analysis and design of direct power control (dpc) for a three phase synchronous rectifier via output...

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18, NO. 3, MAY 2003 823

Analysis and Design of Direct Power Control (DPC)for a Three Phase Synchronous Rectifier via Output

Regulation SubspacesGerardo Escobar, Aleksandar M. Stankovic , Member, IEEE , Juan M. Carrasco , Member, IEEE , Eduardo Galván,

and Romeo Ortega , Fellow, IEEE

Abstract— In this paper, we present a controllerthat directly reg-ulates the activeand instantaneous reactive power in a synchronous three-phase boost-type rectifier . The controller ensures a good regu-lation of the output voltage, and guarantees the power factor closeto one. The controller builds upon the ideas of the well known di-rect torque control (DTC) for induction motors. In our case, theactive and reactive powers replace the torque and flux amplitudeused as the controlled outputs in DTC, thus motivating the name DPC-control . We show that a simple modification to the original al-gorithm makes the selection of the control inputsmore accurate. Toformalize this technique we utilize the concept of output regulation subspaces . A modification is added to the basic controller to dealwith disturbances such as unbalance and distortion in the sourcevoltage. Finally, the proposed controller was tested both in simu-lations and experimentally, and illustrative results are presentedhere.

Index Terms— ac–dc power conversion, nonlinear systems.

I. INTRODUCTION

I N THIS paper we explore direct regulation of active andinstantaneous reactive power in a synchronous three-phase

boost-type rectifier. The use of instantaneous active and reac-

tive power for control purposes was first introduced in [ 6]. Inthat work the authors propose a method to compute the currentreferences based on the computation of the instantaneous activeand reactive power. These current references were later used in ahigh gain (hysteresis) current loop, together with a pulse-widthmodulated (PWM) or space vector modulation (SVM) block.In some applications, this basic method may exhibit disadvan-tages that have been addressed (and in large degree overcomed)in the literature. First, hysteresis controllers cannot guaranteeperfect tracking of a time varying signal, unless arbitrarily fast

Manuscript received February 11, 2002; revised November 1, 2002. This

work was supported by the Consejo Nacional de Ciencia y Tecnología deMéxico (CONACYT) and the Virtual Test Bed (VTB) Project (University of South Carolina) Grant to Northeastern University. Recommended by AssociateEditor F. Blaabjerg.

G. Escobar is with the Department of Applied Mathematics and Com-puter Systems, IPICyT, San Luis Potosí SLP 78210, México (e-mail:[email protected]).

A. M. Stankovic is with the Department of Electrical and ComputerEngineering, Northeastern University, Boston, MA 02115 USA (e-mail:[email protected]).

J. M.Carrasco andE. Galvánare with Escuela Superiorde Ingeniería, Univer-sidad de Sevilla, Sevilla, Spain (e-mail: [email protected]; [email protected]).

R. Ortega is with the Laboratoire des Signaux et Systèmes, CNRS-Supélec,France (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPEL.2003.810862

switching is used. Second, this technique fully applies only inbalanced and sinusoidal operation.

In [7], the authors also utilize the instantaneous active andreactive power for control purposes. They establish first a pro-portional relationship between these variables and the currentsexpressed in the rotational reference (which holds only for sinu-soidal balanced operation). Then, they propose a commutationalgorithm based on the voltage source angular position and the

proportionality between the time derivative of currents in thero-tational reference and the corresponding injected voltage. Thus,a preliminary vector is proposed in such a way that the signof these time derivatives opposes the sign of the errors in realand reactive power. A phase locked loop (PLL) is introduced todetermine the voltage source angular position. Although in thefinal expression of the controller only the active and reactivepowers are involved, the strong use of the properties of the cur-rents makes this method close to the original method proposedin [6]. In addition, the method still needs a PWM block to gen-erate the final control vector. Therefore, this technique cannotbe considered as direct in the terminology that we use.

Later in [9] the authors introduce an algorithm referred as di-

rect power control (DPC). The idea behind this technique con-sists in selecting a control vector from a look up table based onthe error of active and reactive powers as well as on the angularposition of theestimatedvoltagesourcevector. For thelatter, theauthors propose to divide the input space (in the plane) in twelvesectors, and then determine the position of theestimated voltagesource vector with respect to these sectors. They use the factthat dc-bus voltage is regulated by controlling the active power,while the unity power factor operation is achieved by control-ling the reactive power to zero. The look up table is consideredoptimal, although no further explanation is given about the gen-eration of the table. The authors propose to use an estimationof the voltage vector to reduce the number of voltage sensorsand to simplify the implementation. Unfortunately, this modi-fication to the algorithm involves the computation of the timederivative of measured currents. This computation may becomenoisy, especially at low currents, and it is strongly dependent onparameters like the inductance, as pointed out by the authors.

Recently, in [ 8] the authors follow a similar control schemeas in [9]; the main difference is that they propose to estimate avector named virtual flux instead of the voltage source vector.With this modification, the authors try to reduce the extremelylarge sampling frequency required in the original DPC, as wellas the inherent noise introduced in the computation of real and

0885-8993/03$17.00 © 2003 IEEE

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ESCOBAR et al. : ANALYSIS AND DESIGN OF DIRECT POWER CONTROL (DPC) 825

Fig. 2. ORS for the synchronous rectifier.

chosen to be the active and reactive power and , which aredefined as

(3)

(4)

These new outputs are driven toward some desired constant ref-erences, i.e., and , where to guar-antee a power factor close to one, and is a slowly varyingsignal generated by an outer loop to guarantee . Forthe sake of simplicity we have assumed here that the voltagesource is balanced and free of harmonic distortion. The casewhen these disturbances are present requires some further mod-ifications, which are treated in Section VI.

The time derivatives of the active and reactive powers aregiven by

(5)

(6)

where , and we have used .

III. OUTPUT REGULATION SUBSPACES (ORS)

The output regulation subspaces , or simply ORS [ 1], [2] arethe subspaces of the input space where each . Thus weare defining hyperplanes (one hyperplane for each output) withthe characteristic that points “above” the th hyperplane satisfy

while those “below”satisfy . The interestedreaderis referred to [ 1] for a description of such subspaces for (generalvector relative degree ) nonlinear systems.

In the case of a synchronous rectifier, the ORS can be com-puted from (5)–(6) yielding

Fig. 3. Sector definitions for the control algorithm implementation.

TABLE IIUSING ONLY THE ROTATION ANGLE ' TO MODIFY v , AND USING

THE SECTOR DISTRIBUTION ON FIG. 3(a)

with as scalar constants. Clearly, if then . Hence, the ORS define two straight lines dividingthe input space into four quadrants corresponding to differentcombinations of signs for and , as shown in Fig. 2. More-over, both ORS’s are perpendicular to each other all the time, as

is in thedirectionof thevector whileis in the direction of . Note, however, that the intersectionpoint identified by in Fig. 2 (representing the equivalentcontrol or feedback linearizing, decoupling control) is not nec-

essarily at the origin. Our next step is thus to re-examine the useof ORS in the case of a synchronous rectifier.

IV. ORS-B ASED DIRECT POWER CONTROL

A nice feature of the classical DTC is that the selection of thecontrol vector is based only on the knowledge of the positionof the vector of the stator flux relative to the sector definition asgiven by Fig. 3(a). This makes implementationvery simple, as itsuffices to enter this information plus the signs of the output er-rors into a table [see for instance Table II(b)], to immediately getthe control vector that accomplishes the objective at that instant.

Notice that in the case of the synchronous rectifier, under theassumption that both ORS’s are close to the origin (assumptionusually made in DTC), it is enough to determine the position of vector relative to the sectors defined in Fig. 3(a) to obtain acontrol vector. Nevertheless, this simple strategy would exhibitproblems whenever the ORS’s are far from the origin, which iscommon for the synchronous rectifier.

We aim to preserve the same overall philosophy in our algo-rithm, and thus maintain low complexity, but still achieve im-proved accuracy. The idea behind our approach consists in ro-tating the vector by a certain angle just before going intothe table to extract the control vector. We will refer to asthe rotated vector , for instance is a rotation

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826 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18, NO. 3, MAY 2003

Fig. 4. Deformed ORS for the synchronous rectifier.

of rads in the counterclockwise direction, where matrixis described by

This rotation is intended to compensate the error producedwhen considering the approximated ORS. Moreover, the anglefor the rotation of should also be selected. We propose tochose it as a function of the sign of the output errors.

Following the standard DTC, we displace the exact ORS insuch a way that their boundaries cross the origin and remainperpendicular to each other, which yields the dash-dotted linessown in Fig. 4. Next, let us divide the modified ORS into twosegments and rotate each segment in such a waythat they inducethe same partition of the circle in arcs (AB, BC, CD, DA) as the

original exact ORS. The modified ORS (now composed of twosegments each) crosses the circle in the same points (A,B,C,D)as the corresponding original ORS segment. This yields thethick lines shown in Fig. 4, where the segments correspondingto the modified ORS( ) havebeenrotated backward or forwardby depending on the position of the segment, while those cor-responding to ORS( ) are rotated by .

The angles and are computed from the following rela-tions:

(7)

Notice that both angles decrease for larger ; moreover, weobserved that is quite small compared with in typical oper-ation, and it is even smaller more for low currents. This can beeasily seen from the steady state values

(8)In our first scheme, we rotate the vector as follows. First,

we consider only the angle , that is, we assume that ORS( )is very close to the origin.

TABLE IIIUSING BOTH ROTATION ANGLES AND ' TO MODIFY v , AND USING

THE SECTOR DISTRIBUTION ON FIG. 3(a)

TABLE IVUSING THE SECTOR DISTRIBUTION ON FIG. 3(b) TO SELECT THE CLOSEST

CONTROL VECTOR TO THE MODIFIED v

In Table II, , , the minus sign in-

dicates a rotation backward, i.e., in the clockwise direction. Wefirst rotate vector by angle , according to Table II(a) toobtain the modified , i.e., . Now, oncehas been rotated, we look for the position of . Assume atthis point that is located in the sector ; then we proposeto select the control vector according to Table II(b). Notice thatTable II(b) is the standard algorithm used in the DTC technique.We observed in our experiments, however, that this first approx-imation still leads to some distortion in currents.

This distortion can be reduced if angle is also consideredfor the rotation of as we do in our second scheme. In fact,depending on the parameters of the system, we can adjust thecompensating angles (and corresponding tables) to generate thecontrol vectors that may better fit a given application. An ex-ample of modification involving both angles is Table III. 1

When the output voltage is small, a larger equivalent controlis obtained, and problems arise when trying to search for a con-trol vector in the area in Fig. 2 (or in sector as depictedin Fig. 4). If this is the case, a good option would be to select theclosest control vector to the ORS( ). To this end, we proposeto use the distribution of sectors presented in Fig. 3(b) [insteadof the standard distribution of sectors in Fig. 3(a)]. This allowsus to consider the closest control vectors (ahead and behind thevector ) in a more natural way. However, the use of anothersector distribution entails a change in the search algorithm forthe relative position of vector in the new sector distribution.To avoid this modification, we propose to keep the same sectordistribution Fig. 3(a), but to use a rotation of rads to emulatethe sector distribution Fig. 3(b). Finally, the algorithm shown inTable IV is proposed.

With the idea of making more accurate the selection of thecontrol vector in areas and , we propose to increase theresolution of the sector distribution by superimposing two sec-tors in Fig. 3. This yields the sector distribution with enhancedresolution shown in Fig. 5. We note that now each sector hastwo coordinates , being the location on sector dis-tribution of Fig. 3(a), and the one on sector distribution of

1Similar results have been observed if Table III(b) is replaced by Table II(b).



828 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 18, NO. 3, MAY 2003

Fig. 6. Block diagram of the proposed ORS-DPC controller.

VI. CASE OF UNBALANCE AND HARMONIC DISTORTION INSOURCE VOLTAGE

In the case of a source voltage with unbalance and harmonicdistortion, we propose to redefine the outputs as

where represents the fundamental positive symmetric se-quence (component) of . Notice that, if ( a con-stant or slowly varying signal) and then, the cur-rent vector is forced to followa balanced signal proportionalto , guaranteeing a PF close to unity.

In this case the time derivatives are given by

The angles and are now computed from the followingrelations

Notice that the expressions above involve the use of .However, direct access to this quantity is not available. Instead,we propose to estimate using a filter with the structureshown in Fig. 7, which follows very closely the ideas presentedin [5], and referred to as resonant filters.

The filter in Fig. 7 extracts the th harmonic component of a time varying periodic (two-dimensional) signal . Moreover,according to the structure of the filter, it is possible to extractboth positive an negative sequence symmetric componentsand , respectively, of the th harmonic ( for funda-mental), as well as the associated complex Fourier coefficients(phasors) and . The gain is a design parameter thatfor smaller values makes the filter more selective, but slower.

Fig. 7. Resonant filter used to obtain the k th harmonic components of signalx 2 .

Fig. 8. X –Y plot of the steady state operation of (–) the input current i ( t )

compared with ( 1 1 ) its reference current i ( t ) .

VII. EXPERIMENTAL RESULTS

An evaluation of the proposed control policy was com-pleted on a conventional three-phase boost-type rectifier. Theprototype comprises a DSP-based interface, with controllersprogrammed in C language. The following parameters havebeen used in the prototype: grid connection inductancemH, output capacitance mF and an AC grid voltage of

380 V at 50 Hz. The three phase inverter was implementedusing IGBT’s, rated at 15 kW output load. The output voltagecan be regulated from 560 V to 900 V under the full loadconditions. These values correspond to a laboratory prototypethat was designed for different applications, namely, motordrives, static VAR compensator, and synchronous rectifier. Theexperimental setup allows changes in the load from no load to

(corresponding to 11.43 KW at 600 V), and theswitching frequency was fixed to 20 KHz. The experimentswere performed with: the output voltage V, andthe output load . Based on time response criteria,we tuned the parameters of the PI external control loop to

and , and .

In this paper, we only present the experimental results for theORS-DPC controller based on the Table IV. The results for theother controllers based on the other tables are mildly inferior,with exception of Table V, and are omitted here for the reasonof space.

First we show the steady state responses of the currents whenload of is connected. We observe in Fig. 8 that theinput currents follow quite well their corresponding ref-erence currents . The latter has been computed accordingto .

Fig. 9 shows that input currents and and their cor-responding AC source voltages and are in phase,thus guaranteeing operation with a power factor very close to

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ESCOBAR et al. : ANALYSIS AND DESIGN OF DIRECT POWER CONTROL (DPC) 829

Fig. 9. Steady state operation of (–) the input current i and ( 1 1 ) itscorresponding ac voltage v .

Fig. 10. Transient response of the output voltage v ( t ) , fora step load changefrom no load ( R ) to R = 3 1 : 5 .

Fig. 11. Transient response of input currents (–) i ( t ) and ( 1 1 ) i ( t ) , for a stepload change from no load ( R ) to R = 3 1 : 5 .

one. Notice that in this figure thescaleof voltage hasbeen scaledby a factor of 1/10, that is, the figure shows the (scaled) voltages

and .To show the robustness of the proposed controller against

load step disturbances, we abruptly change the load applied tothe system from no load ( ) conditions to .Figs. 10 and 11 show the transient response of theoutputvoltage

and input currents and . Notice that, after a rel-atively short transient, the output voltage is maintained close toits reference value V. Fig. 12 shows the transient re-sponseof the power delivered totheload for the samestepload change.

VIII. S IMULATION RESULTS FOR UNBALANCED CASE

To show theeffectivenessof theproposed controller in case of unbalance and harmonic distortion, we consider the unbalanced

voltage shown in Fig. 13 which is polluted with 3rd and 5th har-monics. These harmonics are unbalanced independently, corre-sponding to approximately 5% THD. Notice that the voltagescale is multiplied by factor 1/10. We used the model (1)–(2)for the system rectifier with the same parameter values as in theexperimental setup. In this case we used the control algorithmbased on Table V. Current reference is computed in the un-balance case according to .

The control parameters for the external PI controller werechosen as follows , , and( , ). The test consists in introducing a stepchange on the load resistance at s going from no load( ) to .

Fig. 12. Transient response of the output powerP ( t )

, for a step load changefrom no load ( R ) to R = 3 1 : 5 .

Fig. 13. Steady state operation of (–) the input current i and ( 1 1 ) itscorresponding AC voltage v , with R = 3 3 : 3 .

Fig. 14. Transient response of the output voltage v ( t ) for a step load changefrom no load ( R ) to R = 3 3 : 3 .

Fig. 13 shows that currents , under a load of ,and the corresponding AC source voltages are in phase toeach other, thus guaranteeing operation with a power factor veryclose to one.

Fig. 14 shows that after a relatively short transient followingan abruptchangeon theload resistance going from no load (

) to , the voltage converges (in the average)toward its desired reference V.

IX. CONCLUSION

In this paper we showed that the basic principles used forDTC can be applied to synchronous rectifiers. The strategy isdenoted as ORS-DPC control, where the name DPC is moti-vated by the fact that active and reactive powers are “directly”controlled, just as the torque and flux amplitude are in DTC.The concept of output regulation subspaces (ORS) is revisited,

and used to formalize the new switching strategies. It was estab-lished in [ 1] that the standard DTC uses an approximated ORS.The strategy proposed here is a modified version of the stan-dard DTC, where we rotate the approximated ORS by a certainangle to consider the exact ORS. Nevertheless, we preserve thebasic ideas of DTC to select the control input vector by meansof a look-up table. Severalmodifications arepossible dependingon the system characteristics and the desired references. We ob-served from our experimental results that the controller guar-antees a good regulation of the output voltage with a near unitpower factor. This holds true even after abrupt load changes,suggesting a robust performance of ORS-DPC against this typeof disturbances.

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