SIX-PHASE BIDIRECTIONAL RECTIFIER: MODELLING, CONTROL ANDEXPERIMENTAL RESULTS IN αβ REFERENCE FRAME

Cesar A. Arbugeri, Samir A. MussaUniversidade Federal de Santa Catarina–UFSC, Departamento de Engenharia Elétrica e Eletrônica–EEL

Instituto de Eletrônica de Potência – INEP, Florianópolis – SC, Brasile–mail: cesar.a@inep.ufsc.br, telles@inep.ufsc.br, samir@inep.ufsc.br

Abstract – Poly-phase machines have been used formany years in motor drives and more recently in windpower generation. In wind generation the most commonpoly-phase system is the six-phase, which consists of a dualthree-phase system, shifted by 30 electrical degrees fromeach other and with no neutral connection. Typically, thissystem employs a twelve-pulse diode rectifier to convertAC to DC voltages. Better results can be achievedemploying an active rectifier, since it can provide lowerdistortion in generator currents and regulated outputvoltage. However, the dynamic and stationary modelof this converter are not reported in the literature,taking that into account, this paper proposes a six-phaseactive rectifier topology for a dual three-phase system.A stationary frame reference for six-phase system isapplied to obtain the converter dynamic model in theαβ coordinates and thus a control strategy is proposedin this frame reference. The proposed topology andcontrol strategy were experimentally verified with a 12kW prototype. The results were appropriated, providinga regulated output voltage, an input current with reducedharmonic distortion and thus high power factor.

Keywords – Control, PFC, Rectifier, Six-phase,Stationary Frame Reference.

I. INTRODUCTION

Poly-phase machines are a viable solution for a wide rangeof applications, not only in motor drives, but also in electricalgeneration. Due to the employment of power electronics inmost of the motor drives and renewable energy, the connectionof these poly-phase machines with three-phase mains are nota restriction. The use of poly-phase systems provides severaladvantages, especially in terms of fault tolerance.

Historically, the use of diode and thyristor rectifiers broughtthe concern about power factor and problems caused in theelectrical grid, due to high Total Harmonic Distortion (THD)of the currents drained from the mains. However, the creationof international standards (for example IEC 1000-3-2 andIEEE 519) has imposed firm restrictions to current harmonicdistortion in modern rectifiers, which narrows the use ofpassive rectifier.

This fact stimulated the study, development andimprovement of several topologies in the field of powerelectronics, while aiming to provide power factor correction.

Manuscript received 17/08/2018; first revision 17/09/2018; accepted forpublication 14/12/2018, by recommendation of Editor Marcello Mezaroba.http://dx.doi.org/10.18618/REP.2019.1.0032

Some works, such as [1]–[6], proposed power convertersemploying diode bridges with some sort of active control,and thus the boost type converters became the most populartopology for operation as power-factor correction (PFC)converter in single-phase systems.

On the other hand, in three-phase PFC rectifier the mostpopular topology is the two-level voltage-source converter(2L-VSC), which provides a reduced harmonic distortion inthe input current and a regulated output voltage. This topologyis widely studied in the literature [7]–[9], and several controlapproaches were proposed to ensure a proper operation asa rectifier, the most common strategies being the classicalcontrol in the stationary frame reference (αβ coordinates) orin the synchronous reference frame (dq coordinates).

Poly-phase systems have become more relevant and moreemployed, initially in motor drives applications, aiming abetter performance in high power applications, being usualsystem of 5, 6, 9 and 12 phases, applied in traction applications[10]–[13]. And more recently in power generation, especiallyin wind systems [14]–[19]. In conventional six-phasegeneration, twelve-pulse diode rectifiers are employed inthe ac-dc voltage conversion, and this solution has beenshowing good results. Nevertheless, the use of active rectifiersthat provide a lower distortion on generator currents and aregulated output voltage might become more attractive.

The work [20] presented the basic operation of a six phaseactive rectifier validating the models and control with numericsimulations, which are experimentally validated in this paper,thoroughly corroborating the simulation results. Besides, amore precise model for the output voltage transfer functionis obtained through the use of the converter’s the powerequations, and a static analysis of the converter is presentedproviding project criteria for the selection of passives andactive components.

