Fractional - order modelling and control for two parallel PWM rectifiers

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1 WeAT1.6 Fratonal - order modellng and ontrol for two parallel PWM retfer Xaohong Wang *. Th Th Gang Hoang **. Zhfeng Pan***. Yangqan Chen**** * Soth Chna Unverty of Tehnology, Gangzho, Chna, (e-mal: xhwang@t.ed.n). ** Soth Chna Unverty of Tehnology, Gangzho, Chna, (e-mal: tranghoanganh2811@gmal.om). *** Soth Chna Unverty of Tehnology, Gangzho, Chna, (e-mal: p.zhfeng@mal.t.ed.n). ****Shool of Eng g,unverty OF Calforna, Mered,CA95343, USA (e-mal: yhen53@mered.ed). Abtrat: For the prpoe of mprovng ontrol qalty, fat repone, tablty and other qalty ndexe for parallel opled PWM retfer, a well a derved from beng able to determne the atal tranfer fnton whh loer to the atal ontrol objet, the fratonal-order tranfer fnton of the threephae PWM retfer. Th paper ed a fratonal proportonal Integral ontroller ntead of a tradtonal PI ontroller. The e of the fratonal-order ontroller objet wll help to degn a hgher preon fratonal-order ontroller of the ytem. Th paper analyze the haratert of the ytem, ontrt the fnton of the fratonal-order ontroller objet, degn the rrent and voltage of the fratonalorder ontroller, perform the mlaton and related experment. Smlaton and experment relt have demontrated that ng the fratonal proportonal ntegral ontroller to ontrol the ytem of two parallel PWM retfer ha relted n deal ontrol, learly mproved ontrol qalty. eyword: Fratonal-order, Modellng, parallel, retfer 1. INTRODUCTION Wth a ngle Ple Wdth Modlaton (PWM) retfer ontrol ytem, the e of tradtonal PI ontroller gve deal ontrol relt, bt when oplng two or more PWM retfer opled n parallel, then the degn parameter are not aboltely arate and entve to hange n ytem parameter. Otpt voltage are too hgh, and the eleton of parameter for the ontroller fflt. To overome thee hortomng, th paper propoe the e of fratonal proportonal ntegral ontroller PI ntead of ng tradtonal PI ontroller. In reent year, the fratonal ontroller ha been nreangly ed n varo feld of engneerng and tehnology, epeally n ytem reqrng hgh-qalty ontrol h a retfer ontrol, nverter, motor, wnd ontrol, boler ontrol ytem, lqd level ontrol and other feld, et C.A. Monje (2010), Ne Bng (2012), Z. Zo (2013), Hang an(2013), Abdelfatah Charef (2013). For the e of fratonal ontroller to ontrol PWM retfer, many entt have alo tded and pblhed the relt, W Zhao Jn (2014) and Nazr Raba(2015), Zhfeng Pan (2017). All pblhed relt have demontrated that the e of fratonal ontroller to replae tradtonal PI or PID ontroller an aboltely relable entf ba. However, the tdy of the applaton of the new fratonal ontroller lmted to the applaton to ngle retfer ytem that are almot not applable to ytem wth two or more parallel retfer. Th paper advane one tep frther and apple a fratonal proportonal ntegral ontroller for parallel PWM retfer ontrol ytem. For eah ontrol ytem and the pef ontrolled objet, eletng the optmal ontroller tll mportant to determne the qalty of ontrol of the ytem. By pef reearh and analy, we know that a 3-phae PWM retfer ntegrated wth a three-phae grd a nonlnear ontrol objet, o t poble to ontrt a tranfer fnton for th objet, whh wll be derbed arately and n proxmty to the atal ontrolled objet. Th of great help for to degn the ontroller n the mot arate and loe way to the ontroller we need to e n the mot pratal ytem. Wth the ontrolled objet beng a fratonal objet, the hoe of a fratonal ontroller wold provde better ontrol relt than ng the tradtonal PI ontroller a before. The analy, allaton, and relt obtaned n the next tep wll prove the orretne of the hoe of the paper. 2. OVERVIEW OF THE FRACTIONA-ORDER CONTROER The overvew trtre of the fratonal order ontroller hown n Fg. 1, where I F() P D (1) 2018 Internatonal Federaton of Atomat Control 54

