Specialization in Ocean Energy

1. Specialization in Ocean Energy MODELLING OF WAVE ENERGY CONVERSION António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2014

2. PART 5 STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION

3. Introduction Theoretical/numerical hydrodynamic modelling • Frequency-domain • Time-domain • Stochastic In all cases, linear water wave theory is assumed: • small amplitude waves and small body-motions • real viscous fluid effects neglected Non-linear water wave theory and CFD may be used at a later stage to investigate some water flow details.

4. Introduction Frequency domain model Basic assumptions: • Monochromatic (sinusoidal) waves • The system (input  output) is linear • Historically the first model • The starting point for the other models Advantages: • Easy to model and to run • First step in optimization process • Provides insight into device’s behaviour Disadvantages: • Poor representation of real waves (may be overcome by superposition) • Only a few WECs are approximately linear systems (OWC with Wells turbine)

5. Introduction Time-domain model Basic assumptions: • In a given sea state, the waves are represented by a spectral distribution Advantages: • Fairly good representation of real waves • Applicable to all systems (linear and non-linear) • Yields time-series of variables • Adequate for control studies Disadvantages: • Computationally demanding and slow to run Essential at an advanced stage of theoretical modelling

6. Introduction Stochastic model Basic assumptions: • In a given sea state, the waves are represented by a spectral distribution • The waves are a Gaussian process • The system is linear Advantages: • Fairly good representation of real waves • Very fast to run in computer • Yields directly probability density distributions Disadvantages: • Restricted to approximately linear systems (e.g. OWCs with Wells turbines) • Does not yield time-series of variables

7. Ouput signal Input signal LINEAR SYSTEM • Random • Gaussian • Given spectral distribution • Root-mean-square (rms) • Random • Gaussian • Spectral distribution • Root-mean-square (rms)

8. Ouput signal Input signal

9. Ouput signal Input signal

10. Linear air turbine (Wells turbine)

11. Linear air turbine (Wells turbine) Average power output

12. Linear air turbine (Wells turbine) Average turbine efficiency

13. Application of stochastic modelling Maximum energy production and maximum profit as alternative criteria for wave power equipment optimization

14. The problem When designing the power equipment for a wave energy plant, a decision has to be made about the size and rated power capacity of the equipment. Which criterion to adopt for optimization? Maximum annual production of energy, leading to larger, more powerful, more costly equipment or Maximum annual profit, leading to smaller, less powerful, cheaper equipment How to optimize? How different are the results from these two optimization criteria?

15. How to model the energy conversion chain Wave climate represented by a set of sea states • For each sea state: Hs, Te, freq. of occurrence . • Incident wave is random, Gaussian, with known frequency spectrum. AIR PRESSURE OWC WAVES TURBINE Linear system. Known hydrodynamic coefficients Known performance curves Random, Gaussian Random, Gaussian rms: p TURBINE SHAFT POWER ELECTRICALPOWER OUTPUT GENERATOR Electrical efficiency Time-averaged Time-averaged

16. = + + + C C C C C struc mech elec other The costs Capital costs Annual repayment Operation & maintenance annual costs Income Annual profit

17. Calculation example Pico OWC plant OWC cross section: 12m 12m Computed hydrodynamic coefficients

18. Calculation example Wells turbine Dimensionless performance curves Turbine geometric shape: fixed Turbine size (D): 1.6 m < D < 3.8 m Equipped with relief valve

19. Inter Calculation example Wave climate: set of sea states Each sea state: • random Gaussian process, with given spectrum • Hs, Te, frequency of occurrence Calculation method: • Stochastic modelling of energy conversion process • 720 combinations  Three-dimensional interpolation for given wave climate and turbine size

20. 0.6 0.55 0.5 D =1.6m 0.45 Dimensionless power output D =2.3m 0.4 0.35 D =3.8m 0.3 0.25 100 150 200 250 300 350 WD (m/s) Calculation example Turbine size range 1.6m < D < 3.8m Turbine rotational speed W optimally controlled. Max tip speed = 170 m/s Plant rated power (for Hs = 5m, Te=14s)

21. Calculation example Wave climates Wave climate 3: 29 kW/m Reference climate: • measurements at Pico site • 44 sea states • 14.5 kW/m Wave climate 2: 14.5 kW/m Wave climate 1: 7.3 kW/m

22. Calculation example Wind plant average Utilization factor

23. Calculation example Annual averaged net power (electrical)

24. Calculation example Costs Capital costs Operation & maintenance Availability

25. Calculation example wave climate 3: 29 kW/m wave climate 2: 14.5 kW/m wave climate 1: 7.3 kW/m Influence of wave climate and energy price

26. Calculation example wave climate 3: 29 kW/m wave climate 2: 14.5 kW/m wave climate 1: 7.3 kW/m Influence of wave climate and discount rate r

27. Calculation example wave climate 3: 29 kW/m wave climate 2: 14.5 kW/m wave climate 1: 7.3 kW/m Influence of wave climate & mech. equip. cost

28. Calculation example 29 kW/m 14.5 kW/m 7.3 kW/m Influence of wave climate and lifetime n

29. CONCLUSIONS

30. Example: Optimization of an OWC sparbuoy for the wave climate off the western coast of Portugal (31.4 kW/m) Optimization involved several geometric parameters

31. Size and rotational speed of air turbine were optimized R.P.F. Gomes, J.C.C. Henriques, L.M.C. Gato, A.F.O. Falcão. "Hydrodynamic optimization of an axisymmetric floating oscillating water column for wave energy conversion", Renewable Energy, vol. 44, pp. 328-339, 2012.

32. END OF PART 5 STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION