PI Controller Design of Grid-side PWM-regulated ac/dc Converters via Stability Analysis based on Passivity

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1 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG 1 P Conroller Deign of Gri-ie PWM-regulae ac/c Converer via Sabiliy Analyi bae on Paiviy Anonio. Alexanrii, Member, EEE, an Michael K. Bourouli, Suen Member, EEE Deparmen of Elecrical an Compuer Engineering niveriy of Para Rion, 65 Para, Greece a..alexanrii@ece.upara.gr, bourouli@ece.upara.gr Abrac PWM-regulae ac/c converer are of grea imporance in iribue power yem applicaion ince hey are wiely ue o inerface a variey of renewable power ource uch a win an phoovolaic generaor wih he gri. Furhermore, hey provie pecial conrol poibiliie of he acive an reacive power. n hi paper, a new abiliy analyi an eign approach for nonlinear P conroller for hi in of evice i propoe, bae on he paiviy analyi of he complee yem. o hi en, fir he nonlinear moel of he yem i obaine, on which uiably moifie P conroller are incorporae. he analyi conuce provie a e of limi, rule an eman ha houl be aifie in orer o achieve fa, accurae an aifacory repone. Furhermore, i i proven ha, in all cae, a cloe-loop yem wih eire amping i guaranee while he conroller ynamic rive he yem a he erence value. he imulaion reul inicae he effecivene of he conroller eign. Keywor- gri-ie PWM ac-c converer; P conroller eign; nonlinear conrol yem; abiliy;. NRODCON hree-phae PWM-regulae (Pule Wih Moulae ac/c (GB converer are freuenly ue in power applicaion ince hey provie ome aracive conrol an operaing feaure, i.e. accurae an fa regulaion (volage, acive an reacive power regulaion, low harmonic iorion, ec. hee feaure offer a ignifican ool in he fiel of moern power yem conrol an abiliy, where iribue generaion play a ey role [1-]. Furhermore, ince iribue generaion i mainly bae on renewable energy ource, opologie ha inclue ac/c converer ogeher wih phoovolaic (PV panel or variable pee win power generaor are wiely ue. n phoovolaic applicaion, ac/c converer are ue a inverer for he inerconnecion of he PV array wih he gri. he conrol a of he inverer, in hi cae, are he Maximum Power Poin racing [4] hrough regulaion of he c volage a an opimal level, a i i eermine from he un irraiaion an emperaure, a well a he gri volage or curren conrol in accorance wih he acive an reacive power reuiremen [5]. n variable pee win power generaion, he ue of wo bac-o-bac ac/c converer wih a c-lin i very common in oay implemenaion. For example, in Doubly-Fe nucion Generaor win applicaion, he roor i connece o he gri hrough a couple of bac-o-bac converer. n hi opology, he Gri-Sie Converer (GSC i reponible for he c volage regulaion a a cerain level while he reacive power i uually regulae o provie operaion wih uniy power facor [6-8]. Similar conrol a for GSC are aope, alo for oher variable pee win applicaion ha may inclue Permanen Magne Synchronou Generaor or Suirrel Cage nucion Generaor. n hi paper, we uy he ynamic performance of he GSC an we propoe a P conroller cheme ha guaranee abiliy. o hi en, fir, he complee moel of he funamenal evice of a gri-ie PWM-regulae ac/c converer i obaine. he moel i evelope on he ynchronouly roaing, volage oriene, erence frame ince in hi frame all he inuoial uaniie are ranforme ino c uaniie in eay-ae. hi mae poible he effecive applicaion of P conroller ha implemen he a ae for hee power evice. More preciely, i i hown ha he original yem i repreene by