Post on 16-Dec-2015
transcript
Electric Drives 1
9.11. FLUX OBSERVERS FOR DIRECT VECTOR CONTROL
WITH MOTION SENSORS
The motor stator or airgap flux space phasor amplitude ma and its
instantaneous position - er+a* - with respect to stator phase a axis have to be
computed on line, based on measured motor voltages, currents and, when available, rotor speed. The torque may be calculated from flux and current space phasors and thus once the flux is computed and stator currents measured, the torque problem is solved.
9.11.1. Open loop flux observers
Open loop flux observers are based on the voltage model or on the current model. Voltage model makes use of stator voltage equation in stator coordinates (from (9.32) with 1 = 0):
(9.95)s
ss
sss
s sirV
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From (9.37) with a = Lm / Lr:
(9.96)
Both stator flux, , and rotor flux, , space phasors, may thus, in
principle, be calculated based on and measured. The corresponding signal
flow diagram is shown in figure 9.29.
sscs
m
rsr iL
L
L
Figure 9.29. Voltage - model open loop flux observer (stator coordinates)
s rsV si
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On the other hand the current model for the rotor flux space phasor is based
on rotor equation in rotor coordinates (b = r):
(9.97)
Two coordinate transformations - one for current and other for rotor
flux - are required to produce results in stator coordinates. This time the
observer works even at zero frequency but is very sensitive to the detuning of
parameters Lm and r due to temperature and magnetic saturation variation.
Besides, it requires a rotor speed or position sensor. Parameter adaptation is a
solution.
The corresponding signal flow diagram is shown in figure 9.30.
0L
sLrirm
rr
rrr
sr
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Figure 9.30. Current control open loop flux observerMore profitable, however, it seems to use closed loop flux observers.
9.11.2. Closed loop flux observers
Figure 9.31. Close loop voltage and current model rotor flux observer
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Many other flux observers have been proposed. Among them, the third flux
(voltage) harmonic estimator [21] and Gopinath observer [22], model
reference adaptive and Kalman filter observers.
They all require notable on line computation effort and knowledge of
induction motor parameters. Consequently they seem more appropiate when
used together with speed observers for sensorless induction motor drives as
shown in the corresponding paragraph, to follow after the next case study.
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9.12. INDIRECT VECTOR SYNCHRONOUS CURRENT CONTROL
WITH SPEED SENSOR - A CASE STUDY
The simulation results of a vector control system with induction motor based
on d.c. current control - are now given. The simulation of this drive is
implemented in MATLAB - SIMULINK. The motor model was integrated in
two blocks, the first represents the current and flux calculation module in d -
q axis (figure 9.32), the second represents the torque, speed and position
computing module (figure 9.33).
The motor used for this simulation has the following parameters: Pn =
1100W, Vnf = 220V, 2p = 4, rs = 9.53, rr = 5.619, Lsc = 0.136H, Lr =
0.505H, Lm = 0.447H, J = 0.0026kgfm2.
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The following figures represent the speed, torque, current and flux responses, for the starting process and with load torque applied at 0.4s. The value of load torque is 4Nm.
Figure 9.34. Speed transient response
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9.13. FLUX AND SPEED OBSERVERS IN SENSORLESS DRIVES
Sensorless drives are becoming predominant when only up to 100 to 1 speed control range is required even in fast torque response applications (1-5ms for step rated torque response).
9.13.1. Performance criteria
To assess the performance of various flux and speed observers for sensorless drives the following performance criteria have become widely accepted:
steady state error; torque response quickness; low speed behaviour (speed range); sensitivity to noise and motor parameter detuning; complexity versus performance.
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9.13.2. A classification of speed observers
The basic principles used for speed estimation (observation) may
be classified as:
A. Speed estimators
B. Model reference adaptive systems
C. Luenberger speed observers
D. Kalman filters
E. Rotor slot ripple
With the exception of rotor slot ripple all the other methods imply
the presence of flux observers to calculate the motor speed.
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9.13.3. Speed estimators
Speed estimators are in general based on the classical definition of
rotor speed :
(9.98)
where 1 is the rotor flux vector instantaneous speed and (S1) is the rotor
flux slip speed. 1 may be calculated in stator coordinates based on the
formula:
(9.99)
or (9.100)
r
^
^
11
^
r
^
S
qrdrs
rs
r1
^
j ;Argdt
d
2sr
sqr
sdr
sdr
sqr
1
^
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are to be determined from a flux observer (see figure 9.32,
for example). On the other hand the slip frequency (S1), (9.23), is:
(9.101)
Notice that is strongly dependent only on rotor resistance rr as
Lm / Lr is rather independent of magnetic saturation. Still rotor resistance is to
be corrected if good precision at low speed is required. This slip frequency
value is valid both for steady state and transients and thus is estimated
quickly to allow fast torque response.
Such speed estimators may work even at 20rpm although dynamic capacity of
torque disturbance rejection at low speeds is limited.
This seems to be a problem with most speed observers.
sqr
sdr
sdr
sqr ,,,
r
2r
mdsqrqsdr
rr2
r
mdsqrqsdr^
1
Lii
Lr/p23
Liip23
S
^
1S
r
^
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9.13.4. Model reference adaptive systems (MRAS)
MRASs are based on comparision of two estimators. One of them
does not include speed and is called the reference model. The other, which
contains speed, is the adjustable model. The error between the two is used to
derive an adaption model that produces the estimated speed for the
adjustable model.
To eliminate the stator resistance influence, the airgap reactive power
qm [25] is the output of both models:
(9.102)
(9.103)
The rotor flux magnetization current equation in stator coordinates
is ((9.15) with 1 = 0):
(9.104)
r
^
dt
idLViq
*s
scssm
s
^
m
^
r
r
^
s
^
m
^
mm
^
ii1
iiLq
m
^
i
r
m
^
s
^
m
^
r
^m
^
1iii
dt
id
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Now the speed adaptation mechanism is:
(9.105)
The signal flow diagram of the MRAS obtained is shown in figure 9.39.
m
^
mi
pr
^
qqs
KK
Figure 9.39. MRAS speed estimator based on airgap reactive power error
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The effect of the rotor time constant r variation persists and influences the speed estimation. However if the speed estimator is used in conjunction with indirect vector current control at least rotor field orientation is maintained as the same (wrong!) r enters also the slip frequency calculator.
The MRAS speed estimator does not contain integrals and thus works even at zero stator frequency (d.c. braking) (figure 9.40.a) and does not depend on stator resistance rs. It works even at 20rpm (figure 9.40.b) [25].
Figure 9.40. a.) Zero frequency b.) low speed operation of MRAS speed estimator
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9.13.5. Luenberger speed observers
First the stator current and the rotor flux are calculated through a full order
Luenberger observer based on stator and rotor equations in stator coordinates:
(9.106)
with (9.107)
The full order Luenberger observer writes:
(9.110)
The matrix G is chosen such that the observer is stable.
(9.111)
s1
sr
ss
2221
1211
sr
ss
V0
Bi
AA
AAi
dt
d
; ;VVV ;iii Tqrdr
sr
Tqd
ss
Tqd
ss
ss
^
s
^^^^
iiGVBxAdt
xd
T
3412
4321
gggg
ggggG
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The speed estimator is based on rotor flux and estimators:
(9.113)
In essence the speed estimator is based on some kind of torque error.
If the rotor resistance rr has to be estimated an additional high reference
current ida* is added to the reference flux current ids
*. Then the rotor resistance
may be estimated [26] as:
(9.114)
sr
^
s
^
i
s
r
^
s
^
si
pr
^
iiagIms
KK
*da
*dsds
^
r
^
r
iii1
dt
d
Remarkable results have been obtained this way with minimum speed down to 30rpm.The idea of an additional high frequency (10 times rated frequency) flux current may be used to determine both the rotor speed and rotor time constant r [27].Extended Kalman filters for speed and flux observers [28] also claim speed estimation at 20 - 25rpm though they require considerable on line computation time.
