Resonant FCS Predictive Control of Power Converter in Stationary Reference Frame

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1 Preprnt of the 9th World Congre The Internatonal Federaton of Automatc Control Cape Town, South Afrca. Augut -9, Reonant FCS Predctve Control of Power Converter n Statonary Reference Frame Lupng Wang K Chun Ng School of Electrcal and computer Engneerng, RMIT Unverty, Vctora 3 Autrala. School of Electrcal and computer Engneerng, RMIT Unverty, Vctora 3 Autrala. Abtract: Th paper propoe reonant fnte control et (FCS)-predctve control of a two level power converter. In the tatonary reference frame, frtly the tradtonal FCS-predctve control ytem hown to be cloed-loop feedback proportonal control ytem. In the preence of contrant, the objectve functon a weghted error functon between the dered optmal and canddate voltage control gnal. The weghtng coeffcent n the objectve functon the rato between the amplng nterval and the nductance. Secondly, t propoed n th paper to degn a dcrete-tme reonant controller for elmnatng the nuodal error ung a cacaded control ytem tructure. The propoed method mple n degn and mplementaton wth an analytcal oluton for the reonant controller gan. Expermental reult are ued to demontrate the effcacy of the propoed approach and much mproved cloed-loop control performance. Keyword: power converter, reonant control, predctve control. INTRODUCTION The predctve control of power converter ha been tuded by many reearcher n lat two decade. Kawabata et al. (99) and Kukrer (99) preent dead beat control wth PWM modulaton for a three phae PWM nverter. Fnte control et predctve control frt ntroduced by Rodrguez et al. () n. Drect predctve control wth explct oluton by Lnder and Kennel (5). To addre ue of the teady-tate error, Agulera et al. (3) ha propoed a cot functon wth ntegral term to mprove teady-tate performance. Th paper organzed a follow. In Secton, the orgnal fnte control et dcued. In Secton 3, a new FCS wth reonant controller preented to addre the nuodal error. Expermental reult are ued n Secton to llutrate the performance of propoed reonant FCS predctve control.. THE ORIGINAL FCS-PREDICTIVE CONTROL OF CURRENT FOR POWER CONVERTER The orgnal FCS-predctve control of current n the α β reference frame degned dentcally to the controller n the d q reference frame. Th mean that the um of quare error between the current reference gnal ( α, β ) and the one-tep ahead predcton of the current gnal ( α, β ) mnmzed at the amplng tme t to obtan the optmal voltage control gnal (v α, v β ). Recedng horzon control prncple appled leadng to feedback control. The author wh to thank Natonal ICT Autrala (NICTA) for fnancal upport on th project. NICTA funded by the Autralan Government a repreented by the Department of Broadband, Communcaton and the Dgtal Economy and the Autralan Reearch Councl through the ICT Centre of Excellence program.. Predctve Control ung One-tep-ahead Predcton The mathematcal model of a power converter n the tatonary frame decrbed by d α (t) = R α (t) v α (t) e α (t) () dt d β (t) = R β (t) v β (t) e β (t) () dt dv dc (t) = 3 dt C (S α(t) α (t) S β (t) β (t)) L (3) where e α and e β are the tatonary frame grd voltage, and α (t) and β (t) are the tatonary frame grd current. v dc the DC bu voltage. The manpulated varable for the current control are the voltage v α, v β n the α - β reference frame, and they are related to S α,s β and v dc (t) va the followng relatonhp: v α = S α(t)v dc (t) v β = S β(t)v dc (t) In the α β reference frame, there are even par of canddate voltage value v α and v β, whch are dependent on the DC bu voltage v dc (t). Ther exact value are characterzed by the value lted below: v dc(t) () The varaton of the canddate voltage value are caued by the change of DC bu voltage v dc. It een from th model that n the α β reference frame, there no nteracton between the current α and β, whch wll () (5) Copyrght IFAC 57

