Behavior of the Predictive DTC based Matrix

Converter under Unbalanced AC Supply

Marco E. Rivera* José R. Espinoza* René E. Vargas** José R. Rodríguez**

Department of Electrical Engineering *

Concepción University, Concepción, CHILE

Tel.: +56 (41) 2203512 - Fax.: +56 (41) 2246999

marcoriv@udec.cl, jose.espinoza@udec.cl

Department of Electronics **

Universidad Federico Santa María, Valparaíso, CHILE

Tel.: +56 (32) 2654214 – Fax.: +56 (32) 2797469

rvargas@elo.utfsm.cl, jose.rodriguez@elo.utfsm.cl

Abstract – A novel control strategy applied to the Matrix

Converter is presented. The strategy combines the advantages of the Direct Torque Control and the Predictive Control, verifying

its robustness through out the analysis of its behavior under an

unbalanced AC supply. The approach selects the best switching state according to an optimizing algorithm that is based on a cost

function. The algorithm ensures unitary input power factor and

zero steady state error in the flux and torque of the AC machine.

The speed is adjusted by a closed loop scheme and the unbalance is naturally mitigated. Several simulations showing the transient

and steady state behavior of the proposed scheme are presented.

I. INTRODUCTION

The Matrix Converter is an alternative that replaces the

rectifier, inverter and the storage energy stages with only one

stage. One of the most popular control techniques used in these

AC drives is the Direct Torque Control (DTC), which obtains a

high quality decoupled behavior of the motor torque and flux,

with fast dynamic response [1]-[4]. On the other hand, Model-

Based Predictive Control is a tool that allows operating with

great efficiency and a high flexibility over the converter and

motor variables. Thus, satisfying multiple and variable

operation criteria in presence of perturbations [4]-[6]. Several

works have been reported considering unbalance in the AC

input voltages. Particularly, for drives based on diode rectifier

as front end converters, where small unbalances have

significant effects. Among the most important ones are the

second harmonic injected into the DC link and the current

unbalance introduced into the distribution system. Also,

applications, based on the Space Vector Modulating (SVM)

technique have been analyzed. The drawbacks of the schemes

have been reported [7]-[10].

This work studies the behavior of the Predictive Control in

combination with DTC applied to an AC drive operating under

AC voltages unbalances. The deterministic characteristics of

the system make possible to implement a Reference Model in

order to predict the state variables evolution. The model

includes naturally the AC input voltage unbalance and thus can

minimize its effects. The algorithm minimizes a cost function,

that implicitly assures unitary input power factor, and flux and

torque equal to a given references. The finite number of states

of the converter simplifies the optimizing algorithm to test the

future condition of the system for all possible switch

combinations. The speed of the machine is controlled by

means of an external closed loop that fixes the internal torque

reference.

II. CONVERTER AND MACHINE MODEL

The topology based on a Matrix Converter is shown in Fig.

1, where a nine switches configuration is considered.

Assuming some restrictions in the topology like no short-

circuits at the input terminals, due to the AC input voltages

presence and no open circuits at the load side, due to the

inductive nature of the AC machine equivalent circuit, this

converter can provide 27 valid combinations, which are

indicated in Table I, in which,

1 switch ON

0 switch OFFKj

S

=

, K ∈ {A, B, C}, j ∈ {a, b, c} (1)

Using the previous expressions, the model of the converter

can be written as,

=oL PhL sPhv T vr r

(2)

where, oLvr

is the AC load phase voltage, sPhvr

is the AC input

phase voltage, and TPhL is the Transfer Matrix of the topology,

that is found to be,

.

=

−−−

−−−

−−−

CaCcBaBcAaAc

CcCbBcBbAcAb

CbCaBbBaAbAa

SSSSSS

SSSSSS

SSSSSS

PhLT (3)

Considering the converter lossless and without storing

energy elements, the converter input currents satisfy,

T=sPh PhPh oPhi T ir r

(4)

Fig. 1. Block Diagram of the topology of the Matrix Converter.

where, sPhir

is the AC input current, oPhir

is the AC output

current, and TPhPh is the instantaneous Transfer Matrix of the

topology that is found to be,

Aa Ba Ca

Ab Bb Cb

Ac Bc Cc

S S S

S S S

S S S

=

PhPhT (5)

and represents the switch gating signals, where each elements

is defined in (1).

