1000
ω m 500
70
75
80
85
90
95
t(s)
6
(rad)
4
θ M 2
& θ G
83.5
83.55
83.6
83.65
83.7
t(s)
)
0.1
m
0
(N.τ G −0.1
70
75
80
85
90
95
t(s)
)
2
m
0
(N.
τ Md −2
70
75
80
85
90
95
t(s)
Fig. 28. PBC-robustness test-(a) Regulated Motor speed and its reference. (b)Generator
speed. (c) DFIG & IM rotor position. (d) Generator torque (e) Motor desired torque.
10
(A)
5
sGb
0
& i
−5
i sGa −10
83.5
83.55
83.6
83.65
83.7
t(s)
10
(A)
5
d
sGa
0
& i
−5
i sGa
−10
83.5
83.55
83.6
83.65
83.7
t(s)
10
(A)
5
d
sGb
0
& i
−5
i sGb −10
83.5
83.55
83.6
83.65
83.7
t(s)
Fig. 29. PBC -robustness test-(a) isGa, isGb (b) id , i
, i
sGa
sGa (c) id
sGb
sGb.
142
Electric Machines and Drives
20
(A)
10
rGb
0
& i
−10
i rGa
−20
83.5
83.55
83.6
83.65
83.7
83.75
83.8
83.85
83.9
83.95
t(s)
50
(A)
d
rGa
0
& i
i rGa −50
83.5
83.55
83.6
83.65
83.7
83.75
83.8
83.85
83.9
83.95
t(s)
50
(A)
d
rGb
0
& i
i rGb −50
83.5
83.55
83.6
83.65
83.7
83.75
83.8
83.85
83.9
83.95
t(s)
Fig. 30. PBC-robustness test-(a) irGa, irGb (b) id , i
, i
rGa
rGa (c) id
rGb
rGb.
50
(V)
rGb
0
& v
v rGa
−50
83.5
83.55
83.6
83.65
83.7
83.75
83.8
83.85
83.9
83.95
t(s)
)m 1500
(rp
hat
M
1000
ω m
& M
500
ω m
70
75
80
85
90
95
t(s)
eters
1
m
0.5
Switch on Para
0
70
75
80
85
90
95
t(s)
Fig. 31. PBC-robustness test-(a) vrGa, vrGb (b) ωmM, ˆ
ωmM (c) Switch.
DFIG, which at the same time controls the speed of the IM making use of the rotor voltage of
the DFIG as a control variable. A complete stability proof for inner loop control is given. The
proof of the overall scheme including the outer speed loop follows verbatim from (1) and is
omitted here for brevity.
The main advantage of the PBC is that it requires the measurement of only two mechanical
positions for the speed tracking. The PI controller applied to the inner loop provides good
performance but saturation in the transient state can be observed. Robustness tests were
performed to observe the behaviour of the controllers to machine parameter variations. All
the proposed controllers were found to be robust towards variation in machine resistances.
Also, a power flow analysis can be undertaken between the generator, the IM and the grid
From Dynamic Modeling to Experimentation of
Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control
143
in order to optimize the efficiency of the overall system. A comparison of the experimental
results of the proposed PBC, PI, PBC+P and PBC+PI algorithm is presented in Table 2. It is
based on the performance obtained practically with the different controllers. In addition to the
comparison criteria of Table 2, it is proposed to check the following to see what their effects
are on the controllers’performance:
2
• eω = 1
ω
indication about the IM speed tracking error. Where
M
nT ∑ n
i=1
M( i) − ωRe f M( i)
n is the length of the sampled data and T is the sampling time;
2
• ei
= 1
( i) indication about the stator current tracking error in the
sGa
nT ∑ n
i=1 isGa( i) − idsGa
phase a;
• Observed magnitude of isGa;
• P avg = 1
[ τ
G
n ∑ n
i=1
G( i) ωG( i)] indication about the rotor average value of the instantaneous
absorbed power in the DIFG;
• P avg = 1
[ τ
M
n ∑ n
i=1
M( i) ωM( i)] indication about the rotor average value of the instantaneous
absorbed power in IM;
Rs(Ω) Rr(Ω) Ls( mH) Lr( mH) Lm( mH) J( Nm 2/ rad) DFIG 0.365 0.559
0.938
0.938
12.975 4.358 × 10 − 3
IM
0.5
0.2
1.2
1.2
9.00
1.1 × 10 − 3
Table 1. The parameters for DFIG and IM
PBC
PBC+P
PBC+PI
PI
ωRef M
500 → 1000
500 → 1000
500 → 1000
500 → 1000
[rpm]
→ 1400 → 800
→ 1400 → 800
→ 1400 → 800
→ 1400 → 800
(1 st order filter) (1 st order filter) (1 st order filter) (1 st order filter) τLM [N.m]
0.5 → 1.45 → 0.5 0.5 → 1.4 → 0.5 0.5 → 1 → 0.5 0.5 → 1.15 → 0.5