Wave energy Extracting power from ocean waves Some basic principles

Transcription

1 Wave energy Extracting power from ocean waves Some basic principles Finn Gunnar Nielsen, Geophysical Institute, University of Bergen

2 Issues to be discussed A reminder on the nature of (deep water) gravity waves The Global picture Examples on wave energy converters Extracting wave energy by an oscillating system To absorb wave energy is about generating waves Theoretical efficiencies Two simple numerical examples

3 Classification of ocean waves

4 Linear, deep water waves (Airy waves) Dispersion relation: c As g tanh kd k k g d, tanh( kd) 1, c, kg k Kinetic = potential energy (per unit surface area): Energy flux: 1 H Ep Ev g H 1 H g P cg Ep Ev g g k 4 k

5 Sf(f) (m sec) Energy flux in a wave spectrum Peak frequency 16= 1/ (7.47 sec) Jonswap spectrum Mean energy frequency =1/ (6.78 sec) Zero upcrossing frequency = 1/ (5.88 sec] Hs = sqrt(m) = 3.5m Frequency [1/sec] Spectral moments: n mn S d Significant wave height Average period Zero up-crossing period Energy mean period Hs T T T m m m 1 m m m m 1 Energy flux: P g m g T m g T H s

6 RESOURCES Waves (From World Energy Council) Average energy densities: 1 kw/m wave front 5 kw/m, 3% efficiency: 13 MWh/ym 1TWh/y: 76km

7 The potential exceeds the demands. Source: IPCC SRREN,

8 Main principles From: Babarit, Introduction to Ocean Wave Energy Conversion,

9 Illustration of main principles Source: Bedard(6)»overview: EPRI Ocean Energy Program» Presented to Duke Univerisity Global Change Centre

10 Key principles for extracting wave energy Remember: Waves are not only quasi-static change in surface elevation Energy absorption requires a force working together with a velocity Falnes and Budal (1978): In order for an oscillating system to be a good wave absorber it should be a good wave generator

11 The Tapchan project at Toftestallen, Øygarden (finished 1985) Principle. Test site

12 How is energy extracted? Key principles Linear considerations Linear oscillator interaction with waves Heaving buoy as example

13 Linear oscillator F(t) M X Dynamic equilibrium: Mx Bx Kx F() t x x x F( t) Harmonic oscillation, stationary solution. i t F t F cos t F Re[ e ] A A K M B = M K B Dynamic equation, frequency domain: M i B K x F

14 Linear oscillator - solution cos x xa t xare e F x M K ib i t abs(x/x ) 4 3 (Deg) / / At resonance: Response controlled by damping!

15 Estimating the energy absorption by a heaving buoy Assumptions: Heave motion only Linearized analysis No viscous effects

16 The heaving point absorber, a linearized approach (1:3) Equation of motion (1DOF): mx Fex FR FP In frequency domain: m A i B B B k K x F 33 r l wl ex Wave radiation force: F A x B x k x Force due to power off-take: Heave motion: x R 33 r wl F Bx Kx P F Aex cos( t) m A k K B B B 33 wl r l

17 The heaving point absorber, a linearized approach (:3) Instantaneous power: P P t F x Kx Bx x P Integrated over one wave period: P P F B 1 Aex 1 m A33 k wl K B Br Bx A Maximum power at resonance

18 The heaving point absorber, a linearized approach (3:3) Optimum damping in power off-take: dpp / db Obtained for B B r Maximum mean power: P p _ max FAex 8B r

19 Theoretical limits for power extraction 3D axisymmetric body (deep water) Invoke Haskind relation (relates wave excitation force to wave radiation damping.) 3 Fex3 3D symmetric, heaving buoy: Br 33 3 g H Falnes and Budal, 1978: In order for an oscillating system to be a good wave absorber it should be a good wave generator. I.e. max power: P 3 H g 3 F Aex g p _ max 3 3 8Br 4 18 H T 3 Capture width : L cap 3 g 3 HT max power extraction 3 18 g 1 g TH 3 power in wave per meter

