Control Scheme of Hybrid Matrix ConverterOperating Under Balanced and Unbalanced

ConditionsAmeer Janabi Bingsen Wang

Department of Electrical and Computer EngineeringMichigan State University2120 Engineering Building

East Lansing, MI 48824, USAjanabiam@msu.edu; bingsen@egr.msu.edu

Abstract—Hybrid matrix converters can potentially enablematrix converter in high-power applications that conventionalmatrix converters would not be able to attain. The hybridmatrix converter consists a main matrix converter that processesthe bulk power conversion and an auxiliary back-back voltagesource converter that improves the terminal power quality. Priorsimulation study has successfully demonstrated that the superiorspectral performance can be achieved. This paper is focused onthe control scheme of the hybrid matrix converter operatingunder balanced and unbalanced conditions.

I. INTRODUCTION

Matrix converters have several advantages over traditionalfrequency converters. They are able to provide sinusoidal inputcurrents and output voltages, smaller size, no large energystorage components required, and provide full control overthe input power factor for any load[1][2].Due to the qualities that the matrix converter provides, ithas been a very attractive area of research over the recentyears. The real development was started by Venturini andAlesina published in 1980[3][4], they first introduced the name”Matrix Converter” and provided a detailed mathematicalmodel describing the behavior of the converter. Another mod-ulation technique is based on a fictitious DC link connectinga current source bridge and voltage source bridge presentedby Rodriguez[5]. This approach is known as the indirecttransfer function approach. In 1989 the method of spacevector modulation technique for matrix converters introducedby Huber[5]. In 1992 it was practically confirmed that thenine switches matrix converter shown in Fig.1, could be usedeffectively in vector control of induction motor[7]. However,the usage of the matrix converter was still limited due tothe difficult current communication of bidirectional switches.Recently, many solutions have been presented to solve thecommunication problem of bidirectional switches, such as thefour step method [8].Matrix converters generate current harmonics that are injected

back into the ac system. These current harmonics can result involtage distortions that affect the operation of the AC system.On the other hand the voltage harmonics on the output side

Fig. 1. Three phase matrix converter

will cause a disturbance to the load which in most cases is aninductive load. In order to reduce these harmonics in the sourcecurrent and the output voltage, passive filters are typically usedto reduce the harmonics generated by the matrix converter.Different configurations of low pass passive filters have beenproposed[9], The size and design of these filters depend onmany factors such as:

1- power quality requirements2- power system harmonic content3- converter switching frequency4- converter modulation technique[10].Passive filters seem to be an easy solution in low power

applications of matrix converters. However, this is not thecase when a matrix converter is introduced for high powerinterfaces, the optimal design of the passive filter would be amajor challenge.

The first high power matrix converter is introduced byYaskawa for wind power applications[11], in which the mod-ular concept is used to cope with the various demands in thepower grid. However, the modular concept is presented to beused in a very high voltage applications such as wind mills.In a prior paper the hybrid matrix converter is introduced to

operate at high power medium voltage demands. The topologyused a conventional 9 bidirectional switch matrix convertercombined with auxiliary back-back voltage source converterto improves the terminal power quality. However, the previouspaper did not cover all the operation conditions that hybridmatrix converter might encounter. In this paper, a controlscheme is provided for hybrid matrix converter operatingunder normal and abnormal conditions, such as unbalancedvoltage source and unbalanced load.

The matrix converter is modulated using the low frequencymodulation algorithm [12] and the active filters are controlledbased on the instantiates reactive power theory[13][14] andsymmetrical component theory. This paper is organized asfollows: following the introduction, section two present thehybrid matrix converter, section three present the modulationalgorithms for the active filters under normal conditions,section four present the control scheme for active filter whenhybrid matrix converter operate under abnormal conditions,section five shows the simulation results, finally conclusionand remarks end the paper.

II. PROPOSED HYBRID MATRIX CONVERTER

The operation of the matrix converter at high power hasseveral advantages in efficiency, sizing, and reliability overthe back-to-back converter inverter system[10]. Fig.2 showsthe proposed topology for matrix converter used for mediumvoltage high power applications.

The topology includes the conventional nine bidirectionalswitches matrix converter with a shunt active filter imple-mented on the input side and series active filter implementedon the output side. Both active filters consist of three phaseinverters connected by a common DC link. The shunt activefilter compensate the input current harmonic produced by theswitching operation of matrix converter. The series active filtercompensate the output voltage harmonics. That means, thematrix converter is considered as a two type of harmonicsources, current harmonic source in the input side, and voltageharmonic source in the output side.

III. ACTIVE FILTER MODULATION UNDER BALANCEDCONDITIONS

A. Input Current Conditioning

Considering the two waveforms at the input side, the threephase input voltage is always sinusoidal and the three phasecurrent is extremely distorted because the matrix converter andthe load are behaving non-linearly. The instantaneous threephase active power at the input side pin, can be given by

pin = Vin · Iin, (1)

where ”·” denotes the internal product of the two vectors.Equation (7) can also expressed in the conventional detentionof power,

pin = IAVA + IBVB + ICVC . (2)

Fig. 2. Proposed Hybrid Matrix Converter

The instantaneous input reactive power vector of the threephase system can be expressed as

qin = Vin × Iin, (3)

where ”×” denotes the cross product of the voltage and currentvectors. qin can also be expressed

qin =

qAqBqC

=

∣∣∣∣ VB VCIB IC

∣∣∣∣∣∣∣∣ VC VAIC IA

∣∣∣∣∣∣∣∣ VA VBIA IB

∣∣∣∣

, (4)

andqin = ‖qin‖ =

√q2A + q2B + q2C . (5)

Respectively we can analysis the source current into anactive component Iin−p, and reactive component Iin−q inwhich

Iin−p =

IA−pIB−pIC−p

=pin · VinVin · Vin

, (6)

Iin−q =

IA−qIB−qIC−q

=qin · VinVin · Vin

, (7)

note that the resultant current vector from the addition ofthe two currents Iin−p and Iin−q is always equal the sourcecurrent Iin. Another thing to notice is that the instantaneouspower produced from Vin · Iin−p equal to the input power pin, and the instantaneous power produced from Vin · Iin−q isalways equal to zero. From this observation we can tell thatIin−q is not contributing to any power transmission from thesource to the load. In fact if we made the instantaneous reactivecurrent Iin−q ≡ 0 the current Iin will be transmitting the sameinstantaneous active power pin with unity power factor.

Respectively, the instantaneous active power pin can beanalyzed into two components,

pin = pin + pin, (8)

where pin is the direct component of the instantaneous activepower and it represents the energy flow in one direction fromthe source to the matrix converter, and pin represents theoscillatory component of the instantaneous active power andit represents the energy exchanged between the source andthe matrix converter. By eliminating the current componentthat produces pin, the source current will be sinusoidal. Itis important to note that we do not need an energy storingcomponent to compensate the reactive power qin, because qin

represents the energy exchange among the three phases, whilean we energy storing component is required for compensatingthe oscillatory real power pin, because pin is the real powerexchanged between the source and the matrix converter.

Knowing this, we can generate the compensation current byselecting the appropriate power portion to be eliminated. Thethree phase compensation current could be expressed as

I∗comp =p∗C · VinVinVin

+q∗C × VinVin · Vin

, (9)

where p∗C and q∗C can be assigned from pin and qin, respec-tively, depending on our compensation objectives.