a nonlinear moel wih a pecial ae-pace form. On hi moel uiable nonlinear P conroller are applie an a i i proven in he paper, hi eign enure ha boh he open-loop an he cloe-loop yem mainain he main propery of paiviy. Furhermore, paiviy analyi i effecively exene o a cloe-loop abiliy analyi ha guaranee convergence o he eire euilibrium while imulaneouly i mae poible ome inereing eign coniion o be exrace, namely ome rule an eman for he elecion of he appropriae gain of he propoe conroller. Following hee eman, uiable P conroller are eigne for he cae of regulaing boh he c-ie volage a a eire value an he reacive power a zero uner large iurbance change of he c-inpu curren injece. he conuce paiviy an abiliy analyi coniue he main conribuion of he preen wor; imilar conroller, wih more or le ifference from he propoe cheme, have been applie o far, raher bae on he eigner experience [9-1] /1/$1./ 1 EEE 41

2 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG 1 Finally, o confirm he effecivene of he analyi an he eign evelope, a really har cae where a large power iurbance occur, i imulae.. MODELNG AND ANALYSS OF HE HREE-PHASE PWM-REGLAED AC/DC CONVERER he yem uner conieraion, hown in Fig.1, coni of a hree phae ac/c converer connece o he gri (repreene by a hree-phae conan ac volage inpu: a, b, c hrough an R-L filer. he c-ie coni of a capacior an a c-curren ource (a variable unnown inpu: which provie he poibiliy of uying he yem in a raher general configuraion. a b c L L L R R R inv S1 S S5 D1 D D5 1 5 a V a b V b c V c S S4 S6 D D Fig. 1. Circui iagram of he gri-ie PWM regulae ac/c converer n hi ecion, he ynamic moel of he yem i preene in he ynchronouly roaing erence frame a he angular freuency ω of he uiliy. Since he original moel of he yem conain iconinuou inpu erm ue o he wiching funcion caue by he PWM operaion, he analyi i uie ifficul. An eay way o hanle hi problem i he ue of averaging analyi [11]. nee, for conrol purpoe, boh he average moel of PWM regulae converer [1] an he circui repreenaion on he erence frame are uiable o be ue [1]. i noe ha he higher he PWM wiching freuency he more accurae (pracically ienical i he reuling moel. A. Average PWM-regulae Converer Moel in he Synchronouly Roaing Reference Frame ner balance hree-phae ac coniion, he moel can be ranforme, uing Par ranformaion [14], ino he ynchronouly roaing erence frame a he angular freuency ω of he uiliy, a follow R + L ωl + V R + L + ωl + V Vc V + CV V + V ( c c D 6 C V c (1 where, are he -axi an -axi componen of a, b, c an V, V are he -axi an -axi componen of he converer ac-ie volage. Alo,, are he -axi an -axi componen of a, b, c an, V c are he c-ie curren an c-volage repecively. A hi poin, he following remar can be mae: - he fir wo ifferenial euaion of (1 are each oher couple hrough he cro-coupling erm ωl an ω L repecively. - he hir euaion of (1 i a nonlinear ifferenial euaion ha repreen he ac-c power balance beween he ac- an he c-ie of he converer. Furhermore, we noe ha he oal acive power appeare a he ac- an he c-ie inpu of he converer (operaing a inverer in he erence frame become P V + V P V ( ac c c while he acive an reacive power a he gri ie inpu correponingly are P + Q ( ( Defining he uy-raio of he converer which are he converer regulaing inpu, in he erence frame a V V m an m Vc Vc he following ae-pace repreenaion of (1 can be obaine L R ωl m L ωl R m C V c m m V c where x [ ] Vc i he ae vecor in he erence frame an u [ ] i he inpu vecor. Since, are conan an can be coniere a a iurbance inpu, he only conrolle inpu are he uy-raio m, m. hu, one can eaily ee ha yem (4 ha a pecial form in which he conrolle inpu appear inie he plan marix in a nonlinear manner. ( ( (4 4

3 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG 1 Furhermore, one can eermine from (4 a ingle euilibrium x a 1 R ωl m ω Vc m m x L R m / for ome m, m. P an P ; hee are ue o compenae he crocoupling erm of he original yem ωl an ω L repecively. Figure how he implemenaion of he propoe conroller ha achieve hee a. V c m. CONROL SYSEM DESGN AND ANALYSS Defining he -axi of he ynchronouly roaing frame o be on he a-phae of he inpu volage we arrive a an conan P P m V c hen he acive an reacive power expreion given by ( are implifie o P an Q (5 A. Reacive Power Regulaion o implemen hi a, i.e. o regulae he reacive power of he yem in a way ha enure operaion wih uniy power facor, i i obviou from (5 ha ha o be regulae o zero a eay ae. o hi en, we propoe he following nonlinear (ue o he iviion by ae V c P conroller for he m uyraio 1 P + τ + P V c m (6 where i a poiive calar repreening he inegraor gain, while P an P are poiive calar repreening he proporional gain. hi conroller can guaranee ha or eually Q a. B. DC-volage Regulaion hi conroller coni of wo cacae P conroller. he inner one i a nonlinear conroller provie he m uy-raio 1 m + ( τ (8 P P V c where i a poiive calar repreening he inegraor gain an P, P are poiive calar repreening he proporional gain. he erence inpu i provie by an ouer P conroller, which regulae he c volage a a eire conan erence a follow ( V V ( V V τ Pc c c c c c (9 he propoe conroller a given by (6 an (8 for m an m repecively, are each oher couple hrough he erm Fig.. Complee yem of he gri-ie PWM regulae ac/c converer an he P conroller C. Paiviy Analyi Conier, now, for yem (4 he following orage funcion [15] 1 H L + L + CVc 4 4 hen he erivaive of H, uing (4, i calculae a H R R + + Vc which implie he following ineualiy H + Vc x Gu y u Pc P where obviouly y G x wih G iag{ /, /,1}. Accoring o ( an ( P c power P in injece ino he yem. P coniue he oal real negraing he la ineualiy for H from zero o, we have in (1 H ( H( x ( τ Gu( τ τ P ( τ τ heore accoring o heorem 1 of Appenix A, (1 prove ha he open-loop yem i paive for oupu y G x. Furhermore, (1 inicae ha H repreen he real energy ore in he yem. Now, aing ino accoun he ynamic of he propoe conroller an efining V an V a τ ( τ (11 4

4 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG 1 he cloe-loop yem ynamic coming from (1, (6, (8 an (11 are given by he following 5 h orer nonlinear yem ( R+ P + L + ( P ωl + ( R+ P + L ( P ωl + (a (b Vc + C ( m + m (c (1 ( ( (e where he new ae vecor i x [ ] Vc. Again, ince, are conan an can be coniere a a iurbance, all hee are no conrolle inpu. n orer o chec he cloe-loop yem paiviy, we now conier a orage funcion H he poiive efinie funcion H L + L + CV + + ( c he erivaive of orage funcion H i calculae a H L + L + CVc c + V + V P P R+ R V c y u (14 where u [ ] an 1 1 y [ Vc ]. he ineualiy hol rue ince gain are efine o be poiive calar. P, P negraing he la expreion of H from zero o, we have H ( H ( y ( τ u ( τ τ Accoring o heorem 1, of Appenix A, la ineualiy prove ha he cloe-loop yem remain paive afer he applicaion of he propoe conroller. V. CLOSED-LOOP SYSEM SABLY ANALYSS Now, coming bac a he cloe-loop yem (1 an aing ino conieraion he volage orienaion of he erence frame an operaion a eay