Object Oriented Backward/Forward Algorithm for Unbalanced and Harmonic Polluted Distribution Systems

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1 Oet Orented Bkwrd/Forwrd Algorthm for Unlned nd Hrmon Polluted Dstruton ystems Bogdn omogă, Mre Chndrş Power ystems Deprtment ehnl Unversty of Clu-po Clu-po, omân Anton udrà-andreu,, Andres umper, Ctlon nsttute for Energy eserh (EC) CCEA-UPC, Deprtment d Engnyer Elètr; EUEB, Unverstt Poltèn de Ctluny Brelon, pn nton.sudr@up.edu, ndres.sumper@up.edu Astrt he optmzton n the operton of dstruton eletr systems hs eome n ute prolem. n order to evlute some essentl rter (e.g., tve power losses) the omputton of power flow s solutely neessry. kng nto ount the rel opertng ondtons (unlne, hrmons) t s of gret nterest for urte stedy stte estmton. he exstent solutons proposed n lterture re omplex nd t s dffult to mplement them n unlned nd hrmon polluted systems. By the orgnl dptton of the kwrd/forwrd sweep lgorthm, the uthors propose novel prdgm to nlyze the propgton of symmetres nd hrmon dstortons through rdl eletr networks. Keywords- unlned eletr networks; hrmon dstorton; lod flow; kwrd/forwrd; oets orented progrmmng. ODUCO he optmzton n the operton of dstruton eletr systems hs eome n ute prolem. n order to evlute some essentl rter (e.g., tve power losses) the omputton of power flow s solutely neessry. Currently, ths omputton onsders tht the system s lned nd the wveforms re snusodl. But t s well known tht the modern dstruton systems re unlned nd, often, hrmon polluted. hus, tkng nto ount the rel opertng ondtons (unlne, hrmons) t s of gret nterest for urte stedy stte estmton. Bslly, power flow lgorthms re tertve nd re sed on dfferent proedures: Guss-edel, ewton-phson, kwrd/forwrd sweep. For dstruton systems whh re operted n rdl onfgurtons, the most reommended pproh re kwrd/forwrd sweep sed lgorthms euse of the smll tertons numer requred nd roust onvergene [1]. Generlly, the lod flow lulton for unlned/symmetrl eletrl systems onssts of the determnton of stedy stte qunttes for eh phse t tme. n the se of unlned dstruton systems operted n rdl onfgurtons hve een developed more kwrd/forwrd sweep sed lgorthms wth sgnfnt results s well s [2-5]. On the other hnd, for hrmon polluted eletr networks, the lod flow lulton onssts of the determnton of stedy stte qunttes for eh hrmon omponent t tme. he exstent solutons proposed n lterture re omplex nd t s dffult to mplement them n unlned nd hrmon polluted systems. Besdes the dt types exstng n the numers theory, the hgh level progrmmng lnguges llow the defnton of new rtfl dt types, for nstne strt dt types. he term type of dt desgntes set of vlues (the domn of type) nd set of opertons tht n e performed wth these vlues [6, 7]. he term type of dt desgntes set of vlues (the domn of type) nd set of opertons tht n e performed wth these vlues. he set of opertons n e dvded n three susets: () opertons mong the sme type of dt; () opertons mong the spefed type of dt nd nother type of dt; () opertons performed on the dt tself. As n exmple, we n onsder dt elongng to the rel numers set (the domn of type). n ths se, they n perform rthmetl opertons wth nother rel numer (opertons mong the sme dt types), rthmetl opertons wth n nteger numer (opertons mong the spefed type of dt nd nother type), nd the extrton of nteger prt (operton ppled to the dt type tself). Bsed on these strt dt types, we propose to model the hrmon omplex qunttes ( set of omplex qunttes orrespondng to eh hrmon omponent) through strt dt types wth omplex prmeters (x 1, x 2,, x n). n sme mnner, we propose to model the three-phse hrmon omplex qunttes (symmetrl, symmetrl, lned or unlned) through strt dt types wth three hrmon omplex prmeters (r, s, t); mong these prmeters, some opertons re defned. By mplementng ths model, ll three-phse hrmon qunttes wll e onsdered s hrmon omplex three-phse oets. As result, eletrl engneerng lws, s Ohm or Krhhoff, re redued to smplfed expressons orrespondng to fundmentl hrmon omponent sngle-phse se. Consequently, we hve ntrodued these oets on the kwrd/forwrd sweep lgorthm wth some prtulr dpttons. hs pper ws supported y the proet "Development nd support of multdsplnry postdotorl progrmmes n mor tehnl res of ntonl strtegy of eserh - Development - nnovton" 4D-PODOC, ontrt no. PODU/89/1.5//5260, proet o-funded y the Europen ol Fund through etorl Opertonl Progrmme Humn esoures Development Content s fnl s pulshed wth the exepton of pgnton. omog, B.; Chndrs, M.; udr-andreu, A.; umper, A., "Oet orented kwrd/forwrd lgorthm for unlned nd hrmon polluted dstruton systems," n Eletrl Power Qulty nd Utlston (EPQU), th nterntonl Conferene on, vol., no., pp.1-6, Ot. 2011, do: UL:

2 Bsed on the ove onsdertons, the uthors propose novel prdgm (y the orgnl dptton of the kwrd/forwrd sweep lgorthm) to nlyze the propgton of symmetry nd hrmon dstorton through rdl eletr systems. he proposed model hs een mplemented n the C++ progrmmng lnguge. n order to test the orretness of the proposed method, the uthors hve studed n EEE system.. BACKWAD/FOWAD WEEP o spefy n strt dt type, t s neessry to ndte the two elements of the type,.e. the domn nd the opertons set: the domn: s spefed s mthemtl set; the opertons set: ny operton s desred y ts mthemtl defnton. he domn of these qunttes (the hrmon omplex numers) s denoted y HC nd spefed s: HC = {( v, v, K, v ) v C; = 1, n; n * (2) 1 2 C. Hrmon Complex hree Phse Qunttes s Astrt Dt ypes hree non-snusodl funtons n e representng s set wth systems of rottng vetors (Fgure ). A. Complex Qunttes s Astrt Dt ypes t s possle to represent snusodl quntty through two rel numers n two mnners (Fgure 1): rottng vetor: wth module () nd ngle/phse (θ); omplex numer: wth rel prt () nd mgnry prt (); Fgure. Eletr three-phse non-snusodl qunttes he domn of these qunttes (the omplex three-phse numers) s denoted wth HC nd s spefed s: HC = {( r, s, t) r, s, t } () Fgure 1. Eletr snusodl qunttes Complex numers re very frequently used n eletrl engneerng. hey hve the followng domn (ndted y C): C = {(, ), } (1) B. Hrmon Complex Qunttes s Astrt Dt ypes A hrmon omplex numer my represent mny non-snusodl physl qunttes (voltges, urrents, mpednes et.). t s possle to represent non-snusodl quntty through set of rottng vetors (omplex numers), orrespondng to Fourer seres omponents (Fgure 2). Fgure 2. Eletr non-snusodl qunttes he omplete model (ontnng lso opertons sets) nd further nformton re gven n [8, 9]. Wth these models, the most reltonshps ontnng hrmon omplex three-phse qunttes n e desred y smplfed expressons. n order to nlyze the propgton of symmetry nd hrmon dstorton through eletr systems, the performng of lod flow (power flow) lulton s solutely neessry. n rdl networks, the lod flow lulton n e performed usng spef method, known s the kwrd/forwrd sweep. Bslly, ths method onssts of two steps [1]: kwrd sweep (rnh urrent updte), where, strtng from the end nodes nd gong towrds the soure node, nd usng the Krhhoff s urrent lw, the urrent t eh lod node, s well s the urrent flowng through ts ngong rnh, re lulted; forwrd sweep (node voltge updte), where, strtng n the opposte dreton, from the soure node (whose onstnt voltge s tken s referene) nd gong towrds the end nodes, usng the Ohm's lw, the voltge drop on eh rnh, s well s the voltge t eh lod node, re lulted. n unlned nd/or hrmon polluted power dstruton systems, the ove-proposed models n e ntrodued. Aordngly, the lod flow lulton lgorthm usng the

3 kwrd/forwrd sweep onssts n the sme steps. n the followng, we wll present the prtulr spets of kwrd/forwrd sweep dptton n order to norporte power trnsformers nd power lnes. D. Hrmon hree-phse Model for Power rnsformers t s ssumng tht y three sngle-phse nteronneted trnsformers three-phse trnsformer s onstruted [2]. We wll present the model for two of the most ommonly used trnsformer onnexons: Wye-Wye nd Delt-Wye. 1) Hrmon hree-phse Model for Wye-Wye Conneted Power rnsformers Bkwrd sweep Gven the urrents t the seondry termnls ( ) of the power trnsformer, = {(,, ),, } (4) the m s to lulte the urrents t the prmry sde ( ) = {(,, ),, } (8) the m s to lulte the urrents t the prmry sde ( ) = ( ) / = ( ) / = ( ) / Forwrd sweep (9) (,, ) (10) =, (11) Gven the voltges ( ) nd the urrents ( ) t the prmry termnls of the power trnsformer, where = (5) C represents the turn rto. Forwrd sweep Gven the voltges ( ) nd the urrents ( ) t the prmry termnls of the power trnsformer, = {(,, ),, } (12) the m s to lulte the voltges t the seondry sde ( ) = ( ) / = ( ) / = ( ) / (1) = {(,, ),, } (6) (,, ) (14) where the m s to lulte the voltges t the seondry sde ( ) Z = (7) Z represents the hrmon three-phse mpedne of power trnsformer. 2) Hrmon hree-phse Model for Delt-Wye Conneted Power rnsformers Bkwrd sweep Gven the urrents t the seondry termnls ( ) of the power trnsformer, Z = (15) E. Hrmon hree-phse Model for el Eletr Power Lnes We wll present the generl model for four wres lne euse t hs neutrl ondutor (Fgure 4): Fgure 4. Model of eletr power lne

4 Bkwrd sweep Gven the urrents t the end sde ( ) of the power lne, = {(,, ),, } (16) = Z + Z + Z = Z + Z + Z = Z + Z + Z (20) the m s to lulte the urrents t the egnnng sde ( ) (,, ) (21) =,, (17) = + Z (22) = = = Z Z + + G + Z G Z +, G + Z G, (18) where: represents the flowng urrent through neutrl ondutor; Z represents the mpedne of neutrl ondutor nd Z G represents the grounded mpedne. Forwrd sweep Gven the voltges ( ) nd the urrents ( ) t the egnnng sde of the power lne, = {(,, ),, } (19). = (2) = (24) UMECAL EUL he proposed lgorthm hs een mplemented n the C++ progrmmng lnguge [10]. n order to test the orretness of the proposed method, the uthors hve studed n EEE system for testng unlned rdl eletr networks (Fgure 5) [11]: the m s to lulte the voltges t the seondry sde ( ) Fgure 5. EEE 4 ode est Feeder ABLE. YEM MPEDACE Brnh Hrmon order Phse r [Ω] x [Ω] r [Ω] x [Ω] r [Ω] x [Ω] r [Ω] x [Ω]

5 ABLE. UBALACED AD O-LEA LOAD (COA CUE) ode 4 Phse Hrmon order Module [A] Angle [ o ] Module [A] Angle [ o ] Module [A] Angle [ o ] Module [A] Angle [ o ] ABLE. LE CUE (UBALACED AD O-LEA LOADG) Brnh Phse Hrmon order Module [A] Angle [ o ] Module [A] Angle [ o ] Module [A] Angle [ o ] Module [A] Angle [ o ] ABLE. ODE OLAGE (UBALACED AD O-LEA LOADG) ode Phse Hrmon order Module [] Angle [ o ] Module [] Angle [ o ] Module [] Angle [ o ] Module [] Angle [ o ] hs s rdl system whh ontns n nfnte us, two lnes, trnsformer nd lod. t s proposed to test the propgton of symmetres nd hrmon dstortons tkng nto onsderton the senro: unlned nd non-lner lod, four wre nd three wre lnes nd Delt-Wye onneted power trnsformer (D-GrY). On the presented se study, system mpednes re gven n le [11], soure node s k lne-to-lne nfnte us [11], power trnsformer turn rto s 12.47/4.16 [11] nd the unlned nd non-lner lod dt re gven n le s onstnt urrents. As results, lnes urrents re gven n le nd nodes voltges re gven n le, where: represents Gr for wye onnetons nd for delt onnetons; (Gr represents ground ) represents Gr for wye onnetons nd for delt onnetons; represents Gr for wye onnetons nd for delt onnetons. he m of the ntl test ses s to mke vlle ommon set of dt tht ould e used y progrm developers nd users to verfy the orretness of ther solutons [11] for unlned rdl dstruton systems. hus, the systems dt nd the solutons do not refer to rel power systems. hese test ses hve the unque purpose to verfy the orretness of dfferent models nd mplementtons for unlned systems. Consequently, on ths pper, the hrmon three-phse lne urrents (gven n le ) nd the hrmon three-phse voltges (gven n le ) for EEE 4 ode est Feeder [11] hve the role to demonstrte the orretness of the proposed lgorthm. Also, these dt n e used s enhmrk for studes wth unlned nd hrmon polluted rdl systems.. COCLUO he pper presents n orgnl soluton to perform lulus wth hrmon omplex three-phse qunttes y usng strt dt types. elted mthemtl models for omplex strt dt type, hrmon omplex strt dt type nd hrmon omplex three-phse strt dt type re developed. hese models n e used s oets n ny progrm wrtten n C++

6 progrmmng lnguge (for ny lulus wth omplex numers, hrmon omplex numers or hrmon omplex three-phse numers). By mplementng these oets n kwrd/forwrd sweep sed lgorthm, novel lgorthm for unlned nd hrmon polluted se (stedy stte) ws proposed. A numerl exmple s gven to onfrm the omputer mplementton of proposed oets By the orgnl dptton of the kwrd/forwrd sweep lgorthm, the uthors propose novel prdgm to nlyze the propgton of symmetry nd hrmon dstorton through rdl eletr networks. he omprtve tests performed on n EEE test system hve demonstrted the orretness of the proposed lgorthm. ACKOWLEDGME he uthors would lke to thnk Prof.. G. ârnovn of the ehnl Unversty of Clu-po for hs support. he uthors lso would lke to thnk the nonymous revewers whose suggestons hve mproved the fnl pper. EFEECE [1] H. D. Chng nd M. Brn, On the Exstene nd Unqueness of Lod Flow oluton for dl Dstruton Power etwork, EEE rns. on Cruts nd ystems, ol. 7, o., Mrh [2] M. E. Brn nd E. A. tton, Dstruton rnsformer Models for Brnh Current Bsed Feeder Anlyss n EEE rnstons on Power ystems, ol. 12, no. 2, My 1997, pp [] W. H. Kerstng, W. H. Phllps, nd W. Crr, A new pproh to modelng three-phse trnsformer onnetons, EEE rns. nd. Appl., vol. 5, pp , Jn./Fe [4] J.-H. eng, A dret pproh for dstruton system lod flow solutons, EEE rns. Power Del., vol. 18, no., Jul. 200, pp [5] M. F. AlHr, M. E. El-Hwry, Explotng the dl Dstruton truture n Developng Fst nd Flexle dl Power Flow for Unlned hree-phse etworks, n EEE rnstons on Power Delvery, ol. 25, no. 1, 2010, pp [6] J. C. Mthell, G. D. Plotkn, "Astrt types hve exstentl type". ACM rnstons on Progrmmng Lnguges nd ystems, ol. 10, no., July 1988, pp [7] J. C. Mthell, Conepts n Progrmmng Lnguges, Cmrdge Unversty Press, [8] M. Chndrş, A. udr, C. Bud nd B. omogǎ, he Lod Flow Clulton n Unlned dl Eletr etworks wth Dstruted Generton, n 9th nterntonl Conferene Eletrl Power Qulty nd Utlston EPQU 07, Brelon, pn, Otoer 9-11, 2007, pper on CD-OM. [9] C. Bud, B. omogǎ, M. Chndrş nd A. udr, he Lod Flow Clulton n Hrmon Polluted dl Eletr etworks wth Dstruted Generton, n 9th nterntonl Conferene Eletrl Power Qulty nd Utlston EPQU 07, Brelon, pn, Otoer 9-11, 2007, pper on CD-OM. [10] B. troustrup, C++ Progrmmng Lnguge pel Edton (n romnn), Buureşt, eor, 200. [11] EEE PE dstruton system nlyss suommttee, Dstruton test feeder workng group. Dstruton test feeders: