Chapter 12
Wave power
12.1 Introduction
Very large energy fluxes can occur in deep water sea waves. The power
in the wave is proportional to the square of the amplitude and to the
period of the motion. Therefore the long period ∼10 s, large amplitude
∼2 m waves have considerable interest for power generation, with energy
−1
fluxes commonly averaging between 50 and 70 kW m width of oncoming
wave.
The possibility of generating electrical power from these deep water
waves has been recognised for many years, and there are countless ideas
for machines to extract the power. For example, a wave power system
was used in California in 1909 for harbour lighting. Modern interest has
revived, particularly in Japan, the UK, Scandinavia and India, so research
and development has progressed to commercial construction for meaningful power extraction. Very small scale autonomous systems are used for
marine warning lights on buoys and much larger devices for grid power
generation. The provision of power for marine desalination is an obvious
attraction. As with all renewable energy supplies, the scale of operation
has to be determined, and present trends support moderate power generation at about 100 kW–1 MW from modular devices each capturing energy
from about 5 to 25 m of wavefront. Initial designs are for operation at
shore-line or near to shore to give access and to lessen, hopefully, storm
damage.
It is important to appreciate the many difficulties facing wave power
developments. These will be analysed in later sections, but may be summarised here:
1
2
Wave patterns are irregular in amplitude, phase and direction. It is
difficult to design devices to extract power efficiently over the wide
range of variables.
There is always some probability of extreme gales or hurricanes producing waves of freak intensity. The structure of the power devices
must be able to withstand this. Commonly the 50 year peak wave is
12.1 Introduction 401
3
4
5
6
7
10 times the height of the average wave. Thus the structures have to
withstand ∼100 times the power intensity to which they are normally
matched. Allowing for this is expensive and will probably reduce normal efficiency of power extraction.
Peak power is generally available in deep water waves from open-sea
swells produced from long fetches of prevailing wind, e.g. beyond the
Western Islands of Scotland (in one of the most tempestuous areas of
the North Atlantic) and in regions of the Pacific Ocean. The difficulties
of constructing power devices for these types of wave regimes, of maintaining and fixing or mooring them in position, and of transmitting
power to land, are fearsome. Therefore more protected and accessible
areas near to shore are most commonly used.
Wave periods are commonly ∼ 5–10 s (frequency ∼ 01 Hz). It is
extremely difficult to couple this irregular slow motion to electrical
generators requiring ∼500 times greater frequency.
So many types of device may be suggested for wave power extraction
that the task of selecting a particular method is made complicated and
somewhat arbitrary.
The large power requirement of industrial areas makes it tempting to
seek for equivalent wave energy supplies. Consequently plans may be
scaled up so only large schemes are contemplated in the most demanding wave regimes. Smaller sites of far less power potential, but more
reasonable economics and security, may be ignored.
The development and application of wave power has occurred with
spasmodic and changing government interest, largely without the benefit of market incentives. Wave power needs the same learning curve of
steadily enlarging application from small beginnings that has occurred
with wind power.
The distinctive advantages of wave power are the large energy fluxes
available and the predictability of wave conditions over periods of days.
Waves are created by wind, and effectively store the energy for transmission
over great distances. For instance, large waves appearing off Europe will
have been initiated in stormy weather in the mid-Atlantic or as far as the
Caribbean.
The following sections aim to give a general basis for understanding wave
energy devices. First we outline the theory of deep water waves and calculate
the energy fluxes available in single frequency waves. Then we review the
patterns of sea waves that actually occur. Finally we describe attempts being
made to construct devices that efficiently match variable natural conditions.
With the complex theory of water waves we have sacrificed mathematical
rigor for, we hope, physical clarity, since satisfactory theoretical treatments
exist elsewhere.
402 Wave power
12.2 Wave motion
Most wave energy devices are designed to extract energy from deep water
waves. This is the most common form of wave, found when the mean depth of
the sea bed D is more than about half the wavelength . For example, an average sea wave for power generation may be expected to have a wavelength of
∼100 m and amplitude of ∼3 m, and to behave as a deep water wave at depths
of sea bed greater than ∼30 m. Figure 12.1(a) illustrates the motion of water
particles in a deep water wave. The circular particle motion has an amplitude
that decreases exponentially with depth and becomes negligible for D > /2.
In shallower water, Figure 12.1 (b), the motion becomes elliptical and water
movement occurs against the sea bottom, producing energy dissipation.
The properties of deep water waves are distinctive, and may be summarised as follows:
1
2
3
4
5
6
The surface waves are sets of unbroken sine waves of irregular wavelength, phase and direction.
