Wave Power Absorption by Arrays of Wave Energy Converters in Front of a Vertical Breakwater: A Theoretical Study
Abstract
:1. Introduction
2. Hydrodynamic Formulation
2.1. The Boundary Value Problem
2.2. Determination of the Velocity Potentials
3. Efficiency of a WEC Array
3.1. Motion Equation System
3.2. Absorbed Wave Power—q Factor
4. Numerical Results
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
L = D | L = 5D | L = 10D | ||||
---|---|---|---|---|---|---|
ω | Modulus | Phase | Modulus | Phase | Modulus | Phase |
0.2000 | 2.0008 | 0.0003 | 1.9613 | 0.0003 | 1.8393 | 0.0004 |
0.4000 | 2.0038 | 0.0012 | 1.8400 | 0.0015 | 1.3562 | 0.0022 |
0.5000 | 2.0064 | 0.0020 | 1.7435 | 0.0025 | 0.9946 | 0.0037 |
0.7000 | 2.0157 | 0.0043 | 1.4618 | 0.0059 | 0.0544 | 0.0122 |
0.8000 | 2.0235 | 0.0059 | 1.2661 | 0.0089 | 0.4973 | −3.1334 |
1.0000 | 2.0516 | 0.0108 | 0.7271 | 0.0199 | 1.5992 | −3.1281 |
1.2000 | 2.1144 | 0.0202 | 0.0672 | 3.1071 | 2.2104 | −3.1234 |
1.3000 | 2.1718 | 0.0279 | 0.5816 | −3.1124 | 2.0358 | −3.1146 |
1.5000 | 2.4166 | 0.0613 | 1.8569 | −3.0789 | 0.0674 | −2.8550 |
1.7000 | 3.3197 | 0.1951 | 3.7393 | −2.9835 | 3.5448 | 0.1435 |
1.9000 | 6.0516 | 0.7489 | 8.5278 | −2.5052 | 3.7537 | 0.6989 |
2.0000 | 5.7637 | 1.8289 | 4.8576 | −1.1047 | 3.3664 | −0.6393 |
2.2000 | 1.2536 | 2.9904 | 0.6132 | 2.6193 | 1.8353 | −0.3297 |
2.3000 | 0.5786 | −3.0874 | 0.8055 | 2.9180 | 0.2097 | −0.4343 |
2.5000 | 0.1110 | −2.8392 | 0.4916 | 2.6394 | 0.4678 | 2.6346 |
2.7000 | 0.0941 | −0.5789 | 0.1694 | −3.0218 | 0.2451 | −0.2786 |
2.8000 | 0.0831 | −0.1999 | 0.0772 | −0.0923 | 0.2160 | −0.4910 |
3.0000 | 0.0447 | −0.0520 | 0.0898 | −0.4302 | 0.0917 | 2.7800 |
L = D | L = 5D | L = 10D | ||||
---|---|---|---|---|---|---|
ω | Modulus | Phase | Modulus | Phase | Modulus | Phase |
0.2000 | 2.0014 | 0.0003 | 1.9439 | 0.0004 | 1.8063 | 0.0005 |
0.4000 | 2.0059 | 0.0019 | 1.7682 | 0.0027 | 1.2295 | 0.0036 |
0.5000 | 2.0093 | 0.0035 | 1.6284 | 0.0048 | 0.8049 | 0.0064 |
0.7000 | 2.0176 | 0.0083 | 1.2240 | 0.0098 | 0.2591 | −3.1390 |
0.8000 | 2.0232 | 0.0111 | 0.9501 | 0.0122 | 0.8494 | −3.1326 |
1.0000 | 2.0453 | 0.0162 | 0.2281 | 0.0236 | 1.8822 | −3.1233 |
1.2000 | 2.1085 | 0.0232 | 0.7503 | −3.1085 | 2.0669 | −3.1163 |
1.3000 | 2.1748 | 0.0303 | 1.3225 | −3.1070 | 1.5060 | −3.1055 |
1.5000 | 2.4347 | 0.0809 | 2.5325 | −3.0725 | 1.1436 | 0.0545 |
1.7000 | 3.1884 | 0.1717 | 3.4615 | −2.9981 | 3.6184 | 0.1323 |
