Talk outline

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