The objective of this paper is to propose a six-phase activerectifier topology, as well as models and a control strategyin stationary frame reference which aims to provide an inputcurrent with reduced harmonic content and regulate the outputvoltage. The proposed methodology can be applied not onlyin six-phase generation systems, but also in motor drivesemploying a back-to-back configuration. The remainderof this paper is organized as follows: the converter basicoperation is presented in Section II, describing the topologyand modulation strategy. The converter dynamic model in astationary reference frame is deducted in the Section III. InSection IV are presented the static analysis and an exampleof project for the converter, including a control proposal.The models and the converter operations are validated by aprototype and the results are shown in Section V. Conclusions

+

-

L1

L2

L3

v1

i1

L4

L5

L6

Vo

Six-phase

Generator

v2

i2 v

3 i3 v4

i4 v5

i5 v

6 i6

Fig. 1. Six-phase two-level voltage-source active PWM rectifiertopology.

are presented in Section VI.

II. SIX-PHASE TWO-LEVELS VOLTAGE SOURCECONVERTER

This section presents the six-phase two-levels voltage-source converter and its principle of operation.

A. Six-Phase Rectifier Topology and OperationThe rectifier is a two-level voltage-source converter

topology, as presented in the Figure 1. Each leg ofthe converter is composed by two active switches drivenby complementary gate signals. The converter is currentbidirectional fed by a six-phase source, e.g., a dual three-phase generator or a wye/delta-wye transformer. In this paper,this source consists of a dual three-phase system where one isshifted by 30 electrical degrees from the other and without aneutral connection. This neutral disconnection is due to themachine windings, which are connected in two sets of three,avoiding common mode current circulation and generating thesix-phase system as a dual three-phase system.

B. Modulator SchemeThe operation of the six-phase converter is similar to

a three-phase two-level converter. The main differenceis the phase quantity. The modulation scheme for theconverter consists of a bipolar PWM – the modulator signalis generated by the controllers and compared with a triangularcarrier, generating the gate signal. This signal controls theupper switch, while the lower switch is controlled by itscomplementary.

Other modulators can also be employed, such as a spacevector modulator. However, choosing the suitable vectors andits application time is very complex, mostly due to the highnumber of vectors and the higher dimension of the systemmodel. Since this work is focused in the converter modellingand control, a simple modulation strategy was adopted.

III. STATIONARY REFERENCE FRAME CONVERTERMODELLING

The dual three-phase system voltages are not independent,i.e, there is a coupling among phases. However, the systemmodels can be represented in stationary reference frame (αβ )or synchronous reference frame (dq). The advantage ofemploying the dq transformation, is representing the voltageand current fundamentals by constant values. Nonetheless,

there is a coupling between the axes in this reference frame.In this paper, an αβ transformation is employed due thedecoupling provided among phases.

At a first moment, the stationary frame transformation isshown, transforming the six-phase reference frame into twoαβ planes and two common mode voltages. Afterwards, thelinearized models for the input current and the output voltagetransfer functions are presented, which are then used for thecontroller design.

A. Alpha Beta Stationary Frame Reference TransformationThis linear transformation was initially proposed in [21] for

three-phase systems and in a second moment generalized forpoly-phase systems in [22], and consists in a transformationmatrix exposed in (1). This transformation is proposed todecompose the original six-dimensional rectifier into threetwo-dimensional orthogonal subspaces (α1,β1), (α2,β2) and(01,02). The fundamental components of the rectifier and theharmonics of order k = 12n± 1, n = 1,2,3, . . . are mappedin the (α1,β1) subspace. The harmonics of order k = 6n± 1,n = 1,3,5, . . . are mapped in the (α2,β2) subspace and thezero-sequence components are mapped in the (01,02) subspace[23].

vα1

vβ1vα2vβ2v01v02

= 1√3

1 − 12 − 1

2

√3

2 −√

32 0

0√

32 −

√3

212

12 −1

1 − 12 − 1

2 −√

32

√3

2 0

0 −√

32

√3

212

12 −1

1 1 1 0 0 00 0 0 1 1 1

v1

v2v3v4v5v6

. (1)

Due to geometry and machine characteristics, such asthe rotating field and positioning of the coils, the voltagegenerated will be mapped in the (α1,β1) subspace, implyingthat only currents with belong to this subspace are capable ofgenerating active power. However, it is important to controlthe currents in the subspace (α2,β2), even though they are notcapable of generating active power, as these components canincrease system losses.

B. Input Current Transfer FunctionTo determine the current transfer function, the converter

is represented by its equivalent circuit model depicted in theFigure 2.

For the purpose of obtaining the current transfer function,the converter output voltage (Vo) is considered constant and theconverter legs are replaced with controlled voltages sources.The instantaneous value of this sources are determined by theoutput voltage and the duty cycle average value in a period Ts,as showed in (2), where di(t) is the instantaneous periodicalaverage value relative to the phase i.

〈vSi(t)〉Ts=Vo

(di(t)− 1

2

), i ∈ 1,2,3,4,5,6. (2)

The equivalent circuit also explicits the common modevoltages between the source neutral points and the groundreference point (VCM1 and VCM2). From the equivalent circuitis possible to obtain the equations described in the matrix form

+

-+ -

+ -

+ -

+ -

+ -+ -

L1

L2

L3

v1 v

s1

vs2

vs3

vs4

vs5

vs6

v2

v3

i1

i2

i3

L4

L5

L6

v4

v5

v6

vCM2

i4

i5

i6

+

- vCM1

Fig. 2. Input equivalent circuit model of the Six-Phase RectifierTopology.

as in (3), assuming an unique the inductance value of L for allthe inductors.

v1(t)v2(t)v3(t)v4(t)v5(t)v6(t)

= L ddt

iL1(t)iL2(t)iL3(t)iL4(t)iL5(t)iL6(t)

+

vS1(t)vS2(t)vS3(t)vS4(t)vS5(t)vS6(t)

+

vCM1(t)vCM1(t)vCM1(t)vCM2(t)vCM2(t)vCM2(t)

. (3)

By substituting (2) into (3) and rewriting the equation in acompact form results in (4).

[vin(t)] = Lddt

[iL(t)]+ vo [d(t)]−[

Vo

2

]+[vCM] . (4)

Applying the stationary frame transformation (1) in (4)results in (5), which represents the system equations in αβ .

vα1(t)vβ1(t)vα2(t)vβ2(t)v01(t)v02(t)

= L ddt

iα1(t)iβ1(t)iα2(t)iβ2(t)i01(t)i02(t)

+ vo

dα1(t)dβ1(t)dα2(t)dβ2(t)d01(t)d02(t)

+

0000

√3(vCM1− vo

2 )√3(vCM2− vo

2 )

. (5)

Equation (5) can also be represented by the equivalentcircuits shown in Figure 3, which accurately represents thethree-phase rectifier in αβ coordinates. By inspection, itis possible to notice that the system is decoupled, i.,e, therelationship between inputs and outputs of the MIMO systemcan be expressed as six different SISO systems.

The transfer function of the system is then obtained byequating the equivalent circuit as in (6) and linearizing aroundan equilibrium point, resulting in (7).

vx(t) = L · ddt ix(t)+ vo ·dx(t), x ∈ α1,β1,α2,β2. (6)

+

-

+

-

+

-

+

-

+-

j 1,2

vαj

iαj

L

Vodαj

+

-v0j

vCMj

i0j

L

Vod0j

+

-vβj

iβj

L

Vodβj

(a)

(c)

(b)

Fig. 3. Input equivalent circuit in αβ coordinates (a) α1 and α2components circuit, (b) β1 and β2 components circuit and (c) 01 and02 components circuit.

L · ddt

ix =−Vodx. (7)

Therefore, each controllable SISO system can be describedby the transfer function described in (8). Since the transferfunction is the same for α1, β1, α2 and β2, the controllerdesign can be replicated for the four current controls.

ix(s)dx(s)

=− Vos·L , x ∈ α1,β1,α2,β2 . (8)

Besides, the 01 and 02 components equivalent circuits areopen, as it is assumed that there is no common mode currentcirculation, which implies that they cannot be controlled. Thisoccurs due to the disconnection between the neutral points.and generates the common mode voltages displayed in theequivalent circuit in the Figure 2, similarly to a three-phasethree-wire converter.

C. Output Voltage Transfer FunctionThe output voltage transfer function is obtained through the

power equation of the converter as in (9).

Pin = L · iLddt

iL +Co · voddt

vo +v2

o

Ro+Losses. (9)

Disconsidering losses and representing the variables in thestationary reference frame the equation results in (10).

vαβ iαβ = L · iαβ

ddt

iαβ +Co · voddt

vo +v2

o

Ro. (10)

Applying the small signal model and the Laplacetransformation in (10) results in (11).

vo(s)(

RoCoVos+2Vo

Ro

)= iαβ (s)

(Vαβ −LIαβ

). (11)

Hence the transfer function can be written as in (12).

vo(s)

iαβ (s)= Ro

Vαβ −LIαβ s2+ sCoRo

. (12)

By substituting the real project parameters in (12) results in(13), where the input current and voltage (Iαβ and Vαβ ) in (13)are the peak values.

vo(s)

iαβ (s)=

√3VpRoVo− LVo√

3Vps

2+ sCoRo. (13)

It is possible to notice that the transfer function has a zeroon the right half plane as occurs on the three-phase boosttype rectifier or in the DC-DC boost converter. The non-minimal phase systems are, in general, harder to control incomparison to minimal phase systems, so it is important totake into account the zero in the voltage transfer function.

IV. STATIC ANALYSIS AND PROJECT EXAMPLE

This section aims to present a criterium for thedetermination of the converter’s parameters, as well as voltageand current stress in the semiconductors. The section alsoexperimentally validates the presented models with a projectexample. The control block diagram of the cascaded controlstrategy is presented in Figure 4, which shows the sampledvariables for the control and both voltage and current controlloops.

Voltage

Control

Current

Control

PWM

α1β1α2β2

123456

α1β1α2β2

123456

L1

L2

L3

v1

v2

v3

L4

L5

L6

v4

v5

v6

i1

i2

i3

i4

i5

i6

+

-

Vo

Fig. 4. Converter and control block diagram.

A. Converter Parameters and Stress of the SemiconductorDevicesThe current ripple depends on the voltage on the input

inductance, given by (14).

∆I =1L

∫∆t

0vLdt. (14)

The maximum ripple occurs at the input voltage peak,which for phase A occurs at ωt = π/2. Hence the inductancecan be determined by equation (15).

L =Vp

∆I · fs

(12−

Vp

2Vo

). (15)

The DC bus capacitor can be determined by the hold-uptime criteria. Since the output power is constant, similar to anypoly-phase system, the dc bus capacitance is given by (16).

Co =Po · tHU

V 2o ∆V%

. (16)

The voltage stress in the switches is given by the DC busvoltage, given by (17).

VSmax =Vo. (17)

Diode average and RMS current values are determined by(18) and (19), respectively.

IDavg =Po(2Vo +πVp)

12πVpVo. (18)

IDrms =Po

6Vp

√16Vp +3πVo

6πVo. (19)

The semiconductor devices average and RMS are describedby (20) and (21), respectively.

ISavg =Po(2Vo−πVp)

12πVpVo. (20)

Switch RMS current:

ISrms =Po

6Vp

√3πVo−16Vp

6πVo. (21)

B. Converter SpecificationsThe parameters obtained for the converter design from the

previously obtained equations are summarized in Table I.

TABLE IConverter Parameters in Simulation

Parameter ValueVoltage Source frequency fg = 60 Hz

Switching frequency fs = 9990 HzAC Voltage input Vin = 220 VRMS

DC Voltage output Vo = 800 VOutput Power P0 = 12 kW

Input Inductance Lin = 2 mHOutput Capacitance C0 = 4700 µF

Load Equivalent Resistance R0 = 53 Ω

For the given design specification, the following designconstraints were adopted: for the inductance, a maximumripple of 20% at the peak current was considered, while a holdup time of 33 ms and a voltage ripple less than 10 % (∆V %)were considered the capacitance design.

