linear and nonlinear modelling of oscillating water column wave energy converter seif eldine m....
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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 and Maritime Transport
ProfessorAtilla Incecik
Head of Naval Architecture and Marine Engineering Dept.
University of Strathclyde, Glasgow
ProfessorHassan El-Gamal
Mechanical Engineering Dept.Alexandria University
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 pneumatic power
through the relative motion
between the water column in the
vertical tube that is open at its
base to the sea and the buoy’s
whole body motion.
E&M Plant
Water Column
Spar Buoy
Vertical Tube
Wave Forces Inertia 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.
Diffraction Regime
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
1.5 2 2.5 3 3.5 4 4.5 5 5.5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Wave Frequency (rad/s)
Horizonta
l F
orc
e (
N)
Horizontal Wave Forces on Vertical Cylinder (Model1)
Inertia force
Drag forceTotal force
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Wave Frequency (rad/s)
Vert
ical F
orc
e (
N)
Vertical Wave Forces on Vertical Cylinder (Model1)
Pressure force
Acceleration forceTotal force
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Wave Frequency (rad/s)
Pitch M
om
ent
(Nm
)
Pitch Moment on Vertical Cylinder (Model1)
Inertia moment
Drag momentTotal moment
Inertia Regime
Drag may be ignored
Diffraction Regime
Froude-Krylov approx. is valid
Results agree with Incecik, 2003 & Chakrabarti, 2005 charts
Wind Forces
15 20 25 30 35 40 45 50 55 600
50
100
150
200
250
300
350
Wind velocity (m/s)
Hor
izon
tal W
ind
For
ces
(N)
Wind Forces on Full Scale Spar
ABS
API
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.)
Captive StructureFloating Structure
OWC Dynamic Models
Simplified 2DOF Model
&
One-way Coupling Model
Modified Szumko Model
Single DOF Model
Following the rigid piston model, captive and floating OWC are best described by considering one and two translational mode in heave direction respectively
Szumko Model
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 half-power 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.
dec .
Half-
power
Log.
dec .
Half-
power
Log.
dec.
Half-
power
Model1 0.041 0.084 0.043 0.09 0.046 0.096
Model2 0.043 0.068 0.059 0.095 0.082 NA
OWC Stiffness (N/m)
WC
Hydrostatic
Air
Compressibility
Model1 27.7371 1.0875
Model2 112.8053 4.4227
Single DOF Model (Captive structure)
1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Wave Frequency (rad/s)
Wate
r C
olu
mn R
AO
Model1
Open tube
Experimental
1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Wave Frequency (rad/s)
Wate
r C
olu
mn R
AO
Model1
Open tube
Experimental
1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Wave Frequency (rad/s)
Wate
r C
olu
mn R
AO
Model1
2orifices
Experimental
1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Wave Frequency (rad/s)
Wate
r C
olu
mn R
AO
Model1
2orifices
Experimental
Good agreement between predicted and measured responses, except around resonance due to the use of viscous damping.
Nonlinearity due to Large Waves
Linearized frequency 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.
Non-linear time domain model
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
0 5 10 15 20 25 30-0.2
-0.1
0
0.1
0.2
Oscill
ation (
m)
Model1
0 5 10 15 20 25 30-1
0
1
2x 10
-3
Oscill
ation (
m)
0 5 10 15 20 25 30-0.2
-0.1
0
0.1
0.2
time (s)
Oscill
ation (
m)
Linear term
Perturbed term
Linearized
1 2 3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Wave Frequency (rad/s)
Wat
er C
olum
n R
AO
Model 1
Nonlinear
LinearizedLinear
Experimental
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.154
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
Wave height (m)
Wat
er C
olum
n R
AO
Linear
LinearizedNonlinear
Perturbation results
Comparison
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
1 2 3 4 5 6 7 8 90.02
0.03
0.04
0.05
0.06
0.07
0.08
Wave Frequency (rad/s)
Equ
ival
ent
Line
ar D
ampi
ng R
atio
Equivalent Viscous Damping Ratio (Model1)
Open tube
4 Orifices2 Orifices
0 1 2 3 4 5 6 7 8 9 10-100
-50
0
50
100
Time(sec)
Wate
r E
levation (
mm
)
Water Elevation Decay
30 35 40 45 50 55 60 65 700.2
0.22
0.24
0.26
0.28
0.3
0.32
Mean Water Elevation (mm)
Wate
r E
levation d
ecre
ase/M
ean W
ate
r E
levation
data 1
data 2
data 3
data 4
linear
1 2 3 4 5 6 70
1
2
3
4
5
6
7
8
Wave Frequency (rad/s)
Wat
er C
olum
n R
AO
Model1
VD
Optimized EVD
EVD
Open tube
Matlab Script for L&Q damping coef. calculations
Optimised damping ratios
Comparison
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.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
5
10
15
20
25
30
Oscillations Amplitude (m)
Max
Stif
fnes
s(N
/m)
Model1
Water column stiffness
Pneumatic stiffnessTotal stiffness
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.
Lar
ge W
aves
Dam
ping
Stif
fnes
s
Renewable Energy Converting Platform
The concentration of several devices on one platform has both economic and operational advantages.
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.
Conclusions (RE Platform)
SummarySeveral 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.