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2 MODELLING THE DOWEC TURBINE
The nodding excitation force, F
top
eq
nd
, eq.(2.17), consists of tower top equivalent components
caused by the turbine thrust force F
a
, eq.(2.2), and the external (hydrodynamic) fore-aft forces
F
hyd
nd
. From bending theory of slender beams [8] , it has been derived that the exitation force
components of F
top
eq
nd
appear as given in eq.(2.20)
F
top
eq
nd
= F
a
+ F
hyd
nd
= F
a
+
3
2
N
t,wave
X
k=1
d − z
k
Z
t
3
2
· F
hyd,k
nd
(2.20)
The variables d and z
k
are related to the tower part below the water surface and defined as the
waterdepth and the distance to the water surface at element k, respectively (see 2.3.2).
The naying excitation force, F
top
eq
ny
,eq.(2.18), consists of a nacelle force caused by the drive
train reaction torque on the tower F
top
eq
nac
and the external sideward (hydrodynamic) forces F
hyd
ny
. From bending theory of slender beams [8] , it has been derived that the exitation force
components of F
top
eq
ny
appear as given in eq.(2.21)
F
top
eq
ny
= F
top
eq
nac
+ F
hyd
ny
=
3
2
T
nac
Z
t
+
3
2
N
t,wave
X
k=1
d − z
k
Z
t
3
2
· F
hyd,k
ny
(2.21)
The definitions of d and z
k
are equal as in eq.(2.20), T
nac
is the reaction torque of the drive
train as given in eq.(2.14)
Rotation effects in sideward direction will disturb the (measured) rotor speed. A rough (linear)
approximation of this rotation angle is given in eq.(2.22).
φ
ny
=
2 · x
ny
Z
t
(2.22)
Both in eq.(2.20) and eq.(2.21), simple addition is allowed, because the tower is assumed to
have a cilindrical shape and the angle of attack of the waves is assumed to be in line with the
axial force.
2.2.4 Electric conversion
The electric conversion part of the DOWEC turbine consists of a voltage source back-to-back
converter, which controls the rotor voltages of a doubly fed asynchronous generator using a
dedicated ’field oriented control’ algorithm. The stator of the generator is directly connected
to the grid. This topology is able to realise cost-effective variable speed control for a limited
speed range (+/- 30%), determined by the capacity of the inverter. High speed switching power
electronics are able to set electric generator torque almost instantaneously with respect to the
mechanical dynamics. For the proposed control design purposes, only the generator side in-
verter of the converter is relevant and dynamic behaviour above 5-10Hz (suppression of shaft
torsion resonance 2-3Hz) are neglected. Therefore, a well fitted second order torque-servo ap-
proach as given in eq.(2.23), represents the overall dynamic behaviour from setpoint to actual
torque, sufficiently.
T
∗
e
=
1
(ω
g
0
)
2
·
¨
T
e
+
2β
g
ω
g
0
·
˙
T
e
+ T
e
(2.23)
The used bandwidth amounted to 6Hz (ω
g
0
= ± 38 rad/s), which is quite conservative, the
damping rate β
g
was set to 0.7 which implicates a just critical damped system. The electric
torque servo controlequuipment will certainly be capable to achieve these requirements.
Further simplification can be achieved by ignoring second order dynamics and take only a first
order dynamics into consideration:
T
∗
e
= τ
T
e
·
¨
T
e
+ T
e
(2.24)
ECN-C--03-111 15