ae, he fir wo euaion (a an (b of (1 become ( + ( ω ( ω ( R L P P P L R+ P which provie ( ω ( ( ( R+ ( R+ + ( ω L L V R+ P P P P P ( + ( ( ω R V L P P ( R + ( R+ + ( ω L P P P (15 (16 (17 n orer o enure operaion wih uniy power facor i houl be, which i aifie when ( P L ( R+ P V ω ( V (18 an ubiuing (16 ino (15 we obain ( R + ( P (19 From he above analyi, one can eaily ee ha an euilibrium of (1 exi a follow x Vc ( R+ P n he following, a abiliy analyi bae on paiviy inicae ha ue o amping an he rucure of he yem, he cloe-loop ae converge o hi euilibrium, i.e. a ( :, which correpon o ( R+ P a c-volage V V. c c From he cloe-loop yem repreenaion, a i i given by (1, one can oberve ha euaion (1-(a, (b, ( an (e are inepenen from (1-(c, i.e. from he V c volage. Hence, hee four euaion can be eparaely wrien in he form x F K x E z + K z (1 where x [ ], z [ V V ] an K iag{, }. 1 1 Marix K i a conan marix: K iag, L L, while marix F can be irecly wrien a an appropriae uare marix. Alo, E i he exernal conan inpu vecor. For he unforce yem, i.e. (1 wih E, we can ue a Lyapunov funcion, he orage funcion 44

5 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG H L + L + +. We eaily conclue ha for E, he erivaive of he orage funcion i calculae a H ( R+ P ( R+ P. Since H i raially unboune, he form of H implie aympoic abiliy of x ( a he origin. Alo, from he form of K which ha poiive iagonal erm, zero-ae obervabiliy i eaily eablihe ince x ( implie x ( an ( ( in (1. heore, he only oluion ha can 4 ienically remain in S {[ x z ] R / x( } i he rivial oluion [ x z ]. Hence, from La Salle nvariance Principle he origin i aympoically able [16]. Furhermore, ue o he fac ha he unforce yem (1 i linear, he origin i uniformly globally exponenially able. hu he yem ecribe by (1 i inpu-o-ae able for any conan inpu vecor E [16]. heore, cloe-loop yem (1 i paive wih a boune oluion. A a reul, all coniion menione in [17] are obviouly aifie for yem (1 an he cloe-loop ae [ x z ] i able an converge o he euilibrium:. However, ince hi euilibrium R+ P coincie wih he complee ae euilibrium x Vc he abiliy an R+ P convergence o x i alo proven. V. DESGN DEMANDS AND RLES FOR HE CONROLLERS GANS i obviou ha abiliy an convergence o he euilibrium i achieve uner he following reuiremen an rule: Rule 1: he ouer loop P conroller (9 houl be much lower han he inner one (6 an (8. hi rule enure ha he ouer conroller ynamic o no affec he inner conroller ynamic which have been aen ino accoun in he analyi of yem (1. Rule : he gain P an P houl be poiive. A reuire by (14, for any value of he reiance R, poiive P an P can alway enure paiviy. Rule : he gain an houl be poiive. A reuire by (14, gain poiivene guaranee poiive efiniene of H. For any value of he reiance R, poiive P an P can alway enure paiviy. From he eay-ae analyi an paricularly from (18, i i obviou ha a he value of P approache ha of ω L, he ae V of he inegraor of (6 become inepenen from he ae V of he inegraor of (8. However, paiviy analyi lea o he following rule. Rule 4: Exac cancelaion of he cro-coupling erm ω L oe no coniue a manaory eman. Since he baic hypohei of he previou eay-ae analyi i ha a fa a poible, he following rule houl hol. Rule 5: Gain P houl be large enough. Rule 6: Gain a well a houl be large enough, a hown by (16, ince curren epen on V an V ae. Finally, from (19 one can ee ha ince i regulae hrough V, he P conroller effor of (8 become maller a he gain P become maller. heore: Rule 7: Gain P can be