Discrete Diagonal State Estimator based Current Control for Grid Connected PWM Converter with an LCL filter

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1 Dicrt Diagonal Stat Etimator ba Currnt Control for Gri Connct PWM Convrtr with an LCL filtr Byong-Hon Kim, Honyoung Kim an Subhahih Bhattacharya * FREEDM Sytm Cntr North Carolina Sta Univrity Raligh, USA hon3@nu.ac.kr, *bhatta4@ncu.u Abtract In thi papr, th control an tat obrvr for gri connct PWM convrtr with LCL filtr ha bn icu. h icrt tat-pac obrvr i ign in th -q ynchronou rfrnc fram. Uing th coorinat chang of th matrix, th altrnativ tat quation, which ha iagonal ytm matrix, i obtain. Dicrtiation of altrnativ tat quation uing Zro-Orr Hol mtho i uggt, an th icrt form i implr than that riv by original ytm matrix. By coniring control tability with rpct to th rlation btwn filtr ronant frquncy an ling frquncy, currnt control chm with fback ignal from tat obrvr i alo propo. h igital lay to th PWM an ling i conir for igital implmntation. Kywor LCL filtr, igital obrvr, icrt tim omain, control tability, ronant frquncy, gri-connct convrtr. I. INRODUCION In gri-connct powr conitioning ytm (PCS, fulfillmnt for currnt harmonic rtriction i on of th critical iu. Accoringly harmonic filtr ar mainly inrt btwn th gri an PWM convrtr to ruc harmonic componnt from PWM witching []. LCL filtr ar mor popularly u in variou application compar to L filtr a highr inuctanc, which incra ovrall filtr volum an wight, i rquir for L filtr than th ca for LCL filtr []-[5]. Howvr, a iavantag of th LCL filtr i th ronant bhavior, an it coul invok intability of th ytm. o uppr th ronanc, paiv or activ amping tchniqu ha wily icu. h ronanc can b amp by a uitabl ritor in paiv amping, but th paiv mtho introuc lo in th ytm []. Activ amping chm ar implmnt by moifying th control loop [4]. Howvr, in mot ca, PCS rquir aitional nor to implmnt activ amping, an it incra cot an cra ytm rliability. Som litratur how that th igital tim lay aft th LCL filtr ytm tability [5]-[7]. Appropriat lction of th poition of currnt ning an th ratio of th LCL ronant frquncy ( f r to ling frquncy ( f allow to oprat ytm in tabl without any amping tchniqu. In hort, if fr i lowr than /6, convrtr-i currnt houl b f-back for tabl opration without amping chm. By contrat, if fr i byon /6, gri currnt for fback ignal will tabili th ytm. Whn uing LCL filtr, th currnt nor for th currnt fback in control can b plac ithr at th convrtr output or th filtr output. Although th output powr of PCS at th gri i houl b controll, in om ca, th currnt nor houl b intall at th output of th convrtr u to th packag limitation. For mall cal powr ytm, hunt ritor can b aopt to n currnt. For larg cal ytm, it i har to plac currnt nor at th gri i bcau th LCL filtr i i gtting largr an th fback ignal path from nor to controllr i gtting longr an i mor nitiv to noi. If uch ytm t fr byon /6 to ruc filtr i, th activ amping houl b aopt an aitional nor houl b intall. h notch filtr ba tchniqu wa alo introuc to mak pha margin without aitional nor, but it houl conir frquncy viation from nominal valu [4]. In [8], a icrt tim obrvr ba tatpac currnt controllr coul oprat ytm in tabl without any amping tchniqu although th fr wa byon /6, an th convrtr currnt wa f-back. Howvr, th analytical gain for currnt control wa much mor complicat than convntional currnt controllr. h activ amping ba on tat obrvr wa prnt for LCL filtr [9] without aitional nor. Howvr, bcau th icrt tat obrvr nglct lo componnt, thy uffr from intrnal tability iu. hi papr prnt control chm ba on tat obrvr to tably oprat LCL filtr ytm although th convrtr currnt i f-back an th fr i byon /6. In th prviou paragraph, it wa figur out that aitional nor wa not manatory to tabili th ytm with untabl conition a crib in prviou litratur. h main contribution i aopting convntional ynchronou rfrnc fram proportional-intgral (PI rgulator for currnt control an uggting l complicat icrt tim omain obrvr ign. h tat obrvr timat gri currnt, an th will b f-back to controllr for tabl opration. o implify icrtiation, th tat quation i chang to th altrnativ iagonal ytm by imilar tranformation. h propo altrnativ iagonal ytm can mak ytm intrnally tabl. h obrvr gain ar t in icrt tim omain. For th /8/$ IEEE 3069