The model of the induction machine referred to stator is

obtained as described in [4]. On the other hand, a balanced

three-phase signal can be defined as a space vector by the

transformation,

22

( )3

a b cx ax a x+ +x =

r (6)

where 2 / 3ja e− π= . Hence, the AC induction machine model in

space vector representation becomes,

s s sR L= +o ov i ψr rr & (7)

r r sR L j= + − ω

rr rv i ψ ψr r rr & (8)

where, Rs, Rr and ω correspond to the stator resistance, rotor

resistance, and rotor angular frequency, respectively. The

stator and rotor fluxes are related with their respective currents

trough the equations,

s mL L=s o rψ i + ir rr

and r m

L L= +r r oψ i ir rr

(9)

where Ls, Lr, and Lm correspond to the leakage and mutual

inductances. Finally, the electric torque can be expressed in

current and flux terms such as,

( )2

3e

T p= ×s oψ irr

(10)

with p as the AC machine number of pole pairs.

III. CONTROL STRATEGY

The proposed control scheme is basically an optimization

algorithm and as such it has to be implemented in a

microprocessor based hardware. Consequently, the analysis

has to be developed using discrete mathematics in order to

consider additional restrictions as delays, sampling time,

approximations, etc.

A. Machine Model in Discrete Time

Due to the first order nature of the state equations that

describe the AC machine model (7)-(8) a first order

approximation for the derivatives provides enough accuracy

for the proposed jobs. The first order approximation is,

( ) ( )1

s

x k x kx

T

+ −=& (11)

where Ts is the sampling period. Hence, the stator and rotor

fluxes can be estimated in the αβ0 stationary axes from (7)-(8)

resulting in,

( ) ( ) ( ) ( )-1o s s o s

k k k T R k T= + −αβ0 αβ0 αβ0 αβ0s sψ ψ v i

rr r r (12)

( ) ( ) ( )-1 s r

m o

m

L L Lrk k L kL L

m

= + −

αβ0 αβ0 αβ0r sψ ψ i

rr r (13)

Thus, considering a given voltage vector ( )1k +αβ0sv

r, it is

possible to obtain a stator flux prediction from (12) as,

( ) ( ) ( ) ( )1 1 1o s s o s

k k k T R k T+ = + + − +αβ0 αβ0 αβ0 αβ0s sψ ψ v i

rr r r (14)

and the stator current prediction equation is,

( ) ( )

( ) ( )( ) ( )

1 1

1

s

o o s

s

s

o r r r

s

r Tk k T

L

Tk k jk k

L

σ

+ = − σ

+ ⋅ + + τ − ωσ

αβ0 αβ0

αβ0 αβ0r

i i

v ψ

r r

rr

(15)

where [4], 2

σ s r rr R R k= + , 1

r sk kσ = − ,

r r rL Rτ = ,

r m rk L L= ,

s m sk L L= (16)

The predicted electrical torque, for the next sample time, is

deduced from (10) and (14)-(15) as,

( ) ( )( )21 1

3e o

T p k k= + × +αβ0 αβ0sψ i

rr (17)

B. Predictive Direct Torque Control Strategy

The total proposed control scheme is shown in Fig. 2. The

strategy consists on computing in the sampling instant k the

gating state that minimizes the input reactive power and, at the

same time, minimizes the torque and stator flux error on the

TABLE I Allowed States of the Matrix Converter.

# SAa SBa SCa SAb SBb SCb SAc SBc SCc

1 1 0 0 0 1 0 0 0 1

2 1 0 0 0 0 1 0 1 0

3 0 1 0 1 0 0 0 0 1

4 0 1 0 0 0 1 1 0 0

5 0 0 1 1 0 0 0 1 0

6 0 0 1 0 1 0 1 0 0

7 1 0 0 0 0 1 0 0 1

8 0 1 0 0 0 1 0 0 1

9 0 1 0 1 0 0 1 0 0

10 0 0 1 1 0 0 1 0 0

11 0 0 1 0 1 0 0 1 0

12 1 0 0 0 1 0 0 1 0

13 0 0 1 1 0 0 0 0 1

14 0 0 1 0 1 0 0 0 1

15 1 0 0 0 1 0 1 0 0

16 1 0 0 0 0 1 1 0 0

17 0 1 0 0 0 1 0 1 0

18 0 1 0 1 0 0 0 1 0

19 0 0 1 0 0 1 1 0 0

20 0 0 1 0 0 1 0 1 0

21 1 0 0 1 0 0 0 1 0

22 1 0 0 1 0 0 0 0 1

23 0 1 0 0 1 0 0 0 1

24 0 1 0 0 1 0 1 0 0

25 1 0 0 1 0 0 1 0 0

26 0 1 0 0 1 0 0 1 0

27 0 0 1 0 0 1 0 0 1

sampling instant k + 1. Then, the system applies this gating

state during the whole k + 1 sampling period.

