Linear and Nonlinear modelling of Oscillating Water Column Wave Energy Converter

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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

Professor Atilla Incecik Head of Naval
Architecture and Marine Engineering
Dept. University of Strathclyde, Glasgow
Professor Hassan El-Gamal Mechanical Engineering
Dept. Alexandria University
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Presentation Layout

• Introduction
• Motivation
• Research Objective
• Numerical tool Methodology
• Wave Wind Forces
• OWC Modelling
• Nonlinear Modeling
• Renewable Energy Converting Platform
• Conclusions

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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.

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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.

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Research Objective

• The main objective of this research is to
  develop an experimentally validated numerical
  wave power prediction tool for offshore SparBuoy
  OWC WEC.

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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)

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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 buoys whole body motion.

EM Plant
Spar Buoy
Water Column
Vertical Tube
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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.

 
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Predicted Wave Forces
Diffraction Regime Froude-Krylov approx. is valid
Inertia Regime Drag may be ignored
Results agree with Incecik, 2003 Chakrabarti,
2005 charts
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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.)
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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 One-way Coupling Model
Modified Szumko Model
Szumko Model
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Equations of Motions

•  

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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) OWC Mass (kg)
  Mass Added mass
Model1 1.1310 0.0360
Model2 4.5996 0.2953
  OWC Damping Ratios OWC Damping Ratios OWC Damping Ratios OWC Damping Ratios OWC Damping Ratios OWC Damping Ratios
  WC (Open tube) WC (Open tube) WC 4 Orifices WC 4 Orifices WC 2 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) OWC Stiffness (N/m)
      WC Hydrostatic Air Compressibility
Model1 27.7371 1.0875
Model2 112.8053 4.4227
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Single DOF Model (Captive structure)
Good agreement between predicted and measured
responses, except around resonance due to the use
of viscous damping.
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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 doesnt 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.

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Nonlinearity due to Large Waves
Perturbation results
Comparison
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Nonlinear Damping

• Iterative (optimised) frequency domain model

• Non-linear time 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.

• 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.

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Nonlinear Damping
Optimised damping ratios
Matlab Script for LQ damping coef. calculations
Comparison
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Experimental vs. Numerical Water Column Decay
Test Results (Damping Model1)
Experimental vs. Numerical Water Column Decay
Test Results (Damping Model2)
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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.

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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.

Large Waves
Damping
Stiffness
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Renewable Energy Converting Platform
The concentration of several devices on one
platform has both economic and operational
advantages.
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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.

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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.

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Thank You