th
The 5 Student Conference on Research and Development –SCOReD 2007
11-12 December 2007, Malaysia
Study on Stability and Performances of DTC
Due to Stator Resistance Variation
Auzani Jidin, N.R.N. Idris, Senior Member, IEEE, A.H.M. Yatim, Senior Member, IEEE
Abstract-- A stator-flux-based direct torque control (DTC) of
induction machine has shown significant interests to researchers
since its offer several advantageous compared to many other
control schemes. This scheme requires only one parameter which
is a stator resistance in order to estimate the stator flux-linkage,
hence the electromagnetic torque can be directly calculated. In
practice, a wide variation in stator resistance can occur due to a
large temperature variation. Thus, the mismatch between
estimated and actual stator resistances will deteriorate the
performance of DTC by introducing errors in the stator flux
estimation particularly at low speed. Consequently, the torque
performance will be degraded. This paper presents the study on
stability and performances of DTC due to stator resistance
variation. The effect on DTC performances due to the stator
resistance variation will be analytically studied with the help of
simulation results.
Index Terms-- Direct torque control (DTC), Hysterisis
controller, Induction machines, Stator flux estimation, Stator
resistance.
I. INTRODUCTION
H
igh performance torque controlled induction machine
drives can be achieved when the proper control scheme
utilizes accurate estimated flux. In general, there are two
methods for flux estimation. One is based on current-based
method and the other is voltage-based method. In the current
based method, the flux estimation can be made at any speed,
including zero speed with the help of speed and current
signals. Because of that, this model requires a speed encoder
which is not suitable for some applications. Furthermore, the
estimation accuracy is strongly affected by the variation of
machine parameters, in which some of the parameters are
difficult to compensate because of inaccessibility.
Alternatively, the voltage-based method is mostly preferred
since it only requires one parameter such that a stator
resistance, Rs , and moreover the motor speed sensor is not
required. Thus, this simple structure of flux estimation offers
Auzani Jidin is with Department of Power Electronics and Drives, Faculty
of Electrical Engineering, Universiti Teknikal Malaysia Melaka, Locked Bag
1200, Hang Tuah Jaya, Ayer Keroh 75450 Malacca. (e-mail:
[email protected]).
Nik Rumzi Nik Idris and Abdul Halim M. Yatim are with Department of
Energy Conversion, Faculty of Electrical Engineering, Universiti Teknologi
Malaysia, 81310 UTM, Johor. (email: [email protected])
1-4244-1470-9/07/$25.00 ©2007 IEEE.
less sensitivity to parameter variations. However,
implementation of flux estimation that utilizes an integrator in
this model is no easy task particularly at low speed operation.
The major problems such that a dc drift and initial value
problems can occur in a pure integrator with small offset in the
measurement system utilized in estimating the flux [1][2]. The
problems have been tackled using many strategies in order to
improve the flux estimate so as to make the control scheme
employed, able to operate for a wide speed range, as proposed
in [3]-[5].
Instead of having the problems due to the inaccurate
integration technique or improper sensor as mentioned above,
other problem that has also shown significant interests to be
solved, is errors introduced in flux estimation due to variation
in stator resistance [6]-[13]. In real practice, a large extent in
temperature variation can cause a wide variation in stator
resistance changes in induction machine [6]. Consequently, it
will deteriorate the performance of DTC by introducing errors
in the stator flux estimation and also in the electromagnetic
torque estimation particularly at low speeds. The sensitivity of
the estimator to variation in stator resistance changes is
extremely affected in low speed region wherein the stator
resistance voltage drops become dominant as the stator
voltages become very small. In this case, the adjustment of the
estimated stator resistance to track a wide variation of actual
stator resistance changes is mandatory for accurate flux and
torque estimations. Numerous papers have been presented on
tuning of stator resistance [6]-[13]. In general, the tuning of
estimated stator resistance can be classified into two purposes.
One is for the thermal monitoring purpose which is one of the
fundamental protections required for induction motors [7]-[9],
and the second one is for the flux estimation.
