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

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 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.

Thank You