The motion of any particle of water is circular. Whereas the surface
form of the wave shows a definite progression, the water particles
themselves have no net progression.
Water on the surface remains on the surface.
The amplitudes of the water particle motions decrease exponentially
with depth. At a depth of /2 below the mean surface position, the
amplitude is reduced to 1/e of the surface amplitude (e = 272, base
of natural logarithms). At depths of /2 the motion is negligible, being
less than 5% of the surface motion.
The amplitude a of the surface wave is essentially independent of the
wavelength , velocity c or period T of the wave, and depends on
the history of the wind regimes above the surface. It is rare for the
amplitude to exceed one-tenth of the wavelength, however.
A wave will break into white water when the slope of the surface is
about 1 in 7, and hence dissipate energy potential.
Figure 12.1 Particle motion in water waves. (a) Deep water, circular motion of water
particles. (b) Shallow water, elliptical motion of water particles.
12.2 Wave motion 403
The formal analysis of water waves is difficult, but known; see Coulson
and Jeffrey (1977) for standard theory. For deep water waves, frictional,
surface tension and inertial forces are small compared with the two dominant forces of gravity and circular motion. As a result, the water surface
always takes up a shape so that its tangent lies perpendicular to the resultant
of these two forces, Figure 12.2.
It is of the greatest importance to realize that there is no net motion
of water in deep water waves. Objects suspended in the water show the
motions of Figure 12.1, which contrasts deep water waves with the kinds
of motion occurring in shallower water.
A particle of water in the surface has a circular motion of radius a equal
to the amplitude of the wave (Figure 12.3). The wave height H from the
top of a crest to the bottom of a trough is twice the amplitude: H = 2a. The
angular velocity of the water particles is (radian per second). The wave
surface has a shape that progresses as a moving wave, although the water
itself does not progress. Along the direction of the wave motion the moving
shape results from the phase differences in the motion of successive particles
of water. As one particle in the crest drops to a lower position, another
particle in a forward position circles up to continue the crest shape and the
forward motion of the wave.
The resultant forces F on water surface particles of mass m are indicated
in Figure 12.4. The water surface takes up the position produced by this
resultant, so that the tangent to the surface is perpendicular to F . A particle
at the top of a crest, position P1, is thrown upwards by the centrifugal
force ma2 . A moment later the particle is dropping, and the position in
Figure 12.2 Water surface perpendicular to resultant of gravitational and centrifugal
force acting on an element of water, mass m.
Figure 12.3 Wave characteristics.
404 Wave power
Figure 12.4 Resultant forces on surface particles.
the crest is taken by a neighbouring particle rotating with a delayed phase.
At P2 a particle is at the average water level, and the surface orientates
perpendicular to the resultant force F . At the trough, P3, the downward
force is maximum. At P4 the particle has almost completed a full cycle of
its motion.
The accelerations of a surface particle are drawn in Figure 12.5(b).
Initially t = 0, the particle is at the average water level, and subsequently:
=
− t
2
(12.1)
and
tan s =
a2 sin a2 sin ≈
2
g + a cos g
(12.2)
since in practice g a2 for non-breaking waves (e.g. a = 2 m,
T period = 8 s, a2 = 12 m s−2 and g = 98 m s−2 ). Let h be the height of
the surface above the mean level. The slope of the tangent to the surface is
given by
dh
= tan s
dx
(12.3)
Figure 12.5 Accelerations and velocities of a surface water particle. (a) Water surface. (b) Particle acceleration, general derivation. (c) Particle velocity.
12.2 Wave motion 405
From (12.1), (12.2) and (12.3),
a2
dh a2
a2
=
sin =
cos
− =
cos t
dx
g
g
2
g
(12.4)
From Figure 12.5(c), the vertical particle velocity is
dh
= a sin = a cos t
dt
(12.5)
The solution of (12.4) and (12.5) is
2 x
h = a sin
− t
g
(12.6)
Comparing this with the general travelling wave equation of wavelength and velocity c, we obtain
2
x − ct
2
= a sin
x − t = a sinkx − t
h = a sin
(12.7)
where k = 2/ is called the wave number.
It is apparent that the surface motion is that of a travelling wave, where
=
2g
2
(12.8)
This equation is important; it gives the relationship between the frequency
and the wavelength of deep water surface waves.
The period of the motion is T = 2/ = 2/2g/1/2 . So
T=
2
g
12
(12.9)
The velocity of a particle at the crest of the wave is
2g
v = a = a
12
(12.10)
The wave surface velocity in the x direction, from (12.7), is
12
g
c=
= =g
2
2g
(12.11)
406 Wave power
i.e.
c=
g
2
1
2
=
gT
2
The velocity c is called the phase velocity of the travelling wave made by
the surface motion. Note that the phase velocity c does not depend on the
amplitude a, and is not obviously related to the particle velocity v.
Example 12.1
What is the period and phase velocity of a deep water wave of 100 m
wavelength?
Solution
From (12.8),
2 =
2g 210 m s−2 =
100 m
= 08 s−1
and so T = 2/ = 80 s.
From (12.11)
10 m s−2 100 m
c=
2
12
= 13 m s−1
So
= 100 m
T = 8 s
c = 13 m s−1
(12.12)
12.3 Wave energy and power
12.3.1 Basics
The elementary theory of deep water waves begins by considering a single
regular wave. The particles of water near the surface will move in circular
orbits, at varying phase, in the direction of propagation x. In a vertical
column the amplitude equals half the crest to trough height at the surface,
and decreases exponentially with depth.
The particle motion remains circular if the sea bed depth D > 05, when
the amplitude becomes negligible at the sea bottom. For these conditions
(Figure 12.6(a)) it is shown in standard texts that a water particle whose
mean position below the surface is z moves in a circle of radius given by
r = a ekz
(12.13)
12.3 Wave energy and power 407
Figure 12.6 Elemental motion of water, drawn to show the exponential decrease of
amplitude with depth.
Here k is the wave number, 2/, and z is the mean depth below the surface
(a negative quantity).
We consider elemental ‘strips’ of water across unit width of wavefront,
of height dz and ‘length’ dx at position x z (Figure 12.6(b)). The volume
per unit width of wavefront of this strip of density is
dV = dx dz
(12.14)
and the mass is
dm = dV = dx dz
(12.15)
Let EK be the kinetic energy of the total wave motion to the sea bottom,
per unit length along the x direction, per unit width of the wave-front.
The total kinetic energy of a length dx of wave is EK dx. Each element
of water of height dz, length dx and unit width is in circular motion at
constant angular velocity , radius of circular orbit r, and velocity v = r
(Figure 12.6(b)). The contribution of this element to the kinetic energy in a
vertical column from the sea bed to the surface is EK dx, where
EK dx =
1
1
mv2 = dz dxr 2 2
2
2
(12.16)
Hence
EK =
1 2 2
r dz
2
(12.17)
It is easiest to consider a moment in time when the element is at its mean
position, and all other elements in the column are moving vertically at the
same phase in the z direction (Figure 12.6(c)).
From (12.13) the radius of the circular orbits is given by
r = aekz
(12.18)
408 Wave power
where z is negative below the surface.
Hence from (12.17),
EK =
1
a2 e2kz 2 dz
2
(12.19)
and the total kinetic energy in the column is
EK dx =
z=0
z=−
2 a2 2kz
1 2 a2
e dz dx = dx
2
4
k
(12.20)
Since k = 2/, and from (12.8) 2 = 2g/, the kinetic energy per unit
width of wave-front per unit length of wave is
EK =
1
1 2 2g a
= a2 g
4
2
4
(12.21)
In Problem 12.1 it is shown that the potential energy per unit width of
wave per unit length is
EP =
1 2
a g
4
(12.22)
Thus, as would be expected for harmonic motions, the average kinetic
and potential contributions are equal.
The total energy per unit width per unit length of wavefront, i.e. total
energy per unit area of surface, is
total = kinetic + potential
E = E K + EP =
1 2
a g
2
Note that the root mean square amplitude is
E = groot mean square amplitude2
(12.23)
√ 2
a /2, so
(12.24)
The energy per unit wavelength in the direction of the wave, per unit
width of wavefront, is
E = E =
1 2
a g
2
(12.25)
From (12.28) = 2g/2 , so
E = a2 g 2 /2
(12.26)
12.3 Wave energy and power 409
Or, since T = 2/
E =
1
a2 g 2 T 2
4
(12.27)
It is useful to show the kinetic, potential and total energies in these various
forms, since all are variously used in the literature.
12.3.2 Power extraction from waves
So far, we have calculated the total excess energy (kinetic plus potential) in
a dynamic sea due to continuous wave motion in deep water. The energy is
associated with water that remains at the same location when averaged over
time. However, these calculations have told us nothing about the transport
of energy (the power) across vertical sections of the water.