Commonly, there are two approaches for estimating Rs ,
namely induction machine model-based Rs estimation [7],[8]
and signal injection-based Rs estimation [9],[10],[13]. The
approaches are widely been implemented for the purpose of
flux estimation to obtain excellent drive performance. Ideally,
the stator resistance estimation should be able to operate over
the entire speed range and independent of the motor control
method. For example, the stator resistance tuning has been
proposed using hybrid flux estimation [13,][14]. The hybrid
flux estimation utilizes a combined model which has smooth
transition from the stator voltage to the rotor voltage based
flux estimations from low speeds to high speeds, and vice
versa. In [9],[10], the stator resistance is calculated by means
of dc components of the voltage and current measurements. In
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this method, a dc bias is injected on-line, into the stator supply
voltage. Although these methods are capable of giving a more
reliable in Rs estimation, the motor torque performance may
be disturbed due to the injected signal on-line. It is also
possible to synthesize the tuning of stator resistance using
intelligent control technique. For instance, [11] proposed
Fuzzy estimator which employs stator current phasor error to
adjust the stator resistance. It has been shown that, among the
other variables of the machine, the stator current is highly
affected by the resistance changes. From that point of view,
the PI adaptive control using stator current phasor error was
presented in [6]. The stator current command was derived with
stator flux and torque references as inputs. However, the
derived stator current command does not consider other inputs
such as load torque and operating frequencies, in case if the
stator flux locus tends to become hexagonal shape at low
speeds. In addition, the way to limit the speed is not
mentioned and questionable.
In this paper, the effect on stability and performances on
DTC will be discussed with the help of simulations. As
reported in [6], the instability occurs particularly when
estimated Rs is greater than its actual value. However, the
explanation about this matter is only limited to the
mathematical analysis based on induction machine equations.
This paper will present some analysis on performances of the
drive control quantities (i.e stator current, torque, flux, stator
voltage vectors) when a step stator resistance change is
applied particularly at low speeds operation. The significance
of the study, is to give more insight to the instability which
occur when there is a mismatch in the stator resistance.
.
II. PRINCIPLE OF DIRECT TORQUE CONTROL
The induction machine equations in terms of space vectors,
written in a stator reference frame, are
dψ s
dt
dψ r
0 = Rr i r +
− j ω rψ r
dt
v s = Rs i s +
structure wherein the torque and flux can be separately
controlled using hysterisis comparators as shown in Fig. 1.
The stator flux can be directly calculated from the stator
voltage equation in the stator reference frame that
(
= ³ (v
)
)dt
ψˆ sd = ³ v sd − Rˆ s isd dt
ψˆ sq
sq
− Rˆ s isq
(5a)
(5b)
where v sd ( i sd ) and v sq ( i sq ) are the respective d and q-axis
of stator voltage (current) components. R̂s is the estimated
stator resistance. Thus the magnitude of stator flux is
calculated by
(
2
2
ψˆ s = ψˆ sd
+ ψˆ sq
)
(6)
and the stator flux angle, φ is calculated by
§ ψˆ sq
© ψˆ sd
φ = arctan¨¨
·
¸
¸
¹
(7)
By considering that, ψˆ s = ψˆ s e jφ and i s = i s e jα , where α
is the angle of the stator current with respect to the direct-axis
of the stator reference frame, the torque can be calculated as
written into the following form
3 p
Tˆe =
ψˆ s i s sin (α − φ )
22
(8)
where p is the number of pole and (α − φ ) is the angle
between the stator flux linkage and stator current space vector.
(1)
(2)
where ωr is the rotor angular speed in electrical radians and
v s is the stator voltage space vector. i s and i r are the stator
and rotor current space vectors, while Rs and Rr are the stator
Te
dTe
*
ψs
dψ s
*
φ
ψˆ e
Tˆe
and rotor resistances, respectively. ψ s and ψ r are the stator
and rotor flux linkages, respectively and are given by
ψ s = Ls i s + Lm i r
(3)
ψ r = Lr i r + Lm i s
(4)
where Ls and Lr are the stator and rotor self-inductance,
respectively.
Unlike in FOC, the DTC scheme offer simple control
Fig. 1. Basic DTC
In this scheme, the command values of flux, ψ s* and
torque, Te* are compared to their respective estimated values.
The errors are fed to a two level comparator, in the case of
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flux, and a three level comparator in the case of torque. The
flux comparator output is defined as
IV
III
010
*
dψ s = 1 for ψˆ s ≤ ψ s −
∆ψ s
(9a)
2
∆ψ s
*
dψ s = 0 for ψˆ s ≥ ψ s +
2
100
001
010
(9b)
V
101
110
011
101
(10a)
(10b)
001
101
010
100
001
010
110
001
101
011
dTe = 0 for
for counter-clockwise (CCW) rotation and
100
100
and torque comparator output as
dTe = 1 for
110
011
110
011
001
Tˆe ≤ Te* − ∆Te
Tˆe ≥ Te*
010
110
011
100
010
110
001
101
011
II
101
100
VI
I
Fig. 2. Six sectors of stator flux plane
dTe = 0 for
Tˆe ≥ Te*
+ ∆Te
(11a)
dTe = −1 for Tˆe ≤ Te*
for clockwise rotation (CW).
(11b)
III. EFFECT OF STATOR RESISTANCE VARIATION ON STABILITY
AND PERFORMANCES IN DTC
The output of the comparators and the stator flux angle are
used to index a look up table of optimum voltage vectors as
proposed in [14], in order to determine the suitable voltage
vectors, which is tabulated in Table 1. The sector of the stator
flux as illustrated in Fig. 2 is divided into six sectors. Fig. 2,
indicates that, the appropriate voltage vector (is taken from the
table of optimum voltage vectors) should be chosen in a
particular sector, either to increase stator flux or to decrease
stator flux and either to increase torque or to reduce torque.
The selection of voltage vector is made so as to restrict the
errors of the stator flux and torque within their respective
hysterisis bands. Consequently the fastest torque response and
highest efficiency at every instant can be obtained [14].
TABLE I VOLTAGE VECTORS LOOK-UP TABLE
dψ s
1
0
TABLE 2 INDUCTION MACHINE PARAMETERS
Sector
Counter clockwise
In DTC, the accurate estimated stator flux is very important
to achieve high performance of torque control. To obtain the
high degree accuracy of the stator flux estimation, the stator
resistance value, which is used in the controller, must match to
its real value. The mismatch between the estimated stator
resistance value and its actual value somehow can deteriorate
the torque and flux control especially when the estimated
stator resistance is higher than its actual value as reported in
[6]. However, the explanations on how the unstable of the
performance occurs were not discussed in details. This section
will perform a study on the effect of the higher estimated
stator resistance than its actual value on DTC. On the other
hand, the discussion of the root cause of the instability, will
also be presented. All the analysis will be performed through
simulations using induction machine parameters as given in
Table 2.
dTs
I
II
III
IV
V
VI
1
0
-1
1
0
-1
110
111
101
010
000
001
010
000
100
011
111
101
011
111
110
001
000
100
001
000
010
101
111
110
101
111
011
100
000
010
100
000
001
110
111
011
Stator resistance, Rs
Rotor resistance, Rr
Stator self inductance, Ls
Rotor self inductance, Lr
Mutual inductance, Lm
Combined inertia, J
No. of pole, p
Rated torque, Trated
Rated stator flux, Ψs,rated
5.5 Ÿ
4.51 Ÿ
306.5 mH
306.5 mH
291.9 mH
0.0165 kg-m2
4
10 Nm
1.2 Wb
For the sake of discussion, a mismatch between the
estimated stator resistance and actual stator resistance is
performed with a step actual stator resistance change from
100% to 75% of its nominal value at 2 s. A half of rated torque
command is applied at 0.1 s and a rated stator flux command
is kept constant. The motor is kept running at low speed which
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is about 20 rad/s by adjusting the load torque to the
appropriate value. From Fig. 3, it can be observed that, the
system becomes unstable when estimated stator resistance is
higher than its actual value. In general, the reason of the
instability of the system is due to the opposite effects between
the controller and the motor. Reference [6], reported that, the
increased current due to lower actual stator resistance value,
cause increased stator resistance voltage drops in the estimator
resulting in lower flux linkages and electromagnetic torque
estimations. The lower estimated flux linkage and
electromagnetic torque result in larger errors when they are
compared with their command values. As a result, the
commanding larger voltage is selected to compensate the
errors. In this case, the commanding larger voltage to
compensate the errors as stated in [6], is cannot totally be
acceptable. The next section will study the reasons of
instability problems by examining the effects on flux, torque,
stator voltage vectors and stator current due to a step stator
resistance change with the aid of simulations.
desired stator flux locus is the root on why the unstable
condition occurred. It can be observed that in Fig. 4(b),
particularly at the dotted line in sector 3, the actual stator flux
start to deviate from the estimated stator flux, just after a step
actual stator resistance change is applied at t=2 s.
The divergence between the actual and estimated stator flux
may be reasoned as follows. Firstly, let us study the change of
the actual and estimated stator flux vectors in the particular
sector as illustrated in Fig. 5. Using (1), the change of the
ˆ s and actual stator flux vector,
estimated stator flux vector, ∆Ψ
∆Ψs can be expressed as given in (12a) and (12b),
respectively.
(
= (V
)
R ).∆T
ˆ = V s ,k − I s Rˆ .∆T
∆Ψ
s
s
∆Ψs
s ,k
−Is
(12a)
(12b)
s
ψs
t ≥ 2s
Rs
ψˆ s
is
t ≤ 2s
ωe
t ≥ 2s
ωe
Te
ψˆ s = ψ s
ψs
ψs
ψˆ s
Tˆe
ψˆ s
(a)
Fig. 4
Fig. 3 Instability due to parameter mismatch for a step stator resistance
change
Actual stator flux
θ ≤ φ ≤ θ + 2π
(b)
ψs
and estimated stator flux
ψˆ s ,
for
, (a) before stator resistance mismatch and (b) after stator
resistance mismatch.
where, I s is the stator current vector and ∆T is the time
A. Effect on Stator Flux
Using simulations, the actual and estimated stator flux
locus before and after a step stator resistance change are
carried out as depicted in Fig. 4. The figure shows the
simulation results for one cycle of rotation of stator flux
trajectories. The trajectories of estimated and actual stator flux
are rotated in anticlockwise. The figure also depicts that the
locus of the estimated and actual stator flux tends to become
hexagonal shape because of the flux weakening at every sector
transition. Note that, the locus of estimated and actual stator
flux are deviated each other when estimated stator resistance is
higher than actual stator resistance.
