behavior of the predictive dtc based matrix converter under unbalanced ac supply
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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.
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