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9.13.6. Rotor slots ripple speed estimators
The rotor slots ripple speed estimators are based on the fact that the
rotor slotting openings cause stator voltage and current harmonics s1,2 related
to rotor speed , the number of rotor slot Nr and synchronous speed :
(9.115)
Band pass filters centered on the rotor slot harmonics are used to
separate and thus calculate from (9.115). Various other methods have
been proposed to obtain and improve the transient performance. The
response tends to be rather slow and thus the method, though immune to
machine parameters, is mostly favorable for wide speed range but for low
dynamics (medium - high powers) applications [27].
For more details on sensorless control refer to [30].
r
^
1
^
1
^
r
^
r2,1s
^
N
2,1s
^
2,1s
^
2,1s
^
r
^
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9.14. DIRECT TORQUE AND FLUX CONTROL
(DTFC)
DTFC is a commercial abbreviation for the so called direct self control
proposed initially [32 - 33] for induction motors fed from PWM voltage
source inverters and later generalized as torque vector control (TVC) in [4]
for all a.c. motor drives with voltage or current source inverters.
In fact, based on the stator flux vector amplitude and torque errors sign and
relative value and the position of the stator flux vector in one of the 6 (12)
sectors of a period, a certain voltage vector (or a combination of voltage
vectors) is directly applied to the inverter with a certain average timing.
To sense the stator flux space phasor and torque errors we need to estimate
the respective variables. So all types of flux (torque) estimators or speed
observers good for direct vector control are also good for DTFC. The basic
configurations for direct vector control and DTFC are shown on figure 9.41.
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Figure 9.41. a.) Direct vector current control b.) DTFC control
As seen from figure 9.41 DTFC is a kind of direct vector d.c. (synchronous) current control.
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9.14.1. DTFC principle
Though figure 9.41 uncovers the principle of DTFC, finding how the
T.O.S. is generated is the way to a succesful operation.
Selection of the appropiate voltage vector in the inverter is based
on stator equation in stator coordinates:
(9.117)
By integration:
(9.118)
In essence the torque error T may be cancelled by stator flux
acceleration or deceleration. To reduce the flux errors, the flux trajectories will
be driven along appropriate voltage vectors (9.118) that increase or decrease
the flux amplitude.
iVs
sss
ss
ss
irVdt
d
is
T
0
sss
ss
s0s
ss
ss TiVdtirV
i
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Figure 9.42. a.) Stator flux space phasor trajectory
b.) Selecting the adequate voltage vector in the first sector (-300 to +300)
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The complete table of optimal switching, TOS, is shown in table 9.2.
Table 9.2. Basic voltage vector selection for DTFC
s(i) s(1)
s(2) s(3)
s(4) s(5)
s(6) T1 1 V2 V3 V4 V5 V6 V1
1 -1 V6 V1 V2 V3 V4 V5
0 1 V0 V7 V0 V7 V0 V7
0 -1 V0 V7 V0 V7 V0 V7
-1 1 V3 V4 V5 V6 V1 V2
-1 -1 V5 V6 V1 V2 V3 V4
As expected the torque response is quick (as in vector control) but it is also rotor resistance independent above 1 - 2Hz (figure 9.43) [2].
Figure 9.43. TVC torque response
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9.15. DTFC SENSORLESS: A CASE STUDYThe simulation results of a direct torque and flux control drive system for induction motors are presented. The example was implemented in MATLAB - SIMULINK. The motor model was integrated in two blocks, first represents the current and flux calculation module in d - q axis, the second represents the torque, speed and position computing module (figure 9.44).
Figure 9.44.The DTFC system
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Figure 9.45. The I.M. model
The motor used for this simulation has the following parameters: Pn = 1100W, Unf = 220V, 2p = 4, rs = 9.53, rr = 5.619, Lsc = 0.136H, Lr = 0.505H, Lm = 0.447H, J = 0.0026kgfm2.