Research on Three-phase Optimal Power Flow for Distribution Networks Based on Constant Hessian Matrix

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1 ge of 7 IE Geerto, rssso & Dstruto hs rtle hs ee epte for pulto future ssue of ths ourl, ut hs ot ee fully ete. Cotet y hge pror to fl pulto ssue of the ourl. o te the pper plese use the o prove o the Dgtl Lrry pge. Reserh o hree-phse Optl ower Flow for Dstruto Networks Bse o Costt Hess Mtrx Zho Fegzh, Zho gtg, Ju Yuto *, M Kg, Zhou Xfe 3 College of Iforto Eletrl Egeerg, Ch Agrulturl Uversty, Beg, Ch, Deprtet of Eletro Eletrl Egeerg, Uversty of Bth, UK, 3 Stte Gr Beg Eletro ower Copy, Beg, Ch. * El: uyuto@u.eu. Astrt: he optl power flow (OF prole for tve struto etworks wth strute geerto (DG vrety of sretely ustle eves (e.g., o-lo tp-hgers, OLCs s essetlly o-ovex, oler, xeteger optzto prole. I ths pper, the qurt oel of three-phse OLCs s propose y g rh urrets s ukow vrles, whh gurtee ostt Hess trx throughout tertos. hs pper proposes three-phse OF oel for tve struto etworks, oserg three-phse DG oel. he OF oel s solve y teror pot etho orportg qurt pelty futo s oppose to Guss pelty futo. Furtherore, voltge regultor s lso orporte to the OF oel to for tegrte regulto strtegy. he ethoology s teste vlte o the IEEE 3-us three-phse ule test syste.. Itrouto he optl power flow (OF for tve struto etworks wth strute geerto (DG vrety of sretely ustle eves s o-ovex, o-ler, xe-teger optzto prole. It volves oth srete otuous vrles. By optzg the operto strtegy of the ustle eves, lug shut ptors, ustle strute eergy resoures, o-lo tp-hgers (OLCs, et., the OF s to ze the operto ost whh tkes to out etwork losses. DG eergy storge uts rg ew hlleges to trtol OF proles [-7]. Referee [6] stue how the uertty w outputs the orrelto og ultple w frs woul ffet the OF. Referee [6] ot oser srete ustle eves, whh re prevlet tve struto etworks. Referee [7] propose OF oel for tve struto etworks wth OLCs, where peewse ler oel s opte to oel OLCs. he OF s solve y seo-orer oe relxto etho [7]. Referee [8] optzes the operto of tve struto etwork where o-ouple oel s opte for DG, yelg pproxte results. Referee [9] propose three-phse stey-stte oel for DG the oel e pple to OF for tve struto etworks. he prl-ul teror pot etho hs ee wely use to solve the OF prole for trtol power systes euse of ts vtges suh s effey fst overgee [-]. Referee [3] propose srete vrle proessg etho se o the Guss pelty futo, ut t oes ot opre the oputto effey of the futo wth tht of other pelty futos, e.g., the qurt pelty futo s troue []. By troug vrtul oe to the OLC oel, referee [4] trsfore the OF oel to qurt optzto the Crtes oorte syste, thus provg oputto effey. hs pper proposes ew syetrl eto oel for DG wth three phse ouplg ew qurt oel for three-phse OLCs. he three-phse OF for tve struto etworks s solve y pretor-orretor prl-ul teror pot etho (CDIM orportg pelty futo. he etwork voltges re fe-tue o eh phse y voltge regultor, tegrte regulto strtegy s propose orporte to the OF prole so tht the voltges re fe-tue o eh phse to further optze the etwork loss. Cse stues eostrte tht the qurt oels for three-phse OLCs DG prove oth the oputto effey the ury of the OF. Cse stues lso opre the vtges svtges of the Guss pelty futo; the qurt pelty futo wth otuous vrle sretzto proess. he pts of the voltge regultor o ol voltges etwork losses re lso eostrte.. hree-hse Dstruto Networks.. Qurt oel of three-phse OLCs I the Crtes oorte syste, the OF oel s oler optzto prole wth orer hgher th qurt f oserg the turs rto of the OLC s otrol vrle. Whe solvg the OF prole y the teror pot etho, the Hess trx s upte eh terto, rgg out sgft oputto ure [4][]. I orer to solve ths prole, ew three-phse OLC oel s propose ths pper. By g the rh urrets s stte vrles, the Hess trx eoes ostt throughout the tertos of the OF. Fg. shows the OLC Wye-elt (Y ofgurto, vrtul oe s e to rh. hs trsfors the OLC to el trsforer (wth ustle turs rto k oete seres wth equvlet pee (wth pee R + X. he ol voltges rh urrets re lso show Fg.. IE Revew Copy Oly

2 Fg.. hree-hse Moel of OLCs Wth the el trsforer, there s U& = k( U& U& = k( U& U& U& = k( U& U& Bse o the oservto of eergy, there s * * ( ( * * ( ( * * ( ( U& I& + ( U& U& I& = I& + ( U& U& I& = U& I& + ( U& U& I& = Susttutg ( ( yels, ki& = I& ki & = I& ki & = I& I the three-phse syste, the le voltge s set s the se voltge. So, for the Y trsforer, the str trsforto rto s : 3 uer the per-ut syste. Coserg the equvlet pee rh, there s U& U& ( R + X U& ( R + X U& U& ( R + X.. Qurt oel of three-phse DGs he oel Fg. represets fferet types of DG uts, lug those wth the terfe of voltge-soure overters, w geerto se o ouly fe uto geertors, syhroous geertors, et. IE Geerto, rssso & Dstruto hs rtle hs ee epte for pulto future ssue of ths ourl, ut hs ot ee fully ete. Cotet y hge pror to fl pulto ssue of the ourl. o te the pper plese use the o prove o the Dgtl Lrry pge. ( ( (3 (4 Where I +I q, I + +I + q, I +I q represet zero sequee urret, postve sequee urret, egtve sequee urret fro DG to the etwork, respetvely. U +U q, U + +U + q, U +U q represet zero sequee voltge, postve sequee voltge, egtve sequee voltge t the ouplg pot, respetvely. R +X R +X represet zero sequee pee egtve sequee pee, respetvely. U r+u rq U r+u rq represet the zero sequee egtve sequee of the extto voltge, respetvely. he DG otrol syste ossts of three prts: tve power otrol, retve or voltge otrol, the otrol of the ule opoets. he tve power otrol keeps the postve sequee tve power uhge. he retve otrol ts ostt retve power eto to the gr. DG extto voltge ots oly postve sequee opoet, wth egtve zero sequee opoets eg zero. Cosequetly, the equlty ostrts re estlshe, s show equtos (5, 6, Re ( U + U q ( I Iq = U I + U q Iq = sp ( I ( U + Uq ( I Iq = Uq I U Iq = Qsp (6 U r = Re ( I + Iq ( R + X + U + U q = U rq = I ( I + Iq ( R + X + U + U q = U r = Re ( I + Iq ( R + X + U + U q = U rq I ( I Iq ( R X U U = q = Where, Re[] I[] orrespo to the rel prt the gry prt of the expresso, respetvely. sp Q sp orrespo to the postve sequee vlues of the trget tve power retve power, respetvely. he gr terfe of three-phse ouple DG s oelle s oth voltge otrolle voltge soure urret otrolle urret soure, s llustrte Fg. 3. he sequee vlues of the DG eto urrets the terl voltges re ukow vrles the qurt oel of DG. he vrles lue +,, U&,, +, &. I + + I& U& (7 ge of 7 DG Fg.3. Sequee-phse ouple terfe Fg.. A geerlze sequee opoet oel of DG he trsfortos fro phse vlues to sequee vlues re gve y IE Revew Copy Oly

3 ge 3 of 7 IE Geerto, rssso & Dstruto hs rtle hs ee epte for pulto future ssue of ths ourl, ut hs ot ee fully ete. Cotet y hge pror to fl pulto ssue of the ourl. o te the pper plese use the o prove o the Dgtl Lrry pge. = + (8 + = + (9 Where, + s phse to sequee trsforto trx. orrespos to oplex quttes. he ostrts (5 (9 oser the ouplg hrtersts of three-phse powers. hey re fferet fro the stey-stte oel [] whh oes ot oser sequee otrol uer ule oto. he equtos volvg DG re ether ler or qurt ths gurtees ostt Hess trx the OF. A populr oel of DG s the three-phse oouplg oel [8], esre (. + Q = ( U + U ( I I ( re re Where, Q orrespo to the tve retve power outputs fro DG o phse, respetvely. U re U orrespo to the rel prt the gry prt of the ol voltge of DG o phse, respetvely. I re I orrespo to the rel prt the gry prt of the eto urret fro DG o phse, respetvely. he three-phse ouple oel of DG s urte oel, the three-phse o-ouple oel s pproxte oel. So, the optl power flow results of the three-phse ouple oel re ore urte th those of the pproxte three-phse o-ouple oel. he ove foruls osttute the DG ostrts the three-phse OF oel. Se the hghest orer of these foruls re qurt, the Hess trx s ostt throughout tertos..3. hree-phse oel of voltge regultor A voltge regultor s oete to the gr for the fe-tug of ol voltges. A three-phse Y-oete voltge regultor ossts of three sgle-phse voltge regultors, eh wth tp hger tht hges the tp posto. he ofgurto of the three-phse voltge regultor s llustrte Fg.4. A B A B B A C C Fg.4. hree-phse str-shpe voltge regultor C he followg ostrts pply: A U& = ku& B U & = ku& C U & = ku& ( k k k A ki& B ki & C ki & = ±.65 p = ±.65 p = ±.65 p ( (3 I forul (3, plus ( + orrespos to step up us ( - orrespos to step ow. p eotes the posto of voltge regultor tp (e.g., 3-level ustets wth rge of ±%. he tp s otrolle to regulte the voltge of lo eter wth le voltge rop opestor. he le voltge rop opestor s ouple wth the struto le through voltge trsforer (turs rto N : urret trsforer (turs rto C : C S, s show Fg.5. he pee of the opestor represets the equvlet pee fro the regultor to the lo eter: C R ' + X ' = ( R+ X (4 N he voltge t the lo eter s U& x = U& x / N ( R ' + X ' I& x / C (5 lo If the voltge level of the lo eter s V the wth s V, the voltge wll hge y.75v eh te the tp of the voltge regultor oves to the ext posto. he step up the step ow tp hges re gve y x 9 U lo px = (6.75 x U lo px = (7.75 voltge regultor voltge trsforer urret trsforer le voltge rop opestor Fg.5. Le voltge rop opestor R+X lo eter 3. Optl ower Flow Moel Algorth 3.. OF oel hs pper estlshes three-phse OF oel for struto etworks y tkg the ol voltges rh urrets s the stte vrles. he oetve futo of the OF oel s the u loss of the etwork, s show equto (8. * loss r, r, r= = { A, B, C} & & (8 f ( x = = U I t IE Revew Copy Oly 3

4 Where, s the uer of rhes the etwork. U r, I r, orrespo to the rh voltge rh urret o phse of the r th rh, respetvely. A the rh voltge s the fferee etwee the ol voltges t oth es of the rh. he equtos lue the KCL KVL ostrts (s show equto (9, trsforer rh ostrts (s show equto ( (3 (4, DG rh ostrts (suh s (5 - (9, the regultor rh ostrts (s show equto ( (. U U r,, re,, re r,,,, = A I = A I = AU = AU r,, re r,, (9 Where, U r,,re U r,, orrespo to the rel prt the gry prt of the rh voltge o phse of the r th rh, respetvely. U,,re U,, orrespo to the rel prt the gry prt of the ol voltge o phse of the th oe, respetvely. I r,,re I r,, orrespo to the rel prt the gry prt of the rh urret o phse of the r th rh, respetvely. A s the oe-rh ee trx he equlty ostrts lue stte vrle ostrts otrol vrle ostrts. he stte vrle ostrts lue the geertor tve retve power ostrts, the oe voltge pltue ostrt, the le trssso power ostrt (s show equto (. he otrol vrle ostrts lug the ostrts of the OLC turs rto K, the voltge regultor tp posto K V, retve power ptor opesto pty Q C, s show equto (. G, G, G, Q Q Q G, G, G, U U + U U,,, re,,, K K K,,, K K K V, V, V, Q Q Q C, C, C, ( ( hus, the equtos the OF oel re ether ler or qurt whh gurtees ostt Hess trx whe the teror pot etho s use to solve the oel. I ths pper, we use the pretor-orretor prlul teror pot etho [3] to solve the optzto oel wth the vtges of effey fst overgee. 3.. Cotuous vrle sretzto proess he OF for tve struto etworks volves srete otrol vrles (suh s OLCs turs rto, ptor opesto pty, et.. he oetve futo orportes pelty futo to ope wth the srete vrles. hs troues vrtul loss to the oetve futo whh wll reue the error rought y roug-off. At preset, the qurt pelty futo [] IE Geerto, rssso & Dstruto hs rtle hs ee epte for pulto future ssue of ths ourl, ut hs ot ee fully ete. Cotet y hge pror to fl pulto ssue of the ourl. o te the pper plese use the o prove o the Dgtl Lrry pge. IE Revew Copy Oly the Guss pelty futo [3] re two wely use pelty futos, s gve y ( (3, respetvely. φ( x = v( x ( = ( x φ( x = vg( x = vexp[ ] (3 = Where v s pelty ftor. he vlue of the futo rops to zero whe the ste fro the vetor x= ( x, x, K, x to the eter = (,, K, reues to zero. he pelty futo s orporte to (8, * loss r, r, r= = { A, B, C} & & (4 = U I t+ φ Whe the pelty ftor s lrge eough, the srete otrol vrle wll e optze to the orrespog srete vlue, so tht the pelty futo vlue eoes zero the oetve futo to reh the u vlue. Copre wth the qurt pelty futo, the Guss pelty futo ereses fster th the qurt pelty futo wth the erese of the ste etwee the vetors x ; the forer s ore sestve to the vrto of srete vrles. However, the Guss pelty futo s o-qurt futo, whh kes the Hess trx hge throughout tertos ths oproses oputto effey Itegrte regultory strtegy I orer to ze etwork losses, ths pper proposes OF tegrte regultory strtegy wth voltge regultors. he tegrte regultory strtegy s efe s OF strtegy for vrety of ustle eves to oorte optze ths pper the voltge regultor fe-tues ol voltges o eh phse, osttutes tegrte regultory strtegy wth the geertor, OLCs, shut ptors DGs. he voltge regultor s spel trsforer wth zero equvlet pee. he voltge regultor s oelle y qurt oel where the tp posto s hose s otrol vrle. he Hess trx res ostt uer the tegrte regultory strtegy. hs s the reso why the tegrte regultory strtegy s sutle for the OF oel propose ths pper. 4. Exple Aorg to the three-phse OF oel for struto etworks se o ostt Hess trx, ths pper uses Syol Mth oolox to verfy the oel o MALAB. A the OF lyss s rre out o the ofe IEEE3 test syste s show Fg.6. he utul-pee the utul-tte re otte the fgure. he otrol vrles the exple re geertor retve power, DG output, the turs rto of the OLC, voltge regultor tp posto ptors ustet, whh the turs rto of the OLC the ptors re srete otrol vrles. he rge of the turs rto s set t.9~., ve to 8 tps, ts step 4 ge 4 of 7

5 ge 5 of 7 sze s.5. he xu retve power of the ptors s. p.u. the step sze s. p.u.. he rge of ll ol voltges s.9 ~. p.u.. he voltge regultor tp hs 3-stge ustet wth ± % regulto rge, the voltge level of lo eter s V, the wth s V. Fg.6. Mofe IEEE3 est Syste 4.. Alyze the qurt oel of three-phse OLCs I orer to verfy the rtolty of the propose three-phse OLCs oel the OF prole for tve struto etworks, the OF lulto se o the pretor-orretor prl-ul teror pot etho eeg qurt pelty futo (CDIM-QF s use to opre the trtol o-qurt oel for three-phse OLCs [] the qurt oel for threephse OLCs propose ths pper. he results re show le. le he optzto results qurt oel of o-qurt Network loss Iterto 7 7 Clultg le shows tht the results (etwork loss of the two ethos usg fferet oels re the se, ut the qurt oel hs fster lulto spee. he reso s tht, the Hess trx of the o-qurt oel of OLCs s ot ostt the OF lulto proess, s upte eh terto, whh kes the lulto spee very slow. he qurt oel of OLCs gurtee ostt Hess trx throughout tertos, thus reug the lulto te gretly. I to, the oputto te of the two oels s reltvely log. hs s euse the Hess trx s geerte usg Mtl s utot fferetto futo, whh s oveet ut ot effet. hs, however, oes ot ffet the oluso of ths pper. he se stuy volves totl of 7 tertos. he oe-te lulto of the Hess trx tkes 6.67s o the oputer wth Itel ore 3-34M CU 3.4GHz, 4GB eory. herefore, oservtve estte of the oputto te of the oqurt oel s 7 tes of the te for lultg the Hess trx,.e., 3.84s. herefore, the oputto te le s ustfe. 4.. Alyze the otuous vrle sretzto proess IE Geerto, rssso & Dstruto hs rtle hs ee epte for pulto future ssue of ths ourl, ut hs ot ee fully ete. Cotet y hge pror to fl pulto ssue of the ourl. o te the pper plese use the o prove o the Dgtl Lrry pge. I orer to verfy the overgee propertes of the qurt pelty futo the Guss pelty futo the OF lulto, the CDIM-QF pretororretor prl-ul teror pot etho eeg Guss pelty futo (CDIM-GF re esge. he OF for the ofe IEEE3 three-phse syste s lulte y the two lgorths. hs pper proves the vtges svtges of eh etho y oprg the optzto results (etwork loss, terto uer lultg te (the ury s e -. le ots the results of the OF lulto. It shows tht, the CDIM-GF s etter th the CDIM-QF ters of the optzto results (the etwork loss s reue y.4%. But the terto tes of the CDIM-GF s slghtly hgher th tht of the CDIM-QF, the CDIM-GF tkes loger te. he reso s tht the Guss pelty futo s hgher orer futo the Hess trx s o-ostt trx throughout tertos. So the Hess trx s upte s the tertos resg, whh reses the oputto te. le he optzto results CDIM-QF CDIM-GF Network loss.5.46 Iterto 7 8 Clultg Fg.7 shows the oprso of the CDIM-QF CDIM-GF. It e see tht the ul gp ereses s the tertos resg, the te of the CDIM-QF s out /5 of the CDIM-GF whe the overgee effet of the two pelty futos rehes the esre vlue. Dul Gp Iterto Nuer Fg.7. he optzto results CDIM-QF CDIM-GF 4.3. Alyze the tegrte regultory strtegy he tegrte regultory strtegy of OF s pleete y the ofe IEEE3 three-phse syste wth the voltge regultor. It verfes the effet of the voltge regultor o eh phse voltges fe-tug tve struto etworks, further reue the etwork loss the OF lulto. he results re show le 3. IE Revew Copy Oly 5

6 IE Geerto, rssso & Dstruto hs rtle hs ee epte for pulto future ssue of ths ourl, ut hs ot ee fully ete. Cotet y hge pror to fl pulto ssue of the ourl. o te the pper plese use the o prove o the Dgtl Lrry pge. ge 6 of 7 le 3 he optzto results Voltge regultor tp Network hse A hse B hse loss No.53 Wth As show le III, the etwork loss of the etwork wth voltge regultor s reue y.3% opre wth tht wthout the voltge regultor. he threephse voltge profle of ll oes, efore fter tegrte regultory, s show Fg8, Fg9 Fg. hus, the voltge regultor hs effet o fe-tug the ol voltges. It e see tht, the voltge regultor regulte the ol voltges t eh phse further reue the etwork loss the OF lulto. A-phse voltge pltue/pu Noe uer Fg.8. Voltge profle of phse-a B-phse voltge pltue/pu Fg.9. Voltge profle of phse-b C-phse voltge pltue/pu Fg.. Voltge profle of phse C No regultor Wth regultor No regultor Wth regultor Noe uer No regultor Wth regultor Noe uer 5. Coluso hs pper proposes qurt oel for threephse OLCs. he oel s orporte to the OF oel, where the Hess trx eoes ostt throughout tertos. hs results oserle proveet oputto effey. hs pper solves the three-phse OF prole for tve struto etworks y usg the teror pot etho. he OF oel orportes oth the qurt oel for the three-phse OLC (s exple ove the syetr oel for the three-phse DG. he ethoology s vlte y se stues. 3 he se stues eostrte the vtges svtges of the teror pot etho orportg the qurt pelty futo the Guss pelty futo. 4 A tegrte regulto strtegy supporte y voltge regultor s orporte to the OF. Cse stues eostrte tht the voltge regultor fe-tue the etwork voltges, thus further reug etwork losses. 6. Akowlegets hs work ws supporte y Ntol Key Reserh Developet rogr (6YFB9 Chese Uverstes Setf Fu (7QC Referees [] E. Sortoe M. A. El-Shrkw, Optl ower Flow for Syste of Mrogrs wth Cotrollle Los Bttery Storge, ower Systes Coferee Exposto, 9. SCE 9. IEEE/ES, 9, pp. 5. [] Y. Ju, W. Wu, B. Zhg, H. Su, hree-phse DFIG stey oel fst three-phse lo flow lgorth for struto power systes, ower Syste ehology (OWERCON, Itertol Coferee o,, pp. 6. [3] S. Cot, R. Nolos, S. A. Rzzo, H. H. Zeel, Optl Dspthg of Dstrute Geertors Storge Systes for MV Isle Mrogrs, IEEE rstos o ower Delvery, vol. 7, o. 3, pp. 43 5, Jul.. [4] Y. Ju, W. Wu, B. Zhg, H. Su, A Exteso of FBS hree-hse ower Flow for Hlg V Noes Atve Dstruto Networks, IEEE rstos o Srt Gr, vol. 5, o. 4, pp , Jul. 4. [5] X. B, L. Qu, W. Qo, Roust AC Optl ower Flow for ower Networks Wth W ower Geerto, IEEE rstos o ower Systes, vol. 3, o. 5, pp , 6. [6] X., M. Zhou, et l. Ipt of W Spee Correlto o Optl ower Flow, Autoto of Eletr ower Systes, vol. 37, o. 6, pp. 37 4, 3. ( Chese. [7] B. Lu, F. Lu, S. Me, et l. Optl ower Flow Atve Dstruto Networks wth O-lo p Chger Bse o Seo-orer Coe rogrg, Autoto of Eletr ower Systes, vol. 39, o. 9, pp. 4 47, 5. ( Chese. [8] Y. Lu, W. Wu, B. Zhg. A Mxe Iteger Seoorer Coe rogrg Bse Atve Retve ower Coorte Mult-pero Optzto for IE Revew Copy Oly 6

7 ge 7 of 7 Atve Dstruto Network, roeegs of the CSEE, 4, 34(6: [9] Y. Ju, W. Wu, B. Zhg. hree-phse Stey-stte Moels for Dstrute Geertors, roeegs of the CSEE, 4, 34(: [] Y.-C. Wu, A. S. Des, R. E. Mrste, A ret oler pretor-orretor prl-ul teror pot lgorth for optl power flows, IEEE rstos o ower Systes, vol. 9, o., pp , My 994. [] A. Sous, G. L. orres, C. A. Czres, Roust Optl ower Flow Soluto Usg rust Rego Iteror-ot Methos, IEEE rstos o ower Systes, vol. 6, o., pp , My. [] R. A. Jr, A. H. Cook, B. J. Cory, A prlul teror pot etho for optl power flow spthg, IEEE rstos o ower Systes, vol. 7, o. 3, pp , Aug.. [3] Z. L, W. Wu, B. Zhg. A Lrge-sle Retve ower Optzto Metho Bse o Guss elty Futo Wth Dsrete Cotrol Vrles, roeegs of the CSEE, vol. 33, o. 4, pp , 3. [4] J. Yu, W. Y, G. Xu. A New Moel of Retve Optzto Bse o retor Corretor rl Dul Iteror ot Metho, roeegs of the CSEE, vol. 5, o., pp. 46 5, 5. [5] J. Xu, X. Dg, Z. Q. A Noler retor- Corretor Iteror ot Metho for Retve ower Optzto ower Syste, ower Syste ehology, vol. 9, o. 9, pp. 36 4, 5. [6] J. Cheg, S. L, Q. Cheg. A retor-corretor Iteror ot Metho for Optl Retve ower, rstos of Ch Eletro tehl Soety, vol. 5, o., pp. 5 57,. [7] M. Lu, X. Wg. A Applto of Iteror ot Metho to Soluto of Optzto roles ower Systes, ower Syste ehology, vol. 3, o. 8, pp. 6 64, 999. [8] C. Y. Chug, W. Y, F. Lu, Deopose retor-corretor Iteror ot Metho for Dy Optl ower Flow, IEEE rstos o ower Systes, vol. 6, o. 3, pp. 3 39, Aug.. [9] Y. Ne, Z. Du, J. L, AC-DC optl retve power flow oel v pretor-orretor prl-ul teror-pot etho, IE Geerto, rssso Dstruto, vol. 7, o. 4, pp , Apr. 3. [] J. Zho, Z. Hou, S. J. A Novel Qurt elty Futo Bse Dsretzto Algorth for Newto Optl ower Flow, Autoto of Eletr ower Systes, vol. 3, o. 3, pp. 37 4, 999. ( Chese. [] W. Wu, B. Zhg. Dervto of Detle rsforer Moels hree-phse ower Flow for Dstruto Syste, Autoto of Eletr ower Systes, 3, 7(4: [] Y. JU, F. Ge, Y. L, J. WANG, ADN-Moel: A Ope Soure Sequee-hse Couple Fre rogr for Atve Dstruto Network Stey Stte Alyss, reprts, 6. [3] B. Lg, S. J, W. g, W. Sheg, K. Lu, A rllel Algorth of Optl ower Flow o Hoop ltfor, ro. of IEEE ES As-f ower Eergy Coferee, pp.-5, 6. IE Geerto, rssso & Dstruto hs rtle hs ee epte for pulto future ssue of ths ourl, ut hs ot ee fully ete. Cotet y hge pror to fl pulto ssue of the ourl. o te the pper plese use the o prove o the Dgtl Lrry pge. IE Revew Copy Oly 7