2 viabc i cabc v Cabc i gabc abc DC Sourc PWM Convrtr L R L fg R fg C f (a v i L i c r L i cq R C f v C L fg r C f v Cq r L fg i cq R fg i g v i L i cq r L i c R C f v Cq L fg r C f v C r L fg i cq R fg i gq q (b Fig.. (a Sytm configuration (b -q ynchronou rfrnc fram quivalnt circuit igital implmntation, th voltag compnation coniring igital lay i icu. In Sction II, altrnativ iagonal ytm quation will b prnt. hi altrnativ ytm will b chang to icrt ytm quation uing Zro-Orr Hol (ZOH. In Sction III, th obrvr gain will b t by th irct pol placmnt in icrt tim omain. In aition, th control chm without amping tchniqu will b propo ba on convntional currnt rgulator an tat obrvr. II. ALERNAIVE DISCREE SAE EQUAION A. Diagonaliation Fig. (a how th ytm configuration for gri connct PCS with LCL filtr, an th quivalnt circuit can b pict in -q ynchronou rfrnc fram a in Fig. (b. Complx pac vctor in -q ynchronou coorinat ar u a in [8] an [9]. In thi papr, th gri voltag vctor i align on th q-axi an th q-axi currnt i fin a an activ powr componnt [0]. h tat quation can b rprnt a (. x = Ax Bu y = Cx Du RL jg L 0 whr, A = Cf jg Cf, 0 Lfg Rfg Lfg j g ( L 0 0 B =, C = [ 0 0], D= 0, 0 0 Lfg x = icq vcq i gq, u = viq q, y = i cq. In th aformntion quation, R an R fg tan for th ritanc of convrtr i an gri i filtr inuctor. L an L fg tan for th inuctanc of filtr. C f tan for th filtr capacitanc. g tan for th gri frquncy. i cq, v Cq an i gq rprnt convrtr i currnt, filtr capacitor voltag, an gri i currnt, rpctivly. v iq an q tan for convrtr output voltag an gri voltag rpctivly. h arrow on variabl rprnt complx vctor. For implmntation in igital ignal procor, th ytm ha to b icrti uing vral mtho uch a Eulr backwar, rapoial, or Zro-Orr Hol mtho, an th mtho maintain tability of th original ytm []. Howvr, th invr or xponntial of th original tat quation, which i ntial factor to gt tabl icrt quation, hav complx form, an it can b obtacl for implmntation in practical. o allviat th computational burn, th original tat quation coul b chang to iagonal matrix uing imilar tranformation [], [3]. With lol aumption, th chang of coorinat matrix can b obtain a ( [8]. h tat quation can b writtn in altrnativ tat quation a (3 by imilar tranformation with th tat chang a (

3 Lfg L Lfg L Q = jplfg 0 jplfg ( x = Ax Bu y = Cx Du (3 whr, = = iag [ λ λ λ ] x = Qx (4 A Q AQ 3, L Lfg Lfg L B = Q B =, C= CQ =, L f, um L Lfg Lfg L D= D= 0, Lf, um = L Lfg, an λ i (i =,, 3 rprnt ignvalu. Howvr, th original ytm with ritor cannot b chang to iagonal matrix by (. In aition, th pol of altrnativ tat quation riv with lol aumption ar plac in imaginary axi of Laplac omain, an th rpon of th ytm woul b ocillat. hi impli th ytm i intrnally untabl. o avoi uch ocillatory rpon in th ytm, aymptot ignvalu ha bn propo. h charactritic quation of th original ytm i rprnt a (5. t ( I A ( j ( j ( j LCc ( LRg jg = g LRc g LRg g LCg whr, LCc = / LC f, LCg = / LfgC f, LRc = R / L, an = R / L. LRg LRg fg fg Bcau th pol by inuctor an ritor, LRc an, ar plac much lowr than th pol by inuctor an capacitor, LCc an LCg, (5 can b implifi to (6. h approximat charactritic quation giv th aymptot ignvalu a (7. t ( I A = ( jg LR, jg LR, p (5 (6 Fig.. ranfr function of ytm charactritic quation λ = j ( λ =jg LR, λ =j( whr, = / L C, L = L L, p p f g p LR, 3 g p LR, p fg LR, = ( LRc LRg. h iagonal lmnt of th ytm matrix ( A can b chang to th valu in (7, an thi iagonal matrix i vali with ubtl rror. Fig. how th tranfr function comparion of ytm charactritic quation btwn original ytm an iagonal ytm uing aymptot ignvalu. h rror i ngligibl, an th aymptot pol can montrat th ytm ynamic without ocillation. B. Dicrtiation uing Zro-Orr Hol Bcau th ronant frquncy of LCL filtr i aft by igital implmntation, th tat mol for obrvr ign in continuou tim i not vali in igital procor. h tat quation houl b igitali, an th tat obrvr houl b ign in icrt omain. h ytm ha bn icrti with th Zro-Orr Hol (ZOH mtho a (8 [8]-[9], []- []. (7 307

