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Statistical linearization of the Morison’s equation applied to wave energy converters

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Abstract

The viscous drag acting on wave energy converters may have a significant effect on the dynamics during high-energetic sea states and large motions experienced due to resonance. The viscous drag is a nonlinear phenomenon of floating systems usually modelled based on the Morison’s equation using the relative velocity between the structure and the wave particle. To include such a nonlinearity into the system dynamics, nonlinear time domain simulations are generally employed, which are computationally expensive compared to frequency domain simulations. To overcome this problem, this work presents the derivation of the viscous drag force/torque under the statistical linearization technique using the frequency domain model. The technique offers a reliable tool for the estimation of the system dynamics while maintaining a low computational cost when compared to time domain simulations. For the proposed nonlinearity, the resulting equivalent linear term can be decomposed into two components: an excitation term and a damping term. To illustrate the applicability of the derivation, two conceptually different wave energy converters are investigated: a heaving point absorber, and an oscillating wave surge converter. The results obtained using statistical linearization are compared to their respective nonlinear time domain simulations to verify the reliability of the technique. Also, a comparison between the statistical linearization results using the relative motion and using only the structure motion is presented to illustrate the importance of including the relative velocity for wave energy applications. Excellent agreements have been obtained between statistical linearization model using the relative motion and its respective nonlinear time domain model for both devices in terms of spectral content, probability density of the velocity components, and energy absorbed by the device.

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References

  • Babarit A, Delhommeau G (2015) Theoretical and numerical aspects of the open source BEM solver NEMOH. In: 11th European Wave and Tidal Energy Conference (EWTEC2015), Nantes, France

  • Babarit A, Hals J, Muliawan MJ, Kurniawan A, Moan T, Krokstad J (2012) Numerical benchmarking study of a selection of wave energy converters. Renew Energy 41:44–63

    Article  Google Scholar 

  • Bacelli G, Ringwood JV (2014) Nonlinear optimal wave energy converter control with application to a flap-type device. IFAC Proc Vol 47(3):7696–7701

    Article  Google Scholar 

  • Bearman PW, Downie MJ, Graham JMR, Obasaju ED (1985) Forces on cylinders in viscous oscillatory flow at low Keulegan-Carpenter numbers. J Fluid Mech 154:337–356

    Article  MATH  Google Scholar 

  • Berge B, Penzien J (1974) Three-dimensional stochastic response of offshore towers to wave forces. In: Offshore Technology Conference, OTC, Houston, Texas

  • Davidson J, Costello R (2020) Efficient nonlinear hydrodynamic models for wave energy converter design A scoping study. J Mar Sci Eng 8(1):35

    Article  Google Scholar 

  • Folley M (2016) Spectral-domain models. In: Numerical modelling of wave energy converters. Elsevier, Berlin, pp 67–80

  • Folley M, Whittaker T (2010) Spectral modelling of wave energy converters. Coast Eng 57(10):892–897

    Article  Google Scholar 

  • Giorgi G, Penalba M, Ringwood J (2016) Nonlinear hydrodynamic force relevance for different wave energy converter types. In: Proceedings of the 3rd Asian wave and tidal energy conference, pp 154–162, Singapore

  • Giorgi G, Ringwood JV (2018) Comparing nonlinear hydrodynamic forces in heaving point absorbers and oscillating wave surge converters. J Ocean Eng Mar Energy 4(1):25–35

    Article  Google Scholar 

  • Housseine CO, Monroy C, de Hauteclocque G (2015) Stochastic linearization of the Morison equation applied to an offshore wind turbine, In: ASME 2015 34th international conference on ocean, offshore and arctic engineering, American Society of Mechanical Engineers, St. John's, Newfoundland, Canada

  • Journée JMJ, Massie WW (2000) Offshore Hydromechanics: Course OE4630, TU Delft

  • Keulegan GH, Carpenter LH (1956) Forces on cylinders and plates in an oscillating fluid: US Department of Commerce. NBS report, 4821, National Bureau of Standards

  • Molin B (2002) Hydrodynamique des structures offshore. Editions Technip, Paris

    Google Scholar 

  • Morison JR, O’Brien MP, Johnson JW, Schaaf SA (1950) The force exerted by surface waves on piles. J Petrol Technol 2(05):149–154

    Article  Google Scholar 

  • Naess A, Pisano AA (1997) Frequency domain analysis of dynamic response of drag dominated offshore structures. Appl Ocean Res 19(5–6):251–262

    Article  Google Scholar 

  • Newman JN (2018) Marine hydrodynamics. MIT Press, Cambridge

    Google Scholar 

  • Ochi MK (2005) Ocean waves: the stochastic approach, vol 6. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Perez T, Fossen TI (2009) A Matlab toolbox for parametric identification of radiation-force models of ships and offshore structures. Model Identif Control 30:1–15

    Article  Google Scholar 

  • Roberts JB, Spanos PD (2003) Random vibration and statistical linearization. Courier Corporation, Mineola, New York

    MATH  Google Scholar 

  • Sergiienko NY, Neshat M, Silva LSP, Alexander B, Wagner M (2020) Design optimisation of a multi-mode wave energy converter, In: ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering, American Society of Mechanical Engineers, Fort Lauderdale, FL, USA

  • Silva LSP (2019) Nonlinear stochastic analysis of wave energy converters via statistical linearization, MSc thesis, University of São Paulo, Brazil

  • Silva LSP, Morishita HM, Pesce CP, Gonçalves RT (2019) Nonlinear analysis of a heaving point absorber in frequency domain via statistical linearization, In: ASME 2019 38th International Conference on Ocean, Offshore and Arctic Engineering, American Society of Mechanical Engineers, Glasgow, Scotland, UK

  • Silva LSP, Sergiienko N, Pesce CP, Ding B, Cazzolato B, Morishita HM (2020a) Stochastic analysis of nonlinear wave energy converters via statistical linearization. Appl Ocean Res 4:5

    Google Scholar 

  • Silva LSP, Sergiienko NY, Cazzolato BS, Ding B, Pesce CP, Morishita HM (2020b) , Nonlinear analysis of an oscillating wave surge converter in frequency domain via statistical linearization, In: ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering, American Society of Mechanical Engineers, Fort Lauderdale, FL, USA

  • Spanos PD, Arena F, Richichi A, Malara G (2016) Efficient dynamic analysis of a nonlinear wave energy harvester model. J Offshore Mech Arctic Eng 138(4):041901

    Article  Google Scholar 

  • Todalshaug JH, Babarit A, Kurniawan A, Moan T (2011) ‘The NumWEC project. Numerical estimation of energy delivery from a selection of wave energy converters’, Report, Ecole Centrale de Nantes & Norges Teknisk-Naturvitenskapelige Universitet

  • Wolfram J (1999) On alternative approaches to linearization and Morison’s equation for wave forces. Proc R Soc Lond Ser A Math Phys Eng Sci 455:2957–2974

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

L. S. P. da Silva acknowledges the Australia–China Science and Research Fund, Australian Department of Industry, Innovation and Science; and the Adelaide Graduate Centre, the University of Adelaide. C.P. Pesce acknowledges a CNPq Research Grant, nr. 308230/2018-3.

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Correspondence to Leandro S. P. da Silva.

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Appendix A: PTO damping selection

Appendix A: PTO damping selection

For both devices, the PTO damping is selected to optimize the power absorption for the condition specified based on the characteristics of the device. It is important to notice that the differences between both SL models (using the relative velocity and only the body velocity) may be observed over the entire range of conditions simulated (see Figs. 14 and 15).

Fig. 14
figure 14

Power absorbed by the PA for a range of \(B_\mathrm{pto}\) (\(T_\mathrm{p}=\) 7.5 s and \(H_\mathrm{s}=\) 3 m)

Fig. 15
figure 15

Power absorbed by the OWSC for a range of \(B_\mathrm{pto}\) (\(T_\mathrm{p}=\) 12 s and \(H_\mathrm{s}=\) 1.5 m)

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da Silva, L.S.P., Cazzolato, B.S., Sergiienko, N.Y. et al. Statistical linearization of the Morison’s equation applied to wave energy converters. J. Ocean Eng. Mar. Energy 6, 157–169 (2020). https://doi.org/10.1007/s40722-020-00165-9

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