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. 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. Introduction

4. Frequency domain model Basic assumptions: Monochromatic (sinusoidal) waves The system (input  output) is linear 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) Historically the first model The starting point for the other models Introduction

5. 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 Introduction

6. 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 Introduction

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

8. Input signal Ouput signal

9. Input signal Ouput signal

10. Linear air turbine (Wells turbine)

11. Average power output

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

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

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 Maximum annual profit, leading to smaller, less powerful, cheaper equipment or 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: H s, T e, freq. of occurrence . Incident wave is random, Gaussian, with known frequency spectrum. WAVES OWC AIR PRESSURE TURBINE TURBINE SHAFT POWER Random, Gaussian Linear system. Known hydrodynamic coefficients Known performance curves Time-averaged GENERATOR ELECTRICAL POWER OUTPUT Time-averaged Random, Gaussian rms:  p Electrical efficiency

16. The costs Operation & maintenance annual costs otherelecmechstruc CCCCC  Capital costs Annual repayment Income Annual profit

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

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

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

20. Calculation example Turbine rotational speed  optimally controlled. Max tip speed = 170 m/s Plant rated power (for H s = 5m, T e =14s) Turbine size range 1.6m < D < 3.8m 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 100150200250300350  D (m/s) Dimensionless power output D=1.6m D=2.3m D=3.8m

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

22. Calculation example Utilization factor Wind plant average

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

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

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