elece o be large enough bu maller han P. V. SMLAON RESLS n orer o verify he effecivene of he propoe eign approach, he repone of he yem i imulae for he cae where a ep change occur on he c-curren inpu. Paricularly, a ime inan.5ec he c-curren value change from 1A o 5A. hi ep change reul in a large power iurbance in he acive power of he c-ie (up o 5%, i.e. from 1W o 4W an coneuenly in he acive an reacive power of he ac-ie. he yem parameer are given in able. Bae on he eman an rule analyze in he previou ecion, he conroller gain value were choen o be: P 8, 1, P 1, 4, for he inner P conroller an Pc 5, c 5 for he ouer P conroller in orer o be much lower han he inner one. Figure repreen he ep change in he c-curren, while he repone of he -axi curren, he c-lin volage V c, he acive power exchange wih he gri P an he reacive power exchange wih he gri Q are preene in Figure 4-7 for P an in Figure 8-11 for P ω L. can be oberve ha boh he c volage V c an he acive power P repone are inepenen from he value of P, ince he yem amping ha been moulae by P an P. Clearly, i i inicae ha a P value reache ha of ω L, he repone of he -axi curren an he reacive power Q exchange wih he gri i effecively enhance (hey en o be inepenen from he P repone. 45

6 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG x (A 5 P (W ime (ec ime (ec Fig.. Sep change of he c-curren Fig. 6. Repone of he gri acive power P ( P (A -. Q (Var ime (ec ime (ec Fig. 4. Repone of -axi curren ( P Fig. 7. Repone of he gri reacive power Q ( P V c (V 1 (A ime (ec ime (ec Fig. 5. Repone of c-lin volage V c ( P Fig. 8. Repone of -axi curren ( P ω L 46

7 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG 1 V c (V ime (ec Fig. 9. Repone of c-lin volage V c ( P ω L 1 x 14 From he repone obaine by imulaion i i eaily conclue ha he elecion of conroller gain bae on he frame analyze in ecion V an V reul in fa regulaion of all he yem variable o heir eay-ae value. A ranien perio proviing aifacory performance wih limie overhoo i oberve. nee, he heoreical analyi for abiliy an gain elecion i verifie. ABLE. SYSEM PARAMEERS Symbol Quaniy Value ω uiliy angular freuency 1π ra/ec -axi gri volage conponen Vol -axi gri volage conponen 7 Vol (rm R gri filer reiance.4 Ohm L gri filer inucance 5 mh C c-lic capacior 1mF V c-lin volage erence 1 Vol c P (W V. CONCLSON A nonlinear P conroller eign for PWM-regulae ac/c converer i propoe. Furhermore, he analyi preene how ha he yem amping a well a he ranien repone can be effecively enhance by elecing uiable conroller gain. o hi en, a yemaic analyi i aree for he gain elecion which i aboluely verifie by he imulaion reul ime (ec Fig. 1. Repone of he gri acive power P ( P ω L A. Paiviy Preliminarie Le he nonlinear yem x f( x + g( x u, y h( x V. APPENDX (A1 Q (Var ime (ec Fig. 11. Repone of he gri reacive power Q ( P ω L n m where x R, u, y R, f, g, h are mooh an f( h(. Le u call w w( u, y he upply rae, uch ha u, x( i hol w ( <, R + i.e. i locally inegrable. Definiion 1. (Diipaive yem Syem (A1 i ai o be iipaive if here exi a orage funcion V( x uch ha u, x( V( x( V( x( w( Paive yem repreen an imporan ube of iipaive yem, wherein he upply rae wu (, y i 47