The divergence of the actual stator flux locus from the
interval for the same stator voltage vector V s ,k is to be
applied either to increase or decrease the stator flux vector. In
such case, to increase the stator flux an appropriate active
voltage vector <010> is selected while to decrease the stator
flux a zero voltage vector <000> is chosen. Note that, the step
change of the actual stator resistance is set at t=2 s (in sector
3). From Fig. 5, the switching actions kept the increasing and
decreasing of the estimated stator flux within its hysterisis
band approximately. It can be considered that, the estimated
stator flux is well regulated at the beginning, since the
magnitude and phase errors of estimated stator flux can be
neglected; hence no incorrect voltage vector is selected.
Taking into account the fact that, the switching is strongly
affected by the torque regulation. Therefore, the larger
commanding voltage vector to compensate the sudden larger
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errors in the estimated flux and torque as stated in [6], is
questionable. It can be observed that the divergence of the
actual stator flux from estimated stator flux is mainly caused
by the difference between the actual and estimated stator flux
change vectors.
As given in (12), the reduction of estimated stator flux
vector during the application of zero vectors is significantly
higher than that of the actual stator flux vector. Note that, the
ohmic drop in (12a) is greater than that in (12b) when the
same zero voltage vector <000> is applied to decrease these
stator flux vectors. While the increasing of the actual and
estimated stator flux vectors are almost similar since the
ohmic drops in both (12a) and (12b) for the same applied
active voltage vector <010> become less significant. In this
case, the higher change of stator flux vector means the higher
change of stator flux magnitude and hence a steeper slope of
the stator flux vector.
B. Effect on Torque
Based on (8), the accurate of stator flux estimation is
mandatory in estimating the torque. The accurate estimated
torque is so important for high performance and proper control
in DTC since the switching action is strongly affected by the
torque controller. From Fig. 6, the error of estimated torque is
restricted within its hysterisis band since the estimated stator
flux is considered well-regulated in this particular sector.
However, the actual torque deviates from its hysterisis bands
since it is strongly influenced by the actual stator flux.
Fig. 5 Effect of estimated and actual stator flux magnitude due to parameter
mismatch for a step stator resistance change.
transition between sectors. Fig. 8, indicates that the magnitude
of the actual stator flux and stator current are steeply
decreased due to the incorrect voltage vector <001> is selected
in sector 3 of the actual stator flux. Suppose that, the voltage
vector <111> or <011> is selected during flux weakening at
this particular sector transition.
In practice, the stator resistance does not change in a step
manner. A linearly increasing estimated stator resistance is
simulated and the performance is shown in Fig. 9. Even for
such a gradual change of stator resistance, note that the system
becomes unstable. The reason of the instability is similar as
mentioned for a step stator resistance change.
IV. CONCLUSION
This paper has discussed on the effects of the mismatch
values between actual stator resistance and estimated stator
resistance on the performance of DTC. A step change in actual
stator resistance is used to show how the instability of the
system occurred. The simulation results are used to study the
performances of the drive system particularly at low speed in
which the stator flux is weaken at every sector transition. This
study reveals that, the mismatch in the stator resistances result
in divergence on the actual from the estimated flux,
consequently introduced phase and magnitude errors. These
errors lead to the incorrect voltage vector is selected especially
at every transition between sectors.
Fig. 6 Effect of estimated and actual torque due to parameter mismatch for a
step stator resistance change
C. Effect on Voltage Vector Selection
To study the effect on voltage vector selection, let us
examine together with the effect on the stator current and
stator flux, at the first transition between sectors after a step
stator resistance change is applied. From Fig. 7, it can be seen
that the actual and estimated sectors are different at the
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Fig. 7 Effect to the voltage vector selection at sector transition. (a) Actual and
estimated sector, (b) switching of phase A, (b) switching of phase B, (c)
switching of phase C.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Fig. 8 Effect to the stator current, actual and estimated stator fluxes
corresponds to the incorrect voltage vector selection as shown in Fig. 7.
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is
Te
ψs
Tˆe
ψˆ s
Figure 9 Instability due to parameter mismatch for a linear stator resistance
change
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