1. Specialization in Ocean Energy MODELLING OF WAVE ENERGY CONVERSION
António F.O. Falcão
Instituto Superior Técnico,
Universidade de Lisboa
2017

2. STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION
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. Frequency domain model
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: Advantages:
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. Gaussian process
Physical random variables that are expected to be the sum of many independent processes have distributions that are nearly Gaussian.
This is the case of the free surface elevation of real irregular waves

7. Introduction Stochastic model Basic assumptions: Advantages:
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

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

9. Ouput signal
Input signal
LINEAR SYSTEM

10. Input signal
Ouput signal
LINEAR SYSTEM

11. Input signal
Ouput signal
LINEAR SYSTEM

12. Linear air turbine (Wells turbine)

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

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

15. AIR TURBINE AND ELECTRICAL EQUIPMENT FOR THE PICO OWC PLANT

16. The Pico plant

17. 9 sea states, and their frequency of occurrence
Simplified version of the wave climate (in deep water)
9 sea states, and their frequency of occurrence

18. How to model the energy conversion chain ELECTRICAL POWER OUTPUT
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.
OWC
AIR
PRESSURE
WAVES
TURBINE
Linear system.
Known hydrodynamic
coefficients
Random,
Gaussian
Known
performance
curves
Random,
Gaussian
rms: p
TURBINE SHAFT POWER
ELECTRICAL POWER OUTPUT
GENERATOR
Electrical
efficiency
Time-averaged
Time-averaged

19. Pico plant

20. Compare two types of air turbines
Biradial turbine
Wells turbine

21. Turbine performance curves versus pressure head
(dimensionless)
Wells
Pressure head

22. Turbine performance curves versus pressure head
(dimensionless)
Biradial
Pressure head
linearize

23. Constraint: blade tip velocity of the turbine rotor should not exceed 180 m/s
WD/2 < 180 m/s
Why?
Centrifugal stresses, shock waves

24. Wells turbine: single stage and two-stages were considered
For each turbine size D and each sea state (Hs,Te), the rotational speed W was numerically optimized for maximum averaged power output of the turbine.
The annual-averaged power output was computed.

25. Turbine type and size optimization
Rotor diameter

26. Turbine efficiency
Incident wave power

27. Rotational speed control
Points fairly well aligned along lines
BIRADIAL TURBINE
But, in average values
Control algorithm (instantaneous values):
Electromagnetic torque:
Rotational speed

28. This affects the plant performance and the rotational speed control
Effect efficiency and rated power of electrical equipment
The efficiency of the electrical equipment (generator and power electronics) decays markedly for load factor less than about 30 – 35%
The electrical rated power should not be exceeded at any time. The power electronics is very sensitive to overheating.
This affects the plant performance and the rotational speed control

30. A B C D E F G H
Turbine size D = 2 m

31. Average rotational speed W versus electrical rated power Prated in the most energetic sea state i = 9, for different turbine sizes D = 2.0 to 3.0 m

32. The annual production of electrical energy depends on turbine size and electrical rated power

33. The final choice of turbine size and electrical rated power will also depend energy tariff and equipment costs.
Maximum profit versus maximum energy production.

34. Maximum energy production and maximum profit
as alternative criteria for
wave power equipment optimization

35. Which criterion to adopt for optimization?
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?

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

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

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

39. Wave climate: set of sea states Each sea state:
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

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

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

42. Calculation example
Wind plant
average
Utilization factor

43. Annual averaged net power (electrical)
Calculation example
Annual averaged net power (electrical)

44. Costs Capital costs Operation & maintenance Availability
Calculation example
Costs
Capital costs
Operation & maintenance
Availability

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

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

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

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

49. CONCLUSIONS

50. 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

52. 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 , 2012.

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