The algorithm actually calculates the 27 possible conditions

that the state variables can achieve in the instant k + 1. This is

done by the Flux Estimator block in the Fig. 2 that then

computes the 27 possible flux and torque based on (14)-(17).

This information is then used by the Switching State Selector,

Fig. 2, based on the cost function minimization as shown later.

C. AC Power Factor Control Strategy

The state variable model of the AC input side is given by,

f f sR L= + +s s ev i i vr rr r&

(18)

fC= +s e ei i v

r r r& (19)

The model is a second order model and as such an exact

discrete state model is best suited that is then used to obtain the

supply current in the sampling instant k + 1. The model is,

( ) ( ) ( ) ( ) ( )1 2 3 41k C k C k C k C k+ = + + +s s e s ei v v i ir r rr r

(20)

The previous expressions are also evaluated for the 27

possible gating states and as a result, the AC input current

( )1k +αβ0sir

is obtained, Fig. 2. This result is used to calculate

the AC input power factor given by,

( ) ( ) ( ) ( )1 1 1 1s s s s s

pf v k i k v k i kα β β α= + + − + + (21)

This information is also used by the Switching State

Selector, Fig. 2, based on the cost function minimization.

D. Cost Function Minimization

The cost function g is defined as,

( ) ( ) ( ) ( )

( ) ( )

1 1

1 1

s s s s

* *s s e e

g A v k i k v k i k

B ψ ψ k C T T k

α β β α= + − + +

− + + − + (22)

where it is clear that for g = 0 unity power factor is achieved as

well as a flux equal to *s

ψ and a torque equal to *e

T . Therefore,

the goal of the Switching State Selector is to achieve g closest

to zero. Thus, the algorithm tests all the 27 possible conditions

and selects the one that achieves the minimum g. This state is

applied at the instant k + 1, when the algorithm is again

evaluated and the result applied at the instant k + 2.

The cost function g contains the parameters A, B, and C that

should be selected in order to priorize the control action. For

instance, a high A coefficient would lead to fast dynamic to

regulate the AC input power factor. The selection of these

parameters has been done so far testing different values.

E. Speed Control

The speed is controlled using an external controller. This

block generates the torque reference that is the used to

generate the gating patterns as illustrated earlier. The controller

is a PI because the integral part is required in order to achieve

zero steady state error. This is due to the fact that the

predictive DTC fast dynamic can be represented just as a unity

gain between the reference and the controlled variables.

IV. RESULTS

A. Open Loop Behavior: Venturini Method

In order to have results to compare with, the topology is

tested in open loop and modulated using the Venturini method.

A 40% unbalance is introduced in the AC input voltages in t =

0.45 s. The results are given in Fig. 3.

0.4 0.42 0.44 0.46 0.48 0.5-1000

0

1000

(a)

0.4 0.42 0.44 0.46 0.48 0.5

-40

-20

0

20

(b)

0.4 0.42 0.44 0.46 0.48 0.5

-1000

0

1000

(c)

0.4 0.42 0.44 0.46 0.48 0.5

-20

0

20

(d)

Time

0

0.2

0.4

0.6

0.8

-2

-1

0

1

2-1.5

-1

-0.5

0

0.5

1

1.5

(e)

Fig. 3. Key waveforms of the matrix converter operating in open loop and

using Venturinis modulating method; (a) Input Voltages vs; (b) Input Current

is, (c) Line Output Voltage voab, (d) Output Currents io, (e) Stator Flux.

Fig. 2. Proposed control strategy scheme.

Clearly, under unbalanced AC supply voltages, Fig. 3(a),

unbalanced AC load currents are obtained, Fig. 3(d). As a

result, the input AC currents are also unbalanced. In order to

quantify the unbalance degree of the resulting waveforms, an

instantaneous unbalance factor is introduced for the

fundamental component. This is based on the symmetrical

components definition and for a general vector xr

is given by,

1

1

1

( )( )

( )

n

x

p

ku k

k=

x

x

rr

r (20)

where, 21 1 1 1( ) ( ) ( ) ( )n a b c

k x k a x k ax k= + +xr

is the

fundamental component negative sequence and

21 1 1 1( ) ( ) ( ) ( )

p a b ck x k ax k a x k= + +x

r is the fundamental

positive component. Naturally, x1a, x1b, and x1c are the abc

fundamental components of the three-phase quantity x. These

are obtained using a FFT algorithm that uses a moving

rectangular window. The results are given in the Fig. 4.