20 Heaving point absorber Theoretical versus Budal limit. - Or the implication of limited physical size. 1 C = = 7.9 kws/m 4 4 g 3 g C = =.44 kw/(m s 3 ) 3 18 Capture width (P A ): L=λ/π (Heave only) L = 3λ/π (Heave & surge) Resonance Semisubmerged sphere From Falnes (7)

21 E4 /TU

22 The idea of latching (1:) A. Babarit, G. Duclos, A.H. Clément: Comparison of latching control strategies for a heaving wave energy device in random sea

23 The idea of latching (:) Tout = Twa Max. W/o control Tout = 3Twa Source: Clement and Babarit (1) DOF system. Power absorbed in harmonic and first sub-harmonic latching modes, compared with uncontrolled, and ideal modes. Solid line, without control; squares, latching, Tout=Twa; circles, latching, Tout=3 Tin; dashed dotted line, max power 3

24 Main principles From: Babarit, Introduction to Ocean Wave Energy Conversion,

25 Interaction between waves and body - D Energy flux in incident wave: 1 g J cge A 4 Net absorbed wave power per unit length: g P A A A 4 D R T A T : amplitude of transmitted wave A R : amplitude of reflected wave

26 Interaction between waves and body D Incident wave plus heaving and surging body. 3 Radiated waves from a heaving D source 3 Radiated waves from a heaving plus surging source Incident waves moving in positive x-direction wave elevation Wave due to heave -3-4 Heave + surge 1 wave elevation x 3 Radiated waves from a surging D source Incident plus radiated waves from a heaving and surging source x Incident wave wave elevation Wave due to surge wave elevation Incident + heave + surge x x Note importance of phasing

27 Example heaving buoy, motions INPUT Radius 1 m Draft 5 m Wave amplitude 1 m Mass kg Water line stiffness, kn/m Power offtake stiffness kn/m Natural period sec 8 Radiation damping at T N/(m/s) 7 Theoretical maximum energy absorbtion, at resonance (kw) Gam = B_ powerofftake / B_ radiation (T ) B_ powerofftake independent of frequency (not optimum) 6 5 Amplitude of motion gam =.1 gam =.5 gam = 1 gam = gam = 5 gam = 1 abs(x 3 )/A 4 3 x 3 1 x Wave period (sec)

28 Heaving buoy - Power production Average power (kw) Power production, wave amplitude 1 m gam =.1 gam =.5 gam = 1 gam = gam = 5 gam = 1 rage power (kw) Budal limit with A= 1m Theoretical maximum versus "Budal limit" Theoretical max "Budal limit" Wave period (sec) Wave period (sec)

29 Power production off resonance - passive system INPUT Radius.5 m Draft 5 m Wave amplitude 1 m Mass kg Water line stiffness, kn/m Power offtake stiffness kn/m Natural period sec Radiation damping at T N/(m/s) Theoretical maximum energy absorbtion, at resonance (kw) Gam = B_powerofftake / B_radiation (T) High damping in power off-take important to extract energy at periods above resonance Average power (kw) Power production, wave amplitude 1 m gam =.1 gam =.5 gam = 1 gam = gam = 5 gam = Wave period (sec)

30 Example: Gabriell Figure. Illustration of the buoy with a horizontal plate attached underneath

31 Toftestallen In the 198 ies..and in

32 Wave power converters - Examples on installations in full / reduced scale (:) Bostrøm et al. (Sweden, 6-) Heaving buoy. Linear generator Fred Olsen, Buldra. (Norway 4-) Array of heaving buoys. Semisubmersible

33 Oscillating water column (OWC) LIMPET LIMPET

34 Relative motion device attenuator Pelamis (75 kw device)

35 Overtopping Wave Dragon (prototype kw, 4-7MW demonstrator)

36 New ideas - floating hose The ANACONDA concept. Ill. EPSRC Water inside the tubes are propagating in longitudinal direction

37 Summary - Vaste amount of wave energy available. Technical availability uncertain. - No convergence on technical solutions - For an oscillating system to extract energy, it has to generate waves - Advanced control needed to enhance power offtake. - Survivability has shown up to be a critical issue