2 9th IFAC World Congre Cape Town, South Afrca. Augut -9, effectvely reduce the current controller n th reference frame to two ngle-nput and ngle- output controller. To calculate the control varable, at the amplng tme t, the objectve functon choen a um of the quared error between the dered and predcted gnal: J = ( α(t ) α (t )) ( β(t ) β (t )) ( ) T = α (t ) α (t ) β(t ) ([ α (t ) β(t ) ] β (t ) ) α (t ) β (t ) where α (t ) and β (t ) are one-tep-ahead predcton of α (t ) and β (t ), repectvely. The one-tepahead predcton of the α (t ) and β (t ) are expreed n matrx and vector form: α (t ) α (t = (I A β (t ) m ) ) β (t ) vα (t B ) m e α e β (7) (8) where I the dentty matrx wth dmenon and the ytem matrce A m and B m are defned a R A m = R ; B m = By ubttutng the one-tep-ahead predcton gven by (8) nto the objectve functon J (7), we obtan, fα (t J = [ f α (t ) f β (t ) ] ) (9) f β (t ) [ v α (t ) ] Bm T fα (t ) f α (t ) [ v α (t ) ] B T mb m [ vα (t ) ] () where the functon f α (t ) and f β (t ) are defned a fα (t ) = α (t ) α (t f β (t ) β(t ) (I A m ) ) β (t ) e α () e β We obtan the optmal control gnal v α (t ) and that mnmze the objectve functon J: vα (t ) opt opt = ( BmB T m ) Bm T fα (t ) f β (t ) = L fα (t ) () f β (t ) Note that n the α β reference frame, the ytem matrx A m dagonal and there no nteracton between the α and β current. The calculaton of v α (t ) opt and opt gnal are calar operaton. More pecfcally, nce the matrx I A m ha the form: I A m = R R from (), we have v α (t ) opt = ( α(t ) ( R ) α (t ) e α) (3) opt = ( β(t ) ( R ) β (t ) e β) () where α (t ) and β (t ) are current reference gnal n the α β reference frame, and α (t ) and β (t ) are the meaured current of the power converter. It clearly een that predctve controller ue a proportonal feedback control wth a feedforward compenaton. Furthermore, the feedback control gan dependent on the amplng nterval of the current control ytem wth the value kfc α = k β fc = ( R ) (5) In order to enure a negatve feedback n the current control, the quantty R >, that R <. Here the feedback controller have a negatve gan becaue of the negatve dagonal element n B m matrx.. FCS Current Control n α β Reference Frame It eay to how that the objectve functon J () can alo be expreed n term of the optmal voltage gnal n the α β reference frame a ( ) vα (t J = ) vα (t ) opt T ( BmB T m ) ([ vα (t ) = L ] opt ) vα (t ) opt opt (v α (t ) v α (t ) opt ) L ( opt ) () Wth both objectve functon and the optmal control gnal defned, the next tep n the FCS predctve control to fnd the control gnal v α (t ) and that wll mnmze the objectve functon ubject to the lmted number of choce of voltage varable a mentoned n (). In FCS current control propoed n the α β reference frame, the even par of v α and v β value from () are ued to evaluate the objectve functon (). The par of v α and v β that ha yelded a mnmum of the objectve functon J wll be choen a the FCS current control gnal n the α β reference frame. It worthwhle to empha that becaue the reference gnal n the α β frame are nuodal gnal, the error gnal α (t) α (t) and β (t) β (t) wll not converge to zero a t. However, a amplng nterval reduce, the α (t) α (t) and β (t) β (t) wll reduce a the feedback controller gan L R ( ) ncreae..3 Generatng Current Reference Sgnal n α β Frame The reference gnal to the FCS current control n the α β reference gnal are nuodal gnal n whch ther frequency determned by the frequency of the grd ω g. In the applcaton, the dered operatonal performance of a power converter n a cloed-loop current control 58

3 9th IFAC World Congre Cape Town, South Afrca. Augut -9, pecfed va the dered value of d and q current. For ntance, the dered value for the d current related to real power n demand and q choen to be for unty power factor. The reference gnal to the d and q current are tranparent to the applcaton and eay to chooe. For thee reaon, the reference gnal to α and β current are calculated ung the nvere Park Tranform: α (t) co ωg t n ω β (t) = g t d (t) n ω g t co ω g t q (t) (7) In the majorty of applcaton where q (t) =, mply, α (t) = co ω g t d (t), and β (t) = n ω g t d (t). Fgure how the confguraton of the feedback control ytem to generate optmal control gnal v α (t) opt and v β (t) opt. 3. RESONANT FCS CURRENT CONTROL To reduce the teady-tate error n the d and q current, the feedback error α (t) α (t) and β (t) β (t) n the current control ytem need to be reduced. 