4 [ k ] = [ k] [ k] [ k] = [ k] [ k] x A x B u y C x D u A whr, iag xp( λ xp( λ xp( λ (8 A = = 3, ( xp( α α 0 B = A B ( λ L ( λ xp xp λ Lfgλ, xp( λ xp( λ = L f, um λ λ xp( λ3 L xp( λ3 λ3 Lfgλ3 C = C, D = D, tan for ling tim. h rult in (8 i implr an mor tabl than th icrti ytm with ZOH mtho from original ytm in [8] or [9]. h intrnal tability an ocillation of icrt ytm of original on will b icu in Sction IV. III. OBSERVER AND CONROLLER DESIGN A. Obrvr ign h gain of tat obrvr can b calculat in both continuou an icrt tim omain. h gain in continuou omain ar prnt in Appnix, an th gain houl b proprly tranform to icrt tim omain. Inta, irct ign in th icrt tim omain i propo. h iffrnc quation of icrt tat obrvr i givn by (9 whr L = [ L L L3], an th charactritic polynomial of th rror ynamic i givn by (0. Although th pol can b irctly lct in -omain, th pol ar lct in - omain in avanc for phyical inight. On ral pol ( o an conjugat complx pol ( β an β * hav bn lct. h complx pol rprnt ynamic of ζ o o o. h -omain pol ar tranlat to th -omain pol uing = xp a (. ( ( [ k ] = [ k] [ k] [ k] [ k] x A x B u L y C x (9 h h h ( = ( 0( ( t I A L C (0 Z L Z W ( 3 ( 3 L Z W λ λ λ λ λ λ = λλ3 λλ3 λ λ L 3 Z0 W 0 whr, Z ( 0 =, W ( 0 0 =, Z = 0 0, =, λ λ λ3 W = λλ λλ3 λ3λ, W0 = λλλ3, λ = xp( λ, λ = xp( λ, λ3 = xp( λ3, an L = L L L, L = L L L. fg 3 fg 3 h pol of obrvr can b plac at a highr frquncy than th currnt control ynamic an lowr than th Nyquit frquncy. But it i not ncary to t highr than th filtr ronant frquncy. B. Controllr ign A mntion bfor, although both convrtr currnt an gri currnt can b u for fback, th ronant frquncy of LCL filtr aft to th tability of th control. If th ronant frquncy i t ovr /6 an unr / of ling frquncy, th gri currnt control i tabl without activ amping. Howvr, in om ca, th currnt nor houl b intall at th output of convrtr u to th packag coniration. A a rult, th ronant frquncy houl b t unr /6 of ling frquncy for tabl currnt rgulation without activ amping. But th i of filtr houl b gtting largr, or th activ amping implmntation can incra numbr of nor. In thi papr, th gri currnt timat by tat obrvr i f back to rgulator. h ronant frquncy of LCL filtr i t ovr /6 of ling frquncy with fulfillmnt for currnt harmonic rtriction. h tabl currnt rgulation i atifi an th filtr i i gtting mallr. Bcau th ronant frquncy i ign much largr than currnt rgulation ynamic, th LCL filtr can b approximat to L filtr in th viw point of currnt control []. h convntional ynchronou rfrnc fram PI (Proportional an Intgral rgulator can b aopt an th gain can b t to achiv th firt orr low-pa filtr ynamic a (3. h cut-off frquncy of currnt controllr, cc, houl b t unr /0 to /6 of obrvr pol. k = L p f, um cc ( k = R R i fg cc (3 = xp 0 0 = xp ( ( β ( β = xp * h obrvr gain can b calculat by olving (. ( C. Coniration for igital implmntation h lay u to PWM an igital control u to b conir for control loop ign, an it aft to th obrvr ign. With a propr compnation uch a in [4], thi lay can b ftivly compnat. In thi ca, th output of PI controllr irctly u for tat obrvr input, an it i not ncary to chang obrvr gain u to th igital lay. h output of controllr at tim k- bcom input of tat obrvr at tim k. 307

i gq * Currnt rgulator v q * Digital Compnation v q * SVPWM S abc - î gq g, q Stat Etimator PLL abc q i cq abc i cabc - v q * DC Sourc i cabc i gabc Fig. 3.

h timat gri currnt of tat obrvr at k- ar f-back to currnt rgulator, an th output of controllr at k- u for tat timation in th nxt l prio. IV. VERIFICAION A.

(a original tat quation in ( in continuou tim omain, (b tat quation in [8] or [9] in icrt tim omain, (c propo tat quation in icrt tim omain.

h input i t to nominal gri voltag bfor applying tp input, o both an q-axi currnt wr kpt ro. At 0., th -axi voltag i appli in avanc, an th q-axi voltag i appli at 0.

h imply that, in th viw point of intrnal tability, propo iagonal tat quation i intrnal tabl vn in icrt tim omain, but th icrt tat quation with lol i untabl intrnally.

5 i gq * Currnt rgulator v q * Digital Compnation v q * SVPWM S abc - î gq g, q Stat Etimator PLL abc q i cq abc i cabc - v q * DC Sourc i cabc i gabc Fig. 3. Propo control chm (a Convrtr currnt h propo control chm coniring igital implmntation i pict in Fig. 3. h timat gri currnt of tat obrvr at k- ar f-back to currnt rgulator, an th output of controllr at k- u for tat timation in th nxt l prio. IV. VERIFICAION A. Opn loop tt h opn loop rpon of propo iagonal ytm ha bn valuat, an Fig. 4 how th rult. hr ytm hav bn tt. (a original tat quation in ( in continuou tim omain, (b tat quation in [8] or [9] in icrt tim omain, (c propo tat quation in icrt tim omain. h valu of ytm (b wr labl a timat variabl (, an th valu of ytm (b a timat variabl (. Input (convrtr voltag wr t to oprat ytm in rat powr conition. h input i t to nominal gri voltag bfor applying tp input, o both an q-axi currnt wr kpt ro. At 0., th -axi voltag i appli in avanc, an th q-axi voltag i appli at 0.. Sytm (b how ocillation aftr input wa appli, but propo ytm (c ha no ocillation for opn loop tt an how th am ynamic to th original ytm. h imply that, in th viw point of intrnal tability, propo iagonal tat quation i intrnal tabl vn in icrt tim omain, but th icrt tat quation with lol i untabl intrnally. Although th tat obrvr can cancl uch untabl pol by untabl ro an th untabl pol m to iappar form th ytm, th untabl pol till thr an can invok intability. h propo aymptot pol mak ytm intrnally tabl, an uch problm will not appar. B. Clo loop control Fig. 5 how th rult of convntional clo loop currnt control with convrtr currnt fback without tat obrvr. (b Filtr capacitor voltag (c Gri currnt Fig. 4. Opn loop tt rult for tat quation h currnt control cut-off frquncy i ot to 00 H. h ytm can oprat tably if th fr i unr /6. In th contrat, th fr byon /6 ma ytm untabl, an th voltag an currnt at PCC (point of common coupling wr ivrg right aftr PCS tart witching opration. 3073

Fig. 6 an 7 how th rult for propo control chm, an th ronant frquncy of LCL filtr i am to Fig. 5 (b. Although th harwar i intical, propo chm can tabili th ytm without aitional nor a in Fig. 5. Fig. 6 how th timat tat ar wll tracking maur tat in opration.

6 Fig. 6 an 7 how th rult for propo control chm, an th ronant frquncy of LCL filtr i am to Fig. 5 (b. Although th harwar i intical, propo chm can tabili th ytm without aitional nor a in Fig. 5. Fig. 6 how th timat tat ar wll tracking maur tat in opration. V. CONCLUSIONS hi papr prnt control chm to oprat PCS with convrtr currnt fback without activ amping. LCL filtr ha th ronant frquncy byon /6 of ling frquncy, an it wa known that thi conition mak convrtr currnt fback ytm untabl without aitional nor. h propo control chm wa aopt tat obrvr, an th timat currnt wr f-back to currnt rgulator. h icrt tat quation with iagonal ytm matrix wa riv by imilar tranformation an ZOH mtho. h aymptot ignvalu wr propo for iagonal lmnt. h can mak ytm intrnally tabl, allviat th ocillation in opn loop ytm, an montrat th original ytm. h irct icrt obrvr ign wa implmnt. (a f r =.5 kh < /6 f (a Convrtr currnt (b f r = 4.5 kh > /6 f Fig. 5. Clo loop currnt control tt rult accoring to th LCL filtr ronant frquncy (b Filtr capacitor voltag Fig. 6. Clo loop currnt control tt rult of propo control chm (f r = 4.5 kh > /6 f (c Gri currnt Fig. 7. Stat timation rult with clo loop currnt control (f r = 4.5 kh > /6 f 3074

7 APPENDIX h tat obrvr can b t by both continuou an icrt tim omain. h obrvr gain matrix, L = L L L3, in continuou tim omain i givn by (A. h charactritic polynomial of th rror ynamic i ign a (A. On ral pol an conjugat complx pol hav bn lct. L 0 L = Lfg p j p L = 0 p L L L L ( 3 = Lfg ( p 3 whr, 0 ( o jg( g o jgζ oo =, ( = 3g o o ζoo jgo ζo, = ζ j3. o o o g ( = ( o ( ζoo o (Α t I A LC (Α REFERENCES [] B.-G. Cho, S,-K. Sul, H. Yoo, an S.-M. L, LCL filtr ign an control for gri-connct PWM convrtr, in Proc. 8 th Int. Conf. on Powr Elctron. an ECCE Aia, 0, pp [] M. Romhanc, M. Naouar, I. Blkhoja, an Eric Monmaon, Robut activ amping mtho for LCL filtr-ba gri-connct convrtr, PE, vol. 3, no. 9, pp , Spt 07. [3] B.-G. Cho, an S,-K. Sul, LCL filtr ign for gri-connct voltagourc convrtr in high powr ytm, ECCE 0. [4] W. Yao, Y. Yang, X. Zhang, F. Blaabjrg, an P. Loh, Dign an analyi of robut activ amping for LCL filtr uing igital notch filtr, PE, vol. 3, no. 3, pp , Mar. 07. [5] J. Wang, J. Yan, L. Jian, an J. Zou, Dlay-pnnt tability of ingl-loop controll gri-connct invrtr with LCL filtr, PE, vol. 3, no., pp , Jan 06. [6] Y. ang, C. Yoon, R. Zhu, an F. Blaabjrg, Gnrali tability rgion of currnt control for LCL-filtr gri-connct convrtr without paiv or activ amping, in Proc. IEEE Enrgy Convr. Congr. an Expo (ECCE, 05, pp [7] S. Parkr, B. P. McGrath, an D. G. Holm, Rgion of activ amping control for LCL filtr, IEEE rna. In. Appl., vol. 50, no., pp , Jan/Fb 04. [8] J. Kukkola, M. Hinkkann, an K. Zngr, Ovrvr-ba tat-pac currnt controllr for a gri convrtr quipp with an LCL filtr: analytical mtho for irct icrt-tim ign, IEEE rna. In. Appl., vol. 5, no. 5, pp , Stp/Oct. 05. [9] V. Mikovic, V. Blako,. M. Jahn, A. Smith, an C. Romnko, Obrvr-ba activ amping of LCL ronanc in gri-connct voltag ourc convrtr, IEEE rna. In. Appl., vol. 50, no. 6, pp , Nov/Dc. 05. [0] B. Kim an S.-K. Sul, Stability orint ign of frquncy rift antiilaning an pha-lock loop unr wak gri, IEEE J. Emrg. Sl. opic Powr Elctron., vol. 5, no., pp , Jun. 07. [] A. Vloni, an N. Miriaki, Digital control ytm: thortical problm an imulation tool, CRC Pr, Boca Raton, 08. [] C. Chn, Linar ytm thory an ign, Oxfor Univ. Pr, NY Oxfor, 999. [3] S. Fribrg, A. Inl, an L. Spnc, Linar algbar, Prntic Hall, 997. [4] B.-H. Ba an S.-K. Sul, A compnation mtho for tim lay of full-igital ynchronou fram currnt rgulator of PWM ac riv, IEEE ran. In. Appl., vol. 39, no. 3, pp , May/Jun

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