0.4 0.42 0.44 0.46 0.48 0.5

0

0.5

1

(a)

0.4 0.42 0.44 0.46 0.48 0.5

0

0.5

1

(b)

0.4 0.42 0.44 0.46 0.48 0.5

0

0.5

1

(c)

Time Fig. 4. Unbalance factor magnitude of case in Fig. 3; (a) of the AC supply

voltage, (b) of the AC supply current, (c) of the AC load current.

As expected the load and supply currents present an

unbalance factor that is not compensated by the modulating

technique.

B. Proposed Control Strategy for a Balanced AC Source

The proposed control strategy is simulated for a balanced

AC source. The strategy uses a Ts = 50 µs sampling time in

order to consider realistic conditions. The test considers the

starting of the AC machine, a load torque step in t = 0.3 and

reversing in t = 0.5 s. The speed reference is followed by the

system as seen in Fig. 5(a) and as expected, the torque

reference generated by the speed controller is different from

zero during the transients and load torque steps. Fig. 5(c)

shows the reactive power during the process and there is not

doubt that remains in zero even during the transients.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-100

0

100

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-50

0

50

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-2

0

2x 10

4

Time

(c)

0.3

0.4

0.5

0.6

-2

-1

0

1

2-1.5

-1

-0.5

0

0.5

1

1.5

(e)

Fig. 5. Proposed control strategy for a balanced AC supply; (a) speed

reference and actual value, (b) torque reference and actual value, (c) reactive

power, (d) stator flux.

Fig. 6 depicts the input and output currents for this case.

Although it is not clear, the currents are sinusoidal quantities

and in t = 0.2 s the currents are very small as the speed has

reached the steady state value and there is no load torque.

Differently is the situation after t = 0.3 s when a load torque

step takes place.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-20

0

20

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-20

0

20

(b)

Time Fig. 6. Proposed control strategy for a balanced AC supply; (a) input currents,

(b) output currents.

The Fig. 7 shows the AC input and load voltages and Fig. 8

shows the unbalance factors for the key waveforms. All the

unbalance factors are close to zero which confirms that under a

balanced AC supply, the load waveforms are also balanced. It

is important to note that the unbalance factor is calculated for

the fundamental component. As such there could be unwanted

harmonics that are not balanced. However, the sinusoidal

shape of the currents, Fig. 6, indirectly confirms that the

proposed control strategy generates sinusoidal shapes.

0.75 0.8 0.85 0.9-1000

0

1000

(a)

0.75 0.8 0.85 0.9

-1000

0

1000

(b)

Time

Fig. 7. Proposed control strategy for a balanced AC supply; (a) input voltages,

(b) output line voltage voab.

0.78 0.8 0.82 0.84 0.86 0.88 0.9

0

0.5

1

(a)

0.78 0.8 0.82 0.84 0.86 0.88 0.9

0

0.5

1

(b)

0.78 0.8 0.82 0.84 0.86 0.88 0.9

0

0.5

1

(c)

Time

Fig. 8. Unbalance factor magnitude of case in Fig. 5; (a) for the supply

voltage, (b) for the supply current, (c) for the load current.

C. Proposed Control Strategy for Unbalanced AC Source

Exactly the same dynamic test as the previous one is

implemented but under an unbalanced AC source. The

unbalance is taken in about 40%. This unrealistic extreme

value is considered just to show the high performance of the

proposed control strategy. The speed reference is followed by

the system as seen in Fig. 9(a) and as expected, the torque

reference generated by the speed controller is different from

zero during the transients and load torque steps. Fig. 9(c)

shows the reactive power during the process and there is not

doubt that remains in zero even during the transients. This

confirms that the predictive DTC is capable of achieving a

similar performance as if no unbalance was present.

Due to the unbalance in the AC mains, the input current

presents unbalance as indicated the Fig. 12.(b). However, the

load current does not present any unbalance as confirmed by

the Fig. 12(c). Despite the unbalance in the input current, the

input power factor remains close to zero as shown in Fig. 9(c).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-100

0

100

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-50

0

50

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-2

0

2x 10

4

Time

(c)

0.3

0.4

0.5

0.6

-2

-1

0

1

2-1.5

-1

-0.5

0

0.5

1

1.5

(e)

Fig. 9. Proposed control strategy for an unbalanced AC supply; (a) speed

reference and actual value, (b) torque reference and actual value, (c) reactive

power, (d) stator flux.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-20

0

20

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-20

0

20

(b)

Time

Fig. 10. Proposed control strategy for an unbalanced AC supply; (a) input

currents, (b) output currents.

0.75 0.8 0.85 0.9-1000

0

1000

(a)

0.75 0.8 0.85 0.9

-1000

0

1000

(b)

Time

Fig. 11. Proposed control strategy for an unbalanced AC supply; (a) input

voltages, (b) output line voltage voab.

0.78 0.8 0.82 0.84 0.86 0.88 0.9

0

0.5

1

(a)

0.78 0.8 0.82 0.84 0.86 0.88 0.9

0

0.5

1

(b)

0.78 0.8 0.82 0.84 0.86 0.88 0.9

0

0.5

1

(c)

Time Fig. 12. Unbalance factor magnitude of case in Fig. 9; (a) for the supply

voltage, (b) for the supply current, (c) for the load current.

V. CONCLUSIONS

The Predictive Direct Torque Control applied to an

induction machine under unbalanced supply voltages has been

presented. The algorithm tests the 27 possible combinations of

the topology and selects the combination that minimizes a cost

function. The ideal minimum of the cost function is zero and

represents the perfect regulation of the controlled variables.

This is unity input power factor, and a given machine torque

and flux. The control scheme can compensate the unbalanced

supply voltages and thus the load voltages and currents remain

balanced. The compensation is achieved without any penalty in

the transient and steady state operation; moreover, the supply

power factor that is actually kept equal to the unity. The

unbalance is shown using an instantaneous unbalance factor

that is based on the symmetrical components decomposition.

Simulated results were obtained using a 50 µs sampling time in

order to consider practical aspects of the implementation. The

results at the load side are comparable with the ideal case of a

perfectly balanced AC supply.

ACKNOWLEDGMENTS

The authors wish to thank the financial support from the

Chilean Fund for Scientific and Technological Development

(FONDECYT) through project 106 0424.

REFERENCES

[1] Hong-Hee Lee and Minh-Hoang Nguyen, “Matrix Converter Fed

induction Motor Using a New Modified Direct Torque Control Method”,

The 30th Annual Conference of the IEEE Industrial Society, November

2-6, 2004, ECON Busan Korea.

[2] J. Boll, and F. Fuchs “Direct Control Methods for Matrix Converter and

Induction Machine”, IEEE ISIE 2005, June 20-23, 2005, Dubrovnik,

Croatia.

[3] C. Ortega, A. Arias, J.L. Romeral and E. Aldabas “Direct Torque Control

for Induction Motors Using Matrix Converters” Proceedings of the

Fourth International Workshop on Compatibility in Power Elecrtonics.

IEEE Poland Section, 2005, p. 1-8.

[4] J. Rodríguez, J. Pontt, C. Silva, P. Cortés, S. Rees, and U. Ammann.

“Predictive direct torque control of an induction machine”. EPE-PEMC

2004(Power Electronics and Motion Control Conference), Riga, Latvia,

2-4 September 2004.

[5] E. P. Wiechmann, A. R. García, L. D. Salazar, J. Rodríguez “High-

Performance Direct-Frequency Converters Controlled by Predictive-

Current Loop”, IEEE Transactions on Power Electronics, vol. 12, no 3,

May. 1997, pp. 547-557.

[6] J. Rodriguez, J Pontt, C. Silva, P.Cortes, U. Amman, S Rees, “Predictive

current of a voltage source inverter”, in Proc. of IEEE-PESC ’04,

Aachen, Germany.

[7] D. Casadei, G. Grandi, G. Serra, and A. Tani, “Analysis of space vector

modulated matrix converter under unbalanced supply voltages,” in

Proc.SPEEDAM’94, Taormina, Italy, June 8–10, 1994, pp. 39–44.

[8] D. Casadei, P. Nielsen, G. Serra, and A. Tani, “Evaluation of the input

current quality by three different modulation strategies for SVM

controlled matrix converters with input voltage unbalance,” in

Proc.PEDES’96, vol. II, New Delhi, India, Jan. 8–11, 1996, pp. 794–

800.

[9] P. Nielsen, F. Blaabjerg, and J. K. Pedersen, “Space vector modulated

matrix converter with minimized number of switching and feedforward

compensation of input voltage unbalance,” in Proc. PEDES’96, vol. II,

New Delhi, India, Jan. 8–11, 1996, pp. 833–839.

[10] A. Ferrero and G. Superti-Furga, “A new approach to the definition of

power components in three phase system under nonsinusoidal

conditions,”IEEE Trans. Instrum. Meas., vol. 40, pp. 568–577, June

1991.