3. Control Sytem Confguraton It known from nternal model control prncple that n order for the feedback control ytem to track a perodc gnal, the gnal generator need to be embedded n the controller. For the cae of the current control n the α β reference frame, becaue the reference current gnal are nuodal gnal, the generator of a nuodal gnal hould be embedded n the feedback control ytem o that the output current gnal α (t) and β (t) would track ther reference gnal wthout teady-tate error. In hort, the controller hould have a polynomal factor co(ω d )z z, whch ( e jω d z )( e jω d z ), contaned n t denomnator where ω d the dcrete frequency of the nuodal reference gnal. In other word, there a par of complex pole contaned n the controller, R α where the locaton of the pole are at e ±jω d complex plane. on the The controller that ha the capablty to track a nuodal reference gnal or to reject a nuodal dturbance gnal called reonant controller. The reonant FCS current controller propoed to have the feedback tructure a llutrated n Fgure. In the propoed control ytem tructure, the feedback controller k α fc and kβ fc derved from the one-tep-ahead predcton and optmzaton hown n Secton are ued n the nner-loop for fat dynamc repone, whle two reonant controller are ued n the outer-loop to provde further compenaton for the trackng error between the reference and feedback current gnal n the α β reference frame. The frequency ω d the dcrete frequency, havng the unt of radan. Aumng that a nuodal gnal ha a perod of T, wth a amplng nterval, the number of ample wthn th perod T N T = T. The dcrete frequency ω d calculated a ω d = π N T = π T Suppoe that the frequency of the grd 5 Hz and the amplng nterval = 8 econd. Then the frequency parameter ω d then ω d = π. =.5 The frequency parameter ω d tme nvarant n the degn becaue the grd frequency generally aumed unchanged. In realty, t may change wth repect to tme. However, when the amplng nterval mall, th varaton ha a mall effect on the locaton of the complex pole n the controller. Let u ay that the grd frequency vare from 5 to 5.5 Hz. When = 8 econd, the correpondng ω d to 5 Hz approxmately.5 rad and 5.5 Hz.5 rad. The controller pole for the former cae are approxmately.9997 ± j.5 and the latter cae approxmately.9997 ±.5. Therefore, there no need to track the change of the grd frequency and ncorporate t n the reonant controller. e α d q - - α L Vα opt dq / αβ L β - V opt β - e β α - - k k z co ω d z z kfc α α Current Model β k k z - co ω d z z k β - fc β R Fg.. Fnte Control Set Current Control n α β frame β Fg.. Fnte Control Set (FCS) reonant current control n α β frame 59

4 9th IFAC World Congre Cape Town, South Afrca. Augut -9, 3. Outer-loop Controller Degn It eay to how that the one-tep-ahead predctve controller for the nner-loop ytem reult n a cloedloop ytem wth a tranfer functon z (Wang and Gan ()). The ame procedure appled here to obtan the ame reult n the α β reference frame. Wth the nner-loop ytem modeled a one ample of delay z, the tak of degnng the reonant controller n the outer-loop become traghtforward. Fgure 3 llutrate the outer-loop ytem for controllng current α wth the reonant controller and the nner-loop approxmated by the tranfer functon z. It clearly een that the cloed-loop ytem from the reference gnal α to α decrbed by the z-tranfer functon k z k z T (z) = coω d z z k z k z (8) Th a econd order dcrete-tme ytem wth two cloed-loop pole. Thu, the two coeffcent from the reonant controller, k and k, can be unquely determned by ung the technque of pole-agnment controller degn. For mplcty, aumng that the dered cloed-loop are dentcal, denoted a λ <, the dered cloed-loop polynomal for the dcrete ytem gven by ( λz ) = λz λ z By comparng the dered cloed-loop polynomal wth the actual cloed-loop polynomal gven by the denomnator of (8), we obtan the followng equalte: k co ω d = λ k = λ Thee equalte lead to the oluton for the gan of the reonant controller where k = co ω d λ (9) k = λ () The performance tunng parameter for the reonant FCS controller the locaton of the par of dered dcrete cloed-loop pole λ <. Th parameter elected n the degn to reflect the cloed-loop bandwdth of the control ytem, dependng on the qualty of the current model and current enor noe level. A maller λ correpond to fater cloed-loop repone for the reonant FCS control ytem, whch on the other hand, t may caue noe amplfcaton and the reulted cloed-loop ytem le robut. α - k k z α co ω d z z z Fg. 3. Outer-loop ytem wth nner-loop approxmated by a ample of tme delay If one whe to ue the cloed-loop performance pecfcaton n contnuou-tme that cloely correpond to the underlyng phycal