8 r EEE nernaional Sympoium on Power Elecronic for Diribue Generaion Syem (PEDG 1 wu (, y y ( u ( Definiion. he yem wih inpu u an oupu y where n u (, y ( R i paive if here i a conan β uch ha y ( τ u( τ τ β for all funcion u an all. heorem 1. [16] Aume here i a coninuou funcion V ( uch ha V ( V( y( τ u( τ τ for all funcion u (, for all an all V (. hen he yem wih inpu u ( an oupu y ( i paive. A poine ou in [18], here i a rong lin beween paiviy an Lyapunov abiliy an uually funcion y ( τ u ( τ τ repreen he energy injece ino he yem, i.e. uually Pin ( y ( u(, where Pin ( i he oal real power injece. n hi cae, paiviy clearly how ha he power injece ino he yem i eual or le han he power ore wih he ifference coniuing real iipaion. B. Sabiliy bae on Paiviy A proven in [17], paiviy can immeiaely lea o Lyapunov abiliy for yem wih exernal inpu in he ene ha paiviy guaranee amping of he original yem an convergence o he euilibrium in eay ae. n paricular Lemma an heorem 4 of [17] eermine he coniion an aumpion guaraneeing abiliy an convergence of he ae rajecorie o he euilibrium. REFERENCES [1] European Commiion - New ERA for elecriciy in Europe. Diribue Generaion: Key ue, Challenge an Propoe Soluion, ER 91, [] N. Haziargyriou, H. Aano, R. ravani an C. Marnay, Microgri, EEE Power an Energy Magazine, vol. 5, no. 4, pp , July/Augu 7. [] A. Yazani an R. ravani, Volage-Source Converer in Power Syem. Hoboen, NJ: EEE/Wiley, Feb. 1. [4] N. Femia, G. Perone, G. Spagnuolo, an M. Vielli, Opimizaion of perurb an oberve maximum power poin racing meho, EEE ran. Power Elecron., vol., no. 4, pp , Jul. 5. [5] R. Kari, J.-P. Gauber an G. Champenoi, An mprove Maximum Power Poin racing for Phoovolaic Gri-Connece nverer Bae on Volage-Oriene Conrol, EEE ran. n. Elecron., vol. 58, pp , Jan. 11. [6] R. Pena, J. C. Clare, an G. M. Aher, Doubly fe inucion generaor uing bac-o-bac PWM converer an i applicaion o variablepee win-energy generaion, Proc. EE- Elec. Power Appl., vol. 14, no., pp. 1 41, May [7]. Acermann, Win Power in Power Syem. Hoboen, NJ: Wiley, 5. [8] G. Michale, A. D. Hanen an. Haropf, Dynamic behavior of a DFG win urbine ubjece o power yem faul, European Win Energy Conference EWEC 7, Milan, aly, May 7. [9] G. Gail, A. D. Hanen an. Haropf, Conroller Deign an Analyi of a Variable Spee Win urbine wih Doubly-Fe nucion Generaor, European Win Energy Conference EWEC 6, BL-159, Ahen, Greece, Feb. 6. [1] A. H. Kaem, E. F. El-Saaany, H. H. El-amaly, an M. A. A.Wahab, An improve faul rie-hrough raegy for oubly fe inucion generaorbae win urbine, E Renewable Power Generaion, vol., no. 4, pp. 1 14, Dec. 8. [11] P.. Krein, J. Benman, R. M. Ba an B. L. Leieure, On he ue of averaging for he analyi of power elecronic yem, EEE ran. Power Elecon., vol. 5, pp , Apr [1] R. Orega, A. Loria, P. J. Niclaon an H. Sira-Ramirez, Paiviy- Bae Conrol of Euler-Lagrange Syem; Mechanical, Elecrical an Elecromechanical Applicaion. Lonon,.K.: Springer-Verlag, [1].-S. Lee, Lagrangian Moeling an Paiviy-Bae Conrol of hree- Phae AC/DC Volage-Source Converer, EEE ran. n. Elecron., vol. 51, pp. 89-9, Aug. 4. [14] P.C. Kraue, O. Waynczu an S.D. Suhoff, Analyi of Elecric Machinery an Drive Syem, n Eiion, EEE Pre,. [15] M. K. Bourouli an A.. Alexanrii, Dynamic Analyi of P Conroller Applie on AC/DC Gri-Sie Converer ue in Win Power Generaion, in E RPG 11, paper 96, pp. 1-6, Einburgh, K, Sep. 11. [16] H. K. Khalil, Nonlinear Syem, r eiion, Prenice-Hall,. [17] G. C. Konanopoulo, an A.. Alexanrii, Sabiliy an Convergence Analyi for a Cla of Nonlinear Paive Syem, in 5 h EEE Conference on Deciion an Conrol an European Conrol Conference EEE CDC-ECC 11, pp , Orlano, FL., Dec. 1-15, 11. [18] A. J. van er Shaf, L-Gain an Paiviy echniue in Nonlinear Conrol, Springer,. 48