Standard texts, e.g. Coulson and Jeffrey (1977), calculate this power
from first principles by considering the pressures in the water and the
resulting displacements. The applied mathematics required is rigorous and
comprehensive, and of fundamental importance in fluid wave theory. We
can extract the essence of the full analysis, which is simplified for deep
water waves.
Consider an element or particle of water below the mean surface level
(Figure 12.7). For a surface wave of amplitude a and wave number k, the
radius of particle motion below the surface is
r = a ekz
(12.28)
The vertical displacement by (Figure 12.7(b)) from the average position
is
z = r sin t = a ekz sin t
(12.29)
The horizontal component of velocity ux is given by
ux = r sin t = a ekz sin t
(a)
(b)
z
p2
p1
ux
dx
dz
∆z
(12.30)
rω
r
Displaced position
of rotating particle
ωt
Average position
Figure 12.7 Local pressure fluctuations in the wave. (a) Pressures in the wave.
(b) Local displacement of water particle.
410 Wave power
Therefore, from Figure 12.7(a), the power carried in the wave at x, per
unit width of wave-front at any instant, is given by
P =
z=0
z=−
p1 − p2 ux dz
(12.31)
Where p1 and p2 are the local pressures experienced across the element of
height dz and unit width across the wavefront. Thus p1 − p2 is the pressure
difference experienced by the element of width y=1 m in a horizontal
direction. The only contribution to the energy flow that does not average to
zero at a particular average depth in the water is associated with the change
in potential energy of particles rotating in the circular paths; see Coulson
and Jeffrey (1977). Therefore by conservation of energy
p1 − p2 = gz
(12.32)
Substituting for z from (12.29),
p1 − p2 = ga ekz sin t
(12.33)
In (12.31), and with (12.30) and (12.33),
P =
z=0
ga ekz sin ta ekz sin tdz
z=−
= ga 2
(12.34)
z=0
e
2kz
2
sin t dz
z=−
The time average over many periods of sin2 t equals 1/2, so
P =
ga2 z=0 2kz
ga2 1
e dz =
2
2 2k
z=−
(12.35)
The phase velocity of the wave is, from (12.7)
c=
=
k
T
(12.36)
So the power carried forward in the wave per unit width across the wavefront becomes
P =
ga2 ga2 c
=
2 2
4T
(12.37)
From (12.23) and (12.37) the power P equals the total energy (kinetic plus
potential) E in the wave per unit area of surface, times c/2. c/2 is called the
group velocity of the deep water wave, i.e. the velocity at which the energy
12.3 Wave energy and power 411
in the group of waves is carried forward. Thus, with the group velocity
u = c/2,
P = Eu = Ec/2
(12.38)
where E = ga2 /2.
From (12.8),
k = 2 /g
(12.39)
therefore, the phase velocity is
c=
g
g
= =
k
2/T (12.40)
This difference between the group velocity and the wave (phase) velocity is
common to all waves where the velocity depends on the wavelength. Such
waves are called dispersive waves and are well described in the literature,
both descriptively, e.g. Barber (1969), and analytically, e.g. Lighthill (1978).
Substituting for c from (12.11) into (12.37) gives
ga2 1 gT
P =
2 2 2
So
P =
g 2 a2 T
8
(12.41)
Therefore, the power in the wave increases directly as the square of the wave
amplitude and directly as the period. The attraction of long period, large
amplitude ocean swells to wave power engineers is apparent. This relationship is perhaps not obvious, and may be written in terms of wavelength
using (12.9),
P =
1
g 2 a2 2 2
8
g
(12.42)
Example 12.2
What is the power in a deep water wave of wavelength 100 m and
amplitude 1.5 m?
Solution
From (12.12) for Example 12.1, c = 125 m s−1 . With (12.38),
u = c/2 = 65 m s−1
412 Wave power
where u is the group velocity of the energy and c is the phase velocity.
The sea water waves have an amplitude a = 15 m H = 3 m; realistic
for Atlantic waves, so in (12.37)
P =
1
−3
−1
1025 kg m 98 m s−2 15 m2 65 m s−1 = 73 kW m
2
Alternatively, P is obtained directly from 12.41b.
From Example 12.2, we can appreciate that there can be extremely large
power densities available in the deep water waves of realistic ocean swells.
12.4 Wave patterns
Wave systems are not, in practice, the single sine wave patterns idealised in
the previous sections. Very occasionally natural or contrived wave diffraction patterns, or channelled-waves, approach this condition, but normally
a sea will be an irregular pattern of waves of varying period, direction and
amplitude. Under the stimulus of a prevailing wind the wave trains may
show a preferred direction, e.g. the south west to north east direction of
Atlantic waves off the British Isles, and produce a significant long period
sea ‘swell’. Winds that are more erratic produce irregular water motion
typical of shorter periods, called a ‘sea’. At sea bottom depths ∼30 m or
less, significant focusing and directional effects can occur, however, possibly producing more regular or enhanced power waves at local sites. Wave
power devices must therefore match a broad band of natural conditions, and
be designed to extract the maximum power averaged over a considerable
time for each particular deployment position. In designing these devices, it
will be first necessary to understand the wave patterns of the particular site
that may arise over a 50-year period.
The height of waves at one position was traditionally monitored on a
wave-height analogue recorder. Separate measurements and analysis are
needed to obtain the direction of the waves. Figure 12.8 gives a simulated
trace of such a recorder. A crest occurs whenever the vertical motion changes
Figure 12.8 Simulated wave height record at one position (with an exaggerated set
of crests to explain terminology).
12.4 Wave patterns 413
from upwards to downwards, and vice versa for a trough. Modern recorders
use digital methods for computer-based analysis of large quantities of data.
If H is the height difference between a crest and its succeeding trough,
there are various methods of deriving representative values, as defined in
the following.
The basic variables measured over long intervals of time are:
1
2
3
Nc , the number of crests; in Figure 12.8 there are 10 crests.
H1/3 , the ‘one-third’ significant wave height. This is the average height
of the highest one-third of waves as measured between a crest and
subsequent trough. Thus H1/3 is the average of the Nc /3 highest values
of H.
Hs , the ‘true’ significant wave height. Hs is defined as
Hs = 4arms = 4
n
h2 /n
12
(12.43)
i=1
4
5
6
7
where arms is the root mean square displacement of the water surface
from the mean position, as calculated from n measurements at equal
time intervals. Care has to be taken to avoid sampling errors, by recording at a frequency at least twice that of the highest wave frequency
present.
Hmax is the measured or most probable maximum height of a wave.
Over 50 years Hmax may equal 50 times Hs and so this necessitates
considerable overdesign for structures in the sea.
Tz , the mean zero crossing period is the duration of the record divided by
the number of upward crossings of the mean water level. In Figure 12.8,
Tz = /3.
Tc , the mean crest period, is the duration of the record divided by the
number N of crests. In Figure 12.8, Tc = /10; in practice N is very
large, so reducing the error in Tc .
The spectral width parameter gives a measure of the variation in wave
pattern:
2 = 1 − Tc /Tz 2
(12.44)
For a uniform single frequency motion, Tc = Tz , so = 0. In our example = 1 − 032 1/2 = 09, implying a mix of many frequencies. The
full information is displayed by Fourier transformation to a frequency
spectrum, e.g. Figure 12.9.
414 Wave power
Figure 12.9 Distribution of power per frequency interval in a typical Atlantic deep water
wave pattern (Shaw 1982). The smoothed spectrum is used to find Te , the
energy period.
From (12.41) the power per unit width of wave-front in a pure sinusoidal
deep water wave is
P =
g 2 H 2 T
g 2 a2 T
=
8
32
(12.45)
where the trough to crest height is H = 2a. The root mean square
√
(rms) wave displacement for a pure sinusoidal wave is amax = a/ 2, so
in (12.45)
P =
g 2 a2rms T
4
(12.46)
Figure 12.10 Scatter diagram of significant wave height Hs against zero crossing period
Tz . The numbers on the graph denote the average number of occurrences
of each Hs Tz in each 1000 measurements made over one year. The most
frequent occurrences are at Hs ∼ 3 m Tz ∼ 9 s, but note that maximum
likely power occurs at longer periods.
- - - - - these waves have equal maximum gradient or slope
1/n the maximum gradient of such waves, e.g. 1 in 20
_____ lines of constant wave power, kW m−1
Data for 58 N19 W in the mid-Atlantic. After Glendenning (1977).
Figure 12.11 Average annual wave energy MWh m−1 in certain sea areas of the world. After NEL (1976).
12.4 Wave patterns 417
In practice, sea waves are certainly not continuous single frequency sine
waves. The power per unit width of wavefront is therefore written in the
form of (12.46) as
P =
g 2 Hs2 Te
64
(12.47)
Here Hs is the significant wave height defined by (12.43), and Te , called the
‘energy period’, is the period of the dominant power oscillations given by
the peak in the power spectrum, see Figure 12.9. For many seas
Te ≈ 112Tz
(12.48)
Figure 12.12 Contours of average wave energy off north-west Europe. Numbers indicate
annual energy in the unit of MWh, and power intensity (bracketed) in the
unit of kW m−1 . Note that local effects are not indicated.