ytem, then the dered cloed-loop polynomal choen a ξw n w n. For ξ =.77, the par of contnuou-tme complex pole are, = ξw n ± jw n ξ. Wth a amplng nterval, the par of pole are converted from contnuou-tme to dcrete-tme va the followng relatonhp: z = e ξwnjwn ξ z = e ξwn jwn ξ When the dered cloed-loop pole are elected th way, the coeffcent of the reonant controller are found by comparng the dered cloed-loop polynomal wth the actual cloed-loop polynomal gven by the denomnator n (8): k = co ω d e ξwn co(w n ξ ) () k = e ξwn () where w n the dered bandwdth for the cloed-loop current control ytem pecfed n the contnuou-tme. 3.3 Reonant FCS Control Sytem The control ytem hown n Fgure the reonant control ytem wthout the contrant of ung the fnte control et n the α β frame. The reonant control algorthm for ung the fnte control et an extenon of the uncontraned control cae. We wll frt ummarze the algorthm, followed by t dervaton. To derve how the control gnal are choen n the preence of contrant, we wll reort to the oluton va predctve control. Ung Euler dcretzaton method, the dfference equaton of current α and β at the amplng tme t can be decrbed a: α (t ) = ( R ) α (t ) v α (t ) e α β (t ) = (3) ( R ) β (t ) e β () Becaue there no nteracton between the varable n the α β reference frame, for mplcty, the predcton for α condered only, and the reult wll be naturally extended to the varable β. Defne the operator D(z ) a D(z ) = co ω d z z wth z a the backward hft operator z f(t ) = f(t ). Applyng the operator D(z ) to both de of (3) yeld α (t ) = ( R ) α (t ) v α (t ) (5) where α (t ) =D(z ) α (t ) () α (t ) =D(z ) α (t ) (7) v α (t ) =D(z )v α (t ) (8) The varable α (t ) and v α (t ) are the fltered current and voltage gnal wth the denomnator of the reonant controller. 5

5 9th IFAC World Congre Cape Town, South Afrca. Augut -9, Algorthm The reonant FCS control gnal n the α β reference frame at amplng tme t, v α (t ) and, are found by fndng the mnmum of the objectve functon J wth repect to the ndex k, J = L (v α (t ) k v α (t ) opt ) L ( k opt ) where the value of v α (t ) k and k (k =,,,..., ) are gven by the fnte control et, V dc and the gnal v α (t ) opt and opt are computed teratvely ung the followng equaton: v α (t ) opt = co ω d v α (t ) opt v α (t ) opt vα opt (t ) vα opt (t ) = kfc[k α ( α (t ) α (t )) k ( α (t ) α (t )) ( α (t ) co ω d α (t ) α (t ))] opt = co ω d v β (t ) opt v β (t ) opt v opt β (t ) v opt β (t ) = k β fc [k ( β (t ) β (t )) k ( β (t ) β (t )) ( β (t ) co ω d β (t ) β (t ))] The feedback control gan ued n the computaton are defned a: kfc α = k β fc = ( R ) k = co ω d λ k = λ λ < the dered cloed-loop pole locaton for the current control ytem. When the operator D(z ) appled to n(ωt(t )), we obtan the reult that D(z ) n(ωt(t )) = (9) By aumng that the grd frequency ω a a contant (ay n prefect grd condton), the lat term of (3) vanhe when the operator D(z ) appled to t. To nclude the reonant acton nto the controller, the weghted current error e e α(t ) = k ( α (t ) α (t )) k ( α (t ) α (t )) choen a the teadytate of the α (t ), where k and k are calculated a n Algorthm. The teady-tate of v α (t ) choen to be zero. Subtractng the teady-tate from the model (5) gve: α (t ) e e α(t ) = ( R )( α (t ) e e L α(t )) v α (t ) (3) After applyng the ame procedure to the β-ax current, we obtan the formulaton for the β varable a β (t ) e e β(t ) = ( R )( β (t ) e e L β(t )) (3) where e e β (t ) = k ( β (t ) β (t ))k ( β (t ) β (t )). The control objectve to mnmze the error functon J, where [ α (t J = ) e e ] T [ α(t ) α (t ) e e ] α(t ) β (t ) e e β(t ) β (t ) e e β(t ) (3) whch to regulate the fltered current gnal α (t ), β (t ) to be a cloe a poble to e e α(t ) and e e β (t ). By ubttutng (3) and (3) nto (3), t can be hown that the optmal oluton of v α (t ) and that wll mnmze the objectve functon (3) are gven by v α (t ) opt = k α fc(e e α(t ) α (t ) ) (33) opt = k β fc (ee β(t ) β (t ) ) (3) where the feedback controller gan are defned a kfc α = k β fc = ( R ) (35) Furthermore, the objectve functon J can be expreed va completng quare a J = L (v α (t ) v α (t ) opt ) ( opt ) (3) L Now, note that v α (t ), v α (t ) opt,, opt are fltered voltage varable. Thu, by defnton of the fltered control gnal, the followng relatonhp are true: v α (t ) opt v α (t ) opt = co ω d z z opt