José Antonio Armesto (armestoja@unican.es), Víctor Ayllón, Carlos Barrera, Raúl Guanche

Environmental Hydraulics Institute, Universidad de Cantabria

DIFFERENT APPROACHES TO COMPUTE THE RESTORING TERM

IN TIME DOMAIN POTENTIAL FLOW MODELS

References:[1] Vissio G, Valrio D, Bracco G, Beiro P, Pozzi N, Mattiazzo G. ISWEC linear quadratic regulator oscillating control. Renewable Energy 2017;103:372 –82.[2] Cordonnier J, Gorintin F, Cagny AD, Clément AH, BabaritA. Searev: Case study of the development of a wave energy converter. Renewable Energy 2015;80(0):40 – 52.[3] Salcedo F, Ruiz-Minguela P, Rodriguez R, Ricci P, Santos M. Oceantec: Sea trials of a quarter scale prototype. In: Proceedings of the 8th European Wave and TidalEnergy Conference 2009. Uppsala, Sweden.[4] Armesto, JA, Guanche, R, Ayllón, V, Barrera, C., Losada, IJ, Vidal, C, Cobo, I. Numerical and experimental study of a pendulum wave energy converter. In:Proceedings of the 11th European Wave and Tidal Energy Conference 2015. Nantes, France.[5] Cummins, W. E. The impulse response function and ship motions. Schiffstechnik, 101-109. 1962[6] DNV. SESAM Users manual, WADAM. DNV.

Acknowledgments:

The authors would like to express their gratitude to the Spanish Ministry of Economy and Competitiveness and in particular to the State Secretariat forResearch, Development and Innovation for funding the ``VAPEO - Ocean Climate Variability influence over Wave Energy Converters Power Production''project (ENE2013-48716-R), within the National Programme for Research Aimed at the Challenges of Society, modality 1, Research Challenge: Research,Development and Innovation.

In the last decade, high energy costs and the need for diversification of energy sources has motivated support for the development of Marine Renewable Energies (MRE) based on the study of different technologies as a key segment of blue growth in coastal communities. The diverse technologies to be tested in laboratories or through sea trials has to be supported by adequate numerical tools that allow for design optimization in order to maximize energy efficiency conversion and to reduce design risks.

In recent years, some projects have developed WECs that convert energy using a PTO forced by the rotation or pitching of the WEC [1-4]; large motions may therefore be expected for the sake of power production. The amplitude of the rotations of these WECs is large, and the classical approximation of the restoring force might therefore not be accurate enough.

The numerical study of wave energy converters in time domain based Cummin’s [5] equation and using potential flow theory for the hydrodynamic coefficients of the body [6], is usually done using a constant restoring matrix. This approach is appropriate for small movements and assuming the flotation area of the hull does not change. A methodology to reduce such limitations is studied with the objective to accurately study larger movements of the platforms using potential theory.

Three different approaches have been studied to compute the restoring term used in Cummin’s equation:• The first approach is to use a constant restoring matrix, as it is usually done in the literature (GM). • The restoration term applied for the rotation of the platform is computed every time step using the GZ curve. The time domain value is obtained

by the interpolation of the GZ curve depending on the instantaneous rotation (GZ). • The third approach is to compute every time step the submerged volume computed from the instantaneous position of the platform (DH).A WEC with bumps which enters the water for large rotation amplitudes [4], which requires increasing the restoring force in such situations, is used to compare the results obtained by each approach.

The three models are used to simulate decay tests in pitch with different initial angle of incidence.

The restoring force and momentum are computed as the combination of weight and buoyancy: • The weight force is constant in time, as it is

given by the mass of the body.• The buoyancy force depends on the

instantaneous submerged volume. • The momentum generated is computed as

product of the force by the distance between the center of gravity and the center of buoyancy.

The submerged volume is computed using a panelization of the hull. Then at each time step:• move the mesh according to the movements of the WEC,• identify the panels below the free surface, and those panels cut by

the free surface,• compute the barycenter, area, and normal vector of each panel

below the free surface,• compute the volume corresponding to the panel • compute the barycenter of the volume corresponding to each panel,• sum all the volumes with their corresponding signs and compute the

center of buoyancy as the weighted average of the volume of the panels.

The three models are used to simulate the behaviourin pitch under regular waves of different periods.

A sensitivity analysis is done for the number of panels in the presented methodology.

Validation of the results obtained by the three models against laboratory results in a decay in pitch.

A sensitivity analysis is done for the number of panels in the presented methodology.The computational cost in seconds of the different

methodologies for the decay tests in pitch is compared in the Table below.

𝝓𝟎 GM GZ DH

5 0.65 0.64 76.66

10 0.70 0.73 92.42

15 0.74 0.78 99.38

20 0.78 0.78 105.63

• The results are clearly improved when the time dependent methodologies are used to compute the restoring force.

• The computational cost increases with the presented methodology and it grows linearly with the number of panels used. The results obtained with the presented methodology improve the results even with a low number of panels.

• However, a single GZ curve would not be enough in a three dimensional problem , as the three rotations would have to be taken into account. This deficit is avoided with the time dependent methodology presented as movements in all DOFs are already considered. The 3-D general case will be addressed in future works.

CONCLUSIONS