F: () the fratonal order ontroller; G (): the ontrolled objet; : the npt gnal; : the otpt gnal; ; ; X() P I D Y () : the fator of proportonal, ntegral and dervatve vale repetvely;,: the potve real

The fratonal order ontroller PI D ha two addtonal adjtable parameter, ompared to the tradtonal ontroller PID, o t nreae the ontrol range for the ytem and the lope of the logarthm haratert not -

2 F: () the fratonal order ontroller; G (): the ontrolled objet; : the npt gnal; : the otpt gnal; ; ; X() P I D Y () : the fator of proportonal, ntegral and dervatve vale repetvely;,: the potve real nmber. The fratonal order ontroller PI D ha two addtonal adjtable parameter, ompared to the tradtonal ontroller PID, o t nreae the ontrol range for the ytem and the lope of the logarthm haratert not - 20dB/de bt -20dB/de or +20dB/de. It eay to ee that tradtonal PID, PI and PD ontroller are jt one of the peal ae of fratonal PI D hoe of a reaonable et of parameter an mprove the ontrol qalty of the ytem. ontroller. Therefore, the expre. In addton, the degn of the ontroller rele heavly on the parameter of the ontrolled objet, o fndng the exat ontrolled objet loet to the atal objet eental. The atal model of the three-phae PWM retfer ytem after deoplng, nldng oordnate tranformaton, SVPWM vetor modlaton, dead-tme nflene are all nonlnear, therefore, defnng the tranfer fnton of the atal ontrolled objet relatvely dfflt. We ame that the ontrolled objet model of the rrent loed loop nknown (Fg. 3). Baed on the npt/otpt haratert of the real objet, t poble to ontrt a replaement ontrolled objet for the atal objet. To verfy the aray of th objet t an be done throgh devaton e (the dfferene between the oberved rrent of the ontrted objet and the rrent of the atal ytem). The fratonal order tranfer fnton of the three-phae PWM retfer expreed a (2). Fg. 1. The overvew trtre of the fratonal order ontroller 3. DESIGN OF FRACTIONA PROPORTIONA INTEGRA CONTROER FOR THE CONTRO SYSTEM COUPING TWO PARAE THREE-PHASE PWM RECTIFIERS In the ytem (Fg. 2), we e two dental opled parallel PWM retfer and ndvdally ontrolled to nreae the ytem performane and mnmze damage aed by the falt. When oplng two PWM retfer n the ytem, there wll be ome problem h a balaned rrent or average rrent method. However, de to the length ontrant, we do not d thee e, bt rather only d the degn of a ontroller for the ytem. For mplty of degn when bldng a fratonal order ontroller for a ytem, beae two PWM retfer are n parallel, theoretally the ontroller degn for two retfer the ame, o we only need to degn a fratonal ontroller for one et, then the other wll be ed the ame ontroller a the frt one. Meanwhle, n order to mprove the qalty ontrol, we degn a rrent ontroller and a voltage ontroller ndependently. 3.1 Fratonal order modellng For the three-phae PWM retfer AC/DC ytem trtred a, we realze that or ontrolled objet a nonlnear objet beae of the ollapng haratert of the IGBT, dead-tme, R ol retor and ntor are nonlnear element. o the ontrol objet of the loed-loop rrent an not only e the frt order ytem 1 to R Fg. 2. The trtre of the ontrol ytem oplng two parallel three-phae PWM retfer by ng dret rrent ontroller 1 B () T 1 T 1 Where Tp the tme ontant of the retfer. p Baed on the expermental method derbed above, we an determne the fratonal order tranfer fnton fnton defned a (3) very loe the ontrolled objet : B ( ) (2) (3) 55

* q * q PI PI q d V d Vq Controlled objet model Blak Box Atal Controlled objet Atal Controlled objet PWM ~ q e From the relt obtaned a hown n Fg. 6 and Fg. 7 we fnd that ng the ontrol objet model 0.

In the next tep th paper wll e th relt to degn a fratonal order ontroller for the ytem. Fg. 3. Sheme of the loed-loop rrent for ontrolled objet to be blt and atal ontrolled objet Fg. 4.

The dfferene between and Fg. 7. The rrent haratert of the real ytem (above) and the ytem ng new ontrted ontrol objet B() (below)when ng PI ontroller degned for ontrol objet From Fg.

