198
200
202
204
(e) t(s)
Fig. 12. PBC-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG & IM
rotor position. (d) Generator torque (e) Motor desired torque.
Figure 12 presents the mechanical IM speed and its smooth reference, the mechanical DFIG
speed, the DFIG and IM rotor positions, the DFIG torque τG and the IM desired torque τMd.
The real IM speed tracks the reference very well, i.e. low overshoot and no steady state error
are observed. Figure 13 shows the stator currents isa and isb, and their references over a suitable period of time. The stator currents do not track exactly their desired values but are
bounded. This is because the goal of the PBC is to track the IM speed and to keep internal
signals bounded.
Figure 14 shows the DFIG rotor currents irGa and irGb, and their references over a period of time. Again, these currents are sinusoidal and bounded.
Figure 15 presents the DFIG rotor voltages vrGa and vrGb, the IM rotor speed ωmM and its estimation ˆ
ωmM, the estimated IM load torque ˆ τML, and the estimated IM speed, given by
From Dynamic Modeling to Experimentation of
Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control
133
20
10
(A)
sGb
0
& i
i sGa
−10
−20
193.74
193.76
193.78
193.8
193.82
193.84
193.86
193.88
193.9
193.92
193.94
(a) t(s)
20
10
(A)
d
sGa
0
& i
i sGa
−10
−20
193.74
193.76
193.78
193.8
193.82
193.84
193.86
193.88
193.9
193.92
193.94
(b) t(s)
20
10
(A)
d
sGb
0
& i
i sGb
−10
−20
193.74
193.76
193.78
193.8
193.82
193.84
193.86
193.88
193.9
193.92
193.94
(c) t(s)
Fig. 13. PBC-(a) isa, isb (b) idsa, isa (c) id , i
sb
sb
.
15
10
(A)
5
rGb
0
& i
−5
i rGa
−10
−15
193.74
193.76
193.78
193.8
193.82
193.84
193.86
193.88
193.9
193.92
193.94
(a) t(s)
50
(A)
d
rGa
0
& i
i rGa
−50
193.75
193.8
193.85
193.9
193.95
194
194.05
(b) t(s)
50
(A)
d
rGb
0
& i
i rGb
−50
193.75
193.8
193.85
193.9
193.95
194
194.05
(c) t(s)
Fig. 14. PBC-(a) irGa, irGb (b) id , i
, i
rGa
rGa (c) id
rGb
rGb.
134
Electric Machines and Drives
60
40
(V)
20
rGb
0
& v
−20
v rGa
−40
−60
193.75
193.8
193.85
193.9
193.95
194
194.05
(a) t(s)
1600
1400
(rpm)
1200
mM 1000
ω
&
800
ω mM 600
400
184
186
188
190
192
194
196
198
200
202
204
(b) t(s)
2
1
0
(N.m)
τ LM −1
τ
−2
−3
184
186
188
190
192
194
196
198
200
202
204
(c) t(s)
Fig. 15. PBC-(a) vrGa, vrGb (b) ωmM, ˆ
ωmM (c) ˆ τML.
(76)-(77), is tracking the real speed. Hence, a good estimation of the real IM load torque is
obtained. It has to be noticed that the IM rated torque is 0.7 Nm.
It can be concluded that the PBC provides good practical performance even when the applied
load torque is twice the magnitude of the nominal load torque of the IM.
7.3 PBC + P
1500
(rpm)
ω mM 1000
&
ref
500
ω mM
55
60
65
70
75
80
(a) t(s)
3000
(rpm)
2000
ω mG 1000
55
60
65
70
75
80
(b) t(s)
6
(rad)
4
θ M
&
2
θ G
68.85
68.9
68.95
69
69.05
69.1
(c) t(s)
0.3
0.2
(N.m)
0.1
τ G
0
55
60
65
70
75
80
(d) t(s)
2
1
0
(N.m)
τ Md −1
−2
55
60
65
70
75
80
(e) t(s)
Fig. 16. PBC+P-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG &
IM rotor position. (d)Generator torque (e) Motor desired torque.
As with the PBC alone, the results obtained with the PBC+P are given in figures 16-19. On the
whole, the system behaviour is the same as the PBC alone. One difference that is noticeable is
From Dynamic Modeling to Experimentation of
Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control
135
15
10
(A)
5
sGb
0
& i
−5
i sGa
−10
−15
68.85
68.9
68.95
69
69.05
69.1
(a) t(s)
15
10
(A)
5
d
sGa
0
& i
−5
i sGa
−10
−15
68.85
68.9
68.95
69
69.05
69.1
(b) t(s)
15
10
(A)
5
d
sGb
0
& i
−5
i sGb
−10
−15
68.85
68.9
68.95
69
69.05
69.1
(c) t(s)
Fig. 17. PBC+P-(a) isGa, isGb (b) id , i
, i
sGa
sGa (c) id
sGb
sGb.
15
10
5
(A)
rGb
0
& i
i rGa
−5
−10
−15
68.85
68.9
68.95
69
69.05
69.1
(a) t(s)
60
40
20
(A)
d
rGa
0
& i
i rGa −20
−40
68.85
68.9
68.95
69
69.05
69.1
69.15
69.2
69.25
69.3
(b) t(s)
60
40
(A)
20
d
rGb
0
& i
i rGb −20
−40
68.85
68.9
68.95
69
69.05
69.1
69.15
69.2
69.25
69.3
(c) t(s)
Fig. 18. PBC+P-(a) irGa, irGb (b) id , i
, i
rGa
rGa (c) id
rGb
rGb.
the small error between the desired stator currents and the real ones thanks to the proportional
controller.
The PBC+P controller exhibits good practical performance but not significantly better than
those obtained with the PBC alone.
7.4 PBC + PI
Again, as for the PBC and the PBC+P controllers, figures 20-23 show the results. It can be
seen in figure 21 that the integral actions on the stator currents do not decrease the error
significantly between the real and desired values in comparison with the results for the PBC+P
136
Electric Machines and Drives
60
40
20
(V)
rGb
0
& v
−20
v rGa
−40
−60
68.85
68.9
68.95
69
69.05
69.1
69.15
69.2
69.25
69.3
(a) t(s)
1600
1400
(rpm)
1200
ω mM 1000
&
800
ω mM
600
400
55
60
65
70
75
80
(b) t(s)
2
1
0
(N.m)
τ LM −1
τ
−2
−3
55
60
65
70
75
80
(c) t(s)
Fig. 19. PBC+P-(a) vrGa, vrGb (b) ωmM, ˆ
ωmM (c) ˆ τML.
1500
(rpm)
ω mM 1000
&
ref
500
ω mM
40
45
50
55
60
65
(a) t(s)
3000
(rpm)
2000
ω mG
1000
40
45
50
55
60
65
(b) t(s)
6
(rad)
4
θ M
&
2
θ G
52.6
52.65
52.7
52.75
52.8
52.85
(c) t(s)
0.3
0.2
(N.m)
0.1
τ G
0
40
45
50
55
60
65
(d) t(s)
2
1
0
(N.m)
τ Md −1
−2
40
45
50
55
60
65
(e) t(s)
Fig. 20. PBC+PI-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG &
IM rotor position. (d)Generator torque (e) Motor desired torque.
From Dynamic Modeling to Experimentation of
Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control
137
(A)
10
sGb
0
& i
−10
i sGa
52.6
52.65
52.7
52.75
52.8
52.85
(a) t(s)
(A)
10
d
sGa
0
& i
−10
i sGa
52.6
52.65
52.7
52.75
52.8
52.85
(b) t(s)
(A)
10
d
sGb
0
& i
−10
i sGb
52.6
52.65
52.7
52.75
52.8
52.85
(c) t(s)
Fig. 21. PBC+PI-(a) isGa, isGb (b) id , i
, i
sGa
sGa (c) id
sGb
sGb.
(A)
10
rGb
0
& i
−10
i rGa
52.6
52.65
52.7
52.75
52.8
52.85
(a) t(s)
50
(A)
d
rGa
0
& i
i rGa −50
52.6
52.65
52.7
52.75
52.8
52.85
52.9
52.95
53
(b) t(s)
50
(A)
d
rGb
0
& i
i rGb −50
52.6
52.65
52.7
52.75
52.8
52.85
52.9
52.95
53
(c) t(s)
Fig. 22. PBC+PI-(a) irGa, irGb (b) id , i
, i
rGa
rGa (c) id
rGb
rGb.
controller (see fig. 17). This is due to the fact that the reference values are sinusoidal and that
the bandwidth of the PI controllers cannot be increased sufficiently experimentally.
It can be concluded that the PI action on the stator currents does not improve significantly the
performance obtained with the PBC+P controller.
7.5 PI
The PI control law (with Kp and Ki are proportional and integral gains) is given below:
BvrG = B Kp( isG − id ) +
)
sG
Ki( isG − idsG
(78)
138
Electric Machines and Drives
(V)
50
rGb
0
& v
−50
v rGa
52.6
52.65
52.7
52.75
52.8
52.85
52.9
52.95
53
(a) t(s)
(rpm)
1500
ω mM 1000
&
500
ω mM
40
45
50
55
60
65
(b) t(s)
2
0
(N.m)
τ τ LM −2
40
45
50
55
60
65
(c) t(s)
Fig. 23. PBC+PI-(a) vrGa, vrGb (b) ωmM, ˆ
ωmM (c) ˆ τML.
)m
(rp
1500
M
ω m 1000
&
ref
M
500
230
235
240
245
250
255
ω m
2500
t(s)
)m 2000
(rp
1500
G 1000
ω m 500
230
235
240
245
250
255
t(s)
6
(rad)
4
θ M 2
& θ G
241.35
241.4
241.45
241.5
241.55
t(s)
)
0
m
(N.
−0.2
τ G −0.4
230
235
240
245
250
255
t(s)
)m 2
0
(N.
−2
τ Md
230
235
240
245
250
255
t(s)
Fig. 24. PI-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG & IM
rotor position. (d) Generator torque (e) Motor desired torque.
From Dynamic Modeling to Experimentation of
Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control
139
10
(A)
5
sGb
0
& i
−5
i sGa
−10
241.35
241.4
241.45
241.5
241.55
t(s)
10
(A)
5
d
sGa
0
& i
−5
i sGa
−10
241.35
241.4
241.45
241.5
241.55
t(s)
10
(A)