Dynamic System Modelling of Multiloop Operation for Paralleled DC/DC Converters

Transcription

1 Dynaic Syte oelling of ultiloo Oeration for Parallele DC/DC Converter * Iri aoura, ** euvo Suntio, * Kai Zenger * Helinki Univerity of echnology Control Engineering aboratory, P.O.Box 54, 5 HU, Finlan el: Fax: Eail: igaoura@cc.hut.fi ** Univerity of Oulu Electronic aboratory, P.O. Box 45, FIN-94 Oulu, Finlan el: Fax: Abtract In thi aer a yte of arallel-connecte DC/DC converter incluing all interconnection reitance i oelle an analye. H otial control i ue in orer to eign a robut controller that guarantee robut tability an robut erforance in ite of ifferent uncertaintie. he oelling, analyi an control eign roceure reente are verifie by a eign exale.. Introuction A ower uly that i ue to ower telecounication equient can be cooe to arallel-connecte DC/DC converter, a hown in Fig.. hee converter require categorical current haring echani to enure roer oeration. he ain avantage of uing arallel converter configuration with current haring control i to increae the ower roce caability that irove a reliability of the whole yte. he current haring chee can be achieve by injecting a ignal roortional to the eire converter ut current into the voltage loo. he increae voltage loo error ignal force the uty cycle an then the ut current of the converter to increae. An exceive work ha been one in the fiel of oelling, analyi, an control eign of arallel-connecte DC/DC converter. Previou tuie on arallelconnecte DC/DC converter have not fully aree the oelling roble of interconnection reitance [,3] an the analyi roble of interaction aong converter [,3,4]. ot current-loo controller are eigne earately to enure tability. However, the interaction affect the tability roertie an the ynaic erforance ha not been tuie. Alo oe oelling roble have been raie when hevenin circuit are coniere to oel arallele converter. hee inclue colexity an itreatent of interaction [,6]. oel of arallelconnecte DC/DC converter, which are reente in [3,4,6] can never be erfect becaue certain characteritic were not oelle at all, e.g. variation of ower tage coonent, change of oerating oint, loa alteration, etc. he uncertaintie cannot eaily be oelle, however they ut be inclue in the analyi. Variou eaure of robutne can be ue; robut tability an robut erforance guarantee that the yte erfor well in ite of iturbance an oel uncertaintie. Previou tuie ehaize the alication of robut control theory to DC/DC converter but never trie to utilize theory in ractical iue. Alication of robut control theory to two-buck converter in arallel i reente in [] where the noinal tability an robut tability in ter of ingular value have been enure. he iulation reult how goo ynaic erforance of the yte. he work ha been extene to eign robut ecentralize control incluing both tructure an untructure uncertaintie a reente in []. he iulation reult how goo iturbance rejection caabilitie. DC/DC Converter DC/DC Converter oule #n/ oule # oule # oule #n oule #n/ oule #n/ Batterie Batterie Fig. eleco ower yte elecounication Switching Syte he ain uroe of thi aer i to oel a yte of arallel-connecte DC/DC converter incluing all interconnection reitance, a hown in the aenix. he uncertaintie, coonent of ower converter, oerating oint, an loa change, are coniere in orer to eign a robut controller that guarantee robut tability an erforance of the yte.

2 H otial control i ue to eign a tabilizing controller that can achieve robut tability an robut erforance. he oelling, analyi an control eign roceure reente i verifie by iulation reult on two 5W-buck converter connecte in arallel. hi aer i organize a follow. he yte ecrition an control objective are reente in Section. he tate-ace rereentation an all tranfer function of arallele buck converter are given in aenix. Uncertainty oel for lant an iturbance oel are obtaine a a ultilicative inut uncertaintie a hown in Section 3. In Section 4, H otial controller i introuce an analye in orer to achieve robut tability an robut erforance. A eign exale i reente in Section 5. Concluion i given in Section 6.. Syte oelling & Control Objective ( hel to erive all tranfer function of the yte incluing interaction i reente in aenix. he cloe-loo tructure of arallele DC/DC converter incluing uncertainty oel an erforance objective i hown in Fig. 3(a where the true arallele DC/DC converter oel i rereente by ahe box. W ( y ( ( ( ( wi ( ( ( Wr ( wi ( _ u ( r ( K ( W ( ( z ( r W ( _ n ( W ( n y ( y ( u (a u ( ( r ( W ( u z ( Switch r v r ( v ( y ( v _ in i i i C C r C r w ( P ( z ( u ( y ( K ( v _ in Switch i r i C C r C v r i r r ( r Fig. he oelling yte In ractice, arallel-connecte DC/DC converter ay be locate far aart an ay require unequal length of cable connecting the to bu bar where the loa they hare i connecte to, a hown in Fig.. he unequal cable length an interconnection oint to bu bar between converter reult in ifferent ieance an contribute to affect on the currenthare itribution. Concluing, all interconnection reitance houl be inclue in the erive oel. In [4,6], the cable reitance ha been taken into account when the coarion between ifferent control technique wa carrie. he converter cable ha ignificant iact on the erforance of arallel-connecte converter an, therefore, houl be inclue in converter eign. he tate-ace rereentation of average an linearize oel that r v _ R i g (b Fig. 3 he control yte of arallele DC/DC converter (a an general control yte (b he ultilicative inut uncertainty ha been ue to rereent the oel uncertainty. he erforance objective i that the nor of tranfer function fro w to z be all for all oible uncertainty tranfer function. W i ue to achieve a eirable erforance of variou frequency range. here are two reliinary control objective to be coniere: Equalize all the ut current of arallele oule. uarantee that the ut voltage tay within ecification in ite of ifferent uncertaintie. In other wor, the control objective i to eign a tabilizing controller K uch that for all the erturbe cloe-loo yte reain table. he erforance requireent can be efine in ter of weighte enitivity a ( = ( ( i < ( S W I I W K for all erturbation [5]. After a few iteration to atify equation (, the erforance weight w ( jω for two-ientical arallele buck converter whoe araeter are

3 hown in Section 5, ha been choen with very all gain that ha been coenate with final controller w ( jω = 9 ( where W = w I n n an n=. he eaureent noie ha been oelle to avoi oe ifficultie in the ynthei roceure. he noie weight wn ( jω for two-ientical arallele buck converter ha been choen a a high-a filter. 5 wn ( jω = (3 5. where Wn = wn I n n. 3. Uncertainty oel A firt rincile et of uncertaintie ab arallele converter oel woul inclue: Uncertainty in aive coonent (i.e., caacitor an inuctor: Accoring to anufacturer, the variation of can be ab ±% of noinal value an of C ab ±%. Uncertainty in araitic eleent (i.e., ESR of an C: hey een on anufacturing of an C. he variation can be ±5% of noinal value. Uncertainty in ynaic loa (i.e., reitive loa: he ut ower of a eleco ower yte can be varie fro % to 9% of it axiu. Uncertainty in line ource (i.e., inut voltage: Due to utility uly icontinuity the variation of line ource can be ab ±% of noinal value. Uncertainty in interconnection reitance (i.e., reitance of cable an interconnection oint between converter: he variation of thee reitance can be ab ±5% of noinal value. hee uncertaintie have been lue together into one full-block uncertainty at the inut of a noinal oel, a hown in Fig. (a. he noinal lant an the uncertainty weight W i araeterie an entire et of lant,, which ut be uitably controlle by the robut controller K. { I W : i table an } = ( i (4 he unknown tranfer function ( i ue to araeterie the ifference between the noinal oel an the actual behaviour of real yte,. o o thi, the weight W i houl be choen o that the noralize erturbation atifie { ( } ax σ w ω (5 i Alo the weight W i houl be choen o that the noralize erturbation atifie { ( } ax σ w ω (6 i he uncertaintie weight, a hown in Fig. 4, are erive by uing equation (5 an (6. w w i i 4 8 = = Wi = wi I n n an Wi = wi I n n Plant oel Uncertainty Diturbance oel Uncertainty Fig. 4 he uncertaintie boun 4. Feeback Control Deign (7 (8 Coniering equation (A8 that reent arallele DC/DC converter a an IO linear-tie invariant yte y = u (9 where the feeback control, a hown in Fig. 3(a, ha been alie to the yte. = ( ( u K W r W n y r n he erforance weight W an W u are aue to be table an iniu hae. he reference r an eaureent noie n are inclue in aition to iturbance a external inut. he weight W, W r, an W n are ue to noralize the inut.