C. Model ValidationTo validate the converter model, the time domain response

of the converter and the linearized model were comparedwhile applying an input disturbance. Figure 5 shows theresponse for a duty cycle disturbance, which demonstrates thesuperposition of both responses.

Figure 6 demonstrates the response of the output voltagefor a current step. A small difference between the transferfunction and the circuit response was observed, especially inthe static gain. Most of the difference is due to the inductanceresistance and a minor difference due to non-ideality of thesemiconductors. However, the difference is very small andthe transfer function is a representative model, which can beemployed in the control design.

t[ms]

iαj[A]

10

-10

15 30

Converter

TF

Fig. 5. Validation of the proposed transfer function of the inputcurrent per duty cycle (8): Comparison of the power converter versusequation 8 during a step on duty cycle.

800

840

TF

Converter

vo[V]

t[s]1,00,5

Fig. 6. Validation of the proposed transfer function of the outputvoltage per input current (13): Comparison of the power converterversus equation 13 during a step on the input current.

D. Control DesignThe control strategy consists in two cascaded control loops,

a faster inner loop that controls the input currents and a slowerloop which is responsible for controlling the DC link voltage.A block diagram of the proposed control strategy is presentedin Figure 7, where the variables indicated with an asterisk arethe references for the control loops.

i1

i2

i3

i4

iα1

iα2

iα2

iβ1

iβ2

iβ2

i5

i6

1

2

3

4

5

6

α1

β1

α2

β2

Cvo

Ci

Ci

Ci

Ci

Vο

*

vo

mα1

mα2

mβ1

mβ2

m1

m2

m3

m4

m5

m6

* *

iβ1

iα1* *

sin (ωt) cos (ωt)

1

2

3

4

5

6

α1

β1

α2

β2

Fig. 7. Stationary reference frame control strategy block diagram forthe six-phase PWM active rectifier.

The voltage controller input is the DC voltage error andits output controls the αβ currents amplitude. The voltagecontroller output is multiplied by sine waves generating theα1β1 currents reference, while the α2β2 currents reference arealways.

The input currents (in αβ coordinates) are subtracted fromthe currents references, resulting in the currents error, which isapplied to the current controllers generating the duty cycle inαβ reference frame. On applying the inverse transformationon it, the duty cycle for each arm is obtained.

For a proper closed loop operation, two compensator weredesigned to control the input current and the load bus voltage.As previously discussed, the current and voltage transferfunctions used for the controller project are presented in (8)

and (13), respectively.The chosen compensator was a proportional integrator (PI)

controller, projected to reject step disturbances, as presentedin (22).

PI(s) = KPI1+ sTPI

sTPI. (22)

To design the PI controller, the root locus method wasemployed, with the parameters arbitrated for the closed loopresponse for the current loop a damping higher than 0.7(ζ > 0.7) and a natural frequency lower than 4π

10Ts(ωn <

4π

10Ts).

The criteria for the voltage loop was ζ > 0.9 and ωn <4π

20Tsobtaining the current and voltage controllers respectively

shown in (23) and (24).

CI(s) = 1.11+10−4

s10−4 . (23)

CV (s) = 281+6 ·10−3

s6 ·10−3 . (24)

V. EXPERIMENTAL RESULTS

In this section the modelling and the structure are validatedby a prototype, presented in Figure 8. The parameterspresented in the Table I were used for the project of theconverter.

DC Bus CapacitorsCurrent and Voltage

Sensors

IGBT Gate Drivers

Power IGBTs

Signal Conditioning

and Control

Auxiliary Power Supply

Fig. 8. Six-phase voltage converter prototype.

A. Converter Prototype and Experimental Results

5 A/div

i1i4

i2i5 i

3i6

30º

Fig. 9. Six-phase input voltages and currents of the converteroperating in steady state.

