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620 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 3, MARCH 2009 A Simple Single-Input–Single-Output (SISO) Modelfor a Three-Phase PWM Rectifier Bo Yin , Member, IEEE , Ramesh Oruganti , Senior Member, IEEE , Sanjib Kumar Panda , Senior Member, IEEE ,and Ashoka K. S. Bhat , Fellow, IEEE Abstract —Thechallengeincontrollingathree-phasepulsewidthmodulation(PWM)rectifierunderbalancedconditionsarisesfromthe fact that the state-space averaged model reported in literaturehas a multi-input–multi-output nonlinear structure and further-more exhibits a nonminimum phase feature. In this paper, a simplesingle-input–single-output model is constructed by separating the d -axis and the q -axis dynamics through appropriate nonlinearfeedforward decoupling while maintaining nearly unity power fac-tor operation. With the proposed model, the nonminimum phasefeature inherent in an ac-to-dc rectifier becomes a simple right-half-plane zero appearing in the small-signal control-to-outputtransfer function. In addition, the model exhibits a close similaritytoadc–dcboostconverterunderbothlarge-signalandsmall-signaloperating conditions. This makes it possible to extend the systemanalysis and control design techniques of dc–dc converters to thethree-phasePWMrectifieralso.Thevalidityoftheproposedmodelhas been verified experimentally in the frequency domain underopen-loop operation of the PWM rectifier. The usefulness of themodel is further demonstrated through closed-loop operation of the rectifier with both voltage mode and inner-current-loop-basedschemes. IndexTerms —Pulsewidthmodulation(PWM)powerconverters,voltage mode control. I. I NTRODUCTION O VER the past several years, considerable research work [1]–[14] has been carried out on the control of ac-to-dcpulsewidth modulation (PWM) rectifiers (Fig. 1), since theseconverters possess many desirable features such as sinusoidalline currents at a required power factor, a nearly constant dcoutput voltage, and bidirectional power delivery capability. Asthe filter capacitor required is generally small under balancedsupply voltage conditions, it may also be believed that theseconverterscanofferexcellentdynamicresponseofthedcoutputvoltage. Manuscript received June 16, 2006; revised August 6, 2007. First publishedMarch 24, 2009; current version published April 8, 2009. Parts of this paperhave been presented at the 31st Annual Conference of the IEEE Industrial Elec-tronics Society (IECON 2005), North Carolina, November 2005, and at the32nd Annual Conference of the IEEE Industrial Electronics Society (IECON2006), Paris, France, November 2006. Recommended for publication by Asso-ciate Editor J. Kolar.B. Yin is with the National University of Singapore, Singapore 117576,and also with Vestas Technology R&D Singapore Pte Ltd., Singapore 189720(e-mail:
[email protected]).R. Oruganti and S. K. Panda are with the Centre for Power Electronics, De-partment of Electrical and Computer Engineering, National University of Sin-gapore, Singapore117576 (e-mail:
[email protected];
[email protected]).A. K. S. Bhat is with the Department of Electrical and Computer En-gineering, University of Victoria, Victoria, BC V8W 2Y2, Canada (e-mail:
[email protected]).Digital Object Identifier 10.1109/TPEL.2008.2012529 Control of three-phase PWM rectifiers in the d – q syn-chronouslyrotatingframe(SRF)hasbeendevelopedfromfield-oriented control techniques [15], [16] for ac drives in early1980s.Normally,thecontrolobjectivesofaPWMrectifieraretoregulatethedcoutputvoltageonthedcside,achieveunitypowerfactor (UPF) operation on the ac side, and also to achieve fastdynamic response to line and load disturbances. A state-spaceaveraged model has been proposed for the three-phase PWMrectifier in the d – q SRF [3]–[5]. However, the model, thoughaccurate, does not give sufficient insight into the controller de-sign and behavior of the three-phase PWM rectifier system duetoitscomplexmultiinput–multioutput(MIMO)nonlinearstruc-ture and the presence of a nonminimum phase feature [5]. Dueto this, designing a proper controller for such a converter hasbeen generally a challenging task.It is shown in [5] that direct control of the dc output voltageby means of input–output linearization is not possible as thedc output voltage is a nonminimum phase output variable. If carried out, the resultant internal dynamics of the d -axis currentwill be unstable. On the other hand, by selecting the d -axis andthe q -axiscurrentsasdummyoutputvariables,input–outputlin-earization can be applied and a first-order stable zero dynamicssystem can be obtained for the dc output voltage. Regulation of the dc output voltage can be achieved indirectly once the d -axiscurrent is well controlled.Inapparentcontrast,directcontrolofthedcoutputvoltagehasbeen realized by means of input–output feedback linearizationin [6], by using a simplified model. It may appear that theprediction in [5] and the approach in [6] contradict each other.However, it is worth noting that the nonminimum phase fea-ture inherent in a three-phase boost-type PWM rectifier is notreflected in the simplified model in [6]. This paper shows thatdesignignoringRHPzeroispossibleprovidedtheswitchingfre-quencyislowsuchthattheRHPzeroisatorbeyondtheNyquistfrequency or the closed-loop system bandwidth is designed tobe low.Several models for PWM rectifiers operating under bal-anced input supply conditions are available in literature [3]–[8].As mentioned earlier, the popular state-space averaged model[3]–[5] does not give sufficient insight into the controller designdue to its complex MIMO nonlinear structure and the presenceof the nonminimum phase feature. The models in [6] and [7] dosimplifythesystemstructureandcontrollerdesignbyoverlook-ing the inherent nonminimum phase feature in a PWM rectifier.Thisisparticularlysoin[7],wheretheestablishmentofaMIMOlinear model theoretically enables arbitrary pole placement inthe closed-loop system. However, although the nonminimum 0885-8993/$25.00 © 2009 IEEE YIN et al. : SIMPLE SINGLE-INPUT–SINGLE-OUTPUT (SISO) MODEL FOR A THREE-PHASE PWM RECTIFIER 621 Fig. 1. Structure of a three-phase ac-to-dc PWM rectifier. phase feature is absent in the models proposed in [6] and [7], itdoes exist in the real boost-type PWM rectifier. The existenceof this nonminimum phase feature imposes a strict limit on theachievable closed-loop bandwidth of the real system. The sim-plified models in [6] and [7] are valid only if the closed-loopsystem operates within this limit. Thus, the information regard-ing the location of the nonminimum phase feature in the modelis required for proper controller design even if the simplifiedmodels in [6] and [7] are being used.Anovelreduced-order(RO)small-signalmodelhasbeenpro-posed in [8]. The three-phase PWM rectifier has been modeledhereasadc–dcconverterwithequivalentpowertransfercapabil-ity and small-signal characteristics. However, the second-orderROmodelproposedisa one-sixthlinefrequencyaveragedmodel thatcapturesonlytheequivalentpowertransfercapabilityoftheconverter.Inaddition,theestablishmentoftheROmodelneitherreducesthenumberofrequiredcurrentcontrollersnorsimplifiesthe control task from a tracking into regulation problem. Threeidentical current controllers are needed to be implemented fortrackingpurposesbasedonthismodel.Besides,thesignificanceof the rooted nonminimum phase feature in the design of thecontroller is not brought out in the RO model.The fact that the nonminimum phase feature of a PWM rec-tifier presents itself as a right-half-plane (RHP) zero in thesmall-signal transfer function from current reference to dc out-put voltage in a control-oriented model has been explored inthe research reported in [11]–[14]. In this approach [11], [12],linear decoupling terms were applied to the overall PWM rec-tifier model in order to reduce cross coupling between d - and q -axis currents. The current loops were then closed with con-ventional p-type average current mode controllers. From theresulting model, the control-oriented model between dc out-put voltage and d -axis current reference was obtained by per-forming a small-signal analysis. The control-to-output transferfunction, G c = v dc / i d ∗ , which contains an RHP zero and astable pole, has been obtained by approximation of the mea-sured frequency response in the low- and mid-frequency range.The control-oriented model presented by this control-to-outputtransfer function facilitates voltage loop design as the model re-duces to a single-input–single-output (SISO) system after clos-ing the current loops [13]. The presence of the RHP zero in botha three-phase PWM rectifier system and a dc–dc boost con-verter shows similarity between them. These results give impor-tant insights into the behavior of a three-phase PWM rectifiersystem.However, the control transfer function developed in [11]and [12] only links the dc output voltage to the d -axis refer-ence current. Also, this link is established through the use of experimental frequency response and also assuming a certainform of current controller (p-type). The control-oriented modeldoes not link the output voltage to the operation of the rectifierswitches as would be expected in a detailed model of a powerconverter.This paper may be viewed as a continuation of the effortin [11]–[14] to obtain a simple and accurate control-orientedmodel for the three-phase PWM rectifier. It proposes a simpleSISO model for a three-phase rectifier. In the proposed model,the MIMO system is first decoupled into two SISO systems inwhich the q -axis model is a first-order linear system determin-ing the regulation of power factor, whereas the d -axis model,which is shown to be similar to that of a traditional dc–dc boostconverter, is a second-order nonlinear system determining