Linear and Nonlinear modelling of Oscillating Water Column
Wave Energy Converter
Seif Eldine M. Bayoumi, Ph.D.
Assistant Professor
Mechanical Engineering Dept.
The Arab Academy for Science, Technology
Professor
Professor
and Maritime Transport
Atilla Incecik
Hassan El-Gamal
Head of Naval Architecture and
Mechanical Engineering Dept.
Marine Engineering Dept.
Alexandria University
University of Strathclyde, Glasgow
Presentation Layout
• Introduction
• Motivation
• Research Objective
• Numerical tool Methodology
• Wave &Wind Forces
• OWC Modelling
• Nonlinear Modeling
• Renewable Energy Converting Platform
• Conclusions
Introduction
Marine renewable energy sources are crucial alternatives
for a sustainable development. Waves are considered as an
ideal renewable energy source since a Wave Energy
Converter has a very low environmental impact and a high
power density that is available most of the hours during a
year.
Motivation
Prior studies proved that the SparBuoy Oscillating Water
Column has the advantage of being axi-symmetrical and
equally efficient at capturing energy from all directions, but
its efficiency (capture factor) is affected significantly by the
incident wave period.
Research Objective
The main objective of this research is to develop an
experimentally validated numerical wave power prediction
tool for offshore SparBuoy OWC WEC.
Numerical Tool Methodology
In order to achieve the objective, the numerical tool developed
should be able to model:
- the environment (Wave & Wind Forces and wave spectrum)
- the WEC structure motions response (Rigid Body Motions)
- the mooring system (Mooring/Structure Interaction in Surge Motion)
- the water column oscillations inside captive structure (1DOF)
- the water column oscillations inside floating structure (2DOF)
- the nonlinearities in frequency and time domain (Large Waves, Damping &
Pneumatic Stiffness)
- the pneumatic power absorber (Device Evaluation)
SparBuoy Oscillating Water Column
The
Spar
Buoy
has
a
predominant heave motion and
generates
through
pneumatic
the
relative
E&M Plant
power
motion
between the water column in the
Spar Buoy
vertical tube that is open at its
Water Column
base to the sea and the buoy’s
whole body motion.
Vertical Tube
Wave Forces
Inertia Regime
Diffraction Regime
It is important to mention that
in the present study the
Morison equation was used to
calculate the forces on the
structure. In this case forces are
assumed to be composed of
inertia and drag components.
On the other hand, considering
preliminary models of WECs,
it is usually assumed that
forces remain within the
diffraction regime. In this case
forces are assumed to be
composed of pressure and
acceleration components.
Predicted Wave Forces
Horizontal Wave Forces on Vertical Cylinder (Model1)
Vertical Wave Forces on Vertical Cylinder (Model1)
0.7
0.8
Inertia force
Drag force
Total force
0.6
Pressure force
Acceleration force
Total force
0.7
0.6
0.5
Vertical Force (N)
Horizontal Force (N)
0.5
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
1.5
0
2
2.5
3
3.5
4
4.5
Wave Frequency (rad/s)
5
5.5
-0.1
1.5
6
2
2.5
Pitch Moment on Vertical Cylinder (Model1)
3
3.5
4
4.5
Wave Frequency (rad/s)
5
5.5
6
2.5
Inertia moment
Drag moment
Total moment
2
Diffraction
Regime
1.5
Drag may be
ignored
Pitch Moment (Nm)
Inertia Regime
1
0.5
Froude-Krylov
approx. is valid
0
-0.5
-1
-1.5
-2
-2.5
1.5
2
2.5
3
3.5
4
4.5
Wave Frequency (rad/s)
5
5.5
6
Results agree with Incecik, 2003
& Chakrabarti, 2005 charts
Wind Forces
Wind forces on the structure are calculated based on
guidelines provided by American Petroleum Institute (A.P.I.) and
American Bureau of Shipping (A.B.S.)
Wind Forces on Full Scale Spar
350
ABS
API
Horizontal Wind Forces (N)
300
250
200
150
100
50
0
15
20
25
30
35
40
45
Wind velocity (m/s)
50
55
60
OWC Dynamic Models
Following the rigid piston model, captive and floating OWC are best
described by considering one and two translational mode in heave
direction respectively
Floating Structure
Captive Structure
Single DOF
Model
Simplified 2DOF Model
Modified Szumko Model
&
Szumko Model
One-way Coupling Model
Equations of Motions
Calculation Assumption &
Results
Structure and water column mass (measured)
Structure and water column added mass (assumed to be frequency independent)
Structure, Water column and PTO damping (measured using logarithmic decrement and halfpower bandwidth methods)
Structure and water column hydrostatic stiffness (corresponds to the water plane area)
Pneumatic stiffness (calculated in term of air properties and chamber dimensions)
OWC Mass (kg)
Mass
Added mass
Model1
1.1310
0.0360
Model2
4.5996
0.2953
OWC Damping Ratios
WC (Open tube)
WC + 4 Orifices
WC + 2 Orifices
Log.
Half-
Log.
Half-
Log.
Half-
dec.
power
dec.
power
dec.
power
OWC Stiffness (N/m)
Model1
0.041
0.084
0.043
0.09
0.046
0.096
WC
Air
Model2
0.043
0.068
0.059
0.095
0.082
NA
Hydrostatic
Compressibility
Model1
27.7371
1.0875
Model2
112.8053
4.4227
Single DOF Model
(Captive structure)
Model1
Model1
4.5
4.5
Open tube
Experimental
3
3
Water Column RAO
3.5
2.5
2
2.5
2
1.5
1.5
1
1
0.5
0.5
1
2
3
4
5
6
Wave Frequency (rad/s)
7
8
Open tube
Experimental
4
3.5
0
0
9
1
2
3
Model1
7
8
9
4.5
2orifices
Experimental
4
3
3
Water Column RAO
3.5
2.5
2
2.5
2
1.5
1.5
1
1
0.5
0.5
1
2
3
4
5
6
Wave Frequency (rad/s)
7
8
2orifices
Experimental
4
3.5
0
4
5
6
Wave Frequency (rad/s)
Model1
4.5
Water Column RAO
Good agreement
between predicted
and
measured
responses, except
around resonance
due to the use of
viscous damping.
Water Column RAO
4
9
0
1
2
3
4
5
6
Wave Frequency (rad/s)
7
8
9
Nonlinearity due to Large Waves
Linearized frequency
domain model
Non-linear time domain
model
Nonlinear oscillations are
analysed asymptotically by
means of perturbation method.
This approach doesn’t require
the wave force to be calculated
in the time domain.
For more accurate prediction
numerical nonlinear approach
is adopted. This requires the
calculation of wave force in
time domain, which is obtained
by taking into account the
instantaneous
Oscillation
amplitude.
Nonlinearity due to Large Waves
Model 1
Model1
4.5
4
0
-0.1
-0.2
3.5
0
5
10
15
20
25
30
-3
Oscillation (m)
2
x 10
Perturbed term
1
0
-1
0
5
10
15
20
25
30
0.2
3
2.5
2
1.5
1
Linearized
0.1
0
0.5
-0.1
-0.2
0
5
10
15
time (s)
20
5
25
0
30
4.8
Perturbation
results
1
2
Linear
Linearized
Nonlinear
4.9
3
4
5
6
Wave Frequency (rad/s)
7
8
4.7
Water Column RAO
Oscillation (m)
Nonlinear
Linearized
Linear
Experimental
Linear term
0.1
Water Column RAO
Oscillation (m)
0.2
4.6
4.5
4.4
Comparison
4.3
4.2
4.1
4
0.06
0.07
0.08
0.09
0.1
0.11
0.12
Wave height (m)
0.13
0.14
0.15
9
Nonlinear Damping
Iterative (optimised)
frequency domain model
This is achieved by assuming
amplitude of motion, the damping
coefficients are calculated and then
the equation of motion is solved.
Motion amplitudes obtained from
these equations can now be used to
determine new damping coefficients
and the equation of motion is again
solved.
Non-linear time domain
model
This requires the calculation of
damping force in time domain,
which is achieved by taking
into account the instantaneous
oscillation amplitude. The
linear and quadratic damping
coefficients are not optimised
in this case but taken as
constants.
Nonlinear Damping
Equivalent Viscous Damping Ratio (Model1)
Water Elevation Decay
0.08
50
0
-50
-100
Water Elevation decrease/Mean Water Elevation
data 1
data 2
data 3
0
1
2
3
4
5
Time(sec)
6
7
8
9
10
0.32
Optimised
damping
ratios
data 4
linear
0.3
0.28
Open tube
4 Orifices
2 Orifices
0.07
Equivalent Linear Damping Ratio
Water Elevation (mm)
100
0.06
0.05
0.04
0.26
0.03
0.24
0.22
0.2
30
35
40
45
50
55
Mean Water Elevation (mm)
60
65
Model1
1
2
3
4
5
6
Wave Frequency (rad/s)
7
8
VD
Optimized EVD
EVD
Open tube
7
6
Water Column RAO
Matlab Script for
L&Q damping coef.
calculations
0.02
70
8
5
4
Comparison
3
2
1
0
1
2
3
4
Wave Frequency (rad/s)
5
6
7
9
Experimental vs. Numerical Water Column Decay Test
Results (Damping Model1)
Experimental vs. Numerical Water Column Decay Test
Results (Damping Model2)
Nonlinear Pneumatic Stiffness
In the current research nonlinear effect due to air
compressibility is modelled in time domain by considering
the instantaneous pneumatic chamber volume in calculations.
Model1
30
Max Stiffness(N/m)
25
Water column stiffness
Pneumatic stiffness
Total stiffness
20
15
10
5
0
0
0.01
0.02
0.03 0.04 0.05 0.06 0.07
Oscillations Amplitude (m)
0.08
0.09
0.1
Stiffness
Damping
Large Waves
Conclusions (Nonlinear modelling)
Linearized (frequency domain) solution is much closer to the
linear solution than the nonlinear (time domain) one, which questions
the suitability of this approach to this type of nonlinearity.
The clear disagreement between the experimental results and the
EVD approach results near resonance is caused by the inaccurate
detection of the linear and quadratic damping coefficients. In
contrast, the adopted iterative procedure used to optimize the
damping coefficients was very successful leading to a very good
agreement with the experimental results and allows the analysis to be
performed in frequency domain.
Results showed that the max pneumatic stiffness is not just small
compared to the water column hydrostatic stiffness but the increase
in the pneumatic stiffness due to the increase in oscillation amplitude
is very small.
Renewable Energy Converting
Platform
The concentration of several devices on one platform has
both economic and operational advantages.
Conclusions (RE Platform)
It is noted that the measured relative RAO inside the four
OWCs are similar to each other and similar to the relative RAO
in case of single SparBuoy. Consequently, the power captured
by the platform is almost four times the power captured by
single SparBuoy OWC WEC. In addition to the wind power
expected to be captured by wind turbine mounted on top of the
platform.
In addition the platform offers a wide area exposed to sun
light and it is equipped with the infra-structure required for
power conditioning and transformation. Therefore mounting
photo voltaic solar panels on this area would be recommended
to increase the output power of the platform.
Summary
Several mathematical model and computer programs have been generated in order to
develop the numerical wave power prediction tool. The proposed tool is able to:
- Calculate the wave spectrum and characteristics (Height & Period)
- Calculate the environmental loads on the structure (Wave & Wind)
- Determine the linear and quadratic damping coefficients from experiments (If Available)
- Predict the structure motion response considering the interaction with the mooring
system in surge and the coupling with the internal water column in heave.
- Model the water column oscillation linearly and nonlinearly in both frequency and time
domain (Large Waves, Damping & Pneumatic Stiffness)
- Calculate the power absorbed and evaluate the WEC.
In addition, experiments have been carried out in order to validate the results.
Finally, the idea of a hybrid renewable energy converting platform has been proposed and
experimentally investigated.
Thank You