418 Wave power
Until modern developments in wave power, only an approximate value
of P could be obtained from analogue recording wave meters such that
P ≈
2
Tz
g 2 H1/3
64
2
≈ 490 W m−1 m−2 s−1 H1/3
Tz
(12.49)
However, with modern equipment and computer analysis, more sophisticated methods can be used to calculate (1) arms and hence Hs and (2), Tz
or Te . Thus
P = 490 W m−1 m−2 s−1 Hs2 Te
= 550 W m−3 s−1 Hs2 Tz
(12.50)
Since a wave pattern is not usually composed of waves all progressing
in the same direction, the power received by a directional device will be
significantly reduced.
Wave pattern data are recorded and tabulated in detail from standard
meteorological sea stations. Perhaps the most important graph for any site
is the wave scatter diagram over a year, e.g. Figure 12.10. This records
the number of occurrences of wave measurements of particular ranges of
significant wave height and zero-crossing period. Assuming the period is
related to the wavelength by (12.9) it is possible to also plot on the diagram
lines of constant wave height to wavelength. Contours of equal number of
occurrences per year are also drawn.
From the wave data, it is possible to calculate the maximum, mean,
minimum etc. of the power in the waves, which then can be plotted on
maps for long-term annual or monthly averages. See Figures 12.11 and
12.12 for annual average power intensities across the world and north-west
European.
12.5 Devices
As a wave passes a stationary position the surface changes height, water
near the surface moves as it changes kinetic and potential energy, and the
pressure under the surface changes. A great variety of devices have been
suggested for extracting energy using one or more of these variations as
input to the device. Included are devices that catch water at the crest of
the waves, and allow it to run back into the mean level or troughs after
extracting potential energy.
The Engineering Committee on Oceanic Resources (2003) described over
40 devices that have reached a stage of ‘advanced development’. Of these
over one-third are, or have been, ‘operational’, but only as one-off pilot
projects. We describe here a representative sample, classified by their general
principles. Wave-power has yet to reach the stage of widespread deployment
of commercial devices that wind-power has attained.
12.5 Devices 419
12.5.1 Wave capture systems
These schemes are probably the simplest conceptually. They develop from
a phenomenon often observed in natural lagoons. Waves break over a sea
wall (equivalent to a natural reef) and water is impounded at a height
above the mean sea level. This water may then return to the sea through a
conventional low head hydroelectric generator. The system thus resembles
a tidal range power system, Figure 13.7, but with a more continual and less
regular inflow of water.
Figure 12.13 is a schematic diagram of the 350 kW Tapchan system
demonstrated successfully in Norway in 1985. In this particular design, the
waves were funnelled in through a tapered channel, whose concrete walls
reached 2–3 m above mean sea level. This allows bigger waves to overtop
the wall early, while smaller waves increase in height as they go up the
channel, so that most of them also overtop the channel walls and supply
water to the reservoir. Most of the engineering work was built into a natural
gully in the rockface – a feature which enabled the system to withstand
several storms over its 5 years of operation, one of which destroyed another
less robust wave power device nearby.
A site for such a system needs to have the following features:
•
•
Persistent waves with large average wave energy
Deep water close to shore so the oncoming waves are not dissipated
cliff face
reservoir
P
turbine house
Q
Figure 12.13 Schematic diagram of the Tapchan wave capture system built in Norway
(see text). Waves flow over the top of the tapered channel into the
reservoir. Water flows from the pipe P near the top of the reservoir,
though the conventional small-head hydro turbines, see Section 8.5,
and then out to the sea at Q.
420 Wave power
•
•
A small tidal range <1 m
A convenient and cheap means of constructing the reservoir, e.g. suitable local natural features.
12.5.2 Oscillating water column (OWC)
When a wave passes on to a partially submerged cavity open under the
water, Figure 12.14, a column of water oscillates up and down in the cavity.
This can induce an oscillatory motion in the air above the column, which
may be connected to the atmosphere through an air turbine. Electrical power
is usually derived from the oscillating airstream using a Wells turbine; such
turbines, once started, turn in the same direction to extract power from air
flowing in either axial direction, i.e. the turbine motion is independent of
the fluid direction – see Problem 12.4 and Figure 12.18.