opt = co ω d z z whch lead to the expreon of v α (t ) opt and opt n an teratve manner: v α (t ) opt = v α (t ) opt co ω d v α (t ) opt v α (t ) opt (37) opt = opt co ω d v β (t ) opt v β (t ) opt (38) By calculatng the actual fltered control gnal ung the ame pat optmal control gnal tate, we obtan v α (t ) = v α (t ) co ω d v α (t ) opt v α (t ) opt (39) = co ω d v β (t ) opt v β (t ) opt () By ubttutng the fltered varable (37)-() nto the objectve functon (3), t become J = L (v α (t ) v α (t ) opt ) L ( opt ) () Wth the fnte control et, the derved objectve functon here dentcal to the one ued n the Algorthm. Th complete the dervaton of the reonant FCS control algorthm. It emphazed that the reonant FCS control algorthm ued the pat optmal control gnal tate (v α (t ) opt, 5

6 9th IFAC World Congre Cape Town, South Afrca. Augut -9, v α (t ) opt ) together wth the current (v α (t )) to predct the fltered v α (t ). If the pat mplemented control gnal tate (v α (t ), v α (t )) were ued n the predcton, then t could reult n accumulated error from the fnte control et and lead to teady-tate error n the reonant FCS control ytem. α α EXPERIMENTAL RESULTS Laboratory prototype of three-phae boot rectfer ued to valdate the control degn a dcued n the prevou ecton. In the expermental et-up, the ytem parameter are V ac = 3V, =.3mH, C dc = 9µF and R load = Ω. The dered cloed-loop pole locaton λ choen a.95 for reonant FCS.. Comparon tudy wth and wthout reonant controller To llutrate the performance of the reonant FCS predctve control, a comparon tudy done between the orgnal FCS and the reonant FCS n repone to tep change on d current from 3A to 5A. The expermental reult hown n Fg. are the d and q current of the orgnal FCS and the reonant FCS n the ynchronou reference frame. It hown that the orgnal FCS had a clear teady-tate error and reponded to a tep change n.9 mllecond, wherea the reonant FCS dd not have a teady-tate error. It took about.85 mllecond for d current to reach the new reference value and mllecond for q current to return to zero. The repone tme of the reonant FCS can be tuned by changng the λ whch the cloed-loop pole value of the controller. The orgnal FCS controller had a mean value of 3.39A and 5.A for the et pont 3A and 5A repectvely, whle reonant FCS controller had a mean value of 3.8A and.9a. d q (a) Orgnal FCS d q (b) Reonant FCS Fg.. d and q current of FCS and Reonant FCS n repone to tep change from 3A to 5A. Fg. 5 how the expermental reult n tatonary reference frame. It een from the fgure that orgnal FCS had a clearly teady-tate error at peak and trough of the nuodal waveform, whle reonant FCS tracked the reference wthout teady-tate error. β (a) Orgnal FCS β (b) Reonant FCS Fg. 5. α and β current of FCS and Reonant FCS n repone to tep change from 3A to 5A. ACKNOWLEDGEMENTS The author wh to thank Natonal ICT Autrala (NICTA) for fnancal upport on th project. NICTA funded by the Autralan Government a repreented by the Department of Broadband, Communcaton and the Dgtal Economy and the Autralan Reearch Councl through the ICT Centre of Excellence program. REFERENCES Agulera, R., Lezana, P., and Quevedo, D. (3). Fnte-control-et model predctve control wth mproved teady-tate performance. Indutral Informatc, IEEE Tranacton on, 9(), do:.9/tii..7. Kawabata, T., Myahta, T., and Yamamoto, Y. (99). Dead beat control of three phae PWM nverter. Power Electronc, IEEE Tranacton on, 5(), 8. do:.9/ Kukrer, O. (99). Deadbeat control of a three-phae nverter wth an output LC flter. Power Electronc, IEEE Tranacton on, (), 3. do:.9/3.8. Lnder, A. and Kennel, R. (5). Model predctve control for electrcal drve. In Power Electronc Specalt Conference, 5. PESC 5. IEEE 3th, do:.9/pesc Rodrguez, J., Pontt, J., Slva, C., Salgado, M., Ree, S., Ammann, U., Lezana, P., Huerta, R., and Corte, P. (). Predctve control of three-phae nverter. Electronc Letter, (9), do:.9/el:37. Wang, L. and Gan, L. (). Integral FCS predctve current control of nducton motor drve. In IFAC 9th World Congre. 5. CONCLUSION Th paper ha nvetgated the degn and mplementaton of a fnte control et predctve control for a two level power converter. Specfcally, the propoed approach ncluded a dcrete-tme reonant controller n the algorthm to elmnate the nuodal error. Expermental reult verfed that the degn algorthm and mplementaton are effcacou. 5