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

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 Inertia Regime Drag may be ignored Diffraction Regime Froude-Krylov approx. is valid 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.)

Captive Structure Floating 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) MassAdded mass Model Model OWC Damping Ratios WC (Open tube)WC + 4 OrificesWC + 2 Orifices Log. dec. Half- power Log. dec. Half- power Log. dec. Half- power Model Model NA OWC Stiffness (N/m) WC Hydrostatic Air Compressibility Model Model

Single DOF Model (Captive structure) 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 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 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.

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

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)

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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