Finite Element Approach to Electric Field Distribution Resulting from Phase-sequence Orientation of a Double- Circuit High Voltage Transmission Line

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1 Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) Ft Elt Approh to Eltr Fld Dstruto Rsultg fro Phs-squ Ortto of Doul- Crut Hgh Voltg Trssso L A. ISARAMOGKOLRAK T. KULWORAWAICHPOG d P. PAO-LA-OR * Powr Sst Rsrh Ut Shool of Eltrl Egrg Isttut of Egrg Surr Uvrst of Tholog 111 Uvrst Avu Mug Dstrt ho Rths 3 THAILAD Astrt: - Ths ppr proposs thtl odl of ltr flds usd hgh voltg odutors of ltr powr trssso ssts usg st of sod-ordr prtl dffrtl qutos. Ths stud hs osdrd th fft of odutor phs-squ ortto o ltr flds ttd roud doulrut xtr hgh-voltg powr trssso l. Sx tpl-usd phs-squ orttos Thld for 5-V doul-rut trssso ls r vstgtd. Coputr-sd sulto utlzg th two dsol ft lt thod th t hro od strutd MATLAB progrg vrot wth grphl rprstto for ltr fld strgth hs vlutd. Th sulto rsults show tht th phs-squ ortto s o og ftors to flu th ltr fld dstruto roud th trssso l. K-Words: - Phs-squ Eltr Flds Ft Elt Mthod (FEM) Trssso L Coputr Sulto 1 Itroduto For dds du to th rsg of ltrl powr dds Thld Eltrt Grtg Authort of Thld (EGAT) dds to lrg trssso pt stllg 5-V xtr hgh-voltg powr trssso ls oth AC d DC. I th AC sst doul-rut trssso ls osst of sx odutors rug prlll. Ortto of th sx odutors rsults ltr fld dstruto tht us so srous hr to ltro qupt or lvg thgs. Fro ltrtur s ltrogt thor [1] or g thor [] r wdl usd for ltr fld lulto hgh voltg powr trssso ls. Ev th stud EPRI [3] th s ltrogt thor ws plod to lz ltr fld strgth rsultg fro ortto of odutor phs-squs. So fr thr s o rport o ltr fld lulto ths sop usg Ft Elt Mthod (FEM). Th FEM s o of th ost populr url thods usd for oputr sulto. Th dvtg of th FEM ovr othr url thods grg ppltos s th lt to hdl olr t-dpdt d rulr gotr prols. Thrfor ths thod s sutl for solvg th prol volvg ltr fld ffts roud th trssso l usd rulr ross-sto of hgh voltg odutors. I ths ppr 5-V doul-rut xtr hghvoltg powr trssso ls r studd wth sx odutor phs-squ ortto. Coputrsd sulto utlzg th two dsol ft lt thod th t hro od strutd MATLAB progrg vrot wth grphl rprstto for ltr fld strgth hs vlutd. Modlg of Eltr Flds volvg Eltr Powr Trssso Ls A thtl odl of ltr flds (E) rdtg roud trssso l s usull xprssd th wv quto (Hlholtz s quto) s Eq.(1) [4-5] drvd fro Frd s lw. E E E σµ εµ (1) t t whr ε s th dltr prttvt of d µ d σ r th gt prlt d th odutvt of odutors rsptvl. Ths ppr hs osdrd th sst govrg usg th t hro od d rprstg t th ltr fld oplx for E E ω [6] thrfor ISS: ISB:

Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) E t ωe d E t E whr ω s th gulr frqu. Fro Eq.(1) susttutg th oplx ltr fld Eq.

Thrfor ths ppr th FEM s hos to pottl tool for fdg pproxt ltr fld solutos for th PDE dsrd Eq.() [7]. 3 Sst Dsrpto wth th FEM 3.1 Dsrtzto Ths ppr dtrs four-udld doulrut 5-V powr trssso l. Fg.

2 Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) E t ωe d E t E whr ω s th gulr frqu. Fro Eq.(1) susttutg th oplx ltr fld Eq.(1) trsford to ltrtv for s follows. E ωσµ E+ ω εµ E Wh osdrg th prol of two dsos Crts oordt (x) h ( ωσ ε) E 1 E 1 E + x µ x µ () Altll thr s o spl xt soluto of th ov quto. Thrfor ths ppr th FEM s hos to pottl tool for fdg pproxt ltr fld solutos for th PDE dsrd Eq.() [7]. 3 Sst Dsrpto wth th FEM 3.1 Dsrtzto Ths ppr dtrs four-udld doulrut 5-V powr trssso l. Fg. 1 shows th powr trssso l wth th lowrt ortto tp. Hght of odutors show th fgur s th xu sg posto. Th lowst odutors r C d A t th hght of 13. ov th groud lvl [8]. Eh phs odutor s 795 MCM (.77 - dtr). Th ovrhd groud wr hs 3/8 h - dtr. Fg. dspls th do of stud dsrtzg usg lr trgulr lts. Fg. Dsrtzto of th sst gv Fg Ft Elt Forulto A quto govrg h lt s drvd fro th Mxwll s qutos drtl usg Glr pproh whh s th prtulr wghtd rsdul thod for whh th wghtg futos r th s s th shp futos [9-1]. Aordg to th thod th ltr fld s xprssd s follows. ( x ) E + E E E + (3) whr s th lt shp futo d th E s th pproxto of th ltr fld t h od ( ) of th lts whh s whr d + x+ s th r of th trgulr lt. Th thod of th wghtd rsdu wth Glr pproh s th ppld to th dffrtl quto Eq.() whr th tgrtos r prford ovr th lt do. Fg.1 5-V doul-rut four-udld powr trssso l wth low-rt ortto 1 E + x µ x 1 E d µ ( ωσ ε) E d ISS: ISB:

Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) or th opt trx for [ + K]{ E} M (4) ( ωσ ε) M ( ωσ ε) 1 1 1 d 1 1 1 1 4 Rsults d Dsusso Ths ppr

3 Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) or th opt trx for [ + K]{ E} M (4) ( ωσ ε) M ( ωσ ε) d Rsults d Dsusso Ths ppr plos MATLAB progrg to sult ltr fld dstruto for sx phssqu orttos. Eltr fld sultd for h tp dptd Fg K ν + d x x + + ν + 4 S whr ν s th trl rlutvt ( ν 1/ µ ). For o lt otg 3 ods th xprsso of th FEM pproxto s 3 3 trx. Wth th out of ll lts th sst of ods th sst quto s szl s th trx. 3.3 Boudr Codtos d Sulto Prtrs I ths ppr 5-V doul-rut xtr hghvoltg powr trssso ls r studd wth sx odutor phs-squ ortto [3] s show Tl 1. Th oudr odtos ppld hr r tht to st zro ltr flds t th groud lvl d th OHGW. I ddto th oudr odto of odutor surf 5-V powr ls r ssgd s gv [3 8 11]. Ths sulto uss th sst frqu of 5 Hz. Th powr ls r rd odutors of Aluu Codutor Stl Rford (ACSR) hvg th odutvt (σ ) S/ th rltv prlt ( µ r ) 3 th rltv prttvt ( ε r ) 3.5. It ots tht th fr sp prlt ( µ ) s 4π 1-7 H/ d th fr sp prttvt ( ε ) s F/ [1]. Tl 1 Sx tps of phs squs tp1 tp tp3 tp4 tp5 tp6 A A A B A A A B A C A G B B B A B C B C B B B G C C C C C B C A C A C G Fg.3 Eltr fld otour (V/) for tp 1 Fg.4 Eltr fld otour (V/) for tp Fg.5 Eltr fld otour (V/) for tp 3 ISS: ISB:

Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) du to uld phs squg spll tp 6 wth ll th groud wrs lotd o th rght hd sd.

4 Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) du to uld phs squg spll tp 6 wth ll th groud wrs lotd o th rght hd sd. Wh osdr t so sltd postos for or dtl str ltr fld dstruto lrl xpld. Fg. 9 d 1 show th ltr fld plot t th hght of.1 low th lowst odutor posto. Slrl Fg. 11 d 1 lso dsr th ltr fld plot t th hght of.1 ov th hghst odutor posto. Th ofr tht tp 1 3 d 5 r str whl tp 4 d 6 r str s oludd [3]. Fg.6 Eltr fld otour (V/) for tp 4 Fg.7 Eltr fld otour (V/) for tp 5 Fg.9 Eltr flds of th str ortto t th hght of.1 low th lowst odutor posto Fg.8 Eltr fld otour (V/) for tp 6 Fro th sultd rsults th ortto tp hs th fft o ltr fld dstruto roud th powr trssso l. B osrvg th ltr fld strgth t spfd hght lvl ov th groud wth 7- horzotl sp tp 1 3 d 5 r str ltr fld dstruto log th vrtl xs. Tp 4 d 6 r str Fg.1 Eltr flds of th str ortto t th hght of.1 low th lowst odutor posto ISS: ISB:

5 Prodgs of th 8th WSEAS Itrtol Cofr o ELECTRIC POWER SYSTEMS HIGH VOLTAGES ELECTRIC MACHIES (POWER '8) Fg.11 Eltr flds of th str ortto t th hght of.1 ov th hghst odutor posto Fg.1 Eltr flds of th str ortto t th hght of.1 ov th hghst odutor posto 5 Coluso Ths ppr hs studd ltr fld dstruto rsultg fro sx tpl odutor phs-squ orttos. 5-V doul-rut four-udld powr trssso ls of Eltrt Grtg Authort of Thld (EGAT) r vstgtd. FEM dvlopd usg MATLAB progrg s plod. As rsult phs-squ ortto s o of ftors to flu ltr fld dstruto ltr powr trssso ls. Wth th ortto of tp 1 3 d 5 th ltr fld dstruto s str whl th lft two orttos (tp 4 d 6) gv str fld dstruto. Rfrs: [1] Ols R.G. Do D. Bsh R.S. Aot J.R. Cot R. Frzr M. Jff K. ls G.B. Stwrt J.R. Wog R. d Zvdl R.M. Mgt Flds fro Eltr Powr Ls Thor d Coprso to Msurts IEEE Trstos o Powr Dlvr Vol.3 o pp [] L L. d Yougg G. Alss of Mgt Fld Evrot r Hgh Voltg Trssso Ls Prodgs of th Itrtol Cofrs o Couto Tholog 1998 pp. S S [3] Eltr Powr Rsrh Isttut Trssso- L Rfr Boo 345 V d Aov Frd Wdr & So Prtrs I. USA [4] Chr M.V.K. d Slo S.J. url Mthods Eltrogts Ad Prss USA. [5] Wr M. Eltrogt Alss Usg Trssso L Vrls World Stf Pulshg Sgpor 1. [6] Chrstopoulos C. Th Trssso-L Modlg Mthod: TLM IEEE Prss USA [7] Po-l-or P. Kulworwhpog T. Sutor S. d Poug S. Dstrutos of Flux d Eltrogt For Iduto Motors: A Ft Elt Approh WSEAS Trstos o Ssts Vol.5 o.3 6 pp [8] P-og P. Th Eltrogt Fld Effts Alss whh Itrfr to Evrot r th Ovrhd Trssso Ls d Cs Stud of Effts Rduto [M.Eg. thss] Shool of Eltrl Egrg Dprtt of Eltrl Egrg Kg Mogut s Isttut of Tholog Ldrg Bgo Thld. [9] Prsto T.W. R A.B.J. d Sgh P.S. Iduto Motor Alss T-Stppg Thqus IEEE Trstos o Mgts Vol.4 o pp [1] K B.T. Kwo B.I. d Pr S.C. Rduto of Eltrogt For Hros Ashroous Trto Motor Adptg th Rotor Slot ur IEEE Trstos o Mgts Vol.35 o pp [11] Iu G.B. d So S.A. Stud of Trssso L Mgt Flds Prodgs of th Twt-Sod Aul orth Ar IEEE Powr Sposu 199 pp [1] Ht Jr.W.H. d Bu J.A. Egrg Eltrogts (7th dto) MGrw-Hll Sgpor 6. ISS: ISB:

Extension Formulas of Lauricella s Functions by Applications of Dixon s Summation Theorem

Extension Formulas of Lauricella s Functions by Applications of Dixon s Summation Theorem Avll t http:pvu.u Appl. Appl. Mth. ISSN: 9-9466 Vol. 0 Issu Dr 05 pp. 007-08 Appltos Appl Mthts: A Itrtol Jourl AAM Etso oruls of Lurll s utos Appltos of Do s Suto Thor Ah Al Atsh Dprtt of Mthts A Uvrst