A Three-Phase AC Power Source using Robust Model Reference Adaptive Control

Transcription

1 A hree-phase AC Power Source usng obust Model eerence Adaptve Control Eerson Govan Carat, Carlos Mendes chter, Hlton Abílo Gründlng UFSM/C/DELC/NUPEDEE CEP: Santa Mara, S Brazl eerson@engneer.co, rchterc@zaz.co.br, ghlton@pequ.ctlab.us.br Abstract hs paper presents the developent o an equlbrated three-phase power source, whch s able to generate snusodal waveors wth adjustable apltudes and requences, as well as several arbtrary waveors. he setup s based on a coputer-controlled three-phase PWM nverter syste, whch odel s rewrtten usng ab transoraton. A robust odel reerence adaptve controller (MAC) s used to assure syste robustness and perorance. Sulaton and experental results are used to show the closed-loop syste perorance, when derent waveors are generated under several operaton condtons. I. INODUCION Several standard tests n electrcal power devces requre the use o a three-phase AC power source at derent voltages and requences. Inducton otors, power and dstrbuton transorers, unnterruptble power supples (UPS) and standby power supples (SPS) are exaples o devces that need these tests. Soe o these devces need addtonal haronc tests such as the njecton o haroncs n power and dstrbuton transorers. In developng an AC power source we ay experence soe probles. Due to haronc dstorton, odel reerence trackng errors ay appear. o copensate these errors, elevated swtchng requences or hgh perorance controllers are needed. Low [1], [], proposes a snglephase AC power source usng respectvely sldng ode control and generalzed predctve control (GPC), wth nverter swtchng requency at 5 khz. However, n hgh power statc converters, swtchng losses ncrease wth the elevaton o the swtchng requency. Eorts have been done or that hgh power converters obtan good perorance even operatng at low swtchng requences [3]-[6]. Such systes deand hgh perorance controllers to copensate eventual or perodc dsturbances n the waveors, such as the ones that appear under nonlnear loads. In addton, odelng errors and unodeled dynacs are qute coon, due to splcatons o the odel and characterstcs o the plant. All these actors nluence drectly the converter perorance, and, n ore crtcal cases, they even coprose the controllers used n the syste, as descrbed n [7]. Gründlng et al [7], ntroduce a robust odel reerence adaptve controller (MAC) appled or sngle-phase unnterrupted power supples, whch even operatng at low swtchng requences presents results wth good transent response. In [8], Carat et al apply MAC or a three-phase UPS structure, assurng robustness and good perorance. In ths work t s pleented a three-phase AC power source that s able to generate equlbrated snusodal voltages wth adjustable apltudes and requences. Moreover, haroncs can be added to the undaental sne wave, so arbtrary waveors can be generated. he threephase syste odel s obtaned usng αβ transoraton. A MAC controller s desgned to guarantee robustness and good perorance. Sulaton and experental results are presented to very the dynac perorance o the resultng closed-loop syste. hs paper s organzed as ollows: the Secton II contans the descrpton o the plant whle n the Secton III presents the closed-loop syste structure. he proposed MAC control law s descrbed n the Secton IV. he Secton V presents the paraeter adaptaton algorth. Sulaton and experental results are gven n the secton VI. he secton VII concludes the paper. II. PLAN DESCIPION he proposed syste shown n Fg. 1 s ored by a threephase nverter, a delta ( ) connected LC lter and three-phase load. Syste state space odel s gven by: and = S 1 3L V V 1 S = S 1 V 1 V S S V V ( ) = + 1 3C 1 1 S UW VW 1 S ( V V ) V = +. S S L L SL, S, (.1) (.) In (.), the load currents L and SL depend o the conguraton and balancng condton o the load. he resstve load s connected n wye (Y) conguraton as shown n the Fg. a. he load currents L and SL, n the Fg. a, are gven by: VS VS =, 3 3 L + VS VS =. (.3) 3 3 SL +

2 algorth coputes separately the orthogonal control varables, u α and u β. hen, the three-phase control varables, u u, u v and u w, obtaned by the -1 transoraton, are converted n PWM sgnals, u, v and w, where (k) = u k, u ( k) E / and s + ( ) E the saplng perod. hen, the PWM nverter generates the LC lter nput voltages, Vu, Vv and Vw, respectvely. Fg. 1. hree-phase PWM nverter syste. When the load s connected n delta ( ) conguraton, as shown n the Fg. b, the load currents L and SL becoe: VS VS VS VS =, SL = +. (.4) L + (a) he nverter swtches S 1-6 are turned on and o once every saplng nterval, so that the lne to lne nverter voltages, V UV, V VW and V WU, are pulses wth apltude E, and E. hese nverter output PWM voltages generate the output syste voltages, V S, V S and V. III. CLOSED-LOOP SYSEM SUCUE he three-phase reerence odel outputs, y r, y s and y t, are converted n αβ coordnates, y α and y β, respectvely, by a transoraton, as ollows: = (3.1) = 1 3 (3.) In ths pont s nterestng to note that y α and y β are ndependent orthogonal varables. Such as shown n the Fg. 4, the control laws u α and u β and the trackng errors e α and e β are coputed separately or each axs α and β, respectvely, usng a sngle MAC controller or each axs. Usng the transoraton, the load easured voltages varables, V r, V s and V t are converted n orthogonal output voltage varables, V α and V β. he syste output errors, e α and e β, are obtaned through a coparson o these varables wth that ones ro the reerence odel, y α and y β, respectvely. Fro these errors, the MAC Fg.. Load conguraton: (a) wye -Y and (b) delta -. (b) Fg. 3. eerence odel o a three-phase UPS syste. Fg. 4. Block dagra o a three-phase UPS syste wth MAC controller.

3 IV. MAC CONOLLE SUCUE Consder a sngle-nput sngle-output plant (SISO) as presented n Fg. 5: y = G(z)u (4.1) y = G (z) 1+ µ (z) + µ (z) wth [ [ ] ] u a Z (z) G (z) = k p (4.) (z) where G(z) s the transer uncton o the plant, G (z) s the strctly proper transer uncton o the odeled part o the plant, µ (z) and µ a (z) are addtve and ultplcatve perturbatons, respectvely. Z (z) and (z) are onc polynoals wth degree and n, respectvely. As n [7] the nput u and the output y, n the dscrete te doan, are used to generate n-1 densonal auxlary vectors, so that 1 ù 1(k) = (zi F) q u(k) (4.4) 1 ù (k) = (zi F) q y(k) where F s a atrx stable and (F,q) s a controllable par. he plant nput s taken as ( k) = q ( k) w( k) + c r( k) u where k) = [ è (k), è (k), θ (k)] (4.5) è ( 1 3 s a (n-1) densonal vector o control paraeters, c (t) s a scalar gan, (k) [ ù (k), ù (k), y(k) ] ù =. he reerence 1 y odel trackng error s e1 (k) = (k)- y(k). V. PAAMEE ADAPAION ALGOIHM here are a nuber o well-known paraeter estaton technques whch have been successully appled to dentcaton probles [1], [1]. In these schee s consdered a recursve least-squares (LS) oded algorth: t sε1(k) P(k) z(k) q(k + 1) = ( I t sσp(k) ) q( k) (5.1) (k) Fg. 5. MAC UPS P(k + 1) = (1 + tsλµ ) P(k) L P(k) z(k) z (k) P(k) L ts + µ (k) P (k) (5.) Concernng the odeled part o the plant G (z) the ollowng assuptons are ade: S1. Z (z) s a onc Hurwtz polynoal o degree ( n-1). S. (z) s a onc polynoal o degree n. S3. he sgn o k p and the values o and n are known. Concernng the unodeled plant part s assued that: S4. (z) s a strctly proper stable transer uncton. a S5. (z) s a stable transer uncton. S6. A lower bound p > or whch the poles o (z-p) and (z-p) are stable are known. a he control objectve s: Gven the reerence odel y = W (z) r ( K D (z))r (4.3) = where (k) P = P s so that < P() λ I, [ (k ] 1 ) µ k µ µ, = 1+ α, z ( k) = W (z)iw(k), ( u(k) + y(k) 1) (5.3) (k + 1) = (1 ts δ)(k) + tsδ1 +, δ1 () >, 1 1 δ δ, (5.4) where α 1, δ, δ 1, λ, µ and are postve constants and δ + δ n p, q, q s such that the poles o W (z-q ) and the egenvalues o F + qi are stable and δ s a postve constant. p > s dened δ satses [ ] n S6 and σ n (5.1) s gven by + where D (z) s a Hurwtz polynoal o degree n * = n - and r(t) s unorly bounded, desgn a adaptve controller so that or soe µ * > and any µ [, µ * ) the resultng closed-loop plant s stable and the output plant y tracks the reerence odel output y as closely as possble or all perturbatons (z) and (z) satsyng S4 - S6. a σ = σ σ q M 1 M q < M q M q M (5.5)

