Waves, WECs, and Arrays
Cameron McNatt
Coastal and Ocean Engineering
MS candidate
Outline
1) Background on waves, constructive and
destructive interference
2) WECs – linear theory, motions and optimal
motions, power, WEC wave field
3) Arrays interactions, Some interesting papers
Waves!
Velocity Potential
• Inviscid, Irrotational Flow
• 𝜙 – scalar, function of time and space
• Velocity is the gradient of the velocity potential
𝜕𝜙
𝜕𝑥
𝑢
𝜕𝜙
𝛻𝜙 =
= 𝑣 =𝑢
𝜕𝑦
𝑤
𝜕𝜙
𝜕𝑧
• 𝜙 also used to compute surface elevation (𝜂) and
pressure (𝑝)
Wave Energy and Wave Energy Flux
1
2
𝐸 = 𝜌𝑔𝑎
2
𝐸𝑓 =
1 𝑇 0
𝐸
𝑇 0 −ℎ
∙ 𝑢𝑑𝑧 𝑑𝑡 = 𝐸𝑐𝑔
(Power)
Linear Wave Theory
𝜙=
𝜙𝑖
Two Waves at Different Frequencies
𝐸𝑓
2
2
1
1
= 𝜌𝑔𝑎1 𝑐𝑔1 + 𝜌𝑔𝑎2 𝑐𝑔2
2
2
Two Waves at Same Frequency and Direction
1
𝐸𝑓 = 𝜌𝑔 𝑎1
2
2
+ 𝑎2
2
+ 2𝑎1 𝑎2 cos 𝜖1 − 𝜖2
∙ 𝑐𝑔
Constructive
Interference
𝜖1 = 𝜖2
1
𝐸𝑓 = 𝜌𝑔 𝑎1 + 𝑎2
2
2
∙ 𝑐𝑔
Destructive
Interference
𝜖1 = 𝜖2 + 𝜋
𝐸𝑓 = 0
Two Waves, Same Frequency and Opposite Directions
1
𝐸𝑓 = 𝜌𝑔 𝑎1
2
2
− 𝑎2
2
∙ 𝑐𝑔
Standing Waves
Fully Standing Wave
𝐸𝑓 = 0
Wave Energy Converters! (WECs)
WEC
Hydrodynamics
𝜙 = 𝜙0 + 𝜙𝑠 +
𝐸𝑓 ≠ 𝐸𝑓0 + 𝐸𝑓𝑠 +
𝐸𝑓𝑖
𝜙𝑖
𝜙0 - Incident wave
𝜙𝑠 - Scattered wave
𝜙𝑖 - Radiated wave each DOF
(𝜙𝑖 = 𝜙1𝑖 𝑋𝑖 )
Wave fields
from WAMIT
Linear WEC Forces (heave)
Hydro Forces
𝜕𝜙
𝑝=−
𝜕𝑡
Body Mechanical
𝑓𝑑 =
𝑓𝑃𝑇𝑂 = −𝑑𝑥 − 𝑘𝑥
𝑝0 + 𝑝𝑠 𝑛3 𝑑𝑠
𝑆
𝑓𝑟 =
𝑝𝑟 𝑛3 𝑑𝑠 = −𝑎𝑥 − 𝑏𝑥
𝑆
𝑓ℎ𝑠 = −𝜌𝑔𝐶𝑠 𝑥 = −𝑐𝑥
WEC Equation of Motion (heave)
𝑚𝑥 = 𝑓𝑑 + 𝑓𝑟 + 𝑓ℎ𝑠 + 𝑓𝑃𝑇𝑂
𝑚 + 𝑎 𝑥 + 𝑑 + 𝑏 𝑥 + 𝑘 + 𝑐 𝑥 = 𝑓𝑑
𝑥 = 𝑋𝑒 𝑖𝜔𝑡 and 𝑓𝑑 = 𝐹𝑑 𝑒 𝑖𝜔𝑡
−𝜔2 𝑚 + 𝑎 𝑋 + 𝑖𝜔 𝑑 + 𝑏 𝑋 + 𝑘 + 𝑐 𝑋 = 𝐹𝑑
𝐹𝑑
𝑋=
𝑘 + 𝑐 − 𝜔2 𝑚 + 𝑎
+ 𝑖𝜔 𝑑 + 𝑏
WEC Power
𝑢 = 𝑥 = 𝑈𝑒 𝑖𝜔𝑡
𝑃 = 𝑅𝑒 𝑓𝑑 + 𝑓𝑟 + 𝑓ℎ𝑠 ∙ 𝑅𝑒 𝑢
Math…
1
1
∗
𝑃 = 𝑅𝑒 𝐹 𝑑 𝑈 − 𝑏 𝑈 2
2
2
When you use the WEC equation of motion…
1
𝑃= 𝑑𝑈2
2
WEC Max Power and Optimal Motions
Again power is…
Math…
1
1
∗
𝑃 = 𝑅𝑒 𝐹 𝑑 𝑈 − 𝑏 𝑈
2
2
1 𝐹𝑑
𝑃=
8 𝑏
2
2
1
1 𝐹𝑑
− 𝑏 𝑈−
2
2 𝑏
2
So…
𝑃𝑚𝑎𝑥 =
1 𝐹𝑑 2
when
8 𝑏
𝑈=
1 𝐹𝑑
2 𝑏
Rather that using equation of motion, prescribe the
velocity (𝑈 is also the velocity of resonance)
WEC and Waves
𝜙 = 𝜙0 + 𝜙𝑠 +
𝜙𝑖
When you
superimpose all the
velocity potential, you
get partial standing
waves
𝐸𝑓−𝐶𝑉
1
=
𝑇
𝑇
0
𝐸 ∙ 𝑢 ∙ 𝑛 𝑑𝑧 𝑑𝑡 = −𝑃𝑊𝐸𝐶
0 −ℎ
Optimal Motions
𝑈=
1 𝐹𝑑
2 𝑏
means 𝑋 =
1 𝐹𝑑
−𝑖
2𝜔 𝑏
Low frequencies (long waves), 𝑋 could be very large.