B. Output Voltage Conditioning

The continuous current requirement of the matrix converterat the output side makes it a perfect fit to apply the seriesactive filter to compensate the output voltage harmonics. Byusing the dual approach of the instantaneous reactive powertheory we can define an instantaneous active output power andinstantaneous reactive output power as

pout = Vout · Iout, (10)

qout = Vout × Iout. (11)

In turn, we define the instantaneous active voltage vectorVout−p and instantaneous reactive voltage vector Vout−q as

Vout−p =

Va−pVb−pVb−p

=pout · I1outIout1 · I1out

, (12)

Vout−q =

Va−qVb−qVc−q

=qout × I1outIout1 · I1out

, , (13)

note that the upper script ”1” denotes the fundamentalcomponent of the output current. In most cases, series activefilters is used in applications were the current is sinusoidal.However, this is not the case in the matrix converter and itis a required further control effort to extract the fundamentalcomponent of the output current.

Similar observation can be made in the series active filter.The addition of the two voltage vectors Vout−p and Vout−q

Fig. 3. Control circuits of the active filters in case of balanced conditions

always equal the output voltage Vout. The instantaneous powerproduced from Vout−p ·Iout is equal to the to the output powerpout, and the instantaneous power produced from Vout−q ·Ioutalways equals zero.

The instantaneous output active power has a similar envelopto the instantaneous input active power, the only differenceis the switching losses. We can also define an oscillatorycomponent of the instantaneous output active power and thecorresponding component of output voltage that causes thisoscillation. By selecting the appropriate portions of power tobe compensated we can write the equation of the compensatingvoltage as

V ∗comp ==p∗c · I1outI1out · I1out

+q∗c × I1outI1out · I1out

, (14)

where p∗c and q∗c can be assigned from pout and qout accordingto our one’s compensation objectives.

The control circuits of the two active filters are shown infig.6, circuit (a) includes computational circuits for the in-stantaneous input reactive power qin, instantaneous oscillatorycomponent of the input active power pin, and instantaneousreactive component of the input current Iin−q , instantaneousoscillatory component of the input active current Iin−p. circuit(b) includes computational circuits for the instantaneous outputreactive power qout, instantaneous oscillatory component ofthe output active power pout, and instantaneous reactive com-ponent of the output reactive voltage Vout−q , instantaneousoscillatory component of the output active voltage Vout−p.Circuit (c) shows the fundamental component extraction fromthe output current. It’s important to mention that using theFourier analysis method of extracting the fundamental com-petent of the output current is not effecting the response ofthe control circuit, because Fourier analysis method has fasterresponse than the instantaneous reactive power theory method.The Fourier analysis method required an accurate informationabout the fundamental component frequency. This will notcause any problem because we always have an accurateinformation about the output fundamental frequency.

IV. GENERAL CASES INCLUDING DISTORTION ANDIMBALANCE IN THE VOLTAGES AND CURRENTS

A. Unbalanced Voltage Source

In the previous section the assumption is made that thesource voltage is balanced, meaning the amplitudes of thethree phase voltages are equal to each other and there is a120o phase shift among them. In case of unbalanced voltagesource, further analysis needs to be considered to obtain thecorrect compensation current for the input stage of the matrixconverter.

The unbalanced voltage source may include positive, nega-tive, and zero sequence components according to the symmet-rical component theory. The symmetrical component transfor-mation is applied on both the input voltage and current todetermine the sequence components.

Vin0Vin+Vin−

=1

3

1 1 11 α α2

1 α2 α

VAVBVC

, (15)

The subscripts ”0”, ”+”, and ”-” correspond to the zero,positive, and negative sequences, respectively. The complexnumber in the transformation matrix corresponds to the phaseshift in the three phase system, α = 1∠120o = ej

2π3 ,

Inin0Inin+Inin−

=1

3

1 1 11 α α2

1 α2 α

InAInBInC

, (16)

where ”n” denotes the harmonic component order.The time domain equivalent voltage and current can be de-

rived from the phasors given by (15) and (16). By synthesizingthe symmetrical components, the a− b− c input voltages canbe written as:

VA =

VA0︷ ︸︸ ︷√2Vin0 sin(ωit+ φ0) +

VA+︷ ︸︸ ︷√2Vin+ sin(ωit+ φ+)