3 * q * q PI PI q d V d Vq Controlled objet model Blak Box Atal Controlled objet Atal Controlled objet PWM ~ q e From the relt obtaned a hown n Fg. 6 and Fg. 7 we fnd that ng the ontrol objet model B ( ) for ontroller degn gve better ontrol qalty, fat repone tme. Th agan demontrate that bldng a fratonal order model for ontrol objet effetve and neeary. In the next tep th paper wll e th relt to degn a fratonal order ontroller for the ytem. Fg. 3. Sheme of the loed-loop rrent for ontrolled objet to be blt and atal ontrolled objet Fg. 4. The oberved rrent rve ~ and the atal rrent Fg. 6. The rrent haratert of the real ytem (above) and the ytem ng the ontrol objet 1 R ng PI ontroller degned for ontrol objet (below) when Fg. 5. The dfferene between and Fg. 7. The rrent haratert of the real ytem (above) and the ytem ng new ontrted ontrol objet B() (below)when ng PI ontroller degned for ontrol objet From Fg. 4,5, the fratonal order tranfer fnton efned a (3) whh very loe to the atal haratert of the ytem. The dferene between and 0. So (3) the fratonal order tranfer fnton of the ontrolled objet we need to fnd. To valdate the orretne of the ontrted ontrol objet model, a well a to degn two PI ontroller by ng 1 tradtonal method throgh frt order hold and the R fratonal ontrol objet B ( ) , then we e the two ontroller that were degned for the atal ytem and ompare the relt. From Shemat dagram of the three-phae PWM retfer we have Wth: R d (4) dt C Saa Sbb S (5) From (4) we an determne the atve power of the ytem 56

4 P d 2 C dt R (6) 3.2 Determne the parameter of the fratonal ontroller for the rrent loop rt,, p I Wth Udr=U+U we obtan p C r d dt d RC (7) For the three-phae PWM retfer opled n parallel, elet the expeted phae and the t-off freqeny, the degned ytem mt meet m the followng ondton: Where, R / U ; () P( ) ( R C 1) T 1 r T. R C (8) atfe arg[ F ( j ) B( j )] (13) m I F ( j ) B( j ) 1 (14) I However, we alo ee that apator C and retor R are nonlnear element, o the ontrol objet of the rrent loop rt an alo be ondered a fratonal order objet, from whh we fnd the mathematal tranfer fnton of the rt (R//C) a expreed n (9) C () T 1 Ung the ame method wth the rrent loop, oneqently, we fgre ot the objet of the voltage loop, below (10) C () (9) (10) Baed on the hemat dgrame of the PWM retfer throgh DCC and Fg. 2, we have three-phae PWM retfer ng the fratonal order ontroller a follow: Ung the rrent ontroller tranfer fnton n (11) we have o 2 p I R m (15) Where n 2 I I m (16) om jnm Rm jim (17) B( j ) For three-phae PWM retfer ytem, we hooe the expeted phae ; 2730 rad / ; let vary n (0.1:1) wth a tep of 0.1, we obtan a ombnaton of 9 otpt haratert orrepondng to dfferent vale. 45 m From (15), (16) we an determne the vale of p ; p Rm Im ot (18) 2 I F () I I p (11) I I m n 2 (19) Eqaton (11) the tranfer fnton of the rrent ontroller. I F () U p (12) Eqaton (12) the tranfer fnton of the voltage ontroller, GI () the tranfer fnton of the rrent loop rt. Fg. 8. Crrent loed loop,voltage loed loop Ue ntegrated tme and abolte error method, we determne the vale of 91.6; 1.6; 0.9 P I F ( ) I 0.9 (20) 3.3 Determne the parameter of the fratonal ontroller for the voltage loop:,, p I A for the voltage loop, a ded above, the apator C and t retane R are nonlnear element, o the tranfer fnton of apator C and retor R parallel oplng C (). In addton, from Fg. 8(b) we fnd that the rrent loop alo part of the ontrolled objet of the voltage loop, and the rrent loop a fratonal element, from whh we have a fratonal ontrol objet of the voltage loop D( ) G ( ). C( ). Callaton are the ame a for the I 57

rrent loop, from the ontrolled objet' tranfer fnton, let and, we obtan a ombnaton of otpt haratert orrepondng to dfferent vale, from expreon (15), (16) we an vary n (0.