Optimal velocity is in phase with diffraction force and
optimal position is 90 degrees out of phase.
Typically, diffraction force is maximum at the wave crest
But for long waves, we expect the device to be
somewhat of a wave-follower, (position in phase with
the wave)
Optimal Motions
Wave Follower
Optimal Motions
NOTE: No actual calculations of forces were made here!
Video just to visualize the phase difference between wave-follower and optimal motions
Capture Width
• Capture width has units of length
• The actual width or the body is not relevant
𝑃
ℒ=
𝐸𝑓 𝑐𝑔
ℒ𝑚𝑎𝑥
𝑃𝑚𝑎𝑥
=
𝐸𝑓 𝑐𝑔
• For a point absorber (no scattered wave) operating in
heave
𝜆
ℒ𝑚𝑎𝑥 =
2𝜋
How is it possible to absorb waves that
are outside the width of the body?
• The radiated wave interacts with the diffracted
wave to create standing wave patterns which
reduce the wave energy flux!
• For optimal motions, the amplitude can be very
very large – this creates a larger radiated wave,
which in turn reduces more of the wave energy
flux
Example of Real v. Optimal Motions
• Cylinder
–
–
–
–
–
–
radius = 5 m
depth = 18 m
12 second period
1 m amplitude
Infinite depth
operating in heave
• Computations done in
WAMIT
Diffracted Wave
• Incident + Scattered
• Body held fixed
6
Real Damping: 𝑑 = 10 𝑘𝑔/𝑠
• Motions
𝐹𝑑
𝑋=
𝑘 + 𝑐 − 𝜔2 𝑚 + 𝑎
+ 𝑖𝜔 𝑑 + 𝑏
𝑋 = 0.68 𝑚
• Power
𝑃 = 𝑑𝑢 𝑢 = 63 kW
• Radiated Wave
𝐴𝑟 ~10−3 𝑚
Optimal Motions
• Motions
𝑖 1 𝐹𝑑
𝑋=−
𝜔2 𝑏
𝑋 = 29. 9 𝑚
• Power
1 𝐹𝑑 2
𝑃=
= 1.6 MW
8 𝑏
• Radiated Wave
𝐴𝑟 ~0.1 𝑚
Arrays!
Array Computation
𝜙 = 𝜙0 + 𝜙𝑠 +
•
•
•
•
𝜙𝑖
But the DOF are now multiple devices
Solution is a matrix
Can be done in WAMIT
Lots of mathematical tricks that I don’t quite
understand (i.e. just compute 𝜙𝑖 for a single
device…)
q Factor
• Key feature of an array is constructive and
destructive interference
𝑃𝑎𝑟𝑟𝑎𝑦
ℒ𝑎𝑟𝑟𝑎𝑦
𝑞=
=
𝑁 × 𝑃𝑑𝑒𝑣𝑖𝑐𝑒 𝑁 × ℒ𝑑𝑒𝑣𝑖𝑐𝑒
• Nondimensional number that indicates the
performance of an array
𝑞 = 1 – no gain from the array
𝑞 < 1 – net destructive interference
𝑞 > 1 – net constructive interference
Maximum q Factor
ℒ𝑎𝑟𝑟𝑎𝑦
𝑞=
𝑁 × ℒ𝑑𝑒𝑣𝑖𝑐𝑒
• Early papers made assumption of device under optimal
𝜆
motions. ℒ𝑑𝑒𝑣𝑖𝑐𝑒 = for point absorber in heave.