The first device of this kind has been developed by Professor Trevor
Whittaker and his team of Queens University Belfast and operated without damage on the Scottish island of Islay for several years, but at less
than expected power output. Based on that experience, a larger 500 kW
device was installed on Islay in 2000, using robust construction techniques
adaptable for other sites; named the ‘Limpet’, after shellfish renowned
for their firm attachment to rocks, it is contributing much of the island’s
power. The Limpet is one of the three initial wave power devices in
Scotland for commercial operation; electricity is exported to the utility grid
within the Scottish Administration’s renewables obligation programme, see
Chapter 17.
sound
baffle
wells
turbine
air
wave motion
water
rock
Figure 12.14 Schematic diagram of an on-shore wave power system using an oscillating water column. Based on the LIMPET device operational on the island
of Islay, west of Scotland, for grid connected electricity generation.
12.5 Devices 421
An advantage of using an oscillating water column for power extraction
is that the air speed is increased by smooth reduction in the cross-sectional
area of the channel approaching the turbine. This couples the slow motion
of the waves to the fast rotation of the turbine without mechanical gearing.
Another advantage is that the electrical generator is displaced significantly
from the column of saline water. The air cavity’s shape and size determine
its frequency response, with each form and size of cavity responding best
to waves of a particular frequency. In principle, the system efficiency is
considerably improved if such devices have active tuning to the wide range
of sea-wave frequencies encountered by using (a) multiple cavities, or (b)
detectors of the incoming waves that feed-forward information for the
cavity shape to be changed and the pitch of the turbine to be adjusted.
A majority of second generation wave-power devices have been shoreline OWC devices, broadly similar to the Limpet. However, the OWC
mechanism is also used in offshore devices sitting on the sea bed, such
as the Osprey (which operates in the near-shore zone off the islands of
Orkney, northern Scotland) or in floating devices such as the Japanese
Whale, Figure 12.15, and the Masuda wave-powered navigation buoys.
12.5.3 Wave profile devices
This class of devices float on or near the sea surface and move in response
to the shape of the wave, rather than just the vertical displacement of water.
Ingenious design is needed to extract useable power from the motion.
The Pelamis, Figure 12.16, is a semi-submerged articulated structure,
which sits as a ‘snake’ aligned approximately head-on to the oncoming
waves. The device consists of cylindrical sections linked by hinged joints,
and wiggles like a snake in both the vertical and horizontal directions as
turbine
air
chamber
buoyancy
chamber
Stabiliser
Figure 12.15 The Whale device – a prototype floating wave-power system deployed off
Gokasho Bay, Japan in 1998. The device is 50 m long and has 120 kWe
generating capacity [after ‘Japan’s Marine Science and Technology Center
( JAMSTEC)’].
422 Wave power
wave
direction
Figure 12.16 Sketch of the Pelamis wave-power device, as seen from the side (not to
scale in the vertical direction). Motion at the hinges produces hydraulic
power fed to electrical generators. Note that the motion is not purely
vertical; the device also ‘wriggles’ from side to side. The device is
loosely moored to the sea floor by the mooring ropes as indicated. See
<www.oceanpd.com/Pelamis/>.
a wave goes past. The wave-induced motion of its joints is resisted by
hydraulic rams, which pump high-pressure oil through hydraulic motors via
smoothing accumulators. The hydraulic motors drive electrical generators
at each joint to produce electricity. Electrical power from all the joints in
parallel is fed down a single umbilical cable to an electricity grid network
connection on the sea bed. Several devices can be so connected in an array
or ‘farm’, which itself is then linked to shore through a seabed cable.
Each Pelamis is held in position by a mooring system which maintains
enough restraint to keep the device positioned but allows the machine to
swing head-on to oncoming prevailing waves as the ‘snake’ spans successive
wave crests. A 750 kWe prototype, 120 m long and 3.5 m in diameter, was
installed in 2004 offshore of the main island of Orkney, northern Scotland.
The length is such that it automatically ‘detunes’ from the longer-wavelength
high-power waves, in order to enhance its survivability in storms.
12.6 Social and environmental aspects
Only about 10 to 15 devices were operational in sea conditions by 2005,
so there is some uncertainty about their social economic and environmental
aspects. Nevertheless, some generalisations can already be made on the
basis of studies and pilot projects.
The grandiose schemes that were promoted by governments, especially in
the UK, in response to the ‘energy crisis’ of the 1970s have faded from view
as such, but did promote much useful research and marine data collection.
The emphasis now is to obtain small to medium scale working systems
that have proven operational experience and that, from the onset, export
power into utility grid networks for commercial income. Development is
12.6 Social and environmental aspects 423
from laboratory and simulation models to prototype installation, typically
of 100–1000 kWe capacity and thence to ‘farms’ of multiple devices. Of
importance is future interconnection and shared management with offshore
wind farms.
From the experience of the initial plants, the projected cost of wave-power
generated electricity power encourages optimism. For example the Limpet
and Pelamis installations both accepted contracts to supply electricity for
15 years at less than 7p/kWh (≈US$0.15/kWh). It is reasonable to project
that with greater deployment, which spreads development costs over multiple units, and with incremental engineering improvements from the pilot
plants, these costs may halve within tens of years (an example of the ‘learning curves’ discussed in Section 17.6). The consequent predicted costs of
∼3p/kWh would compare favourably with most alternatives for the isolated
coastal and island communities, which offer the best initial opportunities
for wave-power. Thereafter, electrical power connection will be to national
power networks, possibly sharing sub-sea grids with offshore wind power.
Reliability and low operational costs are the most critical factors in
achieving low average costs per kWh for systems which are capital intensive
(see Chapter 17). This is particularly true for wave-power systems, which
necessarily operate in vigorous sea conditions. If a system is destroyed by
a storm in its first few years of operation, it will not pay its way, and
power suppliers will not want to invest in further similar devices. Schemes
therefore need to be designed for long lifetimes and with small number of
moving parts to minimise failures. Fortunately, engineers can now draw
on the experience of the offshore oil and wind industry to ‘ruggedise’ their
designs and allow more confident installation and operation.
Onshore and near-shore wave-power devices, in general, are easier and
cheaper to construct, maintain and connect to power networks than fully
offshore devices. Yet offshore devices can tap into waves of greater power,
with consequent larger and more sustained output, yet perhaps greater
danger of malfunction.
Good efficiency requires the device to be matched to a wide frequency
spectrum of waves, yet engineering for survivability may reduce this efficiency. Typically wave-power devices have average efficiencies ∼30% and
capacity factors also ∼30%. (The capacity factor is the ratio of energy output over a period to that which would have been output if the device had
operated throughout at its full rated power.)
As with other energy systems, the cost of energy from a wave-power
system can be reduced if the cost of construction is shared with other
benefits. For example, some systems can be integrated with conventional
breakwaters. Others (particularly a string of floating devices like the Whale,
aligned to face the waves) can in principle absorb energy across a wavefront
to the extent that the water behind them is relatively calm; such wave-power
systems act as breakwaters.
424 Wave power
The main environmental benefit of wave-power systems is, as with all
renewable energy, the mitigation of greenhouse gas emissions by substituting for fossil fuel use. Negative factors for offshore installation include
hindrance to shipping and fishing. For near shore devices, acoustic noise
may cause annoyance – the noise from some OWC devices has been likened
to a rhinoceros giving birth.
Problems
12.1 By considering elements of water lifted from depth z below the mean
sea level to a height z above this level in a crest, show that the potential
energy per unit length per unit width of wave front in the direction of
the wave is
EP =
1 2
a g
4
12.2 Figure 12.17(a) shows a conceptually simple device for extracting
power from the horizontal movement of water in waves. A flat vane
hinged about a horizontal axis at A (about /8 below the mean surface level) oscillates as indicated as waves impinge on it. Experiment
indicates that such a device can extract about 40% of the energy in
the incoming waves; about 25% of the energy is transmitted onwards
(i.e. to water downstream of the vane) about 20% is reflected.
Salter (1974) designed the ‘vane’ shown in Figure 12.17(b) with
a view to minimising these losses. It rotates about the central axis
at O. Its stern is a half-cylinder (radius a) centred at O (lower dotted line continues the circular locus), but from the bottom point the
shape changes into a surface which is another cylinder centred at
O , above O. This shape continues until it reaches an angle to the
vertical, at which point it develops into a straight tangent which is
continued to above the surface. For the case shown OO = 05a and
= 15 .
a
b
By considering the movement of water particles that would occur
in the wave in the absence of the device and relating this to the
shape of the device, show that for wavelengths from ∼4a to ∼12a
the device can absorb ∼70% of the incoming energy.
By 2004, this device, known as a Duck because of its bobbing
motion, had undergone extensive laboratory and theoretical development. Figure 12.17(c) indicates how a full scale (a ∼ 8 m) system
might look in cross-section. The outer body moves (oscillates) relative to the inner cylinder. Suggest and justify (i) a way in which
the inner cylinder could be made into a sufficiently stable reference
point (ii) a way in which the irregular oscillatory motion could be
harnessed into useable energy for distribution to the shore.