Dynamic Behavior of Current Controllers for Selective Harmonic Compensation in Three-phase Active Power Filters

Transcription

1 Dynamc Bhaor of Currnt Controllrs for Slct Harmonc Compnsaton n Thr-phas Act Powr Fltrs Frnando Brz, Pablo García, Mchal W. Dgnr, Dad Díaz-Rgosa, Juan Manul Gurrro Unrsty of Odo, Dpt. of Elc., Computr HEV and FCEV Rsarch Dpt, Rsarch and Adancd & Systms Engnrng Engnrng, Ford Motor Company Gjón, 334, Span Vllag Road, MD 7, Darborn, MI 48-3, T: , E: frnando@sa.uno.s T: , E: mdgnr@ford.com Abstract Currnt rgulators ar a crtcal part of act powr fltrs (APF's). Dsgn of currnt rgulators capabl of compnsatng hgh frquncy harmoncs cratd by non-lnar loads s a challngng task. Slct harmonc currnt compnsaton usng harmonc rgulators s a abl mthod to ach ths goal. Howr, thr dsgn and tunng s not an asy task. Th prformanc and n th stablty of harmonc currnt rgulators strongly dpnds on mplmntaton ssus, wth th tunng of th controllr gans bng crtcal. Furthrmor, th prsnc of multpl currnt rgulators workng n paralll can crat unwantd couplngs wth th fundamntal currnt rgulator, whch can rsult n a dtroraton of APF currnt control,.., oscllatons and sttlng tms largr than xpctd. Ths papr addrsss th dsgn and tunng of slct harmonc compnsators, wth a focus on thr stablty analyss and transnt bhaor. Indx trms Act fltrs, currnt control, slct harmonc compnsaton, dgtal control, rsonant controllrs I. INTRODUCTION Shunt act powr fltrs (APF) ar powr lctronc dcs dsgnd for th compnsaton of harmonc currnt from nonlnar loads, whch also ha th capablty of compnsatng ract powr and unbalancd loads, s Fg. [,4 7]. Th currnt controllr s a crtcal componnt of ths APF's wth th choc aalabl of rgulatng thr th ln currnts [], Fg. and th block dagram shown n Fg. a, or th act fltr currnts, th block dagram shown n Fg. b [,,8]. Th frst opton has th adantag that only th ln currnt nds to b masurd. Th scond opton rqurs masurmnt of both th APF currnts and th ln currnts, whch ar ndd for dtrmnng th harmoncs that nd to b dcoupld, but has th adantag of prodng or-currnt protcton for th APF. From a dynamc pont of w both solutons ar smlar and th frst on wll b usd n ths papr []. A numbr of currnt control stratgs for APF ha bn proposd, ncludng hystrss, lnar (PI), sldng and dadbat controllrs [6,7]. Ths mthods normally ha sgnfcant lmtatons lmnatng harmoncs njctd by nonlnar loads. Slct harmonc compnsaton was dlopd to addrss ths concrns and s a control stratgy n whch sral (normally lnar) harmonc currnt rgulators work n paralll, ach canclng a spcfc harmonc njctd Τhs work was supportd n part by th Rsarch, Tchnologcal Dlopmnt and Innoaton Programs of th Spansh Mnstry of Scnc and Innoaton ERDF undr grant MICINN--ENE-494 and th Mnstry of Scnc and Innoaton undr grant MICINN--CSD9-46. by th load [6,4 7]. An appalng proprty of ths mthods s that thy can totally cancl th harmoncs ncludd n thr dsgn, ncludng th ablty to dynamcally slct th harmoncs to compnsat whn th magntud of th harmonc currnts surpass th APF's capablty. Concrns for th mplmntaton of harmonc currnt rgulators nclud thr computatonal rqurmnts, tunng, and crcut confguraton. Whl th frst on s bcomng lss mportant thanks to fast and rlatly chap dgtal sgnal procssors, slcton of th gans for th controllrs, as wll as crcut confguraton optons, to guarant stabl opraton and adquat dynamc prformanc s a challng. Powr grd l Non-lnar loads a,b,c a,b,c d,q d,q L f, R f s lqd s lqd jθ jθ F ld lqd V dc ld lqd currnt Fqd rg. V dc lq V dc PWM nrtr Fg..- Block dagram of a paralll APF, ncludng ln currnt and DC bus oltag control. lqd G FCR Fqd PWM nrtr l G LPF (optonal) F load lqd load qd Fqd Fqd lqd G F lqd load qd Fqd Fqd PWM Fqd Fqd lqd G FCR nrtr G F G LPF (optonal) Harmonc dtcton Fg..- Schmatc rprsntaton of th APF qd currnt control shown n a synchronous rfrnc fram (outr dc-lnk oltag control not shown). ln currnt s rgulatd, act fltr currnt s rgulatd. G FCR stands for fundamntal currnt rgulator transfr functon, G F for th nduct fltr transfr functon and G LPF for th (optonal) low-pass fltr n th currnt fdback.

2 u, w, wu, (V) - load u, 4 load, load w -4.4 c) Ln oltag harmoncs (pu) h=7,, 3, d) h= h= Load. h=, 3, 7 currnts h=7 harmoncs (pu)...7 tm (s) Fg. 3.- Currnt drawn by an lctrc dr durng an acclraton transnt. Th dr was connctd to th ln through a dod rctfr. ln oltags, dod rctfr (load) phas currnts, c) harmonc contnt of th ln oltag, n pu of th fundamntal componnt (h=), d) harmonc contnt of th load currnt, n pu of th maxmum alu of th fundamntal componnt: h= Ths papr analyzs th dsgn, tunng and mplmntaton of harmonc currnt rgulators. Contnuous modls wll b dlopd frst for ths analyss, snc thy mak t asr to prsnt th concpts nold. Th ffcts du to a dgtal mplmntaton of th harmonc currnt rgulators wll thn b prsntd, wth th goal of mantanng th stablty and prformanc of th rsultng dscrt currnt rgulators. II. PARALLEL ACTIVE POWER FILTER CONTROL An xampl block dagram of an APF s shown n Fg.. Th ln currnt rfrncs ar gnratd from an outr dc-lnk oltag control loop and th supply ln oltags. Ths currnt rfrncs ar th dal ln currnts, fr of any hgh frquncy harmoncs dmandd by th load and wth th ablty to prod any dsrd powr factor, up to th lmts of th APF. Fg. a shows th schmatc rprsntaton of th APF currnt control n a rfrnc fram synchronous wth th ln oltag ctor. It can b notcd from Fg. a that th plant dynamcs ncludd n ths modl rprsnt th dynamc charactrstcs of th APF output fltr. An nduct fltr, modld as an RL load, s usd for ths purposs n Fg. [3,,7-, ], wth th corrspondng transfr functon bng (). It should b notd, howr, that othr optons, lk an LCL fltr, can also b usd wth th approprat changs n th plant modl (). A.- Currnt harmoncs du to non-lnar loads Th ln oltags and load currnts ar dsturbancs to th systm. In a synchronous rfrnc fram, th fundamntal ln oltag bcoms a dc quantty and s, thrfor, asly compnsatd by a synchronous fram currnt rgulator. On th othr hand, som loads,.g., dod rctfrs, crat currnts wth a larg contnt of hgh frquncy harmoncs. Compnsaton of th load currnts s thrfor a much mor challngng task. Th currnt harmoncs cratd by non-lnar loads typcally ha ordrs of h=, 7,, 3, n th statonary rfrnc fram, wth h= corrspondng to th fundamntal frquncy, ω. Ths harmoncs bcom h=±6, ±, ±8..., whn transformd to th synchronous rfrnc fram [7,8]. Fg. 3 shows th xprmntally masurd currnts drawn by an lctrc dr from th ln durng an acclraton transnt, as wll as th ln oltag. Fg. 3c shows th araton or th tm of th harmoncs n th ln oltag, whl Fg. 3d shows th harmoncs cratd by th load, both calculatd usng th Short-tm Fourr Transform. It can b obsrd from th fgur that th harmoncs n th currnts can chang rlatly quckly, rachng notcably larg alus (rlat to th fundamntal currnt). Th ln oltag harmoncs, on th othr hand, show mnmal chang du to th transnt currnts cratd by th lctrc dr durng ts acclraton. Th APF currnt rgulator, thrfor, nds to b abl to cancl currnt harmoncs that can b at frquncs rlatly larg compard to th fundamntal frquncy and has to prod good dynamc rspons, snc ths harmoncs can chang rlatly fast. B.- Synchronous fram PI currnt rgulators Synchronous fram PI currnt rgulators ha bn wdly usd for th control of thr-phas powr conrtrs [3]. Th transfr functon of a synchronous rfrnc fram PI currnt rgulator mplmntd n th synchronous rfrnc fram s shown n (). A pol-zro notaton wll b usd throughout th papr, wth th notaton usng th proportonal and ntgral K gan bng ndcatd n (). qd G FCR = qd = (s /T ) s = K s ; K = K p T To aluat and compar th prformanc of dffrnt currnt rgulators dsgns, t s usful to us th command trackng,.., output to rfrnc, (3), and th dsturbanc rjcton,.. output to dsturbanc, (4), transfr functons, shown n a synchronous rfrnc fram, wth G FCR standng for th l_qd G FCR G F = (3) G FCR G F G LPF l_qd () Fqd G F = = Fqd Lf (sjω)rf lqd () l_qd load_qd = G FCR G F G LPF (4)

3 .. load qd(jω) harmonc ordr h Fg. 4.- Frquncy rspons functon (FRF) magntud, shown n a synchronous rfrnc fram, of th synchronous rfrnc fram PI currnt rgulator. Th ln frquncy s ω = Hz, th harmonc ordr h bng usd for th frquncy axs. Th currnt rgulator was tund for a 4 Hz bandwdth (8 ω ) wth no low-pass fltr. Command trackng FRF (q. (3)), and Dsturbanc rjcton FRF (q. (4)). s lqd s load qd c) s lqd tm (s) Fg..- Tm rspons of an APF wth a synchronous rfrnc fram PI currnt rgulator (smulatd): ln-currnt command, load currnt, consstng of harmoncs h=±6, ±, ±8, ±4 and ±3, c) actual ln currnt. All th currnts ar shown n th statonary rfrnc fram. fundamntal currnt rgulator transfr functon n a synchronous rfrnc fram (), G F () bng th APF nduct fltr transfr functon and G LPF bng th transfr functon of th low-pass fltr n th currnt fd-back, whch can b optonal dpndng on th currnt masurmnt stratgy. A connnt way to analyz th prformanc of APF s through th us of frquncy rspons functon (FRF) analyss. Snc th APF arabls ar modld usng complx ctor quantts, whch can rotat both forward and backwards, both post as wll as ngat frquncs nd to b consdrd for th transformaton s=jω. Fg. 4-a and 4-b show th magntud of th rsultng FRF for (3) and (4), rspctly. Th currnt rgulator was tund to ach pol/zro cancllaton wth a bandwdth of 4 Hz. Th gans of th controllr bng T = Lf /Rf and = 4 π Lf. It can b obsrd from Fg. 4-a that th FRF has a gan qual to on at frquncy,.., dc, whch mans thy APF wll ha zro rror n th stady-stat. As for th dsturbanc rjcton capablty, currnt harmoncs njctd by non-lnar GHCR_ GHCR_ G HCR _n qdh qdh qdhn qdh lqd lqd lqd lqd K p K p K p G FCR G HCR G FCR G HCR c) Fqd hqd Fqd hqd Fg. 6.- Harmonc currnt rgulator consstng of n paralll-connctd rgulators, currnt rgulator consstng of a fundamntal currnt rgulator G FCR and a harmonc currnt rgulator G HCR wth th nput to both currnt rgulators bng th currnt rror, c) currnt rgulator consstng of a fundamntal currnt rgulator and a harmonc currnt rgulator wth th nput to th harmonc currnt rgulator bng th masurd currnt. loads wll ha typcally ordrs of h=±6, ±, ±8,..., n th synchronous rfrnc fram [7,8]. Complt lmnaton of th harmoncs njctd by th load would rqur a gan qual to zro n (4) at th frquncs corrspondng to thos harmoncs. It can b obsrd from Fg. 4-b that synchronous rfrnc fram PI currnt rgulator prods prfct dsturbanc rjcton at frquncy,.., th fundamntal frquncy but th dsturbanc rjcton capablty dcrass rapdly as th frquncy (harmonc ordr) ncrass. Fg. shows an xampl of th smulatd tm rspons of ths currnt rgulator. For ths smulaton, a constant ln currnt rfrnc consstng of a sngl componnt at a frquncy ω was commandd at t=. s, Fg. -a, and a non-lnar load s connctd btwn t=. s and t=. s, Fg. -b. From ths fgur, both ts capablty to follow wthout sgnfcant rror th currnt command, (whn th load dos not crat harmoncs th ln currnt n Fg. -c prfctly matchs th rfrnc of Fg. -, as wll as th mpact of th currnt harmoncs cratd by th load (whn harmoncs cratd by th load ar prsnt thr s a notcabl dffrnc btwn th commandd and th actual ln currnt). III. HARMONICS CURRENT REGULATOR FOR ACTIVE POWER FILTERS Th prncpls of synchronous rfrnc fram currnt rgulators can b xtndd for th cancllaton of harmoncs cratd by non-lnar loads through th us of harmonc currnt rgulators. In ths concpt, sral currnt rgulators, ach dsgnd to cancl a spcfc harmonc, ar connctd n paralll, Fg. 6a. Sral dffrnt dsgn approachs for th slct harmonc currnt rgulators ha bn proposd [6- ] and wll b dscussd n th followng sub-sctons. A.- Harmonc synchronous currnt rgulators dsgn Whl all harmonc synchronous currnt rgulators dsgns plac a pol on th magnary axs at th frquncy to b canclld, dffrnt optons ha bn proposd for th slcton of th rgulator s zro placmnt. Th zro can b

4 tund to ha th sam magnary componnt as th pol at a dstanc /Th from th magnary axs, wth () bng obtand (s Fg.7a-top). A pur ntgrator can also b usd (6), (Fg. 7b top) [8, ]. Fnally, f th zro s placd on th ral axs n th fundamntal frquncy rfrnc fram, (7) s obtand (Fg. 7c top) [8, ]. It should b notd that th gan h n ()-(7) s normalzd by ddng by th gan of th fundamntal currnt controllr. Ths was don for connnc n th root locus analyss prsntd latr n th papr. G HCR _h = G HCR _h = G HCR _h = qd_h qd qd_h qd qd_h = K ph = K ph = K ph (s /Th j h ω) s j h ω /Th s j h ω (s /Th) s j h ω qd As alrady mntond, non-lnar loads typcally crat harmoncs of ordr h=±6, ±, ±8, n a synchronous rfrnc fram, [7,8]. Implmntng th currnt rgulators n a fundamntal synchronous rfrnc fram has th adantag of allowng smultanous cancllaton of post and ngat squnc harmoncs,.., ±h, wth a sngl rgulator (Fg. 6-. Eq. (8)-() show th rgulators that rsult from ()-(7) whn smultanous cancllaton of th post and ngat squnc componnts s mplmntd, wth th corrspondng pol-zro mappng bng shown n th subplots at th bottom of Fg. 7. G HCR _h = G HCR _h = G HCR _h = qd_h qd qd_h qd qd_h = K ph = K ph = K ph () (6) (7) (s /Th s (h ω) ) s (h ω) (8) (/Th s) s (h ω) (9) (s /Th s) s (h ω) () qd Whl all th thr dsgns shown n Fg. 7 ha th sam harmonc rjcton capablty n th stady-stat, dffrncs xst n thr transnt rspons du to th dffrnt zro placmnt [8, ]. Du to spac rstrctons, th analyss prsntd n ths papr wll b lmtd to th dsgn n Fg. 7a but th followng rsults wr found as part of ths rsarch. Th dsgn n Fg. 7-c was found to ha smlar dynamcs rspons to th dsgn n Fg. 7-a, whl th dsgn n Fg. 7b showd slowr dynamc rspons. /Th /(Th) Post squnc harmonc rgulators Im Im Im hω hω hω R R R /Th Post & ngat squnc harmonc rgulators Im Im Im h ω h ω h ω R R R /T h hω hω hω c) Fg. 7.- Pol-zro confguratons for th harmonc currnt rgulators, shown n a fundamntal frquncy synchronous rfrnc fram Ral Ral Fg. 8.- Root locus, shown n a synchronous rfrnc fram, whn th harmonc currnt rgulator G HCR s fd by th currnt rror (Fg. 6. G HCR consstng of f rgulators to compnsat harmoncs ±6 to ±3 (±3 π rad s to ±8 π rad s ). It should b notd that th ral and magnary axs ar scald dffrntly load qd(jω) harmonc ordr h Fg. 9.- FRF shown n a synchronous rfrnc fram, whn th harmonc currnt rgulator G HCR s fd by th currnt rror, Fg. 6-b. Th rst of condtons ar as dscrbd n Fg. 8. Th dashd ln corrsponds to th FRF for th cas of only a PI currnt rgulator bng usd shown n Fg

5 s lqd tm (s) Fg..- Tm rspons of APF wth harmonc currnt rgulators (smulatd): ln currnt rspons to th currnt command and load currnt shown n Fg. a and b. rspctly, whn th harmonc currnt rgulator G HCR s fd by th currnt rror (Fg. 6. Th rst of condtons ar as dscrbd n Fg. 9. Th command trackng and dsturbanc rjcton transfr functons whn th fundamntal currnt rgulator and th harmonc currnt rgulator ar combnd (Fg. 6 ar () and (), rspctly. l_qd (G FCR G HCR ) G F = () (G FCR G HCR ) G F G LPF l_qd l_qd = () (G FCR G HCR ) G F G LPF load_qd Fg. 8 shows th root locus of () as a functon of, wth th pol-zro pars du to th harmonc currnt rgulators bng radly obsrabl. Th root locus wll b usd latr for th analyss of th dscrt form of th harmonc currnt rgulators. Fg. 9-a and 9-b show th command trackng and dsturbanc rjcton FRF's for ths controllr. It can b obsrd from Fg. 9-b that t fully rjcts ach of th harmoncs ncludd n ts dsgn (gan qual to zro). It can also b obsrd from Fg. 9-a that ncludng th harmonc currnt rgulator rsults n a modfcaton of th command trackng FRF compard to th PI currnt rgulator cas (3), wth a sgnfcant ncras of th magntud at frquncs dffrnt from dc. Fg. shows th smulatd tm rspons of ths currnt rgulator. It can b obsrd that a dramatc mpromnt n th cancllaton of th dstorton ntroducd by th load currnt s achd whn compard to th cas of no harmonc currnt rgulator shown n Fg.. B.- Harmonc currnt rgulator usng synchronous PI currnt controllrs placd n th fdback path Snc th harmonc currnt rgulators ar manly ntndd for dsturbanc rjcton, thy could b placd n th fdback path, Fg. 6-c. Th rsultng command trackng transfr functon for ths controllr s (3), wth th corrspondng FRF bng shown n Fg. -a. l_qd l_qd = G FCR G F (G FCR G HCR ) G F G LPF (3) It can b obsrd from ths fgur that ths controllr has a dc gan qual to on, and thrfor no rror rgulatng th fundamntal currnt. Howr, a notcabl dcras of th s load qd harmonc ordr h tm (s) Fg..- Command trackng FRF shown n a synchronous rfrnc fram, whn th harmonc currnt rgulator G HCR s fd by th masurd ln currnt (Fg. 6c). Tm rspons of APF wth harmonc currnt rgulators (smulatd): ln currnt rspons to th currnt command and load currnt shown n Fg. -a and -b rspctly (Fg. 6c). Th rst of condtons ar as dscrbd n Fg out/load Mark load qd(jω) harmonc ordr h Fg..- Frquncy rspons functon (FRF) shown n a synchronous rfrnc fram, whn th controllrs formng th harmonc currnt rgulator G HCR nclud dampng. Th rst of condtons ar as dscrbd n Fg. 9. FRF gan at frquncs nar dc can b obsrd, rsultng a srous dtroraton of ts command-trackng charactrstcs du to th pols of G HCR bng placd n th fdback path. Th dsturbanc rjcton s th sam as for th dsgn n Fg. 6b,.., (), Fg. 9-b. Fg. -b shows th tm rspons for ths controllr. As xpctd, ths confguraton rjcts th harmoncs njctd by th load but has poor command trackng proprts, whch would srly lmts t us. C.- Harmonc currnt rgulator usng synchronous rfrnc fram PI currnt rgulators wth dampng To allat stablty concrns sn durng th dgtal mplmntaton of harmonc currnt rgulators, harmonc currnt rgulators wth dampng wr dlopd [-]. Th transfr functon n th contnuous doman s shown n (4), whr a nw trm has bn addd to th dnomnator that s tund through th slcton of th qualty factor, Q. G HCR _h = h (s /Th s (h ω) ) s (h ω/q) s (h ω) (4) Th corrspondng FRF's ar obtand substtutng (4) n () and () and ar shown n Fg.. Th nw trm addd n th dnomnator of th currnt rgulator transfr functon mos th pols from th magnary axs towards th lft (stabl) half of th s plan, Fg. 8. Ths rsults n a mor

6 wll dampd systm, but at th prc of not fully cancllng harmoncs cratd by th load, Fg. -b. cancllaton, Th = T / and h/=, (8). Th ffcts of changs n ths gans ar studd by th nd of ths scton. h=6 h= Ral Ral Fg. 3.- Root locus currnt control usng harmonc currnt rgulator. Samplng prod T=.ms. samplng dlay of T and samplng dlay T/ (synchronous samplng), no LPF n th currnt fdback, harmoncs dcoupld h=±6, ±, ±8, ±4 and ±3. (zooms n nxt fgur) Samplng dlay=t, no LPF IV. DISCRETE IMPLEMENTATION OF THE HARMONIC CURRENT REGULATOR All th dscusson and analyss prsntd n th prous scton was for th contnuous tm doman, ths scton analyzs th ssus that ar rlant to th dgtal mplmntaton of harmonc currnt rgulators, and prods a crtra for th slcton of th harmonc currnt rgulator gans n th dscrt doman. A.- Dscrtzaton mthods Harmonc currnt rgulators ar dsgnd to accuratly control spcfc, wll dfnd frquncs. It s thrfor mandatory to us dscrtzaton mthods that xactly match th frquncs ntrst from th contnuous to th dscrt doman. Dtald dscusson on th prncpls and prformanc of dscrtzaton mthods can b found n [4], th Tustn transform wth pr-warpng, wll b usd n ths papr. B.- Currnt masurmnt and samplng Currnt masurmnt and samplng s crtcal for th mplmntaton of harmonc currnt controllrs du to th rlatly hgh frquncy of th harmoncs bng compnsatd. Th currnt fdback path n a dgtal currnt rgulator conssts of a snsor, an optonal low-pass fltr, Fg., and a samplng & A/D conrson dc that ntroducs a dlay n th control []. Th currnt snsors normally ha bandwdths gratr than tns of khz and, thrfor, ha rducd mpact. Dffrnt confguratons for th LPF and samplng ar dscussd n th nxt subscton. C.- Analyss of dscrtzd harmonc currnt rgulators Ths scton analyzs th dscrt harmonc rgulators that rsult from th dscrtzaton of th contnuous dsgns from Scton III. Th Tustn transform wth pr-warpng was usd n all th cass, wth a samplng prod of T=. ms (xcpt whn statd othrws), whch concds wth th swtchng prod of th xprmntal stup. In th analyss prsntd n ths scton only th gans and h ar ntally changd, wth =h. Th rst of gans of th fundamntal and harmonc currnt rgulators ar kpt constant, wth T = Lf /Rf mplmntd for pol-zro Samplng dlay=t/ (synchronous saplng), no LPF c) Samplng dlay=t/ (synchronous saplng), LPF of bandwdth. khz Ral Ral d) Samplng dlay=t/, dampng wth Q factor = Fg. 4.- Zoom of th branchs of th root locus corrspondng to harmoncs h=6 (lft) and h=3 (rght), for thr dffrnt confguratons of th harmoncs currnt rgulator. Fg. 3-a shows th root locus of th rsultng dgtal mplmntaton of th controllr prsntd n Sub-Scton III- A, wth th corrspondng contnuous root locus shown n Fg. 8. Th currnts wr sampld at th bgnnng of th swtchng prod, rsultng n a dlay of on swtchng prod T bfor th oltag command s updatd. Th pol-zro pars lyng on th magnary axs n th Fg. 8 ar sn to l on th unt crcl n Fg. 3-a. Fg. 4-a shows a zoom of th branchs n Fg. 3-a corrspondng to th harmoncs h=6 and h=3, rspctly. Som facts can b obsrd form Fg. 3-a and 4-a. A sgnfcant porton of th branch corrspondng to h=3 ls outsd th unt crcl, manng that th systm wll b unstabl for small alus of. On th othr hand, t can b obsrd from Fg. 3-a that, for larg alus of, thr ar

7 6 T=.8 ms, SD=T/, NO LPF T=. ms, SD=T, NO LPF, wth dampng T=. ms, SD=T/, NO LPF T=. ms, SD=T/, LPF bw= khz T=. ms, SD=T/, LPF bw=. khz 4 T=. ms, SD=T, NO LPF K p 3 K p mn K p max Numbr of harmoncs dcoupld Fg..- Stablty lmts for gan K p s. th numbr of harmoncs pars bng dcoupld ( h=±6; h=±6, ±; ; h=±6 to ±36;) for dffrnt confguratons of th harmonc currnt rgulator: T stands for samplng prod, whch s qual to th swtchng prod, SD stands for samplng dlay (T/ corrsponds to synchronous samplng, T for a complt samplng prod dlay), and LPF for th low-pass fltr n th currnt masurmnt. In all th cass, th harmoncs currnt rgulators wr dscrtzd usng Tustn wth prwarpng, and K ph =K p. two branchs that l outsd of th unt crcl. It can b concludd, thrfor, that thr s a lmtd rang of gans mn<<max for whch th harmonc currnt rgulator s stabl. Fg. ( ) shows th alus of mn and max that prod stabl opraton as a functon of th numbr of harmoncs bng compnsatd. It can b obsrd from th fgur th rang of gans for that prod stabl opraton rducs as th numbr of harmoncs bng compnsatd ncrass, wth ths rducng practcally to zro,.., mn max, for th cas whn harmoncs ar ncludd n th currnt rgulator dsgn. Ths lmts th numbr of harmoncs that can b compnsatd to 4. Fg. 3-b and 4-b show th root locus whn th samplng dlay s rducd from T to T/ by samplng th currnts n th mddl of th swtchng prod. Although only slght dffrncs ar obsrd btwn th root locus n Fg. 3-b and Fg. 3-a, notcabl dffrncs xst whn th trajctory of th branch corrspondng to h=3 s zoomd n Fg. 4b, whch shows a sgnfcantly ncrasd rang of alus of that prod stabl opraton. Th bnfts of rducng th samplng dlay can also b sn n Fg. ( ) through th ncrasd rang of alus that allow stabl opraton, ncludng compnsaton up to th 6 th harmonc. In th two cass dscussd so far, thr was no low-pass fltr n th currnt fdback. Th ffcts of a low-pass fltr can b obsrd from th root locus n Fg. 4-c. Th fltr can b sn to ha an ffct smlar to th dlay analyzd n Fg. 4-a, du to th fltr s lag charactrstc. Fg. shows th rang of gans that prod stabl opraton for th cas of two dffrnt bandwdths for th low-pass fltr,. khz ( ) and khz ( ). It can b notcd from ths fgur that a lowpass fltr wth a bandwdth as hgh as khz (half of th swtchng frquncy) has a sbl mpact on th stablty lmts of th harmonc currnt rgulator.. T/Th.6.8 h/.. T/Th.6.8 h/ ε trak ε dst Fg. 6.- Smulaton rsults. command trackng and dsturbanc rjcton rrors as a functon of T h.and K ph.gans, for th cas of a harmonc currnt rgulator usng rsonant controllrs (8) to compnsat for harmoncs h=±6 up to h=±3, wth a samplng dlay of T/ and no low-pass fltr. Th fundamntal currnt rgulator gans and T wr tund for pol-zro cancllaton and a bandwdth of 4 Hz. Fg. 4-d shows th branchs corrspondng to th harmoncs h=6 and h=3 for th cas of th harmonc rgulator wth dampng dscrbd n Scton III-C. Th ffcts of usng dampng ar dnt n th branch for th cas of h=3, whch always ls wthn th unt crcl, manng that thr s no rsk of nstablty du to th gans bng too small. Th rang of alus of that allow stabl opraton bng shown n Fg. ( ). Fnally, th rang of gans that allow stabl opraton, and consquntly, th numbr of harmoncs that can b compnsatd, can b ncrasd by dcrasng th samplng prod T. Fg. ( ) shows an xampl of th mpromnt that s obtand whn T s rducd from to.8 ms (th swtchng frquncy ncrass from to. khz). It should b notd, howr, that ncrasng th swtchng frquncy can ha mportant mplcatons on th nrtr losss, as wll as on th nduct fltr dsgn, that nd to b carfully consdrd. E.- Dlay compnsaton It has alrady bn shown n th prous sctons that th dlay ntrnsc to th dgtal mplmntaton of th currnt controllrs has an adrs mpact on thr prformanc. Th mportanc of ths ffct bcoms mor rlant as th rgulator harmonc ordr ncrass, and also dpnds on th currnt rgulator dsgn [8] and dscrtzaton mthod [4], whch ntually dpnds of th placmnt of th zros of th dscrt currnt rgulator. Dlay compnsaton has bn wdly analyzd n th ltratur, and s not addrssd n ths papr. Proposd mthods nclud adancng th angl of th currnt rgulator output oltag [7], as wll as, modfcatons to th dscrt currnt rgulator [4,6]. Slcton of th xact dlay amount to compnsat s oftn mad ad hoc,.g., two samplng prods compnsaton s rcommndd n [6] and [7], howr, no clar crtra for ths slcton s gn. It should fnally b notd that compnsaton mthods normally assum stady-stat opraton, whr t s possbl to calculat th phas shft that th dlay wll produc for ach harmonc. Howr, ffctnss of th compnsaton wll b rducd.

8 durng transnts [6]. F.-Harmonc currnt rgulator gan slcton In th prous analyss, th controllr s gans Th, T and h wr thr kpt constant or had a fxd rlatonshp to, whch was ard for th controllr tunng. Th nflunc of changs n ths paramtrs on th prformanc of th harmonc currnt rgulators s dscussd n ths subscton. Th ntgral tm constant of th fundamntal currnt rgulator, T, s oftn slctd to cancl th nduct fltr dynamcs (or nar that alu), wth th proportonal gan,, slctd to obtan th dsrd closd-loop bandwdth. Usng ths crtra, th gans, h and Th, of th harmonc currnt rgulators stll nd to b slctd. Gn th hgh ordr of th rsultng transfrs functons nold, t s not possbl to obtan mtrcs for th dynamc rspons (sttlng tm, orshoot, ) from analytcal solutons. Numrcal aluatons wr usd nstad. Two mtrcs wr dfnd to aluat th dynamc prformanc of th command trackng and dsturbanc rjcton, () and (6). In ths quatons, t was assumd that th systm was xctd at t=, wth t bng a tm larg nough so that th ln currnt has rachd stady-stat n all th cass. Th mtrc ε track ntgrats or a tm t th dffrnc btwn th rspons to changs n th fundamntal currnt of th harmonc currnt rgulator () mnus th dal rspons (3),.., th ln currnt for th cas that no harmonc currnt rgulator s prsnt. Ths mtrc masurs th mpact that th harmoncs currnt rgulator has on th command trackng proprts of th APF. Th mtrc ε dst ntgrats th ln currnt or a tm t, aftr th load starts cratng harmoncs at t=. ε trak = t t (lqds lqds dal) dt () ε dst = t t lqds dt (6) Fg. 6 shows ε trak and ε dst as a functon of gans h and Th. Both mtrcs wr normalzd by ddng by th largst alu. Th followng conclusons can b rachd form th fgur. Th gan Th has a small mpact on th command trackng rror (Fg. 6a and 7), but has a notcabl mpact on th dsturbanc rjcton capablty. Ths suggsts usng smallr alus of Th. (ncras of T/Th). Valus too small of Th can mak th harmonc currnt rgulator too snst to nos, alus of Th Th/ wr found adquat. Valus of h clos to prod a good trad-off btwn command trackng and dsturbanc rjcton rrors. Incrasng th alu of h mpros th dsturbanc rjcton capablty, but at th prc of an ncrasd command trackng rror and makng th currnt rgulator mor snst to nos. In all th prous analyss, th sam gans, h and Th, wr usd for ach nddual harmonc currnt rgulator (8) (). It s also possbl to us dffrnt gans for ach harmonc bng compnsatd. If ths opton s chosn, h should ncras proportonally to harmonc compnsatd, as hghr harmoncs rqur hghr gans to gt nto th stabl rgon of th z-plan. Howr, from th analyss carrr out, no rlant dffrncs wr obsrd usng dffrnt h and Th gans wth rspct to th cas of usng th sam gans for all controllrs. V. EXPERIMENTAL RESULTS Ths scton prsnts xprmntal rsults for th analyss from ths papr. Th tst bnch uss 7 A IGBT's wth a swtchng frquncy of khz (T=. ms) and a DC bus oltag of Vdc=7 V. An RL nduct fltr was usd, wth Lf =. mh, Rf = mω, and a maxmum currnt of 3 A (rms). Th APF s connctd to a ln oltag of Hz, 4 V (rms ln-ln). Th APF control was mplmntd on a TMS3F833 DSP. Synchronous samplng was usd, wth a samplng dlay of T/. Du to nos problms, a frst-ordr, low-pass fltr, wth a cut-off frquncy of. khz, was usd. Accordng to th analyss prsntd n ths papr, harmoncs h=±6, ±, ±8 and ±4 could b compnsatd, Fg.. l u, l, l w tm (s) Fg. 7.- Command trackng rspons to a stp un th q-axs ln currnt commnd. Harmonc currnt rgulator consstng of h=±6, ±, ±8 harmoncs. l u, l, l w tm (s) Fg. 8.- Ln currnts whn th load s an uncontrolld rctfr, th harmonc currnt rgulator bng connctd at t=.7 s. Harmonc currnt rgulator compnsats for h=±6, ±, ±8 and ±4 harmoncs. l u, l, l w tm (s) Fg. 9.- Ln currnts whn th APF was act whn an uncontrolld rctfr s connctd at t=.3. Th harmonc currnt rgulator s confgurd as dscrbd n Fg. 8. Th fundamntal currnt was tund for a 4 Hz bandwdth. Th gans usd for th harmonc currnt rgulator wr h=.7 and Th=T/ (s Fg. 6). Fg. 6 shows th command trackng rspons to stp of A n th q-axs (ract currnt) ln currnt command, wth th harmonc currnt rgulator consstng of four harmonc controllrs at h=±6, ±, ±8 and ±4. It can b obsrd

9 from th fgur that th APF shows fast rspons wth th systm bng prfctly stabl. Fg. 8 shows th dsturbanc rjcton capablty of th HCR whn th load s an uncontrolld rctfr and th APF s connctd. Stabl opraton and good dynamc bhaor ar obsrd, confrmng th corrctnss of th gans tunng. Fnally, Fg. 9 shows th dsturbanc rjcton capablty of th HCR whn t s act and an uncontrolld rctfr s connctd to th ln, th APF s sn to ract th to th dstorton rstorng narly snusodal ln currnts. V. CONCLUSIONS Th dsgn and tunng of harmonc currnt rgulators for APF has bn dscussd n ths papr. Harmonc currnt rgulators ar a good soluton to prfctly cancl slctd harmoncs njctd by non-lnar loads. Howr, slcton of th gans for th controllrs, to guarant stabl opraton and adquat dynamc prformanc s a challng, bng oftn mad ad hoc. Furthrmor, t can strongly dpnd on crcut confguraton optons. Stablty analyss of harmonc currnt rgulators has bn prsntd n ths papr. From ths analyss, th mpact that dffrnt mplmntaton ssus lk th samplng stratgy, swtchng prod and us of fltrs, has on th maxmum numbr of harmoncs that can b canclld by th harmonc currnt rgulator, as wll as th rang of gans that can b usd to guarant th stablty of th systm, ha bn stablshd. Exprmntal rsults ha bn prodd to support th analyss. REFERENCES [] Mattall, P.; Marafao, F.P., Rptt-basd control for slct harmonc compnsaton n act powr fltrs," IEEE Trans. Ind. Elctr., ol., no., pp. 8-4, Oct. 4. [] G. Escobar; A. A. Valdz; J. Lya-Ramos; P. Mattall, Rptt- Basd Controllr for a UPS Inrtr to Compnsat Unbalanc and Harmonc Dstorton," IEEE Trans. Ind. Elctr., ol.4, no., pp. 4-, Fb. 7. [3] Garca-Crrada, A.; Pnzon-Ardla, O.; Flu-Batll, V.; Roncro- Sanchz, P.; Garca-Gonzalz, P., Applcaton of a Rptt Controllr for a Thr-Phas Act Powr Fltr," IEEE Trans. Powr Elctr., ol., no., pp , Jan. 7. [4] Ozkaya, H.; Snturk, O.S.; Haa, A.M., Prformanc nhancmnt and comparson of dscrt tm currnt rgulators for paralll act fltrs," 7 Europan Confrnc on Powr Elctroncs and Applcatons, pp.-, - Spt. 7. [] Ozkaya, H.; Slcuk Slnturk, O.; Haa, A.M., Prformanc Enhancmnt of Dscrt Tm Hystrss Currnt Rgulators and Comparson wth Lnar Currnt Rgulators for Paralll Act Fltrs," Elctrc Machns & Drs Confrnc, 7. IEMDC '7, ol., pp.8-87, 3- May 7. [6] Mattall, P., "A closd-loop slct harmonc compnsaton for act fltrs," IEEE Trans. Ind. Appl., ol.37, no., pp.8-89, Jan/Fb. [7] Lascu, C.; Asmnoa, L.; Bolda, I.; Blaabjrg, F., Frquncy Rspons Analyss of Currnt Controllrs for Slct Harmonc Compnsaton n Act Powr Fltrs," IEEE Trans. Ind. Elctr., ol. 6, no., pp , Fb. 9. [8] Lascu, C.; Asmnoa, L.; Bolda, I.; Blaabjrg, F., "Hgh Prformanc Currnt Controllr for Slct Harmonc Compnsaton n Act Powr Fltrs," IEEE Trans. Powr Elctr., ol., no., pp.86-83, Spt. 7. [9] L, S.J.; Sul, S.K., A harmonc rfrnc fram basd currnt controllr for act fltr," Appld Powr Elctroncs Confrnc and Exposton,. APEC., ol., no., pp ol.,. [] Lnwar, W.; Sumnr, M.; Zanchtta, P.; Cula, M., A Hgh Prformanc Harmonc Currnt Control for Shunt Act Fltrs Basd on Rsonant Compnsators," IEEE Industral Elctroncs, IECON 6-3nd Annual Confrnc on, ol., no., pp.9-4, 6- No. 6. [] Lnwar, W.; Sumnr, M.; Zanchtta, P., Dsgn and Analyss of Hgh Prformanc Currnt Control for Shunt Act Fltrs," 3rd IET Intrnatonal Confrnc on Powr Elctroncs, Machns and Drs, 6, pp.9-9, Mar. 6. [] Ladsa, C.; Zanchtta, P.; Sumnr, M., Improd Voltag Harmonc Control for Shunt Act Powr Fltrs Usng Multpl Rfrnc Frams," IEEE Intrnatonal Symposum on Industral Elctroncs. ISIE 7, pp , Jun 7. [3] F.. Brz, M.W. Dgnr and R.D. Lornz, "Analyss and Dsgn of Currnt Rgulators Usng Complx Vctors", IEEE Trans. on Ind.. Appl, ol.36, no.3, pp.87-8, May/Jun.. [4] A.G. Yps, F.D. Frjdo, J.Doal-Gandoy, O. Lopz, J. Malar, and P. Frnandz-Comsaña, "Effcts of Dscrtzaton Mthods on th Prformanc of Rsonant Controllrs", IEEE Trans. on Powr Elctr., ol., no.7, pp. 69-7, Jul.. [] L. Rotrgus-Lmong, R. Bojo, G. Gra, A. Tncon, "Comparng th Prformanc of Dgtal Sgnal Procssor-Basd currnt Controllrs for Thr-phas Act Powr Fltrs", IEEE Ind. Elctr. Magazn, ol.3, no., pp. -3, March 9. [6] Xaomng Yuan; Mrk, W.; Stmmlr, H.; Allmlng, J.; "Statonaryfram gnralzd ntgrators for currnt control of act powr fltrs wth zro stady-stat rror for currnt harmoncs of concrn undr unbalancd and dstortd opratng condtons," IEEE Trans. on Ind. Appl., ol.38, no., pp.3-3, Mar/Apr [7] Bojo, R.I.; Gra, G.; Bostan, V.; Gurrro, M.; Farna, F.; Profumo, F.;, "Currnt control stratgy for powr condtonrs usng snusodal sgnal ntgrators n synchronous rfrnc fram," IEEE Trans. Powr Elctr., ol., no.6, pp. 4-4, No.