MODELLING OF OWC WAVE ENERGY CONVERTERS

1. Short Course on Wave Energy Technology Lisbon, 14-18 July 2014 MODELLING OF OWC WAVE ENERGY CONVERTERS António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2014

2. Basic approaches to OWC modelling will be analized here

3. Basic equations Volume flow rate of air displaced by OWC motion • Decomposeinto • excitationflow rate • radiationflow rate

4. Basic equations

5. Thermodynamics of air chamber Assume compression/decompression process in air chamber to be isentropic (adiabatic + reversible)

6. Aerodynamics of air turbine X X Dependence on Mach number Ma in general neglected, because of scarce information from model testing.

7. Frequency domain analysis • Linear turbine • Linear relationship air density versus pressure Linearize: Wells turbine

8. Frequency domain analysis The system is linear Decompose Note: radiation conductance G cannot be negative

9. PICO OWC PLANT, AZORES, PORTUGAL

10. Frequency domain analysis (deep water) Axisymmetric body (deep water)

12. Frequency domain analysis Power Power available to turbine = pressure head x volume flow rate Regular waves Time average

13. Frequency domain analysis Power Turbine power output Wells turbine

14. Time-domain analysis of OWCs The Wells turbine is approximately linear. So frequency-domain analysis is a good approximation. Other turbines (e.g. impulse turbines) are far from linear. So, time-domain analysis must be used, even in regular waves. This affects specially the radiation flow rate, with memory effects. The theoretical approach is similar to time-domain analysis of oscillating bodies.

15. radiation flow rate memory function

16. STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION

17. 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.

18. Introduction Frequency domain model Basic assumptions: • Monochromatic (sinusoidal) waves • The system (input  output) is linear (e.g. a linear damper and a linear spring) • 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)

19. 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

20. 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

21. Many processes in Nature behave in such a way that the Gaussian probability density function applies. The sum of a large number of independ random variables (without any one being dominat) is Gaussian distributed. The surface elevation at a given point in real ocean waves is approximately a Gaussian random process.

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

23. Ouput signal Input signal

24. Ouput signal Input signal

25. Ouput signal Input signal

26. Linear air turbine (Wells turbine)

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

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

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

30. 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?

31. 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

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

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

34. 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

35. 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

36. 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)

37. 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

38. Calculation example Wind plant average Utilization factor

39. Calculation example Annual averaged net power (electrical)

40. Calculation example Costs Capital costs Operation & maintenance Availability

41. 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

42. 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

43. 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

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

45. CONCLUSIONS

46. Air turbine aerodynamic performance Variables

47. Dimensional analysis in turbomachines

48. Dimensional analysis in turbomachines X X X X X X • In most cases: • effectsofvariations in ReandMa are ignored (are difficult to obtainfrommodeltesting); • airdensityat turbine inletisreplacedbydensity in atmosphere

49. flow rate Biradial turbine (frommodeltesting) pressure instantaneous averaged instantaneous power output averaged pressure pressure

50. Rotational speed affects the performance curves of the turbine Damping 750 1000 1250 Wells turbine 1500