Three-Phase Induction Generator Feeding a Single-Phase Electrical Distribution System - Time Domain Mathematical Model

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1 Three-Phse Induton Genertor Feedng Sngle-Phse Eletrl Dstruton System - Tme Domn Mthemtl Model R.G. de Mendonç, MS. CEFET- GO Jtí Deentrlzed Unty Eletrotehnl Coordnton Jtí GO Brzl 763 L. Mrtns Neto, Dr. J.R. Cmho, PhD. Unversdde Federl de Uerlând Shool of Eletrl Engneerng Eletrl Mhnes L. P.O. Box: 93 - Cmpus Snt Môn Uerlând MG Brzl e.ml: jrmho@ufu.r BSTRCT Ths pper presents tme domn mthemtl model of three phse nduton genertor feedng sngle-phse eletrl dstruton system. The ojetve n ths work s to nlyze the three-phse nduton genertor when workng n the onons ove desred. From suh model t s possle to mke more prese nlyss, sne some prolems should e esly oserved usng the tme domn tehnque, suh s unlnng of mgnetomotve fores (mmf) t the mhne nner mgnet rut. Ths s mpossle to e vsulzed when usng frequeny domn modelng. 2 HP three-phse nduton genertor model s used on dgtl smulton. Through the results t s esy to show tht the proposed model s very effent. Keywords: nduton genertor, sngle-phse, dstruton system 1 INTRODUCTION Countres wth lrge terrtorl extenson, where eletrl energy onsumers re hrterzed y: - smll monthly kwh expenure; - smll numer of onsumers per km of dstruton network; - smll mxmum smultneous demnds; - nd lso smll mount of fnnl resoures to e spent n rurl eletrfton progrms. Hvng the Brzln senery s our exmple, the eletrl energy uthortes turned to sngle-phse (one wre wth erth return) s less expensve opton to eletrl dstruton systems n rurl res. In the produton sde, due to onstnt pressure of the tehnology to eome more effent nd to nrese the mxmum demnd, the rurl onsumer eomes loked to the lmttons of the sngle-phse system However, n some regons where smll hydroeletr potentls re vlle, there's posslty to generte energy wth the use of three-phse nduton mhnes, wth squrrel ge rotor onneted to sngle-phse dstruton system n order to supply typl rurl three-phse lods. Some studes hve lredy een done out the three-phse nduton genertor onneted to sngle-phse energy system. These studes were of gret mportne ut developed through frequeny domn mthemtl models[1]. The use of frequeny domn tehnques doesn't llow the nlyss for the ehvor of the three-phse nduton genertor onernng the eletromgnet unlnes nsde the mhne. So, wre of ths mtter ths pper presents tme domn mthemtl model [2][3][4], who llows suh studes to e mde. These studes show the eletromgnet ehvor n the three-phse nduton genertor nner eletromgnet rut. 2 INDUCTION GENERTOR The proposed three-phse nduton genertor s onneted to sngle phse eletrl dstruton system. It s n ordnry nduton mhne wth squrrel ge rotor nd sttor phses dspled sptlly y 12 o nd sme numer turns/phse. n dequte ptne C p s oupled etween phses B nd C, s n Fgure 1. The ojetve s to otn relle opertng pont for ths unlned onfgurton, where the genertor s delverng nomnl power under nomnl voltge. s desred n referene [1], lned stuton etween genertor phses n e otned, nd ths lne s of fundmentl mportne onernng the onneton of three-phse nduton genertor n prllel wth sngle-phse voltge soure. Through the frequeny domn mthemtl modelng proposed n [1], t s ler the good ehvor for the proposed genertor, n terms of "rms" vlues, where the voltge unlne s not so entuted. However, n nterestng pont not rought n pture n [1] s the ehvor of eletromgnet torque nsde the genertor, for smll voltge unlnng the nternl eletromgnet torque shows swngng ehvor. Those swngs re relted to the urrent unlne n the genertor phses s prt of unlned three-phse system. s onsequene ths genertor hs dfferent mgnetomotve fores (mmf) n eh phse. Ths unlnng llows the deomposton of the mmf dstruton n suh wy to otn stt nd rotry mgnet feld dstruton, where the ltter re of postve nd negtve sequene. The postve sequene mgnet feld, B R1, under the three-phse symmetrl nduton mhne workng prnples, rete ndretly rottng mgnet feld, B S1, of sme sequene, n the mhne's rotor. The sme hppens to the negtve sequene mgnet feld, B R2, retng nother B S2. The ove s llustrted n Fgure 2, nd n e seen lerly tht: - The mgnet felds (B S1, B R1 ) nd (B S2, B R2 ) show onstnt ngulr dsplement etween them, therefore the resultng lgnng eletromgnet torque s onstnt;

2 - The ngulr poston etween mgnet felds (B S1, B R1 ) nd (B S2, B R2 ) hnges perodlly, therefore the eletromgnet torque s n oslltng vlue. rotor BC). The mgnet flux ouplng re gven y the followng generl equton: λ = L + L. + L. (8) ( ) j L + L genertor s phse self ndutne, L j genertor s mutul ndutne etween phses nd j,, j nstntneous urrents of phses nd j, respetvely. j For the three-phse genertor ptured n Fgure 1, self nd mutul ndutnes re gven y [L] mtrx s follows: Fgure 1 Conneton dgrm for the three-phse squrrel-ge nduton genertor. Conneted to sngle-phse dstruton system. [ L] = k (9) The [L] mtrx terms re gven y: Fgure 2 Demonstrton of the nternl rottng mgnet felds nsde three-phse nduton genertor onneted to sngle-phse system. Frequeny domn mthemtl models doesn't llow nstntneous nlyss, therefore n nduton genertor tme domn mthemtl model ws ult n order to do ths reserh. 3 MTHEMTICL MODEL Fgure 1 shows the onneton dgrm for the three-phse nduton genertor onneted to sngle-phse dstruton system. From Fgure 1 we hve: ( V) 1. dv = Cp (1) Cp ( ) + = (2) V V V = (3) (4) () (6) The nduton genertor voltge nd urrent reltonshp s presented y the gener equton: dλ v. = r + (7) v,, r, λ re respetvely voltge, urrent, resstne nd oupled mgnet flux for one of the genertor s phse lled. ndex represents one of the genertor s phse (sttor the L j vlues re gven elow. 11 = L11 2. L13 + L33 12 = L12 L23 L13 + L33 13 = L14 L34 14 = L1 L3 1 = L16 L36 22 = L22 2. L23+ L33 23 = L24 L34 24 = L2 L3 2 = L26 L36 33 = L44 L11 = L22 = L33 = KL 2π L44 = 1 os + K L 3 2π L12 = L13 = L23 = os 3 L14 = L 2= L36 = os (. θ) 2π L1 = L26 = L34 = os θ + 3 2π L16 = L24 = L3 = os θ 3 K L s the mhne mgnet srut onstnt. Introdung the three-phse nduton genertor mehnl equtons, they re gven y:

3 dw R 1.( T T ) = m (1) W J dθ R R = (11) 4.1 Usng µf ptor dθ p =. WR (12) 2 p L Tm =. I.. I 4 θ T [ ] [ ] W R rotor ngulr mehnl speed; θ - rotor eletrl ngulr dsplement; p pole numer; T m genertor eletromgnet torque; T - turne torque. (13) From the unon of eletrl (1) to (9) nd mehnl (1) to (13) equtons, n e otned n equton system whh represents the three-phse nduton genertor onneted to sngle-phse dstruton system. The resultng mtrx equton system s shown elow: d[ I' ] 1 d[ L' ] [ L' ]. [ V '] [ R' ].[ I' ] = + [I ] - represents the urrent mtrx (sttor, rotor nd lod); [L ] - represents the ndutne mtrx (sttor, rotor nd lod); [V ] - represents the voltge mtrx (sttor, rotor nd lod); [R ] represents the resstne mtrx (sttor, rotor nd lod). Tle 1. Three-phse nduton mhne equvlent rut prmeters. Sttor Resstne 3,8±,3Ω Voltge V Fgure3 Genertor output voltge V. Voltge Fgure4 Genertor output voltge. Voltge V Rotor Resstne (referred to the sttor) Loked Rotor Retne ( referred to the sttor) Phse Mgnetzton Retne 3,1±,3Ω 3,1±,3Ω 7,1±,7Ω Fgure Genertor output voltge V. 8 V 4 DIGITL SIMULTION One hvng ll the nduton genertor equtons, dgtl smultons hve een done n order to onfrm the eletromgnet torque oslltons. Therefore, the enhmrk ws the three-phse nduton motor used n [1], three-phse nduton mhne wth smlr hrtersts ws tken, 2 HP, 4 poles, 38/22 Volts, squrrel ge rotor, wth equvlent rut prmeters gven y Tle 1. The followng smultons re otned wth respetvely ptors of µf, 7 µf, 9 µf nd 11 µf n order to hve n de of ts effet on the eletromgnet torque ehvor nsde the genertor. Voltge (V) 6 V Tme (S) Fgure 6 Genertor output voltges - V,, V.

4 8 V 6 V Voltge (V ) Fgure 7 Eletromgnet Torque Fgure 11 Genertor output voltges - V,, V. 4.2 Usng 7µF ptor. 1 8 Voltge V (V) Fgure8 Genertor output voltge V Fgure 12 Eletromgnet torque. 4.3 Usng 9µF ptor Voltge (V ) Voltge V (V ) Fgure 9 Genertor output voltge Fgure 13 Genertor output voltge V Voltge V (V ) Voltge (V ) Fgure 1 Genertor output voltge - V Fgure 14 Genertor output voltge.

5 4 3 Through fgures 7, 12, nd 17 n e lerly oserved the presene of suh oslltng eletromgnet torque, where the oslltons re lwys round the turne nomnl torque (7. N.m). 2 Voltge V (V ) Fgure 1 Genertor output voltge V. CONCLUSIONS From the fgures prevously presented n e lerly oserved the exstene of hevy eletromgnet torque oslltons nsde the genertor; oslltons mnly due to the nternl eletromgnet unlne n the mhne. Ths unlne n e very dmgng, n the mehnl pont of vew, to the genertor performne, sne they re the use of hghly oslltng rottonl speed. V o lt g e (V ) V V Wth the use of frequeny domn modelng t s not possle to see lerly these effets. Therefore, t s evdent tht the mthemtl model presented here, for three-phse nduton genertor onneted to sngle-phse system, gve us onons nd resoures for the nstntneous nlyss for the eletromgnet unlne present nsde the genertor. In nother words t s possle to mke the nlyss of generted voltge t the mhne s termnls. Ths model stll gve us onons to evlute more dequte ptor for the genertor opertng under suh restrtons. So the mthemtl model eome vlule tool for the desgn of suh ptor. Fgure 16 Genertor output voltges - V,, V. 6 REFERENCES Fgure 17 Eletromgnet torque. From fgures 3 to 17 n e seen tht vlle voltges t the genertor s termnls re unlned nd ths unlne s dmped or hghlghted s ptors of dfferent vlues re onneted to the system. The eletromgnet torque urves show lerly tht the unlne from the voltges promote eletromgnet torque oslltons nsde the genertor. [1] T.F. CHN, Performne nlyss of Three-Phse Induton Genertor Conneted to Sngle-phse Power System, IEEE Trnstons on Energy Converson, Septemer 1998, Vol 13 No. 3, pp [2] R.G. MENDONÇ, & L. MRTINS NETO, Comprtve Performne nlyss: Symmetrl Three-Phse Induton Motor gnst symmetrl Three-Phse Induton Motor - Oslltng Torque, Proeedngs of XII Brzln utomt Control Conferene XII CB, Vol.I, pp septemer 14-18, Uerlând, MG, Brzl (In Portuguese). [3] L. MRTINS NETO, L.; R.G. MENDONÇ, J.R. CMCHO, ND C.H. SLERNO, The symmetrl Three- Phse Induton Motor Fed y Sngle Phse Soure: Comprtve Performne nlyss, IEEE-IEMDC - Interntonl Eletrl Mhnes nd Drves Conferene, Mlwukee, Wsonsn, My 1997.] [4] L. MRTINS NETO, J.R. CMCHO, C.H. SLERNO,. & B.P. LVRENG, nlyss of Three-Phse Induton Mhne Inlundng Tme nd Spe Hrmon Effets: The,B,C Referene Frme, PES-IEEE Trnstons on Energy Converson, Volume 14, Nr.1, Mrh rtle no.: PE- 14-EC