Environmental Engineering Reference
In-Depth Information
2
4
0000010000
0000001000
0000000100
0000000010
0000000001
3
5
C ¼
ð 14 : 19 Þ
where:
h r
h g
h l T
State variables : x ¼½ y t
c
ð 14 : 20 Þ
ch r
h g
h l
y t
T g T
Inputs : u ¼½ F T
T r
ð 14 : 21 Þ
h r
h g
h l T
Outputs : y ¼½ y t
ð 14 : 22 Þ
c
According to the rotor aerodynamics and the characteristics of C p and C T (see
[ 2 ] ), the inputs of Eqs. ( 14.16 ) and ( 14.21 ), F T and T r , depend on v 1 , b and X r in a
nonlinear way. If the aerodynamic part of these equations is linearized around a
working point (v 10 , b 0 , X r0 ), and the bias components are ignored, the inputs F T
and T r can be described by a transfer matrix whose elements are just gains, so that,
2
3
¼
X r ðÞ
v 1 ðÞ
b ðÞ
F T ðÞ
T r ðÞ
K FX
K FV
K Fb
4
5
ð 14 : 23 Þ
K TX
K TV
K Tb
where the gains are calculated by using the C T and C p curves. Now, the transfer
matrix description G(s) of the wind turbine is calculated by using the transfor-
mation G ð s Þ¼ C sI A
Þ 1 B for y ð s Þ¼ G ð s Þ u ð s Þ .
ð
2
3
y t ðÞ
c ðÞ
X r ðÞ
X g ðÞ
X l ðÞ
4
5
þ D ð s Þ v 1 ðÞ
b d ðÞ
T gd ðÞ
¼ P ð s Þ
ð 14 : 24 Þ
where the plant matrix and the disturbance matrix are,
2
4
3
5
ð
Þ þ K Fb
l 11 ðÞ l 32 ðÞ K FX K Tb K Fb K TX
l 11 ðÞ l 33 ðÞ K FX
A b ð s Þ
1 l 32 ðÞ K TX A T ð s Þ
1 l 32 ðÞ K TX
ð
Þ þ K Fb
l 21 ðÞ l 32 ðÞ K FX K Tb K Fb K TX
l 21 ðÞ l 33 ðÞ K FX
A b ð s Þ
1 l 32 ðÞ K TX A T ð s Þ
1 l 32 ðÞ K TX
P ð s Þ¼
K Tb
1 l 32 ðÞ K TX A b ð s Þ
l 32 ðÞ
l 33 ðÞ
1 l 32 ðÞ K TX A T ð s Þ
1
K Tb
1 l 32 ðÞ K TX A b ð s Þ
l 42 ðÞ l 33 ðÞ K TX þ l 43 ðÞ l 43 ðÞ l 32 ðÞ K TX
1 l 32 ðÞ K TX
l 42 ðÞ
A T ð s Þ
K Tb
1 l 32 ðÞ K TX A b ð s Þ
l 52 ðÞ l 33 ðÞ K TX þ l 53 ðÞ l 53 ðÞ l 32 ðÞ K TX
1 l 32 ðÞ K TX
l 52 ðÞ
A T ð s Þ
ð 14 : 25 Þ
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