The Power of Many?.....
Coupled Wave Energy Point Absorbers
Paul YoungMSc candidate, University of Otago
Supervised by Craig Stevens (NIWA), Pat Langhorne & Vernon Squire (Otago)
Motivation
The big idea
The physics
Results
Where to next?
Talk outline
WECs… WTF?
World resource
Wave energy flux magnitude (kW per metre of wavefront)
Source: Pelamis Wave Power website
Source: Smith et al (NIWA), Analysis for Marine Renewable Energy: Wave Energy, 2008
Source: Smith et al (NIWA), Analysis for Marine Renewable Energy: Wave Energy, 2008
1. Estimate by UK Carbon Trust
Advantages:
High energy density Low social & environmental impact (?) Reliability & predictability (c.f. wind) Low EROEI (?) Direct desalination
AND...
• Practical worldwide resource ~ 2000-4000 TWh/year1
• (Current global demand~ 17000 TWh/year)
Motivation
The big idea
The physics
Results
Where to next?
Talk outline
Point absorbers
Pros:
Suitable for community scale
Less disruption in event of device failure
Cheaper per kW/h?
Cons:
Non-resonant in typical sea conditions
Lower efficiency
Maybe a linked chain of point absorbers will 'see' long
wavelengths better than a lone device?
Key questions
Is it possible to obtain better power output (per unit) with a linked chain?(Can we improve peak efficiency and/or widen bandwidth?)
How are the mooring forces affected?(Survivability)
What is the interplay between the device spacing and the wavelength?
My scheme: model device
1-D (surge only) idealisation
Motivation
The big idea
The physics
Results
Where to next?
Talk outline
Further assumptions/simplifications
Small-body approximation
Linear, small amplitude waves
Neglect hydrodynamic interaction between devices
Forces
Mooring forcesHydrodynamic forces: excitation, drag and radiation
Master equation:
(not including power take-off)
KKKKK RDEM FFFFma
Technical issues…
Importance of memory effects
Motivation
The big idea
The physics
Results
Where to next?
Talk outline
Validating numerical codeFor lone device with zero drag, easy to solve equation of motion analytically.
Discrepancy between models with and without memory effects noticeable when nonlinear drag introduced, but small.
HOT OFF THE PRESS:Things get interesting with
multiple devices.
Some good agreement...
...some poor agreement...
Motivation
The big idea
The physics
Results
Where to next?
Talk outline
Mooring and linkage forces
F M J , K=−S x J−xK , ∣x J−xK∣d0, ∣x J−xK∣d{
Chacterise as tension-only spring
Spring stiffness
(Linkage force on device J from device K)
Device spacing
Position of device K
Hydrodynamic forces(The tricky part...)
Inline force on small(ish) bodies in oscillatory flow often described by Morison equation:
BUT added mass depends on the oscillation frequency...
F=V s uma u− x −12C d A∣x−u∣ x−u
Dragcoefficient
Area 'seen'by fluidFluid density
Fluid velocity
Added mass
Submergedvolume
But under nonlinear conditions, device response may be over much broader range of frequencies...
Data from Hulme, A.: The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. 1982.
How big is the effect?
Semi-submerged sphere moving in surge
For device with a ≈ 2m, energy-bearing wavelengths in typical sea state are 0.056ka0.126
Falnes' formulation
1. Falnes, J.: Ocean Waves and Oscillating Systems: linear interactions including wave-energy extraction. 2002.
Wave forces are decomposed in frequency domain into excitation and radiation forces.
For surge, under small-body approximation, these are1:
F E≈[V sma i] u
F R≈−ma x( + damping term)
F=V s uma u− x −12C d A∣x−u∣ x−u (c.f.
)
F R=−ma ∞ x−∫0
t
K x t−d
Added mass at infinite frequency
Impulse response function
K t =2∫0
∞
cos t d Added damping
This expression is exact, but added mass and damping depend on body geometry.
Radiation force in time domain
Thankfully...
...can fit an analytic function that isn't horrible
Data from Hulme, A.: The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. 1982.
K t =2∫0
∞
cos t dEvaluate integralswith MATLAB symbolic math toolbox to get:
Master equation
m x J t =F M JF DJF EJF RJ
F M J , K=−S x J− xK , ∣x J− xK∣d0, ∣x J−xK∣d{
F M J=FM J , J−1F M J , J1
F EJ=V sma i u x J , t
F D=−12C d A∣x−u∣ x−u
u=u x J , t n.b.
F R=−ma ∞ x−∫0
t
K x t−d
Solution method
Solve numerically with 4th order Runge-Kutta procedure on MATLAB
Cast as 1st order vector equation for y=[x1v1x2v2⋮xnvn
](n.b. will be 4n entries with internal mass included)
Memory integral giving good agreement for linear motion over wavelength range
