0
-2000
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
8
3
2
4
r
error
1
ation /s ]
on erro
0
0
ati
stim [ rad
e
-1
-4
estimZ
Speed
-2
-8
-3
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( a )
( b )
Fig. 8. Application and removal of load; (a) reference, actual, estimated speeds, and speed
estimation error; (b) actual Z, estimated Z, and Z estimation error.
180
100
Reference speed
Actual Z
Actual speed
80
Estimated Z
Estimated speed
s ] 120
d/
60
d [ ra
Z
40
60
Spee
20
0
0
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
10
2
r ro
5
rro 0
ation er s ]
on er
m d/
0
ti
ati
[ ra
mti
d es
-2
-5
esZ
Spee
-10
-4
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
( a )
( b )
Fig. 9. Full load operation at various speeds; (a) reference, actual, estimated speeds, and
speed estimation error; (b) actual Z, estimated Z, and Z estimation error.
Sensorless Vector Control of Induction Motor Drive - A Model Based Approach
87
The estimator performance and drive response are then verified under fully loaded
condition of the drive at various operating speeds. The fully loaded machine is accelerated
to 150 rad/s at 0.3 s, and then the speed is reduced in steps to 100 rad/s, 50 rad/s and 10
rad/s at 1.5 sec, 3 sec and 4.5 sec respectively. Fig. 9 shows the estimation results and
response of the drive during the operation.
Good speed estimation accuracy was obtained under both dynamic and steady state
conditions under various operating conditions and response of the VC induction motor
drive incorporating the estimation algorithms was found to be good. The speed estimation
algorithm presented in this section depends upon the knowledge of the rotor flux, whereas,
the rotor flux estimator is independent of rotor speed and requiring only the measurable
stator terminal quantities, the stator voltage and current.
3. Flux and speed estimation
Induction machines do not allow rotor flux to be easily measured. The current model and
the voltage model are the traditional solutions, and their benefits and drawbacks are well
known. Various observers for flux estimation were analyzed in the work by Verghese and
Sanders (Verghese & Sanders, 1988) and Jansen and Lorenz (Jansen & Lorenz, 1994). Over
the years several other have been presented, many of which include speed estimation
(Tajima & Hori, 1993; Kim et al., 1994; Ohtani et al., 1992; Kubota et al., 1993; Sathiakumar,
2000; Yan et al., 2000).
Tajima & Hori (Tajima & Hori, 1993) proposed MRAS (Schauder, 1992) with novel pole
allocation method for speed estimation while rotor flux estimation was done using
Gopinath’s observer. Extended Kalman Filter was used in (Kim et al., 1994) for estimating
the rotor flux and speed using a full order model of the motor assuming that rotor speed is a
constant. Ohtani et al (Ohtani et al., 1992) used the voltage model for flux estimation
overcoming the problem associated with integrator and low pass filter while speed was
obtained using a frequency controller. A speed adaptive flux observer was proposed in
(Kubota et al., 1993) for estimating rotor flux and speed. Gopinath style reduced order
observer was used in (Sathiakumar, 2000) for estimating the rotor flux while the speed was
computed using an equation derived from the motor model. Yan et al (Yan et al., 2000)
proposed a flux and speed estimator based on the sliding-mode control methodology.
In this section, we present a new flux estimation algorithm for speed sensorless rotor flux
oriented controlled induction motor drive (Thongam & Ouhrouche, 2006). The proposed
method is based on observing the variable Z introduced in Section 2 which when
introduced makes the right hand side of the conventional motor model independent of
rotor flux and speed. Rotor flux estimation is achieved using an equation obtained after
introduction of the newly defined quantity into the Blaschke equation or commonly
known as the current model; while, speed is computed using a simple equation obtained
using the new quantity Z.
3.1 Estimation of rotor flux and speed
The speed computation equation (22) obtained in section 2.4.2 requires the knowledge of
rotor flux and Z. Here, we present a joint rotor flux and speed estimation algorithm. The
block diagram of the proposed rotor flux and speed estimation algorithm is shown in
Fig. 10.
88
Electric Machines and Drives
v
ˆZ
s
Z
Estimation
ˆω
Rotor Flux
Speed
( Equations
i
Estimation
Computation
s
10 & 11)
ˆ
( Equations
ψr
( Equation 22)
28 & 29)
*
Ψr
Fig. 10. Rotor flux and speed estimator
Rotor flux may be obtained directly using equation (4) which is obtained after introducing
the newly defined quantity Z into the Blaschke equation as
ˆ
ψr = ( 12
A si +
∫
14
A Z ) dt (23)
However, rotor flux computation by pure integration suffers from dc offset and drift
problems. To overcome the above problems a low pass filter is used instead of pure
integrator and the phase error due to low pass filtering is approximately compensated by
adding low pass filtered reference flux with the same time constant as used above (Ohtani et
al., 1992). The equation of the proposed rotor flux estimator is given below
τ
ˆ
ψr =
( A si + A Z)
1
*
12
14
+
Ψ (24)
1 +τ s
1
r
+τ s
where τ is the LPF time constant. The command rotor flux *
Ψr is obtained as follows
*
⎡
⎤
*
*
Ψ
⎡
ρ ⎤
⎡
ρ ⎤
*
rα
* cos
*
cos
Ψ = ⎢
⎥
(25)
r
=Ψ r ⎢
⎥ = m
L sid ⎢
⎥
*
*
*
Ψ
⎢
⎥
⎣ rβ
⎢ sin ρ ⎥
⎢ sin ρ ⎥
⎦
⎣
⎦
⎣
⎦
The command rotor flux angle *
ρ is obtained by integrating the command rotor flux speed
as given by
*
*
*
ρ = ∫ e
ω dt = (ω
∫ sl + ˆω )dt (26)
The command slip speed *
ω sl is given by
*
R i
*
r qs
ω sl =
(27)
*
r
L dis
We know that the equation of the back emf is given by:
L dψ
L
m
r
m
e =
=
( 12
A is + 1
A 4 Z ) (28)
r
L
dt
r
L
Sensorless Vector Control of Induction Motor Drive - A Model Based Approach
89
is
12
A
+
τ
1 +τ s
Z
+
14
A
*
Ψ
+
ˆ
1
+
ψ
r
r
1 +τ s
Fig. 11. Rotor Flux Estimator
G e − axis
τ
({
G
L / L ) e}
1
r
m
+τ s
1
G *
Ψ
1
r
+τ s
ζ
G *
Ψr
G
ˆ G
ψ
ζ
ψ
r − axis
1
G
r
*
Ψ
1
r
+τ s
Fig. 12. Obtaining estimated rotor flux
Now, equation (24) may also be written as
τ ⎛ L
⎞
1
r
*
ˆ
ψr =
⎜
e ⎟ +
Ψ (29)
1 τ s ⎜ L
⎟
+
1
r
⎝ m
+τ s
⎠
Block diagram of the rotor flux estimator is shown in Fig. 11. Fig.12 explains how estimated
flux is obtained using equation (29).
3.2 Simulation results
Simulation is carried out in order to validate the performance of the proposed flux and
speed estimation algorithm. The proposed rotor flux and speed estimation algorithm is
90
Electric Machines and Drives
incorporated into a vector controlled induction motor drive. The block diagram of the
sensorless vector controlled induction motor drive incorporating the proposed estimator is
shown in Fig. 13. The sensorless drive system is run under various operating conditions.
First, acceleration and speed reversal at no load is performed. A speed command of 150
rad/s at 0.5 s is given to the drive system which was initially at rest, and then the speed is
reversed at 3 s. The response of the drive is shown in Fig. 14. Fig. 14 (a) shows reference
( *
ω ), actual ( ω ), estimated ( ˆω ) speed, and speed estimation error ( ω − ˆω ). The module of the actual ( |Ψ
ˆ
ˆ
r | ), estimated ( |Ψr | ) rotor flux, and rotor flux estimation error ( |Ψr | − |Ψr | ) are shown in Fig. 14 (b). Fig. 14 (c) and (d) shows respectively the locus of the actual and
estimated rotor fluxes.
The drive is then run at various speeds under no load condition. It is accelerated from rest to
10 rad/s at 0.5 s, then accelerated further to 50 rad/s, 100 rad/s and 150 rad/s at 1.5 s, 3 s
and 4.5 s respectively. Fig. 15 shows the estimation of rotor flux and speed, and the response
of the sensorless drive system.
Then, the drive is subjected to a slow change in reference speed profile (trapezoidal), the
results of which are shown in Fig. 16.
+
−
dc
V
*
ω
*
*
v
*
+
s
i q +
sq
s
v a
* s
v b
−
*
*
−
*
ψ
v
dq
abc
INVERTER
r +
s
i d
+
sd
*
ˆ
ω
s
v c
−
−
*
ˆ
ψ
s
i
*
q
s
i d
i
i
dq
s
ρ
r
sq
abc
FLUX VECTOR
*
GENERATION
Ψr
is
ROTOR FLUX
&
SPEED
v
ESTIMATOR
s
IM
Fig. 13. Sensorless vector controlled induction motor drive
Further, the performance of the estimator is verified under loaded conditions at various
operating speeds. The fully loaded drive is accelerated to 150 rad/s at 0.5 s and then
decelerated in steps to 100 rad/s, 50 rad/s and 10 rad/s at 1.5 s, 3 s and 4.5 s respectively.
Fig. 17 shows the estimation results and response of the loaded drive system.
Then, we test the performance of the estimator on loading and unloading. The drive at rest
is accelerated at no-load to 150 rad/s at 0.5 s and full load is applied at 1 s; we then remove
the load completely at 2 s. Later, after speed reversal, full load is applied at 4 s, then, the
load is removed completely at 5 s. Fig. 18 shows the estimation results and the response of
the sensorless drive.
Sensorless Vector Control of Induction Motor Drive - A Model Based Approach
91
200
0.4
Reference speed
Actual speed
b ]
rad/s ]
0
W 0.2
Actual flux
d [
Estimated speed
ux [
Estimated flux
pee
Fl
S -200
0
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
rror
ror
10
0.2
ation e
0
ation er b ]
0
tim ad/s ] [ r
tim [ W
-10
-0.2
0
1
2
3
4
5
6
peed es
lux es
0
1
2
3
4
5
6
S
F
Time [ s ]
Time [ s ]
( a )
( b )
b ]
b ] 0.2
[ W 0.2
[ W
0
0
d ψ rβ
-0.2
ate -0.2
tual ψ rβ -0.4
Ac
stim -0.4
-0.4 -0.3
-0.2 -0.1
0
0.1
0.2
0.3
0.4
E
-0.4 -0.3
-0.2 -0.1
0
0.1
0.2
0.3
0.4
Actual ψ [ Wb ]
Estimated ψ [ Wb ]
rα
rα
( c )
( d )
Fig. 14. Acceleration and speed reversal of the sensorless drive at no-load
0.4
150
s ]
Reference speed
]
ad/ 100
Actual speed
b
[ r
Estimated speed
0.2
Actual flux
50
x [ Wu
Estimated flux
peed
Fl
0
S
0
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Time [ s ]
Time [ s ]
rror
ror
10
0.2
er
ation e
0
ation b ]
0
stim ad/s ][ r
tim [ W
d e
es
-10
-0.2
0
1
2
3
4
5
6
pee
lux
0