4 4.. Control Iue ( v ( y ( w ( N( z ( w ( z ( ( u ( y ( K ( F < for all where (6 4.. H otial Control F( F ( (a (b Fig. 5 Configuration of control yte he general control configuration in Fig. 5(a, which i ue for analyi roceure, i obtaine by uing a lower F (linear fractional tranforation of Fig. 3(b. he cloe-loo tranfer atrix fro [ v w] to [ y z] can be written a (, ( N = Fl P K = P P K I P K P N N = N N ( where v [ u ] =, w= [ r n], y [ ] = yu y, an z = [ z z ] = W ( y W r W u r u. o enure goo erforance with a all inut uage, i.e. z i all, we woul like to have N all, i.e. all the IO tranfer function in equation ( are all. he H nor i efine a the axiu ingular value over all frequencie, i.e. = u σ. he uncertain cloe-loo ω ax tranfer atrix F fro w to z i obtaine by uing an uer F (, ( F = Fu N = N N I N N WSW WS W WW O r O I n = WuS IKWr WuS IKW WuS IKWn ( he enitivity an the coleentary enitivity atrice are efine a SO = = K K ( K an S ( I = K ( O an = K ( K I (3 (4 For robut tability, the yte ut reain table for all lant in the uncertainty et. hi houl atify that F( i table for all where, which i equivalent to N < for all ω (5 Robut erforance i achieve if robut tability i atifie an the tranfer atrix fro w to z i all for all lant in the uncertainty et. Conier the tanar etu of Fig. 5(b for ynthei roceure, which i obtaine by uing an uer F of Fig. 3(b. he inial realization of ( can be written a A B B ( = = C D D A B C D C D D (6 By auing that D = the tranfer atrix fro u to y i trictly roer, which i a conition to guarantee exitence of cloe-loo tranfer atrice. Alo the inial realization of a tabilizing controller can be written a AK BK K ( = CK D K (7 By etting w =, the tate-ace rereentation of the controlle yte can be efine a: x& = Ax Bu, y = Cx, x& K = AKxK BKy, an u = CKxK DKy. x& A BDKC BCK x x = B C A x & K K K K Cloe loo A atrix (8 he cloe-loo yte i ai to be internally table if the cloe-loo A-atrix i table, i.e. all it eigenvalue have negative real art. he H otial control roble i to ce an internally tabilizing controller K that iniize. he following conition guarantee the exitence of an otial controller K: ( AB, i tabilizable an ( C, A i etectable. hi i neceary an ufficient for exitence of an internally tabilizing controller. D ha full colun rank to enure that the control ignal u i fully weighte in the ut z. An D ha full row rank to enure that the inut ignal w fully corrut the eaure ignal y. A jωi B A jωi B, C D C D have full colun an row rank, reectively for all ω that ean there are no iaginary axi zero in the cro yte fro u to z an fro w to y. Uner thee aution, the atlab coan hinfyn ce the otial controller earching for

5 iniu γ that guarantee the exitence of a controller, i.e. < γ. 5. Deign Exale he configuration of two-ientical arallel-connecte buck converter, a hown in Fig., i coniere to verify the oelling roceure an the alicability of the H otial control eign. he ecification are given a follow. he inut voltage V in = 4V, ut voltage V o = 54V, ax. ut ower P o = 5W, witching frequency f = khz, inuctor = µh, caacitor C = µf, ut reitance R = Ω, equivalent erie reitor of caacitor r C = 5Ω, equivalent erie reitor of inuctor r = 5Ω, interconnection reitance r = 5Ω, cable reitance r = Ω. og agnitue rank equal to 3.7, hence the ret of it ingular value are le than. he controlle yte achieve noinal erforance, robut tability, an robut erforance, a hown in Fig. 8. In Fig. 9, the iulation reult of ut voltage of two-buck converter in arallel how goo rejection to line an loa iturbance. Alo the ut current of two oule are hown in Fig.. Outut Voltage (V he Outut Voltage of Firt oule ine iturbance: V for firt channel at.4 ec oa iturbance: 5A at. ec.5..5 i e (ec Fig. 9 - he ut voltage of firt oule he Outut Current of wo oule µ(n an µ(n Phae (egree Frequency (raian/ec Frequency (raian/ec Fig. 6 he boe lot of H robut controller σ( Singular Value of Cloe-loo Syte Frequency (ra/ec Fig. 7 he ingular value of cloe-loo yte RS Noinal Perforance an Robut Stability et NP Frequency (ra/ec µ(n an µ(n Robut Perforance et Frequency (ra/ec Fig. 8 NP, RS, an RP Boun Fro Fig. 6, H controller how to be table although it gain i all, however it will be increae when coenate with the erforance weight gain. Fig. 7 how the ingular value of the cloe-loo yte. he axiu ingular value of the reulting cloe-loo yte (4-inut an -ut ha he Outut Current (A Concluion i i i e (ec Fig. - he ut current he aer ha reente an accurate oel of arallel-connecte DC/DC converter incluing all interconnection reitance uing tate-ace rereentation. he uncertaintie are oelle an the H controller i eigne in orer to achieve robut tability an robut erforance. he iulation reult eontrate the effectivene of H controller that rovie robut tability an robut erforance an how goo tracking erforance an iturbance rejection caability in the reence of uncertaintie. Reference [] arabanic, D. S., an. B. Petrovic, oeling Parallel Oerating PW DC/DC Power Sulie, IEEE ran. on Inutrial Electronic, Vol. 4, No. 5, October 995,

6 [] arabanic, D., W. Dunfor, an. Petrovic, Robut Decentralize Control of Parallel DC/DC Converter, Proc. of the 998 Power Electronic Secialit Conference, Vol., [3] Rajagoalan J., K. Xing, Y. uo, F. C. ee, an B. anner, oeling an Dynaic Analyi of Parallele c/c Converter with ater-slave Current Sharing Control, Proc. of the 996 Power Electronic Secialit Conference, [4] Siri, K., an J. Bana, Analyi an Evaluation of Current-Sharing Control for Parallel-Connecte DC/DC Converter aking into Account Cable Reitance, Proc. of 995 IEEE Aeroace Alication Conference, Vol., [5] Skogeta, S. an I. Potlethwaite, ultivariable Feeback Control: Analyi an Deign, John Wiley & Son t., Wet Suex PO9 UD, Englan, Noveber 998. [6] hottuvelil, V. J., an. C. Verghee, Analyi an Control Deign of Parallele DC/DC Converter with Current Sharing, IEEE ran. on Power Electronic, Vol. 3, No. 4, July 998, Aenix A. State-Sace Rereentation Accoring to Fig., the tate equation taking into account all interconnection reitance can be written a follow: I & = V I V i (A in in il vl C gl g V& = I V i (A C ic vc C gc g I = i i, i VC = vc v C v C Vin = vin v in v in Vin when the witch i ON j where vin = j when the witch i OFF in = iag( vc = iag( C( rc r ( ( C rc r C rc r iag( C C C R r R r R r R r R r R r r R r R r R r R r R r r r R r 3 = O R r R r R r R r q q= ic = vc ( iag( r r r gc = vc r, r = [ R R R] ( ( ( ( ( il = iag r r r iag r C r C r C vl = iag iag r C r C r C = iag r C r C r C C C C ic C C C vc gl C C C gc V = iv I vv VC vg ig (A3 I = ii I vi VC ig ig (A4 =, = V v v v I i i i ii = I iag C C C ic = iag C C C = iag C C C ( (, ( ( ( (, ( vi vc ig gc iv = iag rc rc rc I ii = I iag r r r = iag r r r vv C C C vi vg C C C ig After averaging an linearization of the reviou equation, a linear tie-invariant yte i realize an rereente in tate-ace for. I & il vl I inv, in in, D gl V in = V ic vc V C gc i & g C x& = A x Bu E w (A5 = iag( D D D, in D in = iag ( Vin Vin Vin in, Vin in V iv vv I vg V in = I ii vi V C ig i g y = C x Du F w (A6 where D =. B. ranfer Function Fro the tate-ace rereentation, all tranfer function of the yte incluing interaction can be erive a follow. v co co co lo lo lo Z _ v _ v _ v _ v _ v _ v v co_ v co_ v co_ v lo_ v lo_ v lo_ v Z O v in O v v co_ v co_ v co_ v i = lo _ v lo _ v lo _ v Z in co_ i co_ i co_ i _ i lo i lo i lo i i co_ i co_ i co_ i lo _ i lo _ i lo _ i v in i O ig O i co _ i co _ i co_ i lo_ i lo_ i lo _ i i (A7 where lo _ v ji i the line-to-ut voltage, co _ v ji i the control-to-ut voltage, Z j i the loa-to-ut voltage, lo _ i jq i the co i i the line-to-ut current, _ jq control-to-ut current, i j i the loa-to-ut current tranfer function. For ilicity equation (A7 can be forulate in general for a y = u (A8 where y i an ut vector, u i a control vector, i a iturbance vector, i a lant oel, an i a iturbance oel.