATIO OF THE O-LOAD VOLTAGE Havng calculated the currents n all damper bars the no-load voltage n each phase can be obtaned by summng the dervatves of the flux seen by each conductor of the phase. The flux seen by each conductor can agan be expressed as the sum of the contrbutons of the feld wndngs and the damper bars (as n equaton 2). In the case of the conductors on the stator the dfferental nductances and the values of flux caused by the feld wndngs calculated for some rotor postons wthn one stator slot ptch have to be re-assgned to the conductors on the stator after a rotaton of the rotor of one stator slot ptch for takng nto account the new ntal poston of each conductor. Ths technque allows to use the magnetc couplng calculated only for some rotor postons wthn one stator slot ptch for any poston of the rotor as well as for any knd of wndng dstrbuton. The followng formula was used for the numercal dervaton of the flux: Voltage [V] Lne voltage Descrbed method Transent FEM 8000 6000 4000 2000 0-2000 -4000-6000 -8000 0.43 0.432 0.434 0.436 0.438 0.44 0.442 0.444 0.446 0.448 0.45 Tme [s] Fgure 2. Comparson of the no-load voltage waveform d ϕ ( t k ) ϕ ( t k ) ϕ ( t k ) ----------------------------------------- t k t k V. COMPARISO OF THE RESULTS (0) 9500 9000 Lne voltage Descrbed method Transent FEM The damper bar currents and the no-load voltage obtaned usng the descrbed method were compared to the currents and the voltage obtaned from 2D transent fnte element smulatons and n one case also to the measured no-load voltage. Fgures 2 and 3 compare the no-load voltage waveform of an exstng generator (6.3kV MVA 750rpm 50Hz) obtaned usng the descrbed method to the no-load voltage obtaned from transent fnte element smulatons. Fgure 4 compares the no-load voltage harmoncs (n % of the fundamental) of the same generator ths tme also to the harmoncs of the measured no-load voltage. Voltage [V] 8500 8000 7500 7000 6500 6000 0.432 0.433 0.434 0.435 0.436 0.437 Tme [s] Fgure 3. Comparson of the no-load voltage detal
50Hz) for two cases: the real generator wth damper cage τ s shfted by ± --- on the pole shoes ( τ beng the stator slot ptch) 4 s and a hypothetcal case wth damper cage centered on the pole shoes. Fgure 4. Comparson of the no-load voltage harmoncs A very good agreement of the results can be observed not only comparng the descrbed method to transent fnte element analyss but also comparng to the measured values. Fgure 5 shows a comparson of the currents n two adacent damper bars obtaned usng the descrbed method to the current obtaned from transent fnte element smulatons. Also n ths case the agreement s very good therefore the noload losses due to currents n the damper bars can be predcted very precsely. Fgure 6. Comparson of the no-load voltage harmoncs As the descrbed method calculates also the damper bar currents the evoluton of the MMF due to these currents can be calculated allowng better understandng of the nfluence of the damper cage poston on the no-load voltage harmoncs. Fgures 7 and 8 show the spatal dstrbuton as well as the evoluton n tme of the MMF due to the damper bar currents n both cases. Current damper bars 3 and 4 4 Descrbed method bar 3 Transent FEM bar 3 Descrbed method bar 4 Transent FEM bar 4 3 2 Current [A] 0 - -2-3 0.4202 0.4204 0.4206 0.4208 0.42 0.422 0.424 0.426 0.428 0.422 0.4222 Tme [s] Fgure 5. Comparson of the currents n two adacent damper bars Fgure 6 shows a comparson of the no-load voltage harmoncs of another exstng generator (0.6kV 3.5MVA 750rpm
about 20 tmes faster than the transent fnte element smulatons performed for comparson. VI. COCLUSIO Fgure 7. Damper bar MMF centered damper bars The modelng method presented n ths artcle allows the predcton of the damper bar currents and of the no-load voltage of lamnated salent-pole synchronous generators wth almost the same precson as transent fnte element smulatons. At the same tme smulaton tme was reduced by a factor of about 20. The magnetostatc fnte element smulatons necessary for the determnaton of the magnetc couplng of the machnes conductors can be automatzed and the calculaton of the damper bar currents and the no-load voltage based on the results of the fnte element smulatons has been mplemented n a user-frendly graphcal tool allowng comfortable applcaton of the method. Ths tool s currently used by one of our maor ndustral partners. In a next stage the method wll be modfed for analyss of the effects of varous types of rotor eccentrcty condtons and stator deformatons n salent-pole synchronous generators. Fgure 8. Damper bar MMF shfted damper bars All calculatons were performed on a desktop PC wth Pentum 4 CPU (2.6GHz HT) and GB of RAM runnng Wndows XP Professonal. The calculatons applyng the descrbed method (ncludng the magnetostatc fnte element calculatons) were REFERECES [] G. Traxler-Samek A. Schwery E. Schm "Analytc Calculaton of the Voltage Shape of Salent Pole Synchronous Generators Includng Damper Wndng and Saturaton Effects" Proceedngs of the 5th Internatonal Conference on Electrcal Machnes (ICEM) 2002 [2] A. Schwery G. Traxler-Samek E. Schm "Applcaton of a Transent Fnte Element Analyss wth Coupled Crcuts to Calculate the Voltage Shape of a Synchronous Generator" Proceedngs of the 0th IEEE Conference on Electromagnetc Feld Computaton (CEFC) 2002 [3] K. Weeber "Desgn of Amortsseur Wndngs of Sngle-Phase Synchronous Generators Usng Tme-Steppng Fnte Element Smulatons" Internatonal Conference on Electrcal Machnes (ICEM) 998 [4] A. M. Knght H. Karmaker K. Weeber "Use of a Permeance Model to Predct Force Harmonc Components and Damper Wndng Effects n Salent-Pole Synchronous Machnes" IEEE Transactons on Energy Converson Vol. 7 o. 4 2002 [5] H. A. Tolyat. A. Al-uam "Smulaton and Detecton of Dynamc Ar-Gap Eccentrcty n Salent-Pole Synchronous Machnes" IEEE Transactons on Industry Applcatons Vol. 35 o. 999 [6] I. Tabatabae J. Faz H. Lesan M. T. abav-razav "Modelng and Smulaton of a Salent-Pole Synchronous Generator Wth Dynamc Eccentrcty Usng Modfed Wndng Functon Theory IEEE Transactons on Magnetcs Vol. 40 o. 3 2004
Stefan Keller receved hs master degree n 2003 from the Swss Federal Insttute of Technology n Lausanne (EPFL) Swtzerland. Snce then he s workng as a Ph.D. student at the Laboratory for Electrcal Machnes of EPFL. The man felds of hs actvtes are synchronous machne modellng and study of the effects of varous rotor eccentrcty and stator deformaton condtons. Ma Tu Xuan receved hs master degree n 970 and hs Ph.D. degree n 977 from the Swss Federal Insttute of Technology n Lausanne. Snce many years he s senor researcher and lecturer at the Laboratory for Electrcal Machnes of EPFL. Hs man felds of actvtes concern machne modelng optmzaton and testng parameters dentfcaton measurement technques and feld calculatons. Jean-Jacques Smond receved hs master degree n 967 and hs Ph.D. degree n 976 from the Swss Federal Insttute of Technology n Lausanne. Tll 990 he has been workng for BBC / ABB n the feld of large electrcal machnes frst as R&D engneer and later as head of the techncal department for hydro- and Desel-generators. Snce 990 he s full professor at the Federal Insttute of technology and drector of the laboratory for electrcal machnes. He s also consultant for dfferent nternatonal electrcal machnes manufacturers and utltes.