hpter 04.00E Phyicl Prblem fr Electricl Egieerig Simulteu Lie Equti Prblem Sttemet Three-phe ytem e the rm fr mt idutril pplicti. pwer i the frm f vltge d curret it delivered frm the pwer cmpy uig three-phe ditributi ytem d my lger ld e three-phe ld i the frm f mtr, cmprer, r imil. Surce d ld c be cfigured i either wye (where urce r ld e cected frm lie t eutrl/grud) r delt (where urce r ld e cected frm lie t lie) cfigurti d mixig betwee the type i cmm. Figure 1 hw the geerl wirig f wye-wye three-phe ytem mdelig ll f the impedce typiclly fud i uch ytem. Durig the typicl lyi udertke i mt rcuit textbk, it i umed tht the ytem i etirely blced. Thi me tht ll the urce, lie, d ld impedce e equivlet, tht i, b c b c N bn N Uder thi umpti, the rcuit i the typiclly reduced t igle-phe equivlet rcuit mdel d the reultt rcuit i lved with igle lp equti. Wht hppe, hwever, whe the ytem i ublced? Typiclly becue the three ld impedce, d N N N e t equl, which reult i differet curret thrugh ech ld, i fte meured i term f the percetge differece betwee the ld curret. 04.00E.1
04.00E.2 hpter 04.00E Figure 1 Three-Phe Wye-Wye Sytem with Pitive Phe Sequece retig imblce i three-phe ytem i t ll tht difficult. ider mll buie pertig i ilted leg f the pwer grid tht lclized pect f ld e t blced by ther eighbrig ld. Let ume tht the primy ld fr thi ytem i 45 kv et f three-phe mtr t 0.8 pwer fctr lggig d, further, tht the electri tht did the wirig fr the lightig mitkely cected tw bk f light t the phe, e t the phe d e t the phe etig imblce i the ytem. Ech f thee lightig ld i 1500 W. The ld fr thi ytem i hw i Figure 2.
Phyicl Prblem fr Electricl Egieerig: Simulteu Lie Equti 04.00E.3 N Figure 2 Mdel f the Sytem Ld The impedce f ech f the ld c be determied by exmiig the pwer cumed i ech phe f the ytem : 3000 15000 / 36.87 3000 12000 j9000 15000 j9000 17.49 / 30.96 kv : 1500 15000 / 36.87 1500 12000 j9000 13500 j9000 16.22 / 33.69 : 15.00 / 36.87 kv V vertig thee t impedce uig the frmul S with V 120V yield: 0.8233/ 30.96 R j 0.7060 j0. 4236 N N 0.8878/ 33.39 R j 0.7387 j0. 4925 N 0.9600 / 36.87 R j 0.7680 j0. 5760 Fr the ret f thi lyi we will ume tht ech phe f the ytem h equivlet urce d lie impedce f R j 0.0300 j0. 0200 d tht the grud retur wire h impedce f R j 0.0100 j0. 0080. Thi yield the equivlet rcuit f Figure 3. 2 kv
04.00E.4 hpter 04.00E Figure 3 Equivlet ircuit Mdel fr the Wrkig Prblem The rcuit c be lyzed uig three lp equti uig the curret, b, d c hw i Figure 3. Fr lp thi yield the cmplex equti: Lp : V 0 R j R j b c R j 0 with lp d yieldig imil reult. umig tht ur imulteu equti lver i t cpble f hdlig cmplex umber we c tur the lp equti it tw epte -cmplex equti ddreig bth the rel d imgiy pt. Uig j d cllectig term yield: Rel : R R R R R 120 (1) mgiy : R R R R R 0 (2) pplyig the me lyi t the d lp yield the remig equti fr the ytem. Rel : R R R R R 60 (3) mgiy : R R R R R 103. 9 (4) Rel :
Phyicl Prblem fr Electricl Egieerig: Simulteu Lie Equti 04.00E.5 R R R 60 R R (5) mgiy : R R R R R 103. 9 (6) Thi yield ytem f ix lie equti d ix ukw,,,,, d tht c be lved by y cvetil me. Thi i hw i mtrix frm i Figure 4. 0.7460 0.4516 0.0100 0.0080 0.0100 0.0080 0.4516 0.0100 0.0080 0.0100 0.0080 0.7460 0.0080 0.0100 0.0080 0.0100 0.0080 0.7787 0.5205 0.0100 0.0080 0.0100 0.5205 0.7787 0.0080 0.0100 0.0080 0.0100 0.0080 0.8080 0.6040 0.0100 0.0080 0.0100 0.6040 0.8080 Figure 4 mplete Sytem f Equti 120.0 0.000 60.00 103.9 60.00 103.9 (7) Oce the curret e kw it i imple prcedure t determie the vltge the three mtr termil V N, VN, dvn uig Ohm Lw. VN j R j VN j R j VN j R j T better evlute the imblce, the percetge differece i curret thrugh the ctul ld elemet i mre fte dered. The bet wy t thik f thi three peple pullig d puhig tgether. f they d t pull d puh i blce, thig c becme utble. the ce f three-phe mtr, thi c reult i igifict wbble with repdig we i the beig d ther pt. T determie the curret thrugh ech ld i g cmputed uig Ohm Lw. VN VN VN ld, ld ld 3 3 3 3 0.7680 j0. 5760 where 3 w cmputed elier. D t frget tht the lightig ld e epte. QUESTONS 1. Wht wuld be the rmificti f lvig the prblem directly uig the three cmplex lie equti? uld we d it uig pprch like Gu-Jrd Elimiti? Wht but me f the ther umericl methd ued t lve imulteu lie equti? 2. Thi prblem i ly iteretig if the grud retur leg N i -zer. Otherwie, we hve three lp equti tht e cmpletely idepedet f ech ther d c be lved directly. Why i tht the ce?
04.00E.6 hpter 04.00E 3. much mre iteretig d prcticl prblem ccur whe the mtr ld i Delt cfigurti. Sice it de t hve the grud retur lie i the middle it reult i dditil lp equti. Sketch the equivlet rcuit fr ytem with Wye urce d mix f Delt d Wye ld. Write the et f equti tht reult frm thi ytem. Slve them. SMULTNEOUS LNER EQUTONS Tpic Simulteu Lie Equti Summy Three phe ld i ytem Mjr Electricl Egieerig uthr Hery Welch Dte Jue 30, 2005 Web Site http://umericlmethd.eg.uf.edu