thepowerdelivery.Thereafter,asimpleSISOmodelcanbeobtainedfor the d -axis operation, when the q -axis current is controlledto be zero or near zero. It was found that the proposed SISOmodel is similar to a dc–dc boost converter under both large-signal and small-signal operations. In the small-signal model,the complex nonminimum phase feature inherent in an ac-to-dcrectifier becomes a simple RHP zero appearing in the small-signal control-to-output transfer function between the dc outputvoltage and an “equivalent” duty cycle of the system.The proposed SISO system can be operated in a “quasi-open-loop”modewithoutputvariablebeingthedcoutputvoltageandthecontrolinputbeingan“equivalent”dutycycle,whichwillbedefined later. This open-loop characteristic of the SISO modelis examined in frequency domain. It is found that the measuredfrequency response shows an excellent agreement with the pre-dictedresponse.ThefindingvalidatestheproposedSISOmodel.The fact that the SISO model shows a close similarity tothat of a traditional dc–dc boost converter makes it possible toextend the system analysis and control design of dc–dc convert-ers to the three-phase rectifiers. By utilizing the small-signalcontrol-to-output transfer functions, both the voltage mode con-trol scheme and inner-current-control-based scheme, which aresimple and well-documented techniques often used in dc–dcconverters, are applied to a three-phase PWM rectifier in this 622 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 3, MARCH 2009 paper. The steady-state performance and also transient perfor-mance are all experimentally investigated. Experimental resultsshow that the proposed controllers can provide both satisfac-tory steady-state performance and good transient performance,thus further showing the usefulness of the proposed model. Asmay be expected, the inner-current-loop-based scheme resultsin better overall performance.II. D EVELOPMENT OF THE S IMPLE SISO M ODEL In this section, the proposed SISO model of the three-phaseboost rectifier is derived. A. Equivalent Circuit for a Three-Phase PWM Rectifier The voltage-source type PWM rectifier is shown in Fig. 1.Here, e a , e b , and e c represent the source voltages and i a , i b ,and i c represent the input currents. Parameters L and R are theinductance and parasitic resistance values of the synchronousinductance. The system differential equations in d – q SRF are asgiven in [3]–[5] Ldi d dt + Ri d − ωLi q = e d − v d Ldi q dt + Ri q + ωLi d = − v q (1) C dv dc dt =34( u d i d + u q i q ) − i dc . (2)Here, e d , e q and i d , i q denote the input voltages and the inputcurrents in the SRF and v d , v q are the control inputs, and in fact,denote the average voltages at the rectifier input terminals againin the SRF. The variables v d , v q are related to the dc outputvoltage as follows: v d = u d v dc 2 v q = u q v dc 2 (3)where u d , u q are the d -axis and q -axis switching functions,respectively. The range of the switching functions is between 0and 1.Under balanced and steady-state operating conditions, vari-ables in (1) and (2) will all be dc quantities [17].Multiplying (2) by v dc on both sides and applying (3), we canshow that Cv dc dv dc dt + v dc i dc =32( v d i d + v q i q ) . (4)Equation (4) shows the power balance between the dc side andthe ac side of the converter. The equivalent circuit based on (1)and(4)isshowninFig.2.Thisissimilartothecircuitdevelopedearlier by other researchers [17], [18]. B. Nonlinear Feedforward Decoupling Controller InFig.2,thecouplingtermsbetweenthe d -axisandthe q -axisare represented by the two current-controlled dependent voltagesources. Decoupling may be achieved, if the effects of thesetwo voltage sources are nullified by appropriately adjusting the Fig. 2. Equivalent circuit in SRF. control inputs v d and v q v d v q = v d 1 + v d 2 v q 1 + v q 2 (5)where v d 1 and v q 1 represent the feedforward decoupling controlparts with v d 1 v q 1 = u d 1 v dc 2 u q 1 v dc 2 = ωLi q − ωLi d (6)where u d 1 and u q 1 are portions of the switching functions, and u d and u q correspond to the nonlinear decoupling controllerwith u d 1 = 2 ωLi q /v dc and u q 1 = − 2 ωLi d /v dc .With this decoupling control, the differential equation on theac side can be rewritten as follows: Ldi d dt + Ri d = e d − v d 2 Ldi q dt + Ri q = − v q 2 . (7)Decoupling control has been performed in the development of control-oriented model in earlier research [11]–[14] also. Asmentioned earlier in Section I, a control-oriented model hasbeen developed in [12] by first introducing decoupling terms,and then implementing current loops. However, the linear de-coupling terms u d 1 = ωLi q /v dc ref and u q 1 = − ωLi d /v dc ref usedin[12]leadtodecouplingthedynamicsof d -axisand q -axisonly when v dc = v dc ref . At other operating points and undertransients, these decoupling terms do not lead to simplificationof system dynamics.By substituting (5) and (6) into (4), the differential equationon the dc side can be written as follows: Cv dc dv dc dt + v dc i dc =32( v d 2 i d + v q 2 i q ) (8)where v q 2 i q represents the effect of q -axis dynamics on the d -axis. However, under balanced supply voltage and unity ornear UPF conditions, the magnitude of this term will be in-significant due to the zero average value of i q . Thus, this termwill usually be negligible and can be viewed as a small dis-turbance in the d -axis. It has been shown experimentally inSection III of this paper that the presence of a small magnitudeof q -axis current of either polarity has a negligible effect on the YIN et al. : SIMPLE SINGLE-INPUT–SINGLE-OUTPUT (SISO) MODEL FOR A THREE-PHASE PWM RECTIFIER 623 Fig. 3. Equivalent circuit in SRF after decoupling and neglecting of q -axisdisturbance on d -axis dynamics. d -axisdynamics,thusjustifyingtheneglecting oftheterm v q 2 i q in (8).Thus, (8) can be approximated as Cv dc dv dc dt + v dc i dc =32 v d 2 i d (9)with v d 2 = u d 2 v dc / 2 . Here, u d 2 is the system control input andis a portion of switching function u d .The equivalent circuit based on (7) and (9) is shown in Fig. 3.It can be seen that the d -axis and q -axis dynamics are totallydecoupledinthismodel.Thislargesignalmodel,whichisbasedon the ignoring of the effect of the q -axis current on the d -axisdynamics, is true under unity or near UPF conditions. C. Simple SISO Model After the aforementioned decoupling and simplification as-suming near UPF operation, a three-phase boost-type PWMac–dc rectifier becomes a dual SISO system [19]. From (7) and(9), the differential equations can be rewritten as Ldi d dt + Ri d = e d − 12 u d 2 v dc C dv dc dt + i dc =34 u d 2 i d (10) Ldi q dt + Ri q = − v q 2 . (11)The q -axismodelgivenin(11)isafirst-orderlinearsystemre-sponsible for power factor regulation, whereas the d -axis modelisasecond-ordernonlinearsystemthatdeterminesthepowerde-livery. The q -axis behavior can be represented in the frequencydomainbyasimplefirst-ordertransferfunction,whichisshownas i q ( s ) v q 2 ( s )=1 Ls + R. (12)In order to appreciate the similarity of the model with thatof a dc–dc boost converter, we can define an “equivalent” dutycycle d = 1 − u d 2 . The averaged models for both a three-phasePWM rectifier and a dc-to-dc boost converter are given in (13)and (14) in Table I. This suggests an equivalent circuit for theproposed SISO model shown in Fig. 4(b) for the three-phasePWM rectifier. Fig. 4. (a) DC-to-DC boost converter. (b) Proposed d -axis equivalent circuitfor the three-phase PWM rectifier. In Fig. 4(b), the switch S is assumed to operate at an “equiv-alent” duty cycle d resulting in a switching function u d 2 . The dcoutput voltages of the equivalent converter under steady-stateoperating conditions with and without inclusion of the parasiticresistance of the inductors are summarized in Table I. Here, D is the steady-state equivalent duty cycle, V dc is average outputdc voltage, and E d is the d -axis supply voltage. As in a dc-to-dcboost converter, the effect of circuit losses will be to reduce thegain, especially at high duty ratio values.The proposed simple SISO model can be obtained by ap-plying the nonlinear decoupling controllers and ensuring UPFregulation, as shown in Fig. 5. This figure shows the implemen-tationof:1)thedecouplingcontrol;2) abc ↔ dq transformationsthat are needed; and also 3) the q -axis current control loop tokeep i q at zero. In this manner, we can operate the three-phaserectifier in a “quasi-open-loop” mode with the variable d ( u d 2 ) being treated as the control input of the d -axis system and v dc being the system output. The d -axis current output i d may alsobe treated as an output if needed. Another point to be noted isthat the transformed switching functions s a , s b , and s c in the abc frame can be used to switch the six switches of the PWMrectifierindifferentwaysdependingontheactualPWMschemeadopted.It must be noted that the operation of the equivalent switch S in Fig. 4(b) is only partially linked to the operation of the actualsix switches of the converter. The operation of the switches of the converter is in fact determined by the complete switchingfunctions u d , u q and the employed PWM technique.It is also worth noting that the models given in Figs. 3 and4(b) are valid for any kind of loads, namely resistive loads,constant power loads, and constant current load. A three-phasePWM rectifier is often used together with closed-loop dc–dcconverters as loads, in which case, they are said to be loadedwith a constant power load. With such a load, the load’s inputimpedance should be used for system modeling and this willcomplicate the analysis and verification of the use of the sim-ple SISO model. Since our aim is only the verification of theproposed PWM rectifier model, a simple resistive load R dc hasbeen made use of in the rest of the paper.
