Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Extremes, wave asymmetry, and other stochastic
properties of a ocean wave energy system
aspects on model choice in an engineering application
Georg Lindgren
Lund University
Vimeiro, 8 – 11 September 2013
Workshop EVT2013 - in honour of Ivette Gomes
Lindgren
Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
EVT versus basic models
A wave energy example
Basic models versus EVT
Basic models:
(G): Water waves have been modeled as a Gaussian process (field) for
more than 50 years
(L): A Lagrange model can produce more realistic asymmetric waves
(S): Non-linear Schrödinger equations can accurately reproduce real
waves
EVT:
EVT useful to model extreme wave events regardless of (G), (L), