Aalborg Universitet. Published in: Energies. DOI (link to publication from Publisher): /en Publication date: 2016

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1 Aalborg Univritt A Rptitiv Control Schm Aimd at Compnating th 6k + Harmonic for a Thr-Pha Hybrid Activ Filtr Luo, Zhaoxu; Su, Mi; Yang, Jian; Sun, Yao; Hou, Xiaochao; Gurrro, Jop M. Publihd in: Enrgi DOI (link to publication from Publihr):.339/n9787 Publication dat: 26 Documnt Vrion Early vrion, alo known a pr-print Link to publication from Aalborg Univrity Citation for publihd vrion (APA): Luo, Z., Su, M., Yang, J., Sun, Y., Hou, X., & Gurrro, J. M. (26). A Rptitiv Control Schm Aimd at Compnating th 6k + Harmonic for a Thr-Pha Hybrid Activ Filtr. Enrgi, 9(), [787]. DOI:.339/n9787 Gnral right Copyright and moral right for th publication mad accibl in th public portal ar rtaind by th author and/or othr copyright ownr and it i a condition of accing publication that ur rcogni and abid by th lgal rquirmnt aociatd with th right.? Ur may download and print on copy of any publication from th public portal for th purpo of privat tudy or rarch.? You may not furthr ditribut th matrial or u it for any profit-making activity or commrcial gain? You may frly ditribut th URL idntifying th publication in th public portal? Tak down policy If you bliv that thi documnt brach copyright pla contact u at vbn@aub.aau.dk providing dtail, and w will rmov acc to th work immdiatly and invtigat your claim. Downloadd from vbn.aau.dk on: ptmbr 22, 28

2 Articl A rptitiv control chm aiming to compnat th 6k+ harmonic for thr-pha hybrid activ filtr Zhaoxu Luo, Mi Su, Jian Yang, *, Yao Sun, Xiaochao Hou, and Jop M. Gurrro 2 School of Information Scinc and Enginring, Cntral South Univrity, Changha 483, China; luozhaoxu@cu.du.cn (Z.L.); umicu@cu.du.cn (M.S.); 2 Dpartmnt of Enrgy Tchnology, Aalborg Univrity, DK-922 Aalborg Eat, Dnmark; oz@t.aau.dk * Corrpondnc: ian.yang@cu.du.cn (J.Y.); Tl.: Abtract: Th traditional rptitiv controllr ha rlativly wor tability and poor tranint prformanc du to th fact that it gnrat infinit gain at all th intgr multipl of th fundamntal frquncy, and it control action i potpond by on fundamntal priod (T). To improv th diadvantag, many rptitiv controllr with rducd dlay tim hav bn propod, which can lctivly compnat th odd harmonic or 6k± harmonic with dlay tim rducd to T/2 and T/3,rpctivly. To furthr tudy in thi ara, thi papr propo an improvd rptitiv chm implmntd in tationary rfrnc fram, which only compnat th 6k+ harmonic (.g. -5, +7, -, +3) in thr-pha ytm and rduc th tim dlay to T/6. So compard with th arlir rducd dlay tim rptitiv controllr, th robutn and tranint prformanc i furthr improvd, th wat of control ffort i rducd, and th poibility of amplifying and vn incting any harmonic noi into ytm i avoidd to th gratt xtnt. Morovr, th propod rptitiv chm i ud in th control of a thr-pha hybrid activ powr filtr. Th xprimntal rult validat th ffctivn of th propod rptitiv control chm. Kyword: rptitiv control; hybrid activ powr filtr; powr quality; harmonic compnation; PACS: J. Introduction Rcntly, du to th widprad application of ditributd gnration, adutabl pd driv,uncontrolld AC/DC rctifir, and othr nonlinar load, th harmonic pollution in powr ytm i gtting mor and mor riou. Th paiv powr filtr (PPF) and activ powr filtr (APF) ar th two common olution applid to mitigat th harmonic [-2]. PPF hav th advantag of low-cot and high-fficincy. Howvr, thy alo hav om inhrnt drawback. Thir compnation charactritic ar trongly influncd by upply impdanc and thy ar highly ucptibl to ri and paralll ronanc with th upply and load impdanc. Th APF,which ar bad on th powr lctronic, can ovrcom abov drawback of PPF [3-4]. Additionally, APF ar mor flxibl and fficint compard with PPF. But, pur APF uually rquir a PWM invrtr with larg kilovoltampr (KVA) rating. Thu, thy do not contitut a cot-ffctiv harmonic filtring olution for nonlinar load abov 5- kw[5-6]. To addr thi iu, hybrid activ powr filtr (HAPF) hav bn dvlopd, which ar compod of mall ratd APF and PPF in diffrnt configuration. Among th variou viabl hybrid activ filtr topologi, paralll hybrid activ filtr prnt a cot-ffctiv olution for harmonic filtring and ractiv powr compnation of high powr nonlinar indutrial load, du to mall rating of th activ filtr 2% 3% of load KVA rating [6]. Thu, thy hav attractd incraing attntion [7-]. Enrgi26,9,x; doi:

3 Enrgi 26, 9, x 2 of 8 Among variou control tratgi, th rptitiv control, a a kind of control mthod bad on intrnal modl principl, can accuratly track th priodic ignal or rct priodic intrfrnc. Hnc it i widly ud in harmonic compnation chm for activ filtr [2-5]. Th traditional rptitiv control tchniqu can gnrat infinit gain at all th intgr multipl of th fundamntal frquncy, including th odd, vn harmonic and dc componnt. Howvr, in mot ca, th introduction of high gain for all frqunci i not ncary, a it could wat control ffort and rduc th ytm robutn without improving ytm prformanc, vn amplify irrlvant ignal and rinct ditortion to ytm [6]. Morovr, th control action of traditional rptitiv controllr i potpond by on fundamntal priod (T), hnc th tranint prformanc i poorr. To improv th drawback abov, and conidring th fact that vn harmonic componnt do not rgularly appar in powr ytm, litratur [6-7] propo a rptitiv control chm aiming at compnating only th odd harmonic. It u a ngativ fdback array intad of th uual poitiv fdback in th traditional rptitiv controllr. Manwhil, th dlay tim of control action i rducd to T/2. A a qunc, th control prformanc i improvd and th convrgnc rat i nhancd. Furthrmor, among th odd harmonic, th group of 6k ± ( k =, ±, ± 2 ) harmonic componnt in lctric indutry ar dominatd du to th wid u of uncontrolld rctifir and ix-pul convrtr. Thu, many improvd rptitiv control chm aiming at compnating 6k ± harmonic hav bn dvlopd [8-2]. For intanc, in [8] a rptitiv control chm bad on th fdback array of two dlay lin plu a fdforward path i prntd, which can only compnat 6k ± harmonic and rduc dlay tim to T/3; In [9], th author propo a 6k ± rptitiv control chm in thr-pha ynchronou rfrnc fram (SRF). It ha an advantag of T/6 dlay tim. Howvr, it nd complx coordinat tranformation and much mor calculation in both poitiv-rotating and ngativ-rotating SRF. Conidring in thr-pha powr ytm, harmonic of th am frquncy can b dcompod into poitiv qunc, ngativ qunc and zro qunc. Gnrally paking, a normal balancd thr-pha ytm mainly contain 6k+ harmonic (uch a -5, +7, -, +3), and rarly contain 6k- harmonic (uch a +5, -7, +, -3). For thi raon, thi papr propo a rptitiv control chm aiming at compnating th 6k+ harmonic implmntd in thr-pha tationary rfrnc fram with T/6 dlay tim. So that th tranint prformanc i furthr improvd. Th 6k+ rptitiv controllr i xprd with complx-vctor notation, o that th dual-input/dual-output control ytm (in th αβ rfrnc fram) can b implifid into on ingl-input/ingl-output ytm. Manwhil, th gnral dign mthod of Lk+M rptitiv controllr i alo introducd, with which a rptitiv controllr aiming at compnating Lk+M harmonic can b aily dducd. Morovr, taking th tranformrl paralll hybrid activ filtr a controlld obct, a harmonic compnating control ytm bad on th propod 6k + rptitiv control chm i prntd. Finally, th xprimntal rult validat th ffctivn of th 6k + rptitiv control chm. 2. Sytm tructur and mathmatical modling of HAPF 2.. Topological tructur analyi

4 Enrgi 26, 9, x 3 of 8 Th topology of th tranformrl paralll hybrid activ filtr i hown in Figur. It conit of a LC paiv filtr and a thr-pha voltag ourc invrtr (VSI). Th purpo of intalling th LC filtr ar: ) to provid ractiv powr compnating and aborb om harmonic; 2) to utain fundamntal voltag at th point of common coupling (PCC). And th activ filtr (VSI) i rponibl for improving th filtring charactritic of paiv filtr and avoiding th undirabl ronanc with th grid. To minimiz it own KVA rating, VSI don't participat in ractiv compnation, and th grid voltag i almot fully droppd on th capacitor in LC filtr. Thu th fundamntal voltag utaind by VSI i mall. So that th dc bu voltag rating of VSI can b t vry low, th KVA rating and powr lo ar rducd gratly. Du to th prnc of VSI, LC filtr i not ncary to b accuratly tund at a crtain harmonic frquncy. Th dign obctiv of LC filtr i to offr a lowt poibl impdanc path for incting harmonic currnt, on th prmi of nuring ractiv powr compnating. Grid i a v a PCC i la i b i c v b v c i lb i lc + C v Ca + v + v Cb Cc i c LR, Sc v c Sb Nonlinar load S a idc i b v b Cdc vdc i a v a Sc Sb S a Figur. Topology of paralll hybrid activ powr filtr 2.2. Mathmatical modling According to Figur, th mathmatical modl in tat-pac rprntation for th ytm i formulatd a di dt di dt di dt a L = va + vca Ria + va b L = vb + vcb Rib + vb c L = vc + vcc Ric + vc () dv C dt dv C dt dv C dt Ca Cb Cc = i a = i b = i c (2) dvdc Cdc = idc = ( Si a a + Si b b + Si a a ) (3) dt

5 Enrgi 26, 9, x 4 of 8 whr S a, S b and S c ar witching function dfind by, ( whn Sx on, Sx off ) Sx = ( x= abc,, ), ( whn Sx off, Sx on) (4) 2. Sytm control According to (-2), it can b infrrd that th tat quation of output currnt i ( x= abc,, ) i a cond-ordr diffrntial quation. If th output currnt control i implmntd in dq ynchronou rfrnc fram, it nd to ampl and fd back th AC capacitor voltag v ( x= abc,, ) to achiv dcoupling control btwn d-axi and q-axi. Thrfor, in thi papr, th output currnt control i implmntd in αβ tationary fram, which ha th advantag of no nd of complx dcoupling control and AC capacitor voltag ampling. Th ovrall ytm control diagram i hown in Figur 2, it i mainly compod of dc-link voltag control and harmonic currnt tacking control. In thi figur, Bpf() i a band-pa filtr to xtract th fundamntal frquncy componnt of input ignal, and it xprion i givn by γω Bpf () = (5) γω + ω whr ω i th grid frquncy; γ i th control cofficint of paband width and γ >. x Cx * V dc Dc-link voltag control PI V fq d T q q (com from Pha-lockd loop) dq/ αβ v f ( αβ ) v( αβ ) PWM modulation S α S α S c i l ( αβc ) v dc harmonic currnt rfrnc i αβc αβ l( αβ ) - Bpf () ih( αβ ) vh( αβ ) Rptitiv controllr i( αβc) i αβc αβ ( αβ ) ih( αβ ) - Bpf () harmonic currnt tracking inctd harmonic currnt Figur 2. Ovrall block diagram of ytm control 2.. DC-link voltag tabiliation mthod Auming that th VSI don't provid ractiv powr compnation for th load and only aborb activ powr from grid to maintain it powr lo. According to th powr conrvation principl, thr i dv dc P = C v + P (6) in dc dc dt lo

6 Enrgi 26, 9, x 5 of 8 whr and dv P in i th activ powr aborbd from gird, dc dc dt P lo i th powr lo of invrtr. It can b infrrd from (6) that if P lo diturbanc, dc C v i th powr of th dc-link capacitor i rgardd a a v dc could b controlld by aduting P in. In th dq ynchronou rfrnc fram (grid voltag orintation), th activ powr by P in and ractiv powr P Q in in VdVfq = X V V V + V = X 2 2 fd d fd fq Q in aborbd by VSI can b givn whr X dnot th fundamntal frquncy impdanc of LC filtr; V d i th d-axi componnt of grid voltag ( V q = ); V fq, V fq ar th d-axi and q-axi componnt of VSI output fundamntal voltag, rpctivly. It can b infrrd from (7) that P in i only rgulatd by (7) V fq. Gnrally, V fq i mall. Thu, Q could b achivd whn V fq i t to. Thu, th dc-link voltag rgulator can b dignd in a (8), and th corrponding bod diagram i hown in Figur 2. Vfq = ( kp + ki )( Vdc vdc ) V fd = (8) whr * V dc i th ratd valu of dc-link voltag; 2.2. Harmonic currnt tracking control k p, k i ar th paramtr of PI controllr. Harmonic currnt tracking control i th important part of ytm control, which contribut dirctly to th prformanc of harmonic compnating. Th block diagram of currnt control i hown in Figur 2. Conidring th ca that th thr-pha load currnt mainly contain 6k+ harmonic, thi papr prnt a 6k+ rptitiv control chm to compnat th harmonic. Th dtail thortical drivation, analyi and dign of propod 6k+ rptitiv controllr i givn in th nxt ction. 3. 6k+ rptitiv control chm 3.. intrnal modl of 6k+ rptitiv controllr Firtly, th intrnal modl of th wllknown traditional rptitiv controllr i givn by T RCt () = (9) T T whr i th priodic tim dlay unit, and T i th fundamntal priod, i.., T = 2πω. By tting th dnominator in (9) qual to zro, it can b obtaind that T whr = kω ( k =, ±, ± 2, 2, ± ) () pk pk i th pol of (9). Sn from (), it i clar that th traditional rptitiv controllr ha an infinit numbr of pol locatd at kω, which i th raon traditional rptitiv controllr ha ronant pak at vry intgral multipl of fundrmntal frquncy ω.

7 Enrgi 26, 9, x 6 of 8 In ordr to mak rptitiv controllr with pol only locatd at a lctd group of harmonic frqunci, a nw intrnal modl i nd to b tructurd. Aum th ordr h of th harmonic mt th rul: h = Lk + M () whr L, M ar intrgr, and L i not qual to zro. Thn, th pol of th nw intrnal modl hould b locatd at = ( Lk + M ) ω ( k =, ±, ± 2, 2, ± ) (2) pk Morovr, to nhanc th frquncy lctivity, an infinit numbr of zro of th nw intrnal modl that locatd in th midpoint btwn two concutiv pol ar introducd a = ( Lk + M +.5 L) ω ( k =, ±, ± 2, 2, ± ) (3) zk Th zro bring anothr bnfit that allowing biggr gain with improvd prformanc. In ordr to atify (2-3), th gnral intrnal modl for Lk + M harmonic can b tructurd a RC () = ( ω ( M+.5 L)) T L g ( ωm) T L Aftr th ubtitution of L=6 and M= in (4), th 6k+ intrnal modl i givn by + RC() = π T 3 6 π T 3 6 Comparing (5) with th traditional intrnal modl givn by (9), it can b found that th dlay tim of 6k+ intrnal modl i rducd to T/6, which man a much fatr dynamic rpon. what' mor, it hould b notd that th 6k+ intrnal modl i xprd uing th complx-vctor π notation, a it contain th complx cofficint 3. A a conqunc, th input ignal of RC () i rquird to b a complx vctor. Th block of th propod 6k+ intrnal modl i hown in Figur 3. (4) (5) 3 () T 6 y() (a)

8 Enrgi 26, 9, x 7 of 8 () y () 2 T T 6 () y () 2 (b) Figur 3. Block diagram of th 6k+ rptitiv controllr intrnal modl: (a) complx-vctor notation;(b) calar notation Auming that th fundrmntal frqucny f = 5 Hz, i.., T =.2, th bod plot of th 6k+ rptitiv controllr intrnal modl i hown in Figur 4. A xpctd, th amplitud-frquncy rpon curv how that 6k+ intrnal modl ha ronant pak that locatd at frquncy multipl 6k+ of 5 Hz (5, -25, 35, -55, 65 Hz...), and ha notch that locatd at frquncy multipl 6k+4 of 5 Hz (-, 2, -4, 5 Hz...). Th pha-frquncy rpon curv how th pha hift i boundd btwn 9 and - 9 dgr, and zro at th pak and notch. Magtitud (db) Pha (dg) Frquncy (Hz) Figur 4. Bod plot of th propod rptitiv controllr intrnal modl Fractional dlay compnation In a practical application, th implmntation of rptitiv control chm i uually prformd T in th digital form. Uing th tranformation z =, (5) can b dicrtizd and it xprion in dicrt tim domain i givn by RC( z) T z= + = π N 3 6 z π N 3 6 z (6)

9 Enrgi 26, 9, x 8 of 8 whr T i th ampling priod, and N = T T (th numbr of ampl pr fundamntal priod ). In mot ca, th ampling frquncy f ( f = T ) i a fixd rat(.g. khz,2.8 khz, 2 khz), and th grid frqucny dtctd by PLL i variab in a crtain rang (.g. 49~5Hz). Thu, N i uually non-intrgr. 6 Lt N 6 z z z z ( D+ d) D d = = (7) whr D and d ar th intgral and fractional part of N 6, rpctivly. N 6 In a common implmntation, z i approximatly tratd a and prformd by d rrving D mmory location, with th fractional ordr part z nglctd. But, thi will cau th ronant pak to dviat from th harmonic frqunci. A a conqunc, th harmonic compration prformanc could b dgradd. d To addr thi problm, fractional dlay (FD) filtr hav bn ud a approximation of z. Th magnitud-frquncy and pha-frquncy charactritic of d z = d z = dωt D z d z can b givn by Thu, it rquir that FD filtr hould hav a unit gain and linar pha in th low-middl frqunci, and achiv a high attnuation rat in th high frqunci to nhanc th ytm tablity. In th condition of z < (i.. π (3 T ) < ω < π (3 T ) ), with th u of th Taylor xpanion, d z can b xprd a d d dd ( ) ( d n+ ) n z = ( + z ) = + d( z ) + + ( z ) (9) n! Spcifically, choo th firt-ordr Taylor xpanion of Fd( z) d dz Figur 5 how th bod plot of Fd( z ), with d z a a FD filtr, that i (8) = + (2) T = µ, d =.2,.5 and.8, rpctivly. It can b n that Fd( z) ha th low-pa filtr natur. In low frqucni, Fd( z) ha a wll linar pha approximatd to th idal valu. Howvr, th main diadvantag of Fd( z ) ar that th cutoff frqucny i too high (gratr than 3 Hz), and it chang with th valu of d. Only whn d=.5, Fd( z ) achiv th lowt cutoff frqucny and bt linar pha.

10 Enrgi 26, 9, x 9 of 8 Magnitud (db) Pha (dg) db idal valu idal valu d =.8 d =.2,.8 d =.5 d =.5 d =.2 idal valu Frquncy (Hz) Figur 5. Bod plot of Fd(z) To ovrcom abov iu, thi papr prnt a FD filtr Qz ( ) by cacading Fd(z) with a zro-pha digital low-pa filtr, i.., Q( z) = Fd( z) M ( z) (2) whr M( z) i th zro-pha digital low-pa filtr ud to low th cut-off frquncy and Incra th attnuation rat in high frqucni. It xprion i givn a ( ) ( ) n M z = az+ a + az (22) whr a, a > and a + 2a = ; n i th ordr of filtr. Although Qz ( ) i noncaual, th tim dlay trm fractional dlay compation, (6) hould b rvid a RC( z) = π + 3 ( ) Qzz π 3 ( ) Qzz D D D z mak it applicabl. Aftr th (23) 3.3. Dign of 6k+ rptitiv controllr Figur 6 how th block diagram of th harmonic currnt tracking control. Thi papr adopt a plug-in rptiv controllr tructur in th control loop,whr th PI controllr i ud to nhanc th tability and improv dynamic rpon, and th rptiv controllr i ud to liminat th tady-tat rror. * i h PI(z) RC(z) G ( z) f P () z k Pz () c v h i h Figur 6. Block diagram of harmonic currnt tracking control In Figur 6, P(z) i th plant of currnt control. According to (-2), it xprion in continuou domain can b obtaind a

11 Enrgi 26, 9, x of 8 P () = L + R + C (24) Obviouly, P() i a cond-ordr ytm. To modify th charactritic of P(), a mthod of output currnt tatu fdback i ud. Accoding to Figur 6, th modifid plant xprion i givn by kp c () P () = = + kp () L ( kc + R) c + + k k k C c c c (25) Equation (25) rval that P () can b viwd a th R bcom to ( kc R ), whil L, C bcom k c and k c tim of it original valu in P(), rpctivly. Th bod plot of P() and P () ar hown in Figur 7, with L = 3 mh, C = 9 µ F, R =. Ω. A n, P () ha canclld th ronant pak appard in P(), and prnt th charactritic of a band-pa filtr. Th paband width dpnd on th valu of k c. A largr k c lad to biggr paband width and mallr pha lad/lag. Magnitud (db) P () P ( ) with k c = 3 P ( ) with k c = 6 Pha (dg) P () P ( ) with k c = 3 P ( ) with k c = Frquncy (Hz) Figur 7. Bod plot of P() and P () In Figur 7, without th rptitiv controllr, th tracking rror btwn th rfrnc * i h and output i h i + PI( z) P ( z) h * ( z) = i (26) whr PI( z) hould b dignd to guarant th tability of ( z). With th propod 6k+ rptitiv controllr, th tracking rror can b writtn a

12 Enrgi 26, 9, x of 8 = z = z ( ) ( ) + H ( zg ) f ( zrz ) ( ) π ( 3 -D - Qzz ( ) ) ( z) π 2-3 f + -D (- H( zg ) ( z))( Qzz ( ) ) 2 (27) whr G ( z) i th compnation function, and f P ( z) H( z) = + PI( z) P ( z) (28) By th mall gain thorm, th ufficint condition for nuring (27) tabl can b givn a π (- ( ) ( ))( 3 -D H zgf z Qzz ( ) ) 2 + < (29) π Clarly, ( 3 D + Qzz ( ) ) 2 i tru. To mak (29) tru, it only nd - H( zg ) ( z ) < f bing atifid. Thu, G ( z) can b chon a f Gf () = H() τ + (3) whr Gf () and H() ar th function of Gf ( z) and H( z) in Laplac domain, rpctivly; ( τ + ) i a low-pa filtr. Morovr, on th prmi of ytm tability, it can b drivd that th numrator of (27) ha uch a tady-tat rlationhip: π 3 (6k+ ) ωt (6k+ ) ωdt ( ) Q = (3) Equation (3) indicat that th 6k+ rptitiv control chm can liminat th tady-tat rror of 6k+ harmonic tracking in D+d T (i.., T/6), which man th propod rptitiv control chm could ha a much fatr tranint tat rpon than th traditional on. 4. Exprimntal rult To validat th corrctn and ffctivn of th propod 6k+ rptitiv control chm, a prototyp of thr-pha paralll hybrid APF i built in lab, which i hown in Figur 8. Th control ytm i ralizd by a combination of digital ignal procor TMS32F28335 and fild programmabl gat array FPGA EP2C8T44C8N. Th powr witch u thr Infinon IGBT modul and th driv circuit u M57962L drivr chip. Th non-linar load ud in th xprimnt i a thr-pha diod rctifir bridg with ritiv load. Th ovrall xprimntal paramtr ar givn in Tabl.

13 Enrgi 26, 9, x 2 of 8 VSI PPF Nonlinar load Figur 8. Photograph of th prototyp 4.. Controllr paramtr Tabl. Exprimntal paramtr Paramtr Symbol Valu Unit grid pha voltag v 6 V (rm) grid frquncy f 5 Hz Inductanc L 3 mh capacitor C 9 µ F dc bu voltag V dc 8 V load ritanc R L 6.6 Ω witching priod T µ In th implmntation of xprimnt, th paramtr of controllr ar givn a follow. ) Dc-link voltag PI controllr: k p = 2, k i = 5. 2) In th harmonic currnt tacking loop: a. Th zro-pha low-pa filtr M( z ) i givn a M( z) (.25z.5.25 z ) 2 = + + ; b. Th numbr of dlay ampl i 42, and th FD filtr i givn a Fd( z) d dz z = + = + (32) c. Output currnt tat fdback gain k c = 3 ; d. PI controllr in th plug-in rptitiv controllr: k p2 =, k 2 =. i. Th compnation function Gf ( z) i givn a z z Gf ( z) = (33) 2.677z z 4.2. LC filtr parmatr

14 Enrgi 26, 9, x 3 of 8 A th non-linar load ud in thi papr i a thr-pha diod rctifir bridg with ritiv load,th 5 th, 7 th, th, 3 th harmonic currnt ar dominatd in load currnt. Aum that th load harmonic currnt ar fully compnatd by th hybrid APF, th voltag drop acro th LC filtr by th inctd compnating harmonic currnt i v = i ( ω L ) ( I ω L ) ω C (34) m m h lh m lh m m> ω mc m> m whr i i th m-th ordr harmonic componnt of load currnt, and I m i th amplitud of m lh lh m i lh. For th hybrid APF, th dign obctiv of LC filtr i to offr a lowt poibl impdanc path for incting harmonic currnt, in othr word, to minimiz th voltag drop v h. Thu, th dc-link voltag rating of VSI can b minimizd. Thn, an optimization function can b givn a fmin = ( I ω L ) = I ( HD ω L ) ω C (35) m h m f m m m= 5,7,,3 ωmc m= 5,7,,3 m whr HD i th individual m-th ordr harmonic ditortion rat, and m I lf i th amplitud of fundamntal componnt in load currnt. Th capacitor C in LC filtr can b cho by th rul a follow: Q C = 3ω CV (36) 2 whr Q C i th ractiv powr dmandd by load, ω i th grid frquncy, V i th grid voltag amplitud. Aum th capacitor C ha bn dtrmind, uch a C=9uf. According to Fig.(b), it can obtaind that HD 5 = 22.4%, HD 7 = 8%, HD = 5.7% and HD 3 = 2.6%. Subtituting th abov paramtr into (32), th optimal inductor L can b obtaind a L=2.8 mh. So w choo L=3 mh for th hybrid APF xprimntal prototyp without lo of much prformanc, and th ronant frquncy of LC filtr i 36 Hz Exprimntal rult Figur 9 how th dynamic bhaviour of dc-link capacitor voltag in tart-up proc. To avoid th inruh currnt caud by capacitor, th ri-ritanc oft-tart mod i ud in xprimnt. Spcifically, whn v V = 6 V, th IGBT ar turnd off, th capacitor ar dc t chargd up with mall currnt du to th ri-ritanc ; Whn vdc > Vt, th ri-ritanc i bypad and thn th PWM pul will b activatd. v dc rach th tting valu 8 V in th tady tat, which vrifi th corrctn of th dc voltag control tratgy.

15 Enrgi 26, 9, x 4 of 8 v dc V t Figur 9. Start-up proc of dc-link voltag To validat th tatic and dynamic prformanc of th propod 6k+ rptitiv control chm, th rlatd xprimntal rult ar hown in Figur -3. For th ak of implicity, only th a-pha wavform ar diplayd. Figur how th harmonic compnation rult whn nonlinar load i diconnctd ( i la = ). A n, i a = i a, and i a i almot th ractiv powr currnt providd by LC filtr. Th wavform of i a i inuoidal with l ditortion, which indicat that th propod rptitiv control chm can wll uppr th undird harmonic componnt. v a i a i a i la Figur. Harmonic compnation rult with nonlinar load diconnctd Figur -2 how th tady-tat harmonic compnation rult with and without fractional dlay compnation whn th nonlinar load i connctd, rpctivly. In Figur, th total harmonic ditortion (THD) of th ourc currnt i a i rducd to 3.8% from 24.8% (THD of th load currnt), and th ditortion ratio of 5 th, 7 th, th and 3 th harmonic in i a ar rducd to 2.3%,.3%,.6% and.2%, rpctivly. A a contrat, th THD of i a i 4.9% in Figur 2. Th comparion xprimnt rult dmontrat th good tatic prformanc of 6k+ rptitiv controllr and ffctivn of th fractional dlay compnation.

16 Enrgi 26, 9, x 5 of 8 i a i a i la (a) (b) Figur. Harmonic compnation rult with FD compnation: (a) ourc (L), compnating (L2), load (L3) currnt wavform ; (b) harmonic ditortion rat graph. i a i a i la (a)

17 Enrgi 26, 9, x 6 of 8 (b) Figur 2. Harmonic compnation rult without FD compnation: (a) ourc (L), compnating (L2), load (L3) currnt wavform ; (b) harmonic ditortion rat graph. Alo, to highlight th ffctivn of th 6k+ rptitiv control chm, th harmonic compnation rult by only th LC filtr i hown in Figur 3. A n, th ourc currnt i till highly ditortd aftr th compnation of th LC filtr, with a THD of 5.9%. Th main raon ar that th ronant frquncy of LC filtr i not prcily tund at a domain harmonic frquncy, and th prformanc of LC filtr riouly dpnd on th intrnal ritanc of grid ourc. i a i a i la (a) (b) Figur 3. Harmonic compnation rult by only th LC filtr: (a) ourc (L), compnating (L2), load (L3) currnt wavform ; (b) harmonic ditortion rat graph. To vrify th dynamic prformanc of th 6k+ rptitiv control chm, Figur 4 how th comparion xprimntal rult of th propod and traditional rptitiv control chm in tranint proc. A n, bfor th tim t, th harmonic compnation function i not nabld, i a i only th ractiv powr currnt providd by LC filtr with inuoidal wavform, and ia i ditortd by th load harmonic. At th tim t, th harmonic compnation function i nabld.

18 Enrgi 26, 9, x 7 of 8 Th 6k+ rptitiv control chm can tak ffct aftr T/6 tim, and liminat th tady-tat rror of harmonic tracking quickly. A a contrat, th traditional rptitiv control chm tak ffct aftr T tim, and nd vral T priod to liminat th tady-tat rror. Th xprimntal rult dmontrat that th 6k+ rptitiv control chm ha a much bttr dynamic prformanc than th traditional rptitiv control chm, which i conitnt with thortical analyi. i a T 6 i a i la t (a) i a T i a i la t (b) Figur 4. Dynamic prformanc comparion: (a) 6k+ rptitiv control chm; (b) traditional rptitiv control chm 5. Concluion In thi papr, a 6k+ rptitiv control chm for HAPF i propod, which aim at compnating th 6k+ harmonic in thr-pha powr ytm. Th intrnal modl of th 6k+ rptitiv controllr i contructd by th gnral mathmatical principl of traditional rptitiv controllr, and xprd uing th complx-vctor notation. A FD compnating mthod for 6k+ rptitiv controllr i alo prntd. Through thortical analyi and xprimnt, it i dmontratd that th 6k+ rptitiv control chm can achiv a fat tranint rpon with dlay tim of T/6, and good prformanc for compnating or uppring th 6k+ harmonic. Furthrmor, du to th abov fatur, th 6k+ rptitiv control chm i alo uitabl to ud in th currnt or voltag control for othr thr-pha grid-connctd invrtr. Rfrnc

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