NONLINEAR CONTROL OF BOOST AC/DC CONVERTERS OUTPUT VOLTAGE REGULATION AND POWER FACTOR CORRECTION. Abdelmajid ABOULOIFA, Fouad GIRI, Ibtissam LACHKAR

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1 NONLINEAR CONTROL OF BOOST AC/DC CONVERTERS OUTPUT VOLTAGE REGULATION AND POWER FACTOR CORRECTION Abdelmajd ABOULOIFA, Fuad GIRI, Ibtssam LACHKAR GREYC, ISMRA, 6 Bd Maréchal Ju, 4050 Cae Abstract: We are csderg the prblem f ctrllg AC/DC swtched pwer certers f the Bst type. The ctrl bjectes are twfld: () regulatg the utput ltage t a desred referece alue, () assurg a utary pwer factr by efrcg the ltage ad the curret delred by the electrc etwrk t be phase. The csdered prblem s dealt wth by desgg a lear ctrller lg a cascade-structure. The er lp regulates the acte pwer; t s bult-up usg the backsteppg desg apprach. The uter lp regulates the certer squared utput ltage usg a fltered PI regulatr. The ctrller thus btaed s shw, usg tls frm the aeragg thery, t achee ts bjectes. Cpyrght 005 IFAC Keywrds: Pwer certers, Vltage ctrl, pwer factr crrect.. INTRODUCTION Statc pwer certers hae a ery wde dma f applcats. Hweer, these certers stll hae a mprtat drawback as they ctrbute t the pllut f the electrc etwrk. Ideed, these certers csttute lear lads fr the dstrbut etwrk ad, csequetly, geerate harmc currets that may cause sme ayg effects such as etra pwer lsses the etwrk. Therefre, certer ctrllers shuld t ly hae as bjecte utput ltage regulat, but als reject f the meted curret harmcs. Mst f preus wrks hae fcussed ly ltage regulat (Sra, et al., 997). I the preset paper, we are csderg the prblem f ctrllg a whle AC/DC certer (Mech ad Fuabk, 99;). We wll partcularly fcus AC/DC certers wth bst chpper (Fg.). Our bjecte s t regulate the utput ltage whle esurg a utary pwer factr (PF). The last bjecte amuts t rejectg the whle curret harmcs at the certer put. O the ther had, AC/DC certers are featured by ther arablestructure ad ther lear dyamcs. T deal wth the csdered ctrl prblem a lear ctrller cludg tw lps s bult-up. The er lp s frst deelped, usg the backsteppg techque, such a way that the certer put curret be susdal ad phase wth the etwrk supply ltage. The certer arable-structure feature s cped wth basg the abe regulatr desg up a aerage mdel f the system. It s wrth tg, that mdel aeragg s wdely used the lterature (Kre, 990). The atural purpse f the uter lp wuld be the regulat f the certer utput ltage. Hweer, we wll chse t perfrm regulat f rather tha. Actually, uderges a (frst-rder) lear dfferetal equat whle uderges a lear equat. Usg ths arable trasfrmat ( ), referece trackg ca be acheed usg a fltered PI regulatr. A theretcal aalyss, lg tls frm the Lyapu stablty ad the aeragg thery, shws that the lear cascade ctrller thus cstructed actually achees, the mea, ts bjectes. The

2 ctrller perfrmaces ad rbustess (wth respect t lad chages) are further llustrated by may smulated eamples.. MODELLING OF THE CONVERTER The certer uder study s represeted by Fg. It cludes three ma parts, amely a -flter, a dde brdge rectfer ad a bst chpper. The latter perates accrdg t the Pulse Wh Mdulat (PWM) prcple, (Mahda, et al., 997). Ths meas that tme s shared terals f legth T. Wth ay perd, the IGBT-swtch s ON durg αt, fr sme 0 α. The, eergy s stred the ductace L ad the dde D s blcked. Durg the rest f the perd,.e. (-α)t, the swtch IGBT s OFF ad, csequetly, the ductace dscharges the lad resstace R. The alue f α ares frm a perd t a ther ad ts arat law determes the trajectry f utput ltage. The arable α, called duty cycle, the turs ut t be the ctrl put fr the certer. Mathematcal mdelg f the certer s cmpleted applyg Krchhff s laws. S dg, e gets: d rect L L () d rect rect C C () d L IGBT rect L L () d C = D R (4) The curret rect flwg the put rectfer s btaed frm (5): = sg( ) rect rect (5) The ltage IGBT takes uderges dfferet equats depedg the state f the IGBT-swtch. These equats ca be ge a uque mathematcal epress by trducg a bary arable µ : f IGBT s ON µ = 0 f IGBT s OFF (6) The, e has fr IGBT the fllwg epresss: L C C rect rect IGBT ( ) = µ (7) D D4 D D L rect L IGBT D D C IGBT C R R Smlarly, the curret D the dde D uderges dfferet laws, depedg the states f the IGBTswtch. It s ge by: D = ( µ ). L (8) Substtutg (5), (7) ad (8) ()-(4), yelds the fal frm f the (stataeus) certer mdel: d rect (9a) L L d sg( ) rect L rcet = + (9b) C C ( ) d µ d L rect = (9c) L L ( µ ) = (9d) L C R C Ths mdel s useful t buld-up a accurate smulatr fr the certer. Hweer, t cat be based up t desg a ctuus ctrl law as t les a bary ctrl put, amely µ. T ercme ths dffculty, t s usually resrted t the aeragg prcess er cuttg terals, (Kre, 990). Ths prcess s shw t ge rse t aerage erss (f the abe mdel) lg as a ctrl put the mea alue f µ whch s thg ther tha the duty cycle α. The aerage mdel turs ut t be the fllwg: d rect L L (0a) d sg( ) rect L rcet = + C C (0b) d ( α) d L rect = (0c) L L ( α) = (0d) L C R C. CONTROLLER DESIGN The ctrller sythess wll be perfrmed tw majr steps. Frst, a curret er lp s desged t cpe wth the PFC ssue. I the secd step, a uter ltage lp s bult-up t achee ltage regulat.. Curret er lp desg The PFC bjecte meas that the certer put curret shuld be susdal ad phase wth the etwrk supply ltage. We therefre seek a regulatr that efrces the curret t track a def referece sgal f the frm ref = β. At ths pt the parameter β s ay real umber. The regulatr wll w be desged usg the backsteppg techque (Krstc, 995), based the (partal) mdel (0a-c). Fg.. AC/DC Bst certer

3 .. Step : Stablzat f the sub-system (0a) z& = z c z () Let us trduce the fllwg trackg errr: Replacg () z& by ts epress (9) ad slg the resultg equat wth respect t z = ref () sg( ) L rect, yelds the fllwg stablsg def where ref = β dete the crrespdg fuct: referece sgal. Usg (0a), tme-derat f () dσ yelds the fllwg errr dyamcs: δ = z c z + + () dz d rect ref L L () sg( ) L rect As s t the actual ctrl put, a I (), ( rect L) stads as a (rtual) ctrl ew errr arable z betwee the rtual ctrl ad arable. The, z ca be regulated t zer f ts desred alue δ s trduced: ( rect L) = σ where σ s a stablsg fuct sg( ) L rect defed by: z = δ (4) d ref σ c z () The, equat (9) becmes, usg () ad (4): L z & = z c z + z (5) Ideed, ths chce wuld mply that: z& z Als, the Lyapu fuct derate becmes: (where c >0 s a desg parameter) whch clearly establshes asympttyc stablty f () wth respect W z c z + z z t the Lyapuu fuct : (6) W = 0.5z (4) Because:.. Step : Stablzat f the sub-system (0a-c) W z < 0 Tme-derat f z As ( rect L) ges, usg (4) ad (0c): s t the actual ctrl put, a ew dz sg( ) ( ) rect errr arable z betwee the rtual ctrl ad ts rect α dδ = (7) desred alue σ (stablsg fuct) s trduced: L L z = L σ (6) The actual ctrl arable, amely α, appears fr the rect The, equat () becmes, usg () ad (6): frst tme equat (7). A apprprate ctrl law fr geeratg α, has w t be fud fr the z & z + z (7) system (7), (5), (7) whse state ectr s (z, z, Als, the Lyapu fuct derate becmes : z ). Let us csder the Lyapu fuct W : W W = W + 0.5z (8) z + z z Its tme-derate alg trajectry f (6) ad (7) s: dw.. Step : Stablzat f the sub-system (0a-b) z c z sg( ) rect rect ( α) dδ (9) Acheg the PFC bjecte amuts t efrcg + z ( ) + z L L the errr arables (z, z ) t ash. T ths ed, e eeds the dyamcs f z. Derg (6), t fllws frm (0b) that: Ths shws that, fr the (z, z, z )-system t be glbally asympttcally stable, t s suffcet t dz sg( ) L rect dσ chse the ctrl α s that W& z c z c z = (9) whch ew f (9) amuts t esurg that: sg( ) L rect sg( ) I the abe equat, the quatty rect rect ( ) α dδ + z z (0) L L stads as a rtual ctrl put. We w eed t Slg the resultg equat wth respect t α, select a Lyapu fuct W fr the (z, z )-system. yelds the fllwg backsteppg ctrl law: As the bjecte s t dre ts states (z, z ) t zer, t s atural t chse the fllwg fuct: L sg( ) rect dδ rect α = z + c z W = W + 0.5z (0) () Usg (8) ad (9), ths mples: W& z + z (z z ) () Prpst.. Csder the system, et called er clsed-lp, csstg f the subsystem (0ad) ad the ctrl law (). If the rat β ad ts three Ths shws that, fr the (z, z )-system t be glbally asympttcally stable, t s suffcet t chse the rtual ctrl put s that W z c z frst derates are aalable, the the er clsedlp system uderges, the (z, z, z )-crdates, c >0). The, t fllws frm () that: the fllwg equat:

4 d z c z = c z 0 0 z z c z () Csequetly, the errr ectr (z, z, z ) s glbally asympttcally ashg.. Outer ltage lp desg The am f the uter lp s t geerate a tug law fr the rat β such a way that the utput ltage be regulated t a ge referece alue ref. Relat betwee β ad. The frst step desgg such a lp s t establsh the relat betwee the rat β (ctrl put) ad the utput ltage. Ths s the bject f the fllwg prpst. Prpst.. Csder the pwer certer descrbed by (0a-d) ad the er ctrl lp defed by (). Uder the same assumpts as Prpst., e has the fllwg prpertes: ) The utput ltage ares respse t the tug rat β accrdg t the fllwg equat: d ˆ ( cs(ω t) )β () R C C where ˆ detes the magtude f the etwrk (susdal) ltage. ) The squared ltage ares, respse t the tug rat β, accrdg t the lear equat: dy = ay + k ( cs(ω t) )β (4) where R C a = ad k = Prf. ) The frst step cssts replacg the crcut part abe the set C -R, by a equalet curret geeratr, as shw by Fg.. I ew f equat (0d), the uderlyg curret alue equ ccdes wth ( α). S, (0d) becmes: d Fg.. Equalet curret geeratr L ˆ equ = (5) C R C equ C 0 R 0 C0 R0 The equalet curret equ wll w be epressed fuct f the tug rat β, usg pwer cserat argumets. Usg the fact that =β (because f Prpst.), the stataeus pwer eterg the certer turs ut t the fllwg: C 0 ( cs(ω.t)) βˆ P = βˆ s ( ω.t) = (6) O the ther had, the pwer that s actually trasmtted t the lad s P =. But, the lad equ eterg stataeus pwer s tegrally trasmtted t the lad (whch s the ly dsspate elemet). The, the quatty P lad des ccde wth P, ths ˆ yelds: = β ( cs(ω t), whch tgether wth equ (5) establshes (). ) Let us trduce the arable chage y = (). Derg y wth respect t tme ad usg (), yelds the frst-rder lear mdel (4) ad cmpletes the prf f Prpst.. Squared utput ltage ctrl. The rat β stads as a ctrl put the system (4). The prblem at had s t desg fr β a tug law s that the squared ltage y = tracks a ge referece def sgal y =. Igrg tme-aryg feature, f ref the ctrlled (frst rder) system, a PI ctrl law shuld be suffcet. Bearg md the fact that the thrd frst derates f β shuld be aalable (Prpsts. ad.), we rather use a fltered PI, b.e., = ( k e + k e ) β (7) p b + s e = y y, e = e, (8) 0 where s may dete as well the derate peratr (s=d/). At ths pt, the regulatr parameters (b, k p, k ) are ay pste real cstats.. Ctrl system aalyss I the fllwg Therem, t s shw that the ctrl bjectes are acheed, the mea, fr a specfc class f referece sgals, wth a accuracy that depeds the etwrk frequecy ω. T frmulate the results the fllwg tats are eeded: = ω ε, a = k k b, a = ( a + k k ) ab b a + 0 p b =, = ab+ b, a = ( a b) a 4 + Therem. (ma result). Csder the AC/DC Bst pwer certer shw by Fg.. tgether wth the ctrller csstg f the er-lp regulatr () ad the uter-lp regulatr (7-8). The, the resultg clsed-lp system has the fllwg prpertes: ) The errr ref ashes asympttcally, ) Let the referece sgal y be egate ad perdc wth perd Nπ/ω, where N ay pste teger. Furthermre, let the regulatr parameters t

5 (b, k p, k ) be ay pste real cstats that satsfy the fllwg equaltes: ( a a a ) a ( a a a ) a > 0 (9) [( ) ( ) ]( ) ( 0 a a a a a a a a a a a a a ) a 0 a > (40) The, there ests a pste ε such that fr ε ε, the trackg errr s a harmc sgal whch depeds ctuusly ε,.e. = e (t, ). Mreer: e ε lm e (t, ε) = 0 ε 0 Remarks.. a) The frst part f the Therem says that the pwer factr bjecte s actually acheed. b) The secd part shws that the trackg bjecte s acheed, the mea, wth a accuracy that depeds the ltage etwrk frequecy (ω ). The larger ω the better trackg bjecte. c) The perd f the referece sgal s ay multple f that f the pwer etwrk (whch s equal t π/ω ). That s, the referece sgal s slwer tha the etwrk ltage. Ths partcularly cludes cstat refereces. Prf f Therem. (Outle). Part f the Therem s a drect csequece f Prpst., usg (7) whch shws that β ad ts derates (up t the thrd rder) are aalable. Ths als guaratees that equat (4), Prpst., hlds t. I rder t pre the secd part f the Therem l, let us trduce the fllwg state arables: = e, e, = ( kpe + k e),, 4=, 5= 4 = b b + s b b + s b b + s The, t fllws frm (4) ad (7-8) that the state ectr X=[,,, 4, 5,] T uderges the fllwg state equat: where: f (X, t) = - a k dx = f(t, X) (4) b ( cs(ω t) ) + ay dy + ( + k p + k ) b( + ) ( ) 4 b (4) Stablty f the abe system wll w be dealt wth resrtg t aeragg aalyss tls. Frst, as y s perdc wth perd N/ω, t wll pre t be useful trducg the fllwg aulary referece fuct: yr (t) = y N t (4) ω 5 Ths readly mples that y = ω (t) yr t ad that y r N s perdc, wth perd π. Let us w trduce the tme-scale chage: τ = ω t. The the term y (4) becmes: ( ) ( τ dy dyr (t) ) ay (t) = yr τ ω N dτ N (44) It als fllws frm (4-4) that X uderges the dfferetal equat: dx =εg( τ, X, ε) (45) dτ where: r τ dy r τ ( ) + ω N a k cs( τ) + ay 5 N dτ g( τ, X, ε) = b - ( + k + k ) p b( + ) 4 ( ) b + (46) Nw, let us trduce the aerage fuct π N G (X) = g( τ, X, 0)dτ. It fllws frm (45) that: πn 0 r - a k ay 5 G (X) = ( ) b + k + k p b( + ) 4 ( ) b (47) r where y detes the mea alue f y r, whch the same as that f y. Nte that the mea alue, er [0, πn], f the derate the frst le f (46), s zer because y r s perdc wth perd π. The aeragg thery suggests t get stablty results regardg the system f terest (4-4), frm aalyzg the fllwg aeraged system: dz = ε G(Z) (48) T ths ed, tce that (48) has a uque equlbrum at: T r r r r - ay - ay - ay - ay Z = 0,,,, k k k k k (49) O the ther had, as (48) s lear, the stablty prpertes f ts equlbrum are full determed by the fllwg state-matr: a k A = bk bk b 0 0 (50) p 0 0 b b b b The equlbrum Z wll be glbally asympttcally stable f all egealues f A hae egate real parts. The meted egealues are the zers f the fllwg characterstc plymal: 5 4

6 5 4 λ I A) = λ + a λ + a λ + a λ + a λ a (5) 4 0 det( + It ca be checked, applyg fr stace the well kw Ruth s algebrac crtera, that all zers f plymal (5) wll hae egate real parts f ts ceffcets (a 0 t a 4 ) satsfy (9-40). Nw applyg aeragg thery (see e.g. Therem 4. (Zh-fe, et al 99)), e ccludes that there ests a ε > 0 such that fr ε < ε, the dfferetal equat (4-4) has a harmc slut, X = X(t, ε), that depeds ctuusly ε. Mreer, e has: lm X(t, ε) = Z ε 0 Ths, tgether wth (48), yelds partcular that m e (t, ε) = 0. The Therem s thus establshed. ε 0 4. SIMULATIONS Perfrmaces ad desg aspects f the ctrller wll w be llustrated by smulats perfrmed the Matlab/Smulk ermet. The ctrlled AC/DC certer has the fllwg characterstcs: ˆ = 60V, L = mh, C = 0µ F, L = 0mH, C = 4000µ F R = 0Ω ad t perates at the cuttg frequecy f c = 0khz. The referece squared utput ltage s a step sgal f ampltude 500 ref (Vlts). The alues c = 0000, c =0000 ad 5000 pred t be apprprate fr the er lp desg parameters. Bearg md Remark a, the uter lp parameters hae bee chse as fllws: k p = 0.005, k = ad b = 000. Fg. 5. curret Fg. 6. ad T aalyze the rbustess capablty f the prpsed ctrller a lad chage s perated accrdg t Fg.7. Ecept fr ths chage, the rest f the certer characterstcs are the same as preusly. The resultg clsed-lp system behar s llustrated by fgures 8 t 0. The frst fgure shws that the effect f the resstace chages the utput ltage s well cmpesated. Fg. 9 shws that the PFC prperty s presered. Fally, Fg. 0 shws that lp mde separat s stll satsfed. Fg. 7. Lad resstace R Fg. 8. ltage Fgures t 6 llustrate the ctrller perfrmaces. As epected (Remark b), cerge the mea t ts referece alue (see Fg..). Furthermre, t s checked that the bsered ltage rpple scllate at the frequecy ω (Remark b) ad s much smaller tha the aerage alue f the sgals. Cmparg Fg. ad Fg. 5, e partcularly sees that the magtude arat f the put curret s crrelated t the (mea) alue f the squared utput ltage. Ths cfrms the pwer cserat thrugh the crcut. Fg. 4 shws that the uter-lp ctrl β s practcally uaffected by the rpple phemea. Fally, Fg. 6 shws that the put curret ad the utput ltage are phase, esurg a utary pwer factr. Fg.. squared ltage Fg. 4. sgal β Fg. 9. curret ad ltage REFERENCES Fg. 0. sgal β Kre P.T., J. Betsma, R.M. Bass, ad B.Leseutre (990). O the use f aeragg fr aalyss f pwer electrc system. IEEE Trasact Pwer Electrcs, Vl. 5,. Krstc, M., I. Kaellakpuls ad P.V. Kktc (995). Nlear ad adapte ctrl desg ( Jh Wlly & Ss, NY). Mech A. Ad S.Fuabk (99). Step-up/dw ltage PWM AC t DC cectr wth e swtchg dece. IEE prcedgs-b, l. 40,., pp.5-4. Sra-Ramrez H., R. Perez-Mre, R. Ortega, ad M. Garca-Esteba (997). Passty-based ctrllers fr the stablsat f DC-t-DC pwer certers, Autmatca,. Zh-fe Z., D. Tg-re, H. We-za ad D. Zhe- (99). Qualtate thery f dfferetal equats (Amerca Mathematcal Scety, Rhde Islad, USA).