COMPARATIVE ANAYSIS OF SIMUATION RESUTS We mlate and ompare the relt of the ytem that nlde two parallel PWM retfer ng the tradtonal PI ontroller and the fratonal order ontroller PI, the ytem mlated n

5mH R 30, We Fg. 9. Harmon FFT of grd rrent nder IO-PI ontroller (a) and FO-PI ontroller (b) 5mH ntor of the retfer ha a vale of otpt apator C=280F, load retor have the mlaton relt a hown n Fg. 9~11.

Partal enlarge of DC voltage nderio-pi and FO-PI ontroller when load hange From the mlaton relt t an be een that when ng the fratonal proportonal ntegral ontroller (FO-PI), the mlaton relt gven n the

59%), the power fator approxmately 1, when the load hanged after a 0.3 eond nterval, the ytem ng the PI ontroller (FO-PI) retrn to a table tate wth an otpt voltage abot 700V.

5 rrent loop, from the ontrolled objet' tranfer fnton, let and, we obtan a ombnaton of otpt haratert orrepondng to dfferent vale, from expreon (15), (16) we an vary n (0.1:1) wth 80 m ; 100 rad / determne the vale of From the ombnaton of the obtaned haratert, we determne the vale of p 0.244; I 28.83; 0.95 p I F ( ) U 0.95 (21) 4. COMPARATIVE ANAYSIS OF SIMUATION RESUTS We mlate and ompare the relt of the ytem that nlde two parallel PWM retfer ng the tradtonal PI ontroller and the fratonal order ontroller PI, the ytem mlated n Matlab Smlnk oftware wth the followng parameter: 220V grd voltage, 50HZ freqeny, the npt ntor of the retfer ha a vale of and the otpt apator C=3300F, load retor have the mlaton relt a follow: a) b) 5mH R 30, We Fg. 9. Harmon FFT of grd rrent nder IO-PI ontroller (a) and FO-PI ontroller (b) 5mH ntor of the retfer ha a vale of otpt apator C=280F, load retor have the mlaton relt a hown n Fg. 9~11. R 70 and the, We Fg. 11. Partal enlarge of DC voltage nderio-pi and FO-PI ontroller when load hange From the mlaton relt t an be een that when ng the fratonal proportonal ntegral ontroller (FO-PI), the mlaton relt gven n the form of voltage and phae rrent A deal nodal, the otpt voltage of the PWM retfer qkly reahe to the dered vale of 700V, the overhoot vale mall (7%), the hgh order harmon rrent very mall (2.59%), the power fator approxmately 1, when the load hanged after a 0.3 eond nterval, the ytem ng the PI ontroller (FO-PI) retrn to a table tate wth an otpt voltage abot 700V. Whle the otpt voltage of the onverter ng a onventonal PI ontroller (IO-PI) aymptot to the dered otpt voltage of 700V, the overhot vale relatvely hgh (25%), the hgh order harmon rrent reman 6.8%, the three-phae rrent waveform obtaned only loe to the deal nodal waveform. 5. COMPARATIVE ANAYSIS OF EXPERIMENTA RESUTS Experment are done wth both oplng parallel PWM retfer ytem ng onventonal PI ontroller and ytem ng fratonal ontroller PI, the expermental relt are a follow: Fg. 10. DC voltage wth IO-PI and FO-PI ontroller Smlate and ompare the relt of the ytem that nlde two parallel PWM retfer ng the tradtonal PI ontroller and the fratonal order ontroller PI, the ytem mlated n Matlab Smlnk oftware wth the followng parameter: 220V grd voltage, 50HZ freqeny, the npt Fg. 12. Crrent and voltage of two parallel PWM retfer nder IO-PI ontroller 58

ACNOWEDGMENT Th work a pported by the Projet of Gangdong Provne, Chna No.2015A010106004; 2016B090911003; the Projet of Gangzho ty, Chna, No. 201508030040. a) b) Fg. 13.

From the expermental relt hown n Fg. 12, the rrent waveform of the two retfer loe to the nodal hape, bt there a relatvely hgh rrent harmon omponent of 6.6% (Fg. 13a).

13b), the total rrent of the two retfer (I) alo the deal nodal rrent (Fg. 14b) and n both ae, the retfer DC voltage table at the dered voltage vale (Fg. 12, Fg. 14). The power fator fatorzed o 1.

Th relt demontrate the orretne of the argment preented above. 6.