1.9000 | 6.8404 | 1.0488 | 2.8544 | −2.0463 | 2.6023 | −2.5470 |
2.0000 | 3.8435 | 1.6568 | 2.5416 | 2.7659 | 7.8437 | −0.8686 |
2.2000 | 1.2622 | −3.1016 | 1.5394 | 2.7891 | 0.4806 | 0.4493 |
2.3000 | 0.5502 | −2.9091 | 1.1892 | 2.8538 | 0.8812 | 2.8927 |
2.5000 | 0.2347 | −2.2849 | 0.1617 | 3.0906 | 0.0818 | −3.0315 |
2.7000 | 0.0881 | −0.6917 | 0.2237 | −0.1849 | 0.1847 | −0.4820 |
2.8000 | 0.1035 | 0.2892 | 0.2167 | −0.6014 | 0.0788 | 3.1056 |
3.0000 | 0.0724 | −0.0201 | 0.0052 | −0.0492 | 0.0231 | 2.7385 |
L = D | L = 5D | L = 10D | ||||
---|---|---|---|---|---|---|
ω | Modulus | Phase | Modulus | Phase | Modulus | Phase |
0.2000 | 2.0019 | 0.0006 | 1.9622 | 0.0007 | 1.8401 | 0.0007 |
0.4000 | 2.0057 | 0.0024 | 1.8415 | 0.0024 | 1.3577 | 0.0026 |
0.5000 | 2.0097 | 0.0037 | 1.7460 | 0.0040 | 0.9969 | 0.0045 |
0.7000 | 2.0204 | 0.0078 | 1.4655 | 0.0081 | 0.0571 | 0.0215 |
0.8000 | 2.0308 | 0.0112 | 1.2705 | 0.0122 | 0.4965 | −3.1324 |
1.0000 | 2.0628 | 0.0197 | 0.7355 | 0.0239 | 1.6085 | −3.1218 |
1.2000 | 2.1451 | 0.0182 | 0.0578 | 3.1055 | 2.2208 | −3.1241 |
1.3000 | 2.2337 | 0.0545 | 0.5778 | −3.1087 | 2.0866 | −3.0936 |
1.5000 | 2.4939 | 0.1083 | 1.9252 | −3.0429 | 0.0983 | −2.9389 |
1.7000 | 3.2472 | 0.2510 | 3.8320 | −2.8967 | 3.6565 | 0.2412 |
1.9000 | 7.1944 | 1.1433 | 8.8875 | −2.1145 | 5.2251 | 0.9882 |
2.0000 | 4.1945 | 2.2283 | 4.0644 | −0.8197 | 2.4805 | −1.0937 |
2.2000 | 1.1219 | −3.0935 | 0.4064 | 2.7441 | 1.8300 | −0.2992 |
2.3000 | 0.5504 | −3.0650 | 0.7905 | 2.9267 | 0.2236 | 0.0693 |
2.5000 | 0.1351 | −1.9768 | 0.5312 | 2.6718 | 0.5642 | 2.6916 |
2.7000 | 0.0765 | −0.4791 | 0.1220 | 3.1242 | 0.2068 | −0.0698 |
2.8000 | 0.0490 | 0.2306 | 0.0303 | −0.2045 | 0.2361 | −0.3223 |
3.0000 | 0.0905 | 0.0835 | 0.1137 | 0.0724 | 0.1148 | −3.0764 |
References
- Falnes, J. A review of wave-energy extraction. Mar. Struct. 2007, 20, 185–201. [Google Scholar] [CrossRef]
- McCormick, M.E. Ocean Wave Energy Conversion; Courier Corporation: North Chelmsford, MA, USA, 1981. [Google Scholar]
- Pelc, R.; Fujita, R.M. Renewable energy from the ocean. Mar. Policy 2002, 26, 471–479. [Google Scholar] [CrossRef]
- Aderinto, T.; Li, H. Ocean wave energy converters: Status and challenges. Energies 2018, 11, 1250. [Google Scholar] [CrossRef] [Green Version]
- MacGillivray, A.; Jeffrey, H.; Hanmer, C.; Magagna, D.; Raventos, A.; Badcock-Broe, A. Ocean Energy Technology: Gaps and Barriers. SI Ocean: 2013. Available online: www.si-ocean.eu (accessed on 1 February 2020).
- Belibassakis, K.; Bonovas, M.; Rusu, E. Novel method for estimating wave energy converter performance in variable bathymetry regions and applications. Energies 2018, 11, 2092. [Google Scholar] [CrossRef] [Green Version]
- Bonovas, M.; Belibassakis, K.; Rusu, E. Multi-DOF WEC performance in variable bathymetry regions using a hybrid 3D BEM and optimization. Energies 2019, 12, 2108. [Google Scholar] [CrossRef] [Green Version]
- Mustapa, M.A.; Yaakob, O.B.; Ahmed, Y.M.M.; Rheem, C.K.; Koh, K.K.; Faizul, A.A. Wave energy device and breakwater integration: A review. Renew. Sustain. Energy Rev. 2017, 77, 43–58. [Google Scholar] [CrossRef]
- Vicinanza, D.; Margheritini, L.; Kofoed, J.P.; Buccino, M. The SSG wave energy converter: Performance, status and recent developments. Energies 2012, 5, 193–226. [Google Scholar] [CrossRef] [Green Version]
- Mavrakos, S.A.; Katsaounis, G.M.; Nielsen, K.; Lemonis, G. Numerical performance investigation of an array of heaving wave power converters in front of a vertical breakwater. In Proceedings of the 14th International Offshore and Polar Engineering Conference (ISOPE 2004), Toulon, France, 23–28 May 2004. [Google Scholar]
- Mavrakos, S.A.; Katsaounis, G.M.; Kladas, A.; Kimoulakis, N. Numerical and experimental investigation of performance of heaving WECs coupled with DC generators. In Proceedings of the 9th European Wave and Tidal Energy Conference, Southampton, UK, 5–9 September 2011. [Google Scholar]
- McIver, P.; Porter, R. The motion of a freely floating cylinder in the presence of a wall and the approximation of resonances. J. Fluid Mech. 2016, 795, 581–610. [Google Scholar] [CrossRef] [Green Version]
- Teng, B.; Ning, D.Z. Wave diffraction from a uniform cylinder in front of a vertical wall. Ocean Eng. 2003, 21, 48–52. [Google Scholar]
- Teng, B.; Ning, D.Z.; Zhang, X.T. Wave radiation by a uniform cylinder in front of a vertical wall. Ocean Eng. 2004, 31, 201–224. [Google Scholar] [CrossRef]
- Zheng, S.; Zhang, Y. Wave diffraction from a truncated cylinder in front of a vertical wall. Ocean Eng. 2015, 104, 329–343. [Google Scholar] [CrossRef]
- Zheng, S.; Zhang, Y. Wave radiation from a truncated cylinder in front of a vertical wall. Ocean Eng. 2016, 111, 602–614. [Google Scholar] [CrossRef]
- Konispoliatis, D.N.; Mavrakos, S.A. Theoretical performance investigation of a vertical cylindrical oscillating water column device in front of a vertical breakwater. J. Ocean Eng. Mar. Energy 2019. [Google Scholar] [CrossRef]
- Schay, J.; Bhattacharjee, J.; Soares, C.G. Numerical modelling of a heaving point absorber in front of a vertical wall. In Proceedings of the 32nd International Conference on Ocean, Offshore and Arctic Engineering, 9–14 June 2013; Volume 8. Paper No. OMAE2013-11491. [Google Scholar]
- Spring, B.W.; Monkmeyer, P.L. Interaction of plane waves with a row of cylinders. In Proceedings of the 3rd Specialty Conference of Civil Engng in Oceans (ASCE), Newark, DE, USA, 9–12 June 1975; pp. 979–998. [Google Scholar]
- Yeung, R.W.; Sphaier, S.H. Wave-interference effects on a truncated cylinder in a channel. J. Eng. Math. 1989, 23, 95–117. [Google Scholar] [CrossRef]
- Mavrakos, S.A. The scattered wave field by vertical cylinders in a narrow tank. In Proceedings of the 4th National Symposium on Theoretical and Applied Mechanics, Xanthi, Greece, 26–29 June 1995; Volume II, pp. 819–829. [Google Scholar]
- Butler, B.P.; Thomas, G.P. The diffraction of water waves by an array of circular cylinders in a channel. Ocean Eng. 1993, 20, 295–311. [Google Scholar] [CrossRef]
- McIver, P. The wave field scattered by a vertical cylinder in a narrow wave tank. Appl. Ocean Res. 1993, 15, 25–37. [Google Scholar] [CrossRef]
- Loukogeorgaki, E.; Chatjigeorgiou, I. Hydrodynamic performance of an array of wave energy converters in front of a vertical wall. In Proceedings of the 13th European Wave and Tidal Energy Conference (EWTEC, 2019), Naples, Italy, 1–6 September 2019. [Google Scholar]
- Loukogeorgaki, E.; Boufidi, I.; Chatjigeorgiou, I. Performance of an array of oblate spheroidal heaving wave energy converters in front of a wall. Water 2020, 12, 188. [Google Scholar] [CrossRef] [Green Version]
- Zhao, X.L.; Ning, D.Z.; Zhang, C.W.; Liu, Y.Y.; Kang, H.G. Analytical study on an oscillating buoy wave energy converter integrated into a fixed box-type breakwater. Math. Probl. Eng. 2017, 3960401. [Google Scholar] [CrossRef]
- Zhao, X.L.; Ning, D.Z.; Liang, D.F. Experimental investigation on hydrodynamic performance of a breakwater-integrated WEC system. Ocean Eng. 2019, 171, 25–32. [Google Scholar] [CrossRef]
- Howe, D.; Nader, J.R. OWC WEC integrated within a breakwater versus isolated: Experimental and numerical theoretical study. Mar. Energy 2017, 20, 165–182. [Google Scholar] [CrossRef]
- He, F.; Huang, Z.; Law, A.W.K. Hydrodynamic performance of a rectangular floating breakwater with and without pneumatic chambers: An experimental study. Ocean Eng. 2012, 51, 16–27. [Google Scholar] [CrossRef]
- Zheng, X.; Zeng, Q.; Liu, Z. Hydrodynamic performance of rectangular heaving buoys for an integrated floating breakwater. J. Mar. Sci. Eng. 2019, 7, 239. [Google Scholar] [CrossRef] [Green Version]
- Ning, D.Z.; Zhao, X.L.; Chen, L.F.; Zhao, M. Hydrodynamic performance of an array of wave energy converters integrated with a pontoon-type breakwater. Energies 2018, 11, 685. [Google Scholar] [CrossRef] [Green Version]
- Martins-rivas, H.; Mei, C.C. Wave power extraction from an oscillating water column at the tip of a breakwater. J. Fluid Mech. 2009, 626, 395–414. [Google Scholar] [CrossRef]
- Martins-rivas, H.; Mei, C.C. Wave power extraction from an oscillating water column along a straight coast. Ocean Eng. 2009, 36, 426–433. [Google Scholar] [CrossRef]
- Naty, S.; Viviano, A.; Foti, E. Wave energy exploitation system integrated in the coastal structure of a Mediterranean port. Sustainability 2016, 8, 1342. [Google Scholar] [CrossRef] [Green Version]
- Konispoliatis, D.N.; Mavrakos, S.A.; Katsaounis, G.M. Theoretical evaluation of the hydrodynamic characteristics of arrays of vertical axisymmetric floater of arbitrary shape in front of a vertical breakwater. J. Mar. Sci. Eng. 2020, 8, 62. [Google Scholar] [CrossRef] [Green Version]
- Mavrakos, S.A.; Koumoutsakos, P. Hydrodynamic interaction among vertical axisymmetric bodies restrained in waves. Appl. Ocean Res. 1987, 9, 128–140. [Google Scholar] [CrossRef]
- Mavrakos, S.A. Hydrodynamic coefficients for groups of interacting vertical axisymmetric bodies. Ocean Eng. 1991, 18, 485–515. [Google Scholar] [CrossRef]
- Mavrakos, S.A.; McIver, P. Comparison of methods for computing hydrodynamic characteristics of arrays of wave power devices. Appl. Ocean Res. 1997, 19, 283–291. [Google Scholar] [CrossRef]
- Falnes, J. Ocean Waves and Oscillating Systems: Linear Interactions Including Wave-Energy Extraction; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Faltinsen, O.M. Sea Loads on Ships and Offshore Structures; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Konispoliatis, D.N.; Chatijigeorgiou, I.K.; Mavrakos, S.A. Near trapped wave phenomena in an array of truncated cylinders in a perpendicular arrangement in front of a vertical breakwater. J. Appl. Math. Model. 2020. [Google Scholar] [CrossRef]
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Konispoliatis, D.N.; Mavrakos, S.A. Wave Power Absorption by Arrays of Wave Energy Converters in Front of a Vertical Breakwater: A Theoretical Study. Energies 2020, 13, 1985. https://doi.org/10.3390/en13081985
Konispoliatis DN, Mavrakos SA. Wave Power Absorption by Arrays of Wave Energy Converters in Front of a Vertical Breakwater: A Theoretical Study. Energies. 2020; 13(8):1985. https://doi.org/10.3390/en13081985
Chicago/Turabian StyleKonispoliatis, Dimitrios N., and Spyridon A. Mavrakos. 2020. "Wave Power Absorption by Arrays of Wave Energy Converters in Front of a Vertical Breakwater: A Theoretical Study" Energies 13, no. 8: 1985. https://doi.org/10.3390/en13081985
APA StyleKonispoliatis, D. N., & Mavrakos, S. A. (2020). Wave Power Absorption by Arrays of Wave Energy Converters in Front of a Vertical Breakwater: A Theoretical Study. Energies, 13(8), 1985. https://doi.org/10.3390/en13081985