The operation of six-phase rectifier using the controlstrategy proposed is experimentally verified and the resultsare presented as the following. Figure 9 shows the six-phaseinductor currents which are drawn from source, the currentsare approximate to a sinus wave indicating a high power factor.

5 A/div100 V/div

v1

i1

Fig. 10. Input voltage and current for phase 1 of the converteroperating in steady state.

Figure 10 shows the input voltage and current for the phase 1.In this case the converter operates with 50% of the nominalpower and presents a power factor of 0.995 and a T HDi =8.96%. Most of the harmonic distortion in the current is dueto harmonic components around the switching frequency.

The dynamic response for a load step of 50% of thenominal power is displayed in the Figure 11, the output voltageis regulated and the control provides an adequate dynamicresponse, in accord to the control project criteria. The currentloop present a fast transient having a settling time of circa 2ms, while the voltage loop is slower having a settling time ofproximately 30 ms.

2 A/div100 V/div

2 A/div

i6

io

vo

i1

Fig. 11. Input current and output voltage dynamic response for a loadstep of 50%.

If compared with the twelve pulse diode rectifier, which isthe commercial solution for power rectification in poly phasewind power generation [15], the proposed converter presentsa regulated voltage output with higher voltage level, sinceit is a boost type converter. Besides it presents a reducedharmonic distortion in the currents drained from the generator,the experimental prototype currents presented a T HDi =8.96%, while twelve pulse diode rectifier current THD havetypical values circa 30%. The proposed PWM rectifierreduces significantly the current lower harmonic components,emphasizing the 12n± 1 harmonics, which cause oscillatingtorque in the machine and the 12n±5, that have no influencein the torque, however contribute to the system losses.

VI. CONCLUSION

This paper proposed a six-phase two-level voltage sourcerectifier, its model and a control scheme in stationary frame

reference. The presented converter model is a transfer functionaiming the employ of classical control technics. The modelwas validated by simulation and it presented suitable results.

A stationary frame control strategy for the rectifierwas proposed and validated through simulation. Thecontrol strategy demonstrated appropriated results, providinga regulated output voltage, an input current with reducedharmonic distortion and a high power factor. The proposedtopology may be employed in six-phase wind generationsystems when a lower current distortion or a regulated outputis desired.

The main contribution of this paper is the proposal ofan active rectifier for the six-phase system presenting themodelling and a control strategy for the converter as a singlesystem, and not as a dual three-phase system, providing abetter control and operation of the system, while focusing ontechnical aspects and the operation of the power converter.

ACKNOWLEDGEMENTS

The authors would like to thank Prof. Telles B. Lazzarin forthe review of the manuscript and for the technical discussions.

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Cesar Augusto Arbugeri Nascido em Joaçaba, SantaCatarina, Brasil, em 1990. Recebeu o grau de Engenheiroeletricista e mestre pela Universidade Federal de SantaCatarina (UFSC), Florianópolis, Brasil, em 2014 e 2016,respectivamente. Atualmente é doutorando no programade pos-graduação em Engenharia Elétrica da UniversidadeFederal de Santa Catarina (UFSC) e pesquisador no Institutode Eletrônica de Potôncia (INEP).

Samir Ahmad Mussa Recebeu o grau de Engenheiroeletricista pela Universidade Federal de Santa Maria em 1988,recebeu grau de Mestre e de Doutor pela Universidade Federalde Santa Catarina em 1994 e 2003 respectivamente e Pós-Doutorado no Imperial College London, Inglaterra entre 2015e 2016. Possui graduação em Matemática e habilitação emFísica pela Unifra, Santa Maria-RS (1986). Atualmente ocupao cargo de professor no Departamento de Engenharia Elétricae Eletrônica (EEL) da Universidade Federal de Santa Catarina(UFSC) e pesquisador no Instituto de Eletrônica de Potência(INEP). Seus interesses de pesquisa incluem retificadoresPFC, processamento de sinais digitais e controle aplicado emeletrônica de potência, sistemas baseados em DSP, FPGA emicroprocessadores. Dr. Mussa é membro da SociedadeBrasileira de Eletrônica de Potência (SOBRAEP) e do IEEESociety.