6 ACNOWEDGMENT Th work a pported by the Projet of Gangdong Provne, Chna No.2015A ; 2016B ; the Projet of Gangzho ty, Chna, No a) b) Fg. 13. IO-PID ontrol (a) and FO-PID ontrol (b) parameter nder parallel program of eletr meter reord expermental data a) b) Fg. 14. Tet Crrent and voltage of two parallel PWM retfer nder FO-PI ontroller. From the expermental relt hown n Fg. 12, the rrent waveform of the two retfer loe to the nodal hape, bt there a relatvely hgh rrent harmon omponent of 6.6% (Fg. 13a). When ng the PI ontroller, the obtaned three-phae rrent of the two retfer an deal nodal rrent, the harmon rrent drop dramatally by 2.9% (Fg. 13b), the total rrent of the two retfer (I) alo the deal nodal rrent (Fg. 14b) and n both ae, the retfer DC voltage table at the dered voltage vale (Fg. 12, Fg. 14). The power fator fatorzed o 1. U Expermental relt demontrate that the degned ontroller not only meet the ontrol reqrement of the parallel PWM retfer ytem bt alo mprove many of the ytem qalty ontrol pefaton. Th relt demontrate the orretne of the argment preented above. 6. CONCUSIONS From the relt of the mlaton and the experment, t wa fond that the fratonal order PI ontroller broke the vewpont of the tradtonal PI ontroller. Bldng a fratonal-order ontroller objet that replae the prevo mathematal tranfer fnton of the ontrolled objet of great gnfane n bldng a more arate and effent ontroller for the atal ytem. The ytem more adaptable, the ontrol qalty enhaned, the harmon rrent reed, and the power fator mproved. Moreover, f the ytem parameter are ontrolled n a ertan range, the fratonal order PI ontroller an tll be ontrolled wthot adjtng the parameter of the ontroller. Th the trength of the fratonal order ontroller. REFERENCES Z. Zo,., Zho, Z., Wang, M. and Cheng.( 2013) Fratonal-order repettve ontrol of programmable AC power ore. IET Power Eletron, Vol 7( no. 2). pp Hang an., Zho Xao ang. and Xang Jan Hong. ( 2013). Self adjtng degn on parameter of the fratonal oder PID ontroller. Sytem Engneerng and Eletron, Vol 35( no. 5) Nazr Raba., Zho e ang., Waton Nevlle. and Wood Alan. (2015). Analy and ynthe of fratonal order repettve ontrol for power onverter. Eletr Power Sytem Reearh, no C.A. Monje., Y. Chen., B.M. Vnagre., D. Xe. and V. Fel. (2010.) Fratonal-order ytem and ontrol Fndamental and applaton. Sprnger. Abdelfatah Charef., Mohamed Aabaa., Samr ada., and Jean-Jaqe oea.( 2013). Fratonal order adaptve ontroller for tabled ytem va hgh-gan feedbak. IET Control Theory and Applaton, vol 7(no. 6) , Wang Xan.,Zhang Xao feng.,zho Feng. And Zhang Ha xa. (2013) Dret Power Control Baed on Fratonal- order PID Controller of Three- phae Voltage Sore PWM Retfer. Eletr drver. Vol 43( no. 4) Ne Bng.,Zhao H Mn.,Go Yong Xang. and Wen. (2012)Applaton of Fratonal Order PI Controller n Speed Adjtment Sytem of AC Motor. Jornal of Dalan Jaotong Unverty. Vol 33( no. 2). W Zhao Jn. and Zhang Yan Bn.( 2014). Vetor ontrol of three-phae voltage ore PWM retfer baed on fratonal-order ontroller. 26th Chnee Control and Deon Conferene- CCDC. no Th Th Gang Hoang., Xaohong Wang., Xaozhang Zho., anfang Tan. and WeChang Chen (2017). Fratonal Proportonal Integral ontroller appled nto two parallel 3-phae PWM Retfer IEEE 2nd Advaned Informaton Tehnology, Eletron and Atomaton Control Conferene (IAEAC 2017) Zhfeng Pan., Xaohong Wang., Th Th Gang Hoang., Yng o., Yangqan Chen. and anfang Tan. (2017). Degn and Applaton of Fratonal Order PI D Controller n Grd-Conneted Inverter ytem. ASME 2017 nternatonal Degn Engneerng Tehnal Conferene & Compter and Informaton n Engneerng Conferene (IDETC/CIE 2017). V009T07A

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