2𝜋
• Just as with a single device, there is a maximum power for
an array
∗
1 ∗ −𝟏
1
1 −𝟏
1 −𝟏
𝑷 = 𝑭 𝒅 𝑩 𝑭𝒅 −
𝑼 − 𝑩 𝑭𝒅 𝑩 𝑼 − 𝑩 𝑭𝒅
8
2
2
2
𝑷𝒎𝒂𝒙 =
1 ∗
𝑭 𝒅 𝑩−𝟏 𝑭𝒅
8
when 𝑼 =
1 −𝟏
𝑩 𝑭𝒅
2
q Factor
q
2
1.5
q is a function of
both frequency
and direction
1
0.5
kd
Budal (1977)
• Row of N evenly
spaced point
abosbers operating
at optimal phase
(equal amplitude)
• Point absorber makes
analytical
computation
tractable
• q asymtotes to 𝜋 for
infinite N
Mavrakos and Kalofonos (1997)
Fitzgerald and
Thomas (2007)
Also discovered that
(for a point absorber)
1 2𝜋
𝑞 𝛽 𝑑𝛽 = 1
2𝜋 0
Point absorber approximation and
optimal motions
Found optimal arrangements of 5
buoys using constrained nonlinear
optimization
𝑞𝑚𝑎𝑥 = 2.77
Didn’t really have an explanation
Child and
Venugopal (2010)
• Full analytical solution
for an array of 5
cylinders in heave
• Optimal motions and
motions with real PTO
• Looked at maximizing
and minimizing q
• Noticed the formulation
of parabolas where the
radiated or scattered
wave is in and 90o out
of phase with the
incident wave
1) Place device 1 in the wave field
2) Place device 2 on a parabola
from 1, so that the parabola
from 2 goes through 1
3) Place device 3 on the
intersection of parabolas 1 and 2
4) Repeat for devices 4 and 5
Parabolic
Intersection Method
Child and Venugopal Results
• Parabolic intersection method
– Real device: 𝑞 = 1.136
– Optimal motions: 𝑞 = 1.787
– Worst case: 𝑞 = 0.453
• Genetic Algorithm Optimization
– Real device: 𝑞 = 1.163
– Optimal motions: 𝑞 = 2.010
– Worst case: 𝑞 = 0.326
Take Away…
• Array interactions lead to constructive and
destructive interference
• Be wary of very high q factors – probably used
optimal motions (maybe point absorber)
• Read devices – q is much lower
• q is highly sensitive to wave direction
• There may be no net gain – whatever is gained
from one direction is lost at another direction
McIver (1994)
• “…part of a practical strategy for the design of
wave-power stations with large numbers of
devices might be to reduce destructive
interference effects,…, rather than attempt
large increases in power absorption through
constructive interference.”
?
Haller et al (2011)
• Laboratory Observations
of Waves in the Vicinity of
WEC-Arrays
• Proc. Of the 9th European
Wave and Tidal Energy
Conf., Southampton, UK
• Wave field measurements
from the Columbia Power
array tests in Hinsdale last
winter.
• Mick and Aaron Porter are
using SWAN to try to
reproduce the results
• I am will use WAMIT to try
and reproduce the results
Thank you!
Questions, comments, suggestions?
References
•
•
•
•
•
•
•
•
•
Budal, K. (1977). Theory of absorption of wave power by a system of interacting
bodies. Journal of Ship Research, 21, 248-253
Child, B.M.F and V. Venugopal. (2010). Optimal configurations of wave energy
device arrays. Ocean Engineering, 37, 1402-1417
Cruz, J., R. Sykes, P. Siddorn, R. Eatock Taylor. (2010). Estimating the loads and
energy yield of arrays of wave energy converters under realistic seas. IET
Renewable Power Generation, 4-6, 488-497.
Evans, D. V. (1980). Some analytic results for two and three dimensional waveenergy absorbers. Power from Sea Waves - Proc I.M.A. Conf., Edinburgh, 213-249
Falcao, Antonio F. de O. (2010). Wave energy utilization: A review of the
technologies. Renewable and Sustainable Energy Reviews, 14, 899-918
Fitzgerald, C., and G. Thomas. (2007). A preliminary study on the optimal
formation of an array of wave power devices. Proc. Of the 7th European Wave and
Tidal Energy Conf., Porto, Portugal
Haller, M. C., A. Porter, P. Lenee-Bluhm, K. Rhinefrank, E. Hammagren, H. T. OzkanHaller, & D. Newborn. (2011). Laboratory observations of waves in the vicinity of
WEC-arrays. Proc. Of the 9th European Wave and Tidal Energy Conf., Southampton,
England.
Mavrakos, S. A., and A. Kalofonos. (1997). Power absorption by arrays of
interacting vertical axisymmetric wave-energy devices. Applied Ocean Research,
19, 283-291.
McIver, P. (1994). Some hydrodynamic aspects of arrays of wave-energy devices.
Applied Ocean Research, 16, 61-69