+

VA−︷ ︸︸ ︷√2Vin− sin(ωit+ φ−)

VB =√

2Vin0 sin(ωit+ φ0) +√

2Vin+ sin(ωit+ φ+ −2π

3)

+√

2Vin− sin(ωit+ φ− +2π

3)

VC =√

2Vin0 sin(ωit+ φ0) +√

2Vin+ sin(ωit+ φ+ +2π

3)

+√

2Vin− sin(ωit+ φ− −2π

3),

(17)Similarly, the instantaneous input line currents are found to

Fig. 4. Control circuit of the shunt active filter in the case of unbalancedsource voltage

be

InA =

InA0︷ ︸︸ ︷√2Inin0 sin(ωit+ φ0) +

InA+︷ ︸︸ ︷√2Inin+ sin(ωit+ φ+)

+

InA−︷ ︸︸ ︷√2Inin− sin(ωit+ φ−)

InB =√

2Inin0 sin(ωit+ φ0) +√

2Inin+ sin(ωit+ φ+ −2π

3)

+√

2Inin− sin(ωit+ φ− +2π

3)

InC =√

2Inin0 sin(ωit+ φ0) +√

2Inin+ sin(ωit+ φ+ +2π

3)

+√

2Inin− sin(ωit+ φ− −2π

3),

(18)The input current is the result of adding the results of all thetime domain currents from each harmonic.

Ik =

∞∑n=1

Ink k = (A,B,C). (19)

The above description allows us to analyze the three phaseunbalanced system in to two three phase balanced systemspulse zero sequence component. In the matrix converter casewe will not consider the zero sequence component to becompensated in order to maintain a 0.866% voltage transferratio of the matrix converter.

Fig.4, shows the two balanced systems (positive sequencesystem Vin+ = [VA+ VB+ VC+]T , Iin+ = [IA+ IB+ IC+]T

and the negative sequence system Vin− = [VA− VB− VC−]T ,Iin− = [IA− IB− IC−]T ) can be compensated in two differentcontrol loops, then the total compensating current will be theaddition of the compensating current of the positive sequencesystem and the compensating current of the negative sequencesystem.

I∗comp+ =p∗C+ · Vin+Vin+ · Vin+

+q∗C+ × Vin+Vin+ · Vin+

, (20)

I∗comp− =p∗C− · Vin−Vin− · Vin−

+q∗C− × Vin−Vin− · Vin−

, (21)

I∗comp = I∗comp+ + I∗comp−. (22)

B. Unbalanced load

In case of when different loads are connected to the matrixconverter, each load will draw a different amount of currentleading to a linearly independent three phase output current.The decomposition of the output voltage and current into it’ssymmetrical component is as follows: V n

out0

V nout+

V nout−

=1

3

1 1 11 α α2

1 α2 α

V na

V na

V na

, (23)

I1out0I1out+I1out−

=1

3

1 1 11 α α2

1 α2 α

I1aI1bI1c

, (24)

The time domain equivalent voltage and current can be de-rived from the phasors given by (23) and (24). By synthesizingthe symmetrical components, the a−b−c voltage and currentcan be written as:

V na =

V na0︷ ︸︸ ︷√2Vout0 sin(ωit+ φ0) +

V na+︷ ︸︸ ︷√2Vout+ sin(ωit+ φ+)

+

V na−︷ ︸︸ ︷√2Vout− sin(ωit+ φ−)

V nb =

√2Vout0 sin(ωit+ φ0) +

√2Vout+ sin(ωit+ φ+ −

2π

3)

+√

2Vout− sin(ωit+ φ− +2π

3)

V nc =

√2Vout0 sin(ωit+ φ0) +

√2Vout+ sin(ωit+ φ+ +

2π

3)

+√

2Vout− sin(ωit+ φ− −2π

3),

(25)Similarly, the instantaneous line currents are found to be

I1a =

I1a0︷ ︸︸ ︷√

2Inout0 sin(ωit+ φ0) +

I1a+︷ ︸︸ ︷√

2Inout+ sin(ωit+ φ+)

+

I1a−︷ ︸︸ ︷√

2Inout− sin(ωit+ φ−)

I1b =√

2Inout0 sin(ωit+ φ0) +√

2Inout+ sin(ωit+ φ+ −2π

3)

+√

2Inout− sin(ωit+ φ− +2π

3)

I1c =√

2Inout0 sin(ωit+ φ0) +√

2Inout+ sin(ωit+ φ+ +2π

3)

+√

2Inout− sin(ωit+ φ− −2π

3),

(26)then the output voltage is the result of adding all the harmonicstogether

Vk =

∞∑n=1

V nk k = (a, b, c). (27)

The same consideration of zero sequence voltages will bemade here to maintain the highest voltage transfer ratio. The

Fig. 5. Control circuit of the series active filter in the case of unbalancedload

control process is similar to the one we have in case of unbal-anced source voltage, utilizing the output voltage and currentinto it’s symmetrical component leaving us with two balancedsystems, (positive sequence system Vout+ = [Va+ Vb+ Vc+]T ,I1out+ = [I1a+ I1b+ I1c+]T and the negative sequence systemVin− = [Va− Vb− Vc−]T , I1in− = [I1a− V 1

Ib− V 1Ic−]T ). The

two balanced systems can be compensated into two differentcontrol loops as shown in Fig.5. The total compensatingvoltage will be the addition of the compensating voltage ofthe positive sequence system and the compensating voltage ofthe negative sequence system.

V ∗comp+ =p∗C+ · I1out+I1out+ · I1out+

+q∗C+ × I1out+I1out+ · I1out+

, (28)

V ∗comp− =p∗C− · I1out−I1out− · I1out−

+q∗C− × I1out−I1out− · I1out−

, (29)

V ∗comp = V ∗comp+ + V ∗comp−. (30)

V. SIMULATION RESULTS

A simulation model of the hybrid matrix converter shownin Fig.2 is built using MATLAB Simulink. In the shuntactive filter a hysteresis current controller is used to track theinstantaneous change of the inverter current and compare itback with the reference current.

It is necessary to control the voltage of the DC link capacitorby adding the power loss caused by the inverter switches. Theactive filter generates harmonics at its switching frequency,and it is necessary to filter out these harmonics, typically asmall coupling inductor is connected in series with the inverteroutput to eliminate these high frequency harmonics.

Fig.6, shows the input current and the output voltage afterthe compensation in the case of a balanced source voltageand load, in the same phase with the input voltage whichmeans that all the reactive power has been compensatedeffectively. The compensation of the oscillatory componentof the input active power results in the sinusoidal shape ofthe input current. The same explanation can be made for theoutput voltage. Fig.7, shows the output result of compensatingthe source current when the supply voltage is not balanced.The simulation shows that a sinusoidal input current canbe achieved. Fig.8, shows the simulation in the case of anunbalanced load being fed by the matrix converter.

Fig. 6. Output results under balanced conditions

Fig. 7. Output results under unbalanced voltage source conditions

Fig. 8. Output results under unbalanced load conditions

VI. CONCLUSION AND REMARKS

In this paper, a general control scheme for the hybrid matrixconverter control operating under different conditions (normalcondition, unbalanced source voltage, and unbalanced load).

The analysis of the input and output power shows thatthe instantaneous reactive power theory can be applied indetermining the compensation current of the input side andthe compensation voltage for the output side of the matrixconverter. Further analysis needs to be considered when theHybrid Matrix Converter operates under abnormal conditions,those analyses include the symmetrical component theory.

The proposed topology is very efficient in medium voltagehigh power applications in which the conventional solution ofpassive filters is not effective. The Hybrid Matrix Converterreduces the energy storing components, size, and provideshigher reliability. The proposed topology can be utilized formitigating the effect of voltage sag, especially when the matrixconverter is used to drive sensitive loads.

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