4 > µ where M > q and σ + are desgn paraeters. he oded error n (5.1) s dened n [7] and s gven by or ε ( k) = e ( k) + è (k) z( k) W ( z) è ( k) ( k) 1 1 w ( k) = (k) z( k) + (k) ε1 µη (5.6) he recursve least-squares (LS) used as the paraeter adaptaton algorth has ast convergence copared wth others algorths. VI. ESULS In order to very the analyss and to deonstrate the perorance o the proposed three-phase AC power source usng MAC controller (Fg. 5), both sulaton and experental results have been carred out. able I presents the paraeters o the reerence odel, three-phase PWM nverter, LC lter and load, used to n the sulaton and experental results. the regulaton capablty o the syste. Fg. 8 shows α and β voltages, where sold lnes represent the output voltages and dashed lnes represent reerence odel voltages. As t can be seen, trackng error s very sall. o show trackng capablty to arbtrary waveors, Fg. 9 presents a coplex equlbrated three-phase set o reerence odel voltages, generated by the three shted reerences wth the ollowng waveor: 5 (.6 sn (ωt) +. sn(4ωt) +. sn (5ωt)), where ω = π1 rad/s. espectve output voltages are presented n Fg. 1. Alost no derence can be seen between the waveors n Fg. 9 and Fg. 1. DC coponents were added to the rst and the thrd waveors ntendng to ake clear the results. ABLE I EFEENCE MODEL, 3φ PWM INVEE, LC FILE AND LOAD PAAMEES eerence Model Paraeters Syste Paraeters LC lter nductance L = 1 H LC lter capactance C = 6 µf Load = 1 Ω LC lter nductance L = 5.4 H LC lter capactance C = 75µF Inductors resstance =.1 Ω Load = 17 Ω Output recter capactor C rec = 33µF Saplng nterval t s = 1/18 s Fg. 6. Paraeter adaptaton (q). A. Sulaton esults Sulaton results were obtaned to deonstrate the eectveness o the control schee to track reerence odel outputs n the three-phase AC power source, even wth snusodal or ore coplex waves. eerence odel, wth the paraeters presented n able I, has the ollowng transer uncton: W ( z).194 z =. z z he three-phase AC power source was sulated wth a lnear delta ( ) connected load, such as n Fg. b. Sulated results are presented n Fgs Fg. 6 show the varaton o the dented paraeters n the drect MAC control appled to the syste. he recursve least-squares algorth guarantees ast paraeters convergence. hree-phase output voltages are presented n Fg. 7. An apltude step s appled to show Fg. 7. hree-phase 6 Hz output voltages wth a step reerence. Fg. 8. α and β output (sold) and reerence odel (dashed) voltages.

5 Fg. 9. hree-phase reerence odel output voltages ollowng 5 (.6 sn (ωt) +. sn( 4ωt) +. sn (5ωt)). Fg. 11. hree-phase output voltages ollowng 75 (.6 sn (ωt) +. sn( 4ωt) +. sn (5ωt)). Fg. 1. hree-phase output voltages ollowng 5 (.6 sn (ωt) +. sn( 4ωt) +. sn (5ωt)). B. Experental esults In the pleented three-phase AC power source prototype t was used a syste as n Fg. 1, wth paraeters o able I. A lnear wye (Y) connected load was used, as llustrated n Fg. a. Fg. 11 shows the three-phase AC power source response or a three-phase set o reerence odel voltages, ollowng 75 (.6 sn (ωt) +. sn(4ωt) +. sn (5ωt)), where ω = π1 rad/s. esults are slar to that obtaned n the sulaton shown n Fg. 1. o deonstrate syste capablty to generate arbtrary waveors Fg. 1 presents the generaton o a undaental plus second and seventh haroncs waveor n each phase. Spectral analyss was ade and s presented n Fg. 13, where t can be seen the presence o second and seventh haroncs. A non-lnear load s ored by a three-phase ull-brdge recter wth an output lter capactor (C rec ) as shown n Fg. 14. Connectng the non-lnear load together wth the prevous load (Fg. b), the three-phase AC power source response s presented n Fg. 15. For splcty, just one phase voltage and current responses are presented. Fg. 1. hree-phase output voltages ollowng 75 (.6 sn (ωt) +. sn( ωt) +. sn (7ωt)). Fg. 13. Spectral analyss o one waveor o Fg. 1, showng nd and 7 th haroncs.

6 syste. he proposed three-phase AC power source s partcularly applcable to hgh power AC systes that requre testng under varable apltudes and requences, as well as the presence o haroncs. VIII. EFEENCES Fg. 14. hree-phase ull-brdge recter wth output lter capactor (C rec) and resstve load (). Fg. 15. One phase output voltage (up) and lne current (down) or a nonlnear load. he presented experental results are lted n requency and apltude due to the swtchng requency capabltes and the apltude voltage prototype ltatons. Better results are obtaned a hgher nverter swtchng requency and a hgher LC lter cuto requency are used. In ths way, the proposed schee can generate better qualty waveors as well as ore haroncs can be added to the waveor. At ths te a new prototype wth 6 khz swtchng requency s beng developed, that can generate a hgh qualty 6 Hz snusodal waveor, wth up to haroncs. VIII. CONCLUSIONS hs paper proposes a hgh perorance three-phase AC power source ored by a coputer controlled PWM nverter wth a LC lter. he schee s controlled by a robust odel reerence adaptve controller (MAC), whch guarantees robustness. he syste s able to generate three equlbrated phases wth pure snusodal or arbtrary waveors. Sulaton and experental results deonstrate the eectveness o the proposed schee, generatng several waveors wth good response or derent loads. Moreover, ths schee can be desgned or a reduced order plant, wthout the a pror knowledge o exact odel o the load, LC lter and PWM nverter [ 1 ] K.-S. Low, A DSP-Based Varable AC Power Source, IEEE rans. on Instru. and Measur., vol. 47, no. 4, p , aug. 1998; [ ] K.-S. Low, A DSP-Based Sngle-Phase AC Power Source, IEEE rans. on Ind. Electron., vol. 46, no. 5, p , oct [ 3 ] K. P.Gokhale, A. Kawaura,. G. Hot, Dead Beat Mcroprocessor Control o PWM Inverter or Snusodal Output Waveor Synthess, IEEE rans. on Ind. Appl., vol. IA-3, no. 5, p. 9-99, sep./oct [ 4 ]. Zhang, F. C. Lee, D. Boroyevch, H. Mao, New Hgh Power Hgh Perorance Power Converter Systes, IEEE Ind. Electr. Con., p. 8-14, [ 5 ] S.-Y. Lee, Y.-M. Chae, G.-H. Choe, H.-S. Mok, D.-H. Jang, A New Control Strategy or Instantaneous Voltage Copensator Usng 3-Phase PWM Inverter, IEEE Ind. Elect. Con., p , [ 6 ] A. Kawaura,. Haneyosh,. G. Hot, Deadbeat Controlled PWM Inverter wth Paraeter Estaton Usng Only Voltage Sensor, IEEE Power Electr. Spec.Con., [ 7 ] H. A. Gründlng, E. G. Carat, J.. Pnhero, A obust Model eerence Adaptve Controller or UPS Applcatons, IEEE Ind. Electr. Con., p , [ 8 ] E. G. Carat, H. A. Gründlng, J.. Pnhero, obust Model eerence Adaptve Control Appled to hree- Phase Power Supples, Brazlan Con. on Power Electroncs, p , sep. 1999; [ 9 ] H. W. V. D. Broeck, H. C. Skudenly, G. V. Stanke, Analyss and ealzaton o a Pulsewdth Modulator Based on Voltage Space Vectors, IEEE rans. on Ind. App., vol. 4, no. 1, p , jan. 1988; [ 1 ] P. Ioannou and K. sakals, A obust Drect Adaptve Controller, IEEE ransactons on Autoatc Control, Vol. AC-31, n o 11, pp , [ 11 ]. Lozano-Leal, J. Collado and S. Mondé, Model eerence obust Adaptve Control Wthout A Pror Knowledge o the Hgh Frequency Gan, IEEE ransactons on Autoatc Control, Vol. AC-35, n o 3, pp , 199.