Model Predictive Control of a Direct Three-to-Seven Phase Matrix Converter

Sk Moin Ahmed1, Haitham Abu-Rub2 1,2Department of Electrical and Computer Engineering

Texas A&M University at Qatar Doha, Qatar

moin.sk@qatar.tamu.edu; haitham.abu-rub@qatar.tamu.edu

Zainal Salam3 1,3Department of Electrical Engineering

University Technology Malaysia Johor Bahru, Malaysia zainals@fke.utm.my

Abstract— This paper proposes a model predictive control technique for a direct multi-phase matrix converter with three phase input and seven phase output. The proposed predictive control strategy simultaneously controls the active and reactive power in addition to the source and load current. The proper switching states are predicted according to the optimization algorithm of the cost function. The cost function is developed to control the input power factor along with seven phase load current tracking. The model predictive algorithm is presented in the paper, and the viability of the proposed solution is proved using analytical, simulation and experimental approach.

I. INTRODUCTION The matrix converter has recently developed paramount attention among researchers, as matrix converters contains some distinct advantages such as controlled bidirectional power flow, operation at unity power factor for any load and sinusoidal input and output currents. An overview of the progress in the research field of matrix converter is presented in [1]. Most common topology of the matrix converter discussed in the literature is the three-to-three-phase [2,3]. Small attention has been given to the development of the matrix converters with greater than three outputs, except in [4-7]. Although the three-phase based systems are widely used, certain niche applications have given rise to the need for multi-phase (more than three phases) converters. Multi-phase systems have been widely investigated in the literature and their reviews are presented in [8]. Multi-phase systems are shown to offer several advantages over the more common three-phase systems. In the last few years, researchers have shown some keen interest in learning multi-phase motor drive systems and its various applications. Literature highlights many benefits of multi-phase drive systems such as it being highly controllable, high system redundancy, tolerant to faults, low per-leg conversion rating, low dv/dt of common mode voltages, independence of control of two or more series/parallel connected machines, etc. A two-level voltage

source inverter is the most frequently used power electronics converter, providing multi-phase motor drives.

Now a days matrix converter is utilized in different industrial applications due to several intrinsical advantages over traditional back-to-back converter. As the matrix converter control complexity is higher than the traditional converters, different control techniques emerging from classical Alesina/Venturuni method [9] to carrier-based PWM [10] and space vector PWM [11] are developed in recent times. Recently model based predictive control is also applied to matrix converter [14-17] based drive systems. Due to the advancement of fast digital signal processors and switches, predictive control technique is rapidly become popular in power electronic converters and drives [12,13]. The MPC technique relies on the minimization of a cost or objective function that determines the future footprint of the system input sequences at each sampling instant. One of the most important advantages of this control technique is that the system non-linearites and mathematical constraints can be explicitly included in the cost/objective function formulation.

This paper presents a predictive control technique applied to a multi-phase matrix converter with 3-phase input and 7-phase output. Predictive current control is introduced along with active and reactive power control to achieve the target of simultaneous current and power control. No modulation or linear controllers are needed in the predictive control technique as the evaluation of the objective function exclusively depends on the model of the drive system. The total number of possible switching states for this topology of matrix converter is 2187. The proposed predictive algorithm successfully evaluates the appropriate switching state at every sampling time that yields the minimum value for the formulated cost function in the next sampling instant.

II. DIRECT THREE-TO-SEVEN PHASE MATRIX CONVERTER

The power circuit topology of a three-phase input to seven-phase output matrix converter is presented in Fig. 1. Each leg has three bidirectional power switches and they

978-1-4799-5776-7/14/$31.00 ©2014 IEEE 1059

operate in such a way that input side is not short circuited and the output side is not open circuited. These are the two fundamental constraints that are imposed in the operation of matrix converter to protect the source and the load. With these imposed constraints the maximum valid switching states for three to seven phase matrix converter can be 2187. The switching function is defined as;

Sjk = {1 for closed switch, 0 for open switch}, j = {A,B,C} (input), k= {a, b, c, d, e, f & g} (output). The switching constraint is SAk + SBk + SCk = 1.

sAi

sBi

sCi

Ai

Bi

Ci

Av

Bv

Cv

11S

21S

31S

17S

27S

37S

ai gi

Fig. 1. Three-phase to Seven-phase Matrix converter topology. A matrix converter is analogous to three-level inverter

where the output lines are synthesized by either phase ‘A’, or phase ‘B’ or phase ‘C’. The duty ratios are represented with;

gfedcbajCBAidij ,,,,,, ,,, ∈∈ (1)

The input and output voltages are related as;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

C

B

A

CgBgAg

CfBfAf

CeBeAe

CdBdAd

CcBcAc

CbBbAb

CaBaAa

g

f

e

d

c

b

a

VVV

ddd

dddddddddddddddddd

V

VVVVVV

(2)

Similarly the input and output currents are related as;

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

g

f

e

d

c

b

a

Cg

Bg

Ag

CfCeCdCcCbCa

BfBeBdBcBbBa

AfAeAdAcAbAa

C

B

A

i

iiiiii

ddd

dddddddddddddddddd

III

(3)

The dynamic model of the matrix converter can also be written using the filter inductor current, filter capacitor voltage and load current as;

( ) ( ) ( ) ( )( )tvtvL

tiLR

dttid

iss

ss

ss −+−= 1 (4)

( ) ( )( )titiCdt

vdis

s

i −= 1 (5)

( ) ( ) ( ) ( )tEtvL

tiLR

dttid

oo

oo

oo −+−= 1 (6)

Where ss vi , and ii vi , are the input side current and voltage after filtering and before filtering respectively.

sss CLR ,, are the input side filter parameters and E,L,R oo are the load parameters.

The dynamic model of the seven phase RLE load can be written as:

( ) ( ) ( )( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=+

o

sooo

so L

TRkikekvLTki 1ˆ1ˆ (7)

Where Ro and Lo are the resistance and inductance of the load, Ts is the sampling interval, io is the load current space vector, vo is the inverter voltage space vector used as a decision variable and e is the estimated back emf obtained from [8] as

( ) ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−−+=

s

oooo

so T

LRkikiTLkvke 1ˆ (8)

III. PREDICTIVE CONTROL STRATEGY

The total number of valid switching states is 37or 2187 for the three to seven phase matrix converter topology. The predictive control choses the best possible switching state of the converter based on the evaluation criteria and future predictions of the response of the system. Hence, a cost function is developed that is optimized by means of predictions of control variables using the dynamic model of the controlled variables. For that purpose two aspects are important, optimization of the appropriate cost function and the accurate model of the controlled variables. The control objectives for the system are; a) Precise load current control. b) The source side unity power factor with sinusoidal source currents. c) Simultaneous active and reactive power control. The cost function is formulated according to the control objectives targeted for the system under operation. The switching losses reduction and system efficiency improvement will automatically be achieved by these predefined control objectives. Thus the choice of the cost function is an intelligent step leading to an optimum results meeting all the set control objectives.

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Hence a new cost function is formulated and presented in this paper which incorporates the active and reactive power control with simultaneous current tracking. The tuning parameter λ and γ dictates the emphasis of control. This is highly important in grid connected converter system for wind generation and other applications requiring power factor control. The formulated cost function is given in equation (9) as;

( )

ssssoYoX

oYoXoooo

PPQQii

iiiiiikg

−+−+++

++−+−=+

**22

11**1

γλ

ββαα

(9)

Where *

oiα , *oiβ , *

sQ and *sP are the commanded alpha, beta,

reactive and active power components whereas oiα , oiβ ,

oXi 1 , oYi 1 , oXi 2 , oYi 2 are actual alpha, beta, X1, Y1, X2, Y2 orthogonal current components and sQ and sP are the actual reactive and active power respectively. The seven phase output current references are maintained as sinusoidal, the reactive power reference are put to zero for unity power factor at the source side and can regulate some positive or negative value depending upon the power factor requirement. The overall MPC schematic can be seen from Fig. 2. A total of sixteen ADC channels are used to calculate the optimal switching configuration in the proposed MPC scheme of three to seven phase matrix converter.

Fig. 2. Overall MPC schematic.

IV. SIMULATION AND REAL-TIME RESULTS

To validate the modulation method, Matlab/Simulink model is developed with the following parameters: Three-phase input voltage magnitude = 300V peak, Load element: R =10 ohm; L = 10 mH, Input supply frequency = 50 Hz. Output frequency = 70 Hz, The load is considered as simple R-L load. The results of the seven phase matrix

converter with model predictive control are investigated clearly. The resulting waveforms from the simulation model for seven phase open end R-L load is presented in Fig. 3-7. The seven phase reference and actual load currents are depicted in Fig. 3. Initially the current reference is set to 5A and after few cycle it is stepped to 8A to observe the transient behavior of the MPC system. The load current seems to be perfectly tracking the reference. The currents are sinusoidal, and are correctly displaced 51.40 with each other. A zoomed view at the transient point is also shown in the same figure to observe the robustness of the proposed MPC algorithm. The output orthogonal current components (Alpha, Beta, X1, Y1, X2, Y2) are shown in Fig. 4. The loci of these components are also plotted on the same figure. It clearly shows that all the undesirable components(X1, Y1, X2, Y2) are controlled and suitably eliminated by this control algorithm.

Fig. 3. Seven phase actual and reference load currents.

Fig. 4. Alpha-Beta, X1-Y1 and X2-Y2 components of output currents

The input side currents before filtering are shown in Fig. 5. The source currents after filtering and input phase-A voltage are shown in Fig. 6, where the sinusoidal nature of source

0 0.02 0.04 0.06 0.08 0.1-10

-5

0

5

10

15

Time (s)

Seve

n Ph

ase

Load

refe

renc

e an

d A

ctua

l Cur

rent

s (A

)

0.059 0.06 0.061

-505

0.02 0.04 0.06 0.08 0.1

-10

-5

0

5

10

Time (s)

Out

put C

urre

nt m

agni

tude

(A)

-5 0 5

-5

0

5

X-Axis---->

Y-A

xis-

---->

-0.4 0 0.4-0.5

0

0.5X1, Y1 Locus

X2, Y2 Locus

Alpha, Beta Locus

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current is evident. Unity power factor operation is evident from the voltage and current waveform. Again the transient portion is zoomed to observe the robustness of the system. Fig. 7 shows the transient operation in case of frequency variation. The reference frequency is changed to 10 Hz from 70 Hz and the typical tracking of the actual current can be observed. The transient behavior is also plotted on the same figure showing the healthy operation of the converter in case of frequency variation.

Fig. 5. Three phase unfiltered input currents.

Fig. 6. Three phase source currents and input phase-A voltage

Fig. 7. Output currents in case of frequency variation.

A modular, reconfigurable three-to-seven phase MC prototype is developed. The proposed MPC algorithm is implemented using dSPACE-1005 interface. The dSPACE-1005 is directly connected to all dSPACE I/O boards via

PHS and PHS++ bus. The MC prototype comprises of a diagonal IGBT (Part number: IXA27IF1200HJ, manufactured by XYS) and four anti-parallel ultra-fast recovery diodes (Part number: STTH812FP manufactured by ST Microelectronics). The maximum voltage-blocking capability of the IGBT is 1200 V and the current limit is 43 A. Additional line inductances are applied for the safe operation, in the case of overlapping of the current commutation. The values of passive components used in the experimental setup are listed as follows: input filter inductance, Lf : 200 µH, input filter capacitance, Cf : 20 µF , input filter resistance, Rf : 0.1 Ohm, clamp circuit capacitance, Cc : 3.3 nF , clamp circuit resistance, CR : 34 Ohm. The results are taken at 20 Hz output frequency operation. The reference value is tuned to 5.1 A to obtain noise free currents. The output currents are shown in Fig. 8. However, only four out of seven output currents are shown in the figure, due to limited number of oscilloscope channels (MSO8104A Infiniium Mixed Signal Oscilloscope). The input voltage and current can be observed from Fig. 9. The source current is slightly distorted which can be further improved by introducing active damping.

Fig. 8. Output at 20 Hz operation. 5A/div

Fig. 9. Input voltage and current. 100V/div and 2A/div.

V. CONCLUSION The presented model predictive control for the three to seven phase matrix converter effectively controls the source side current, the load side current and the active and reactive powers. The MPC control offers very attractive results with the approach of minimizing the objective function with the

0 0.02 0.04 0.06 0.08 0.1-20

0

20

Phas

e-A

0 0.02 0.04 0.06 0.08 0.1-20

0

20

Phas

e-B

0 0.02 0.04 0.06 0.08 0.1-20

0

20

Time (s)

Phas

e-C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-15

-10

-5

0

5

10

Time (s)

Thre

e ph

ase

sour

ce c

urre

nts (

A)

&

Inpu

t Pha

se-A

Vol

tage

(V/3

0)

0.06-505

Input Phase-A Voltage

0 0.05 0.1 0.15 0.2

-20

-10

0

10

Time (s)

Out

put C

urre

nts (

A)

-10

0

10

10 Hz Operation70 Hz Operation

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system prediction. The control objectives are properly selected to obtain the load current with very good tracking record. The investigation results shown validate the achievement of the control objectives to larger extent.

ACKNOWLEDGEMENT This publication was made possible by an NPRP grant

No. 04-077-2-028, from the Qatar National Research Fund (a member of The Qatar Foundation). The statements made herein are solely the responsibility of the authors.

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