Download II. Modeling of the PMSM

Performance Analysis of Direct Torque Control
(DTC) for Synchronous Machine Permanent Magnet
(PMSM)
Badre Bossoufi, Mohammed Karim, Ahmed Lagrioui
Badre Bossoufi, Silviu Ioniţă (Membre IEEE)
Laboratory of Data processing, Imagery and Analyzes
Numerical (LIIAN), Faculty of Sciences Dhar El Mahraz.
Fez, Morocco
[email protected], [email protected],
[email protected]
Center of Modeling and simulation of the systems, Faculty
of Electronics, Communications and Computers, University
of PITEŞTI
PITEŞTI, Romania
[email protected], [email protected]
Abstract -- The Direct Torque Control (DTC) is a technique
increasingly used to control the invertors for synchronous
machines. The process is complex including the analog
parameters and discrete events. This system can be seen as a
hybrid dynamic system where the analog part is the Permanent
Magnet Synchronous Machine (PMSM) and the discrete part is
the voltage inverter. In this paper, we propose a particular model
of the direct torque controller for a PMSM presenting the results
of simulation for a certain machine.
Keywords- Direct Torque Control (DTC), Permanent Magnet
Synchronous Machine (PMSM).

Elimination of the current controllers.

Elimination of rotor position sensor.

Inherent delays.
II.
MODELING OF THE PMSM:
The motor considered in this paper is a revolving-field
(interior) PMSM which consists of a three phase stator
windings and a PM rotor.
I. INTRODUCTION
Permanent magnet synchronous motor drives (PMSM)
offers many advantages over the induction motor, such as an
improved overall efficiency, the effective use of reluctance
torque, the smaller losses and the compact built size. Recently,
many studies point out different solutions for the PMSM drive
control providing good capabilities for the quick and precise
torque response, and avoiding the inconveniences related on
the complexity of field-oriented control (FOC) algorithms [1],
[2], [3]. The DTC technique has been recognized as a viable
and robust solution to achieve these requirements. In the
existing literature, many algorithms have been suggested for
the DTC control implementation [1], [4], [5], [6]. Basically,
DTC manages the suitable voltage-vectors according to the
magnetic field flux of the stator and to the difference between
the real torque and the prescribed one, named reference. For
the PMSM drive, the classical eight voltage-vector switching
scheme seems to be suitable only for high speed operation of
the motor while at low speed the six voltage-vector switching
scheme, avoiding the two zero voltage-vectors, seems to be
appropriate.. The voltage-vector strategy using switching table
is widely approached by researchers and it is also a very
common commercial technique, because it is very simple as the
principle and very easy to be implemented respectively. The
stator fluxes linkages are calculated from voltage and current
models PMSM drive. The DTC is very attractive for designers
because of certain features as follows:

Simplicity of its structure.
Figure 1. The basic scheme of the synchronous machine
The voltage equations in a synchronous reference frame can
be derived as follows [4]:
u sd  rs .isd 
usq  rs .isq 
d sd
 . sq
dt
d sq
dt
 . sd
(1)
(2)
Where the direct and quadrate axis flux linkages are:
 sd  Ld .isd   f
(3)
 sq  Lq .isq
(4)
The electromagnetic torque of the motor can be evaluated
as follows,
Ce 

3
p  f .I q  ( Ld  Lq ).I d .I q
2

(5)
The motor dynamics can be simply described by the
equation (6).
Ce  Cr  J .
d
 f .
dt
(6)
The symbols denote the parameters as follows:
Ω - rotation's speed mechanical of the PMSM,
 - Rotation’s speed electric,
p - Number of pairs of poles,
J - Total moment of inertia about the shaft of the PMSM,
f - Coefficient of viscous friction,
Cr - Load torque,
 f - Flux produced by the permanent magnet,
 sd - d axis stator magnetic flux,
 sq - q axis stator magnetic flux,
Ld - d axis stator leakage inductance,
Lq - q axis stator leakage inductance,
rs - Stator winding resistance,
Ce - electromagnetic torque.
III.
DIRECT TORQUE CONTROL. PROBLEM STATEMENT
DTC was proposed in the middle of 1980's by
Depenbrock and Isao Takahashi [9], [11]. The basic idea of
DTC for induction motor is slip control, which exploits the
relationship between the slip and the electromagnetic torque
[2]. In the 1990's, DTC for was developed for PMSM [7],
[12], [13]. By comparison for instance the Rotor Field
Oriented Control, the DTC has some essential advantages such
as less dependency of machine parameter, simpler
implementation and quicker dynamic torque response. There
is no current controller needed in DTC, because it selects the
voltage space vectors according to the errors of flux linkage
and torque. The most common way to carry out the DTC is a
switching table and hysterics controller, as is largely described
in [8] and [14]. In Fig. 2 is depicted a typical DTC system. It
includes flux and torque estimators, flux and torque hysteretic
controllers and a switching table. Usually a DC bus voltage
sensor and two output current sensors are needed for the flux
and torque estimator. Speed sensor is not necessary for the
torque and flux control. The switching state of the inverter is
updated in each sampling time. Within each sampling interval,
the inverter keeps the state until the output states of the
hysteretic controller change. Therefore, the switching
frequency is usually not fixed; it changes with the rotor speed,
load and bandwidth of the flux and torque controllers.
Figure 2. Control of DTC
A. Transformation abc-αβ (Clark):
As control DTC is a vectorial control, it is necessary to
have the components of Concordia of the currents and stator
voltages of the PMSM. One thus breaks up the three stator
currents isabc and the three stator voltages vsabc into components
direct (vs) and quadratic (vs) such as:
 xs 
x  
 s 
1  0.5  0.5  xsa 
2
x  ,
sb
3 0 3  3   
2
2   xsc 

where the parameter
x
(7)
can be Vs or is.
Since the model of the PMSM is expressed in the reference
model (d-q), a transformation from the two-phase system (α-β)
to the two-phase system (d-q) (Concordia) proves to be
essential:
isd  cos( )
 
isq   sin(  )
sin(  )  is 
.  (8)
cos( ) is 
B. The Estimator
1) Flux Estimator
The stator electric equations in the reference model (α-β)
are given by the following equations:
d s 

Vs  rs .is  dt

V  r .i  d s
s s
 s
dt
(9)
where the flux is expressed as follows:
t

ˆ s  (Vs  rs .is ).dt

0

t
ˆ
 s  0(Vs  rs .is ).dt

ˆ 
ˆ  j.
ˆ

s
s
s

(10)
For the high speeds, the voltage drop is neglected and the
equations become:
t

ˆ  V .dt
s
s

t

t
ˆ
 s  t Vs .dt

 s   2s   2s

(11)
2) Torque Estimator
The electromagnetic torque is given an equation derived
from (4) as follows:
Ce 
3p
( s .is   s .is )
2
(12)
where p is the number of pairs of poles
C. Inverter switching control
The switches of the voltage inverter, depicted in Fig.3, must
be ordered so as to maintain the flow and the torque of the
machine. The vector of the stator voltage can be written in the
form:
Vs 
2
.U 0 .( S a  Sb .e
3
2
j
3
 Sc .e
4
j
3
)
(13)
Where (Sa, Sb, Sc) are the logical state of the three switches:
Si = 1 means that the high switch is closed and the low switch
is open (Vi = +U0) and Si = 0 mean that the high switch is
opened and the low switch is closed (Vi = -U0).
The eight voltage vectors are expressed as follows:

V1  V8  0


2
V2  3 .U 0


2
3
.U 0 .(0.5  j
)
V3 
3
2


2
3
.U 0 .(0.5  j
)
V4 
3
2


2
V5   .U 0
3


2
3
V6 
.U 0 .(0.5  j
)
3
2


V7  2 .U 0 .(0.5  j 3 )

3
2
(14)
The eight voltage vectors Vi can be represented on the
complex plane in certain positions as is depicted in Fig.4.
Figure 3. Typic voltage inverter
The machine’s flow and torque control are done by the
discrete events via selecting of the vector of voltage from the
switches of the inverter. As we have three switches, 23 = 8
possibilities results for the Vs vector. Based (13), two of these
eight possibilities (Sa,Sb,Sc)=(0,0,0) and (Sa,Sb,Sc)=(1,1,1)
correspond to the null vector (i.e. the vectors V1 and V8).
Different states of the switches are provided in Tab.1
TABLE I. THE LOGICAL STATES OF THE INVERTER SWITCHING
Figure 4. The voltage vectors and their area of detection
D. Control vector flux
In order to obtain good dynamic performances, a corrector
with hysteresis with two levels is the simplest and best solution
adapted to the DTC (see Fig. 5). This type of controller is able
to easily control and maintain the terminus of the vector flux Φs
in a circular ring, that suggesting its name: flow corrector. The
output of the corrector represented by a Boolean variable eΦ
(=0 or 1) must indicate if the module of flow must decrease
(eΦ=0) or increase (eΦ=1) by such kind to always maintain
ˆ   s
 sref  
Figure 7. Corrector of the torque with hysteresis on 3 levels
F. Law of control
Basically, the law of control implements the rules of choice
of vector VS. There are three parameters that matter for
choosing the adequate vector VS:
Figure 5. The flow corrector with hysteresis on 2 levels
As the direction of the vector flow Φ s is given by the
selected vector of voltage Vi, the sequence of the vectors
voltage maintains the flow in an annulus equal to the width of
hysteresis. In the Fig. 6 is depicted the effect of the correct
flow.
1) The sector determined by the phase of estimated
flow   arctg    . There are six sectors which the


phase can belong.
2) The state from the 2-level hysteretic flow corrector.
3) The state from the 3-level hysteretic torque corrector.
This typical look-up-table control technique is given for
PMSM in Tab.II.
TABLE II. DECISION TABLE FOR PMSM CONTROL
Figure 6. The sequence of the vectors voltage
E. Control Torque
The corrector of the torque has three levels. It makes it
possible to control the motor in both directions of rotation in
terms of the positive or negative torque. The output of the
torque corrector is the Boolean variable eCe which must limit
the torque to a value such as Ceref  Cˆ e  Ce . As it can be
seen in Fig.7 the output eCe can take three values according to
the following rules:

If the error of the torque: Ceréf Ce >0, then it is
necessary to increase the torque: eCe= 1 ;

If the error of the torque: Ceréf Ce<0, then the torque
should be decreased: eCe= 1 ;

If the error of the torque: ∆Ce ≤Ceréf Ce ≤∆Ce it is
necessary to keep the same value of the torque: eCe= 0.
For example if the flow is in sector 1, and the flow Φs trend
to increase (eΦ=1) and the torque also is increasing (eCe=1), the
vector voltage to be applied to the PMSM will be V3. This
choice will have the effect of decreasing the motor’s torque.
Based on Fig.8, different cases for the vector voltage
choosing when the vector flux is located in the certain sector.

If V6 is selected flow must decrease (eΦ=0) and also
torques (eCe = 1).

If V1 or V8 is selected flow must remain constant (eΦ=
the precedent state) and the torque decreases (eCe=0).

If V3 is selected flow must increase (eΦ=1) and also
torques (eCe=1).

If V7 is selected flow must increase (e Φ =1) and the
torque must decrease (eCe= 1).

If V4 is selected flow must decrease (eΦ=0) and the
torque must increase (eCe=1).
Figure 8. The vector voltage when the vector flux is located in certain sector
IV. SIMULATION AND RESULTS
A. Model Verification
In Fig.9 is presented the entire diagram of an original
structure of the DTC for PMSM in the reference model d-q.
The currents isd, isq, Vsd and Vsq are the subject of the Clark’s
transformation in order to obtain the components isα, isβ, Vsα
and Vsβ. The components are going to the estimator of torque
and flow as well as to the detector of the sector. Thereafter, the
estimated values are compared with the references to be
applied to the hysteretic correctors. Their outputs together the
reported sector from detector are introduced into a table of
commutation where is decided the right combination of
switches to the inverter. The inverter will generate the voltage
on three-phase current path.
After that they will be
transformed into coordinates d-q, and the output voltages Vsd
and Vsq are applied in average values at the stator’s terminals
of the PMSM. The MATLAB/Simulink environment was used
to implement the model of the permanent magnet synchronous
motor according to the d-q model and to develop all the
required functional blocks.
B. Results of Simulation
A certain PMSM was used in our study with the following
parameters: Torque nominale=14.2Nm, p=4, rs=0.4578Ω,
Φf=0.171Wb, Lsd=3.34mH, Lsq =3.58mH, J = 0.001469kgm2.
Figure 9. Blocks for the simulation of the DTC under Matlab/Simulink
The inverter dc bus voltage is 300V. We supposed that
the stator magnetic flux amplitude value is the same as the
value of the permanent magnet flux. Also at t=0.07s, a
differential torque step from 8Nm to 0Nm and at t=0.14s
from 0Nm to 8Nm is applied as the reference torque value.
The simulation results are graphical represented in Fig.
10 to 14 and reflects right behavior of the machine with
DTC. The characteristics obtained for the simulated motor
are similar with the other reported in literature. They prove
the efficiency of DTC system.
Figure 10. Trajectory of flux
Figure 11. Electromagnetic Torque
performances and the quality of the control are reflected in
hardware implementation. Machine control is a complex
subject involving issues from the embedded systems and
microelectronics to the power electronics. This kind of
electronics working with analog and digital signals and also
with voltages from milivolts to kilovolts in the same module
is a challenge for packaging and the reliability. The way to
the high capable electronics starts with the appropriate model
and functional simulation of the systems. In this article we
have tested a model of the system PMSM-voltage inverter
DTC controller for a certain machine.
REFERENCES
[1]
[2]
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Figure 12. Stator voltage (V)
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[7]
[8]
[9]
Figure 13. Evolution of the Amplitude of Φs
[10]
[11]
[12]
[13]
[14]
Figure 14. Rotor Speed
CONCLUSIONS
The technology of PMSM control becomes essential for
many applications including the electric vehicles. The
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