Semi-Active Pendulum to Control Offshore Wind Turbine Vibrations

1. Semi-Active Pendulum
to Control Offshore
Wind Turbine Vibrations
Suzana M. Avila
Pedro V. B. Guimarães
University of Brasilia – Brazil
May 27th 2015

2. Presentation Topics
• Justification;
• Problems relating;
• Tuned mass damper;
• Goals;
• Problem description;
• Analysis and results;
• Final remarks.

3. Justification
• Wind farms located at
the seaboard, the so
called offshore wind
turbines, have some
advantages compared to
the onshore ones;
• Wind turbine towers
generally are slender and
flexible due to its high
altitude and can present
excessive vibrations;

4. Structural Control
• An alternative widely studied in the last years
to reduce excessive vibration is the structural
control.
• It consists in the addition of external devices
such as dampers or application of external
forces that change properties of stiffness
and/or damping.

5. Semi-active control
• Semi-active structural
control do not add energy to
the structure and its
properties can be varied
dynamically.
• The semi-active systems are
more reliable and more
robust than active systems.
• These are controllable
passive devices since they
don’t apply any additional
force to the structure.

6. Objective
• a semi-active tuned mass damper (TMD)
pendulum is proposed to control excessive
vibration of an offshore floating wind turbine.
• A bang bang control strategy was considered,
TMD stiffness and damping values were
calculated trough optimal control theory.

7. Floating Offshore Wind Turbine
Hywind, Norway

8. Musial W, Butterfield S, Ram B. Energy from offshore wind. In: Offshore technology conference, Houston, Texas; 2006
Floating Offshore Wind Turbine

9. Structural Model
• The structure is modeled as an inverted
pendulum discrete model.
• This model is presented as a preliminary
model for structural control alternatives
studies, the results serve as a basis for real
structures design with a more careful
modeling.

10. Wind force
Structural Model

11. Simplifying Assumptions
• Angular amplitude is kept within boundaries
for a linear behavior;
• A two dimension vibration system is
considered;
• Wind loading is considered as a concentrated
force applied at the tower’s top;
• Wave loading and blade’s influence are
disregarded;

12. Mathematical Formulation
𝑀1,1 𝑀1,2 𝑀1,3
𝑀2,1 𝑀2,2 𝑀2,3
𝑀3,1 𝑀3,2 𝑀3,3
𝜃
𝜃 𝑑
𝑢
+
𝐶1,1 𝐶1,2 𝐶1,3
𝐶2,1 𝐶2,2 𝐶2,3
𝐶3,1 𝐶3,2 𝐶3,3
𝜃
𝜃 𝑑
𝑢
+
𝐾1,1 𝐾1,2 𝐾1,3
𝐾2,1 𝐾2,2 𝐾2,3
𝐾3,1 𝐾3,2 𝐾3,3
𝜃
𝜃 𝑑
𝑢
=
𝐹 𝑡
0
0
𝑴
𝜃
𝜃 𝑑
𝑢
+C
𝜃
𝜃 𝑑
𝑢
+ 𝑲
𝜃
𝜃 𝑑
𝑢
=
𝐹 𝑡
0
0
Equations of Motion

13. Mathematical Formulation
Space-State Equations
)()()()( tttt EfBuAzz 







)(
)(
)(
t
t
t
x
x
z
 






 
CMKM
A 11
0 I






 
DM
B 1
0






 
HM
E 1
0

14. Bang Bang Control
• The Bang Bang control, also called control ON / OFF control, is a feedback
controller that suddenly changes between two limit values.
• This device compares the input with a target value, so that if the output exceeds
the input, the actuator is switched off, otherwise, the actuator is now on.
• Low cost controller, further its simplicity and convenience.

15. Control Strategy
• The control strategy is to control structural response using
bang bang control, varying pendulum TMD stiffness kd and
damping cd, switching from one extreme set of values to the
other.
• The optimal parameter values (kd and cd) are obtained based
on linear optimal control algorithm (linear quadratic
regulator – LQR)

16. Control Strategy
• First a LQR controller is designed assuming an active
pendulum TMD system and neglecting the actuator
dynamics.
• The optimal actuator force u(t) is defined by the gain matrix
G.
• The actuator force is not really applied at the TMD, this force
is applied through a semi-active damper.

17. Linear Optimal Control
• The linear optimal control problem consist in finding the control vector
u(t) that minimizes the performance index J subject to state equations
constraint.
• In structural control, the performance index is usually chosen as a
quadratic function in z(t) and u(t), as follows
  
ft
t
dtttttJ
0
)()()()( RuuQzz TT

18. Numerical Results
Parametric Study
Rating 5 MW
Rotor, hub diameter 126, 3 m
Hub Height 90 m
Rotor mass 110,000 kg
Tower Mass 347,460 kg
Stewart, G.M., Lackner, M.A., 2011, “The effect of actuator dynamics on active structural
control of offshore wind turbines”, Engineering Structures 33 (2011) 1807-1816
Offshore Wind Turbine Properties
𝑲 𝒅 𝑪 𝒅
OFF 5.9 x 106 N/m 4.4 x 105 Ns/m
ON(HSA) 5.9292 x 106 N/m 2.1654 x 106 Ns/m
ON(WNSA) 5.9112 x 106 N/m 1.1886 x 106 Ns/m

19. A time domain analysis was performed, four
situations were considered for analysis:
1. structure without control
2. system with passive TMD (sTMD) considered as
the semi-active turned OFF
3. system with semi-active ON, with optimum
parameter for harmonic loading (HSA)
4. system with semi-active ON, with optimum
parameter for white noise loading (WNSA).
Numerical Results

20. • Efficiency relation of the semi-active device in
ON and OFF position (sTMD). This efficiency is
measured by rms values
Numerical Results
𝐸𝐹𝐹 =
𝜃 𝑂𝐹𝐹 −𝜃 𝑂𝑁
𝜃 𝑂𝐹𝐹
x 100 %

21. Numerical Results
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-60
-40
-20
0
20
40
60
ON(HSA) VS OFF
%ReductionofON(HSA)
Wind Frequency
0.66 rad/s
0.74 rad/s

22. Numerical Results
0 50 100 150 200 250 300
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-4
time (s)
(rad)
Wind frequency = 0.66 rad/s
ON(HSA)
OFF
0 50 100 150 200 250 300
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x 10
-4
time (s)
(rad)
Wind frequency = 0.74 rad/s
ON(HSA)
OFF

23. Numerical Results
0 50 100 150 200 250 300
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-4
time (s)
(rad)
Wind frequency = 0.66 rad/s
Without Control
ON(HSA)
OFF
0 50 100 150 200 250 300
-2
-1
0
1
2
3
x 10
-4
time (s)
(rad)
Wind frequency = 0.74 rad/s
Without Control
ON(HSA)
OFF

24. Numerical Results
0 50 100 150 200 250 300
-1.5
-1
-0.5
0
0.5
1
1.5
2
x 10
-4
White noise wind force
Time (s)
(rad)
ON(HSA)
OFF

25. Numerical Results
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-20
-10
0
10
20
30
40
50
ON(WNSA) VS OFF
%ReductionofON(WNSA)
Wind Frequency
0.66 rad/s
0.74 rad/s

26. Numerical Results
0 50 100 150 200 250 300
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-4
time (s)
(rad)
Wind frequency = 0.66 rad/s
ON(WNSA)
OFF
0 50 100 150 200 250 300
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x 10
-4
time (s)
(rad)
Wind frequency = 0.74 rad/s
ON(WNSA)
OFF

27. Numerical Results
0 50 100 150 200 250 300
-1.5
-1
-0.5
0
0.5
1
1.5
2
x 10
-4
White noise wind force
Time (s)
(rad)
ON(WNSA)
OFF

28. Bang Bang Simulation
0 50 100 150 200 250 300 350 400 450 500
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
x 10
4
Harmonic Wind Force
Time (s)
(rad)
𝜔 = 0.66 𝑟𝑎𝑑/𝑠 𝜔 = 0.74 𝑟𝑎𝑑/𝑠 𝜔 = 1.5 𝑟𝑎𝑑/𝑠

29. Bang Bang Simulation
0 50 100 150 200 250 300 350 400 450 500
-4
-3
-2
-1
0
1
2
3
4
5
x 10
-4
Time (s)
(rad)
Harmonic Wind Force
BangBang
OFF
ON
T

30. Bang Bang Simulation
0 50 100 150 200 250 300 350 400 450 500
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x 10
5
White Noise Wind Force
Time (s)
(rad)

31. Bang Bang Simulation
0 50 100 150 200 250 300 350 400 450 500
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x 10
-4
Time (s)
(rad)
White Noise Wind Force
BangBang
OFF
ON

32. Conclusions
• A bang bang (ON/OFF) control strategy was considered, semi-active
TMD stiffness and damping values were calculated trough optimal
control theory.
• Since the excitation is ignored in control algorithm, two loading
cases were considered: harmonic and white noise, leading to a set
of two kd and cd parameters (HSA and WNSA).
• It was concluded that HSA parameters are the best choice for
setting the semi-active TMD ON position.
• Satisfactory results were found out compared to those of passive
TMD pendulum.
• Semi-active controller presents a good performance on reducing
excessive vibration amplitudes.
• Further studies would be necessary in order to test other control
strategies to semi-active controller design.

33. Thank you!
• Contact:
Suzana M. Avila – avilas@unb.br
Pedro V. B. Guimarães – pedrobarca@hotmail.com