J. Milli Monfared K. Abbaszadeh E. Fallah Assistant Professor P.H.D Student P.H.D Student

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1 olng an Smulaton of Dual h Pha Inucton achn n Fault conton wo Pha cut off) an Popo A Nw Vcto Contol Appoach fo oqu Ocllaton Ructon J. ll onfa K. Abbazah E. Fallah Atant Pofo P.H.D Stunt P.H.D Stunt Amkab Unvty Of chnology, han, Ian Fax: E-mal: onfa@cc.aku.ac. ABSRAC: Fom th ngy convon pont of vw, tuctu of all lctcal machn can b mplf to an quvalnt twopha machn. At nomal opaton th quvalnt machn ha a balanc tuctu whch th ba fo vcto contol quaton at th mo. Smlaly at pha - cut off conton th quvalnt mol ha an unbalanc tuctu, whch can b u to obtan vcto contol quaton un fault conton. In th pap, ft a ual th-pha nucton machn ha bn mol wth two pha of tato cut off, thn vcto contol quaton hav bn obtan fom -q mol of machn. On th ba of th quaton a nw Roto Fl Ont R.F.O) contol appoach pnt whch ncat how th convntonal R.F.O contol can b aapt fo fault conton. Fnally, a comput mulaton ha bn pnt to compa th opaton of convntonal R.F.O contol wth mpov R.F.O contol, whl two pha of machn cut off. Rult how conabl mpovmnt n v pfomanc pcally n ucton of toqu ocllaton ucton. Kywo: Fault Vcto contol Dual th pha- Inucton machn. I) INRODUCION: Hgh-pow lctc machn v ytm hav foun many applcaton uch a pump, fan, compo, ollng mll, cmnt mll, mn hot, jut to nam a fw. At pnt, th mot uccful typ of th hgh pow v ytm cycloconvt-lctc machn v an ynchonou machn f by cunt ouc thyto nvt. Voltag ouc nvt, pt th avantag n ln pow facto ov a cycloconvt a wll a th avantag of bng abl to u low cot nucton machn, a tll lmt to th low n of th hgh pow ang u to lmtaton on th gat - tun - off typ mconucto pow vc atng. A an appoach n achvng hgh pow atng wth voltag ouc nvt, mult - pha machn v ytm hav mg []. In th mot common uch tuctu two t of th- pha wnng a patally pha hft by lctcal g an ach t of th th - pha tato wnng xct by a th-pha nvt, thfo th total pow atng of th ytm oubl.in aton to nhancng pow atng, t alo blv that v ytm wth uch mult - pha tuctu wll mpov th lablty at th ytm lvl [-5]. In patcula wth lo of on o mo of tato wnng xctaton t, a mult - pha nucton machn can contnu to b opat wth an aymmtcal wnng tuctu an unbalanc xctaton [], [7]. h mot commonly u analytcal tool fo th analy of unbalanc opaton of lctc machn ha bn th wllknown ymmtcal componnt mtho. Although th mtho ha bn u uccfully n th tay tat analy on nuoal xctaton, howv, a fa a th ynamc of machn concn th mtho lo t utlty. A th ynamc bhavo of an lctc machn ctcal n a mon v ytm t ncay to vlop analytcal tool whch can hanl th ynamc of lctc machn un tuctually unbalanc opaton conton. R.zhao an.lpo vlop a molng an contol appoach fo a ual th-pha nucton machn wth on tato wnng opn. h popo mtho wa ctly ba on th aymmtcal wnng tuctu []. Dcouplng of ngy-convon-lat componnt fom non- ngy-convon-lat componnt, th mpotant avantag of th mol, whch u n contol applcaton. Fom th pat ca Vcto contol mtho known a on of th bt mtho n contollng th toqu an p of ac machn. Btwn th vaou typ of Vcto contol appoach, Roto fl ont contol mo utabl fo nucton machn [7]. Howv, n th pha cut off conton, convntonal R.F.O contol-whch gn fo nomal opaton of machn-can not b u to contol th machn. h poblm pcally obv n ocllaton n lctomagntc toqu. Lpo an Zhao pnt a contol appoach fo contollng th nvt, whch f a ual thpha nucton machn wth on pha opn [8]. In th pap, a ual th-pha nucton machn ha bn mol at th tato fault conton of two-pha cut off. atx tanfomaton mtho u to obtan a coupl mol fo machn. Engy-convon-lat componnt of machn vaabl, n th couplng appoach, a mapp to a o-call -q ubpac, an th non-lat ngy convon componnt a mapp to a o-call Z-Z ubpac of mol. On th ba of th mol, tato voltag quaton fo R.F.O vcto contol obtan an a mtho ha bn popo to aapt th convntonal block of vcto contol, fo opatng un fault conton. o how th capablt of th popo mtho, a comput mulaton ha bn pnt. It nclu qzz mol of moto un fault conton, f fom a SPW voltag ouc nvt. h opaton of th popo R.F.O contol n ucng machn toqu ocllaton ha bn compa wth th opaton of

2 convntonal mtho. Smulaton ult how utabl pfomanc fo th popo contol mtho un fault conton. II) ODEL OF DUAL HREE-PHASE INDUCION ACHINE WHILE WO PHASES OF SAOR ARE CU OFF: Suppo that a pha cut off fault occu n pha an f of a mult - pha v ytm nclung a ual th pha nucton machn, a hown n fg : Equaton ) an ) tat that th - ax an q - ax flux a pojcton of th tato flux vcto n a fou mnonal pac, on anoth t of vcto n that pac. h vcto nam an q a th ba vcto of th nw foumnonal pac: 5 = co θ o) co θ o ) co θ o ) co θ o ) 5 q = n θ o) n θ o ) n θ o ) n θ o ) Fom th othogonalty of ba vcto, θ o obtan:. q = q. = θ o = Fg.: Dual th pha nucton machn v Wth two pha opn. Snc, fou npnnt cunt can flow n th gnal ca, t xpct that machn mol to b fou-mnonal. h fou mnonal pac pann by vcto of al machn vaabl can b xp a th ct um of two othogonal ubpac wth on of two-mnonal ubpac pntng th ngy convon popty of th machn - q ubpac) an th oth th non - lctomchancal ngy convon poton Z -Z ubpac). Aumng nuoal wavfom fo wnng patal tbuton, Stato an oto wnng flux, ax can b hown a follow: h ubpac pann by vcto an q pnt th ngy convon popty of th machn. h vcto, whch pan th two-mnonal non- lctomchancal ngy convon ubpac, can b tmn mathmatcally. Dfnng two vcto a Z an Z, fom th othogonalty of ba vcto, th two vcto can b obtan. Ung fou ba vcto to fom th nw ba fo th fou mnonal pac, th followng nomalz compoton tanfomaton ult: [ ].57.4 = h compoton matx fo oto vaabl of th machn [] man th am a th tanfomaton fo balanc opaton, bcau th oto tll mantan a balanc wnng tuctu []. Applyng th tanfomaton [] an [] to th voltag quaton of tato an oto fl, yl: ) achn mol n - q ubpac: a) Fg.: a) Stato wnng flux ax b) Roto wnng flux ax. If th angl btwn - ax of a gap flux plan an a wnng flux ax θ o, -ax an q-ax flux can b wttn a follow: φ a ) 5 φ b φ = co θ o) co θ o ) co θ o ) co θ o ) φ c φ φ a φ ) 5 b φ q = n θ o) n θ o ) n θ o ) n θ o ) φ c φ b) v vq Wh: Lq q q = L L q q L L = Ll 8. Lm, Lq = Ll 4. Lm, L = Ll Lm, = 844. L =. 94 Lm, q m achn mol n Z -Z ubpac: Stato voltag quaton: v z = v z L l z z L l 4) 5) Roto voltag quaton:

3 = L l z L l z o L l o L l Elctomagntc toqu: Pol = q q ) 7) Equaton 4) to 7) compltly pnt qzz mol of ual th-pha nucton machn wth two pha opn. Wtng th quaton n th fom of tat quaton, th comput mulaton of mol can b pfom. III) EQUAIONS OF R.F.O VECOR CONROL IN PHASE CU OFF ODE: Fo obtanng vcto contol quaton t ncay to ncat machn quaton n oto fl ont fnc fam. Fo th unbalanc opatng tuaton nvtgat n th pap, th tanfomaton of tato vaabl ung th balanc otaton tanfomaton woul ult n ac componnt n th R.F.O fnc fam [7]. hfo th followng unbalanc tanfomaton popo fo tato vaabl [7]: [ ] coθ q = nθ nθ co θ Wh θ th angl btwn tatonay fnc fam an an abtay otatng fnc fam. Applyng th appopat tanfomaton on oto an tato voltag quaton an lctomagntc toqu th followng quaton ult, whch xp machn quaton n th abtay otatng fnc fam. Stato voltag quaton: Lq Lq L Lq Lq v q q q v = ) L L L L q Lq Lq q ) ) ) ) ) ) ) ) ) ) q q q q q q ) ) ) ) q q q q Roto voltag quaton: = q ) L ) L q q ) q ) L L Elctomagntc toqu: 9) ) Pol = q q ) ) Wh ubttl " - " ncat backwa otatng vaabl. On th ba of quaton 9) to ) R.F.O vcto contol quaton can b obtan. h quaton a mla to th R.F.O vcto contol quaton n th balanc mo. h quaton a compa n tabl ). abl. Compang R.F.O vcto contol quaton n fault mo an balanc mo Dcpton Balanc mo q. Fault mo q. Pol oqu Equaton. Roto flux Equaton. Angula Vlocty of Roto flux D-ax Componnt Of tato Voltag Equaton. D-ax Componnt Of tato Voltag Equaton. = λ q L Pol = λ q L λ λ= λ λ = q q q = = λ λ f v = U U U = L q f U = L L L ) λ v = U U U f U f Lq U = ) q Lq U= q L ql Lq ) λq Lq f U = ) Lq Lq ) λ Lq f U = Lq Lq λ f vq = Uq U vq = Uq q Uq Uq f Uq f Uq= L Lq Uq= ) L L ) λ LqL Lq ) λ Lq Uq= L ql Lq ) λ f U L L q = q Lq q f U q = q ) q Lq f Uq = q Lq Lq λq It obv that n th fault mo oto an lctomagntc toqu quaton a mla to tho n balanc mo, an only ubttut by q. In tato voltag quaton n aton to fowa otatng componnt - whch a mla to balanc mo quaton - backwa otatng componnt hav appa a wll. In fact, th pat of quaton u to fowa otatng componnt a th am a balanc mo quaton xcpt that machn nuctanc a ubttut

by th man valu of -ax an q-ax nuctanc n th fault mo.

Conng th amount of nuctanc on an q - ax cton, t pobl to nglct th ffnc of th nuctanc n compaon to th um of th nuctanc an popo a contol appoach, whch ult n

In balanc mo In fault mo q L Lq L L L q atx tanfomaton to accong to fnc atx tanfomaton 4 to accong to q. ) [].

m b) Loang: Nf= pm, loa = 5N.m c) Sp fnc vaaton: Nf=5pm, loa = 5N.

Smulaton ult fo mof R.F.O vcto contoll hown n fg.4: a ) IV) SIULAION RESULS: In th cton comput mulaton ult a pnt.

O vcto contoll an th con th am contoll wth popo chang accong to tabl. Smulaton ult of th convntonal contoll hown n fg.: b ) c ) a ) b ) c ) ) Fg.

4 by th man valu of -ax an q-ax nuctanc n th fault mo. h pat u to th backwa otatng componnt a almot mla to th fowa otatng componnt but um of th nuctanc ubttut by th ffnc. Conng th amount of nuctanc on an q - ax cton, t pobl to nglct th ffnc of th nuctanc n compaon to th um of th nuctanc an popo a contol appoach, whch ult n th followng chang n th vcto contol block. abl. Ncay chang n convntonal R.F.O contol block fo opatng at fault conton. In balanc mo In fault mo q L Lq L L L q atx tanfomaton to accong to fnc atx tanfomaton 4 to accong to q. ) []. Balanc mo otatonal Fault mo otatonal tanfomaton accong tanfomaton accong to []. to q. ) ) Fg. : Smulaton ult of convntonal R.F.O vcto contoll a) No loa tatng: Nf= pm, loa = N.m b) Loang: Nf= pm, loa = 5N.m c) Sp fnc vaaton: Nf=5pm, loa = 5N.m ) Stato flux vcto chaacttc Rult how conabl ocllaton n lctomagntc toqu, wth ampltu of about 4N.m fo a loa toqu of 5 N.m. Stato flux vcto chaacttc not utabl th. Smulaton ult fo mof R.F.O vcto contoll hown n fg.4: a ) IV) SIULAION RESULS: In th cton comput mulaton ult a pnt. Smulaton nclu a ual th-pha nucton machn wth two pha opn, an f fom a SPW voltag ouc nvt. wo contoll a u fo machn p contol. h ft th convntonal R.F.O vcto contoll an th con th am contoll wth popo chang accong to tabl. Smulaton ult of th convntonal contoll hown n fg.: b ) c ) a ) b ) c ) ) Fg.4) Smulaton ult of mof R.F.O vcto contoll. a) No loa tatng: Nf = pm, loa = N.m b) Loang: Nf = pm, loa = 5N.m c) Sp fnc vaaton: Nf = 5pm, loa = 5N.m ) Stato flux vcto chaacttc It obv that wth th popo mofcaton, toqu ocllaton uc conably an th ocllaton ampltu ca to about 4 N.m fo th am fault conton. Alo th tato flux vcto chaacttc mpov. Extnc of mall toqu ocllaton n th ca u to th fact that th ffct of backwa otatng componnt nglct, nc popo mofcaton n tabl- mplctly hav uch an aumpton.

5 V) CONCLUSION: Vfcaton, IEEE an. Engy Conv., No., pp , Spt.99. h mtho, popo n th pap fo molng of a ual th-pha nucton machn wth two pha opn how th capablt of matx tanfomaton mtho n molng vaou typ of pha-cut-off fault n th tato of nucton machn. h mpotanc of th molng mtho n couplng ngy-convon-lat componnt fom nonngy-convon lat componnt of machn vaabl uch that utabl contol laton can b obtan fo contol of th machn. It hown that un fault conton ung an unbalanc otatonal tanfomaton, vcto contol quaton can b obtan whch a mla to th quaton n th balanc mo. On th ba of th mlat a mtho popo wh applyng a fw chang n th convntonal R.F.O vcto contoll block, t can b capabl of contollng th machn un fault conton. Smulaton ult how utabl pfomanc of popo contol mtho. Epcally, wth th mtho, toqu ocllaton a conably uc. Howv toqu ocllaton a not compltly lmnat nc th ffct of backwa otatng componnt a nglct. Gvn that th tuctu of two pha opn-ual th pha nucton machn mla to th tuctu of on pha opn convntonal th pha nucton machn th popo mtho of molng an contol can b u fo on pha opn-convntonal th pha nucton machn a wll. VI) REFERENCES: [] Y.Zhao,.A.Lpo, Spac Vcto PW contol of Dual h-pha Inucton achn Ung Vcto Spac Dcompoton, IEEE an. In. Appl., Vol., No.5, pp.-9, Spt./Oct []..Jahn, Impov Rlablty n Sol-Stat ac Dv by man of ultpl Inpnnt Pha-Dv Unt,IEEE an. In. Appl.,Vol. IA-, No., pp.-, ay/jun 98 [] E.A.Klnghn, Hgh Pha O Inucton oto, Pat I-Dcpton an hotcal Conaton, IEEE an. on Pow Appaatu an Sytm, Vol. Pa-, No., Januay 98, pp [4] E.A.Klnghn, Hgh Pha O Inucton oto, Pat II-Expmntal Rult,IEEE an. on Pow Appaatu an Sytm, Vol. Pa-, No., Januay 98, pp [5] J.R.Fu,.A.Lpo, Dtubanc F Opaton of a ultpha Cunt Rgulat oto Dv wth an Opn Pha, IEEE an. In. Apppl. Vol., No.5, pp.7-74, Spt./Oct., 994. [] Y.Zhao,.A.Lpo, olng an Contol of a ultpha Inucton achn wth Stuctual Unbalanc, Pat I-achn olng an ult-dmnonal Cunt Rgulaton, IEEE an. Engy Conv., No., pp , Spt.99. [7] Vcto Contol of AC achn, Pt Va, Oxfo cnc publcaton 99. [8] Y.Zhao,.A.Lpo, olng an Contol of a ultpha Inucton achn wth Stuctual Unbalanc, Pat II-Fl-Ont Contol an Expmntal

Applications of Lagrange Equations

Applications of Lagrange Equations Applcaton of agang Euaton Ca Stuy : Elctc Ccut ng th agang uaton of oton, vlop th athatcal ol fo th ccut hown n Fgu.Sulat th ult by SIMI. Th ccuty paat a: 0.0 H, 0.00 H, 0.00 H, C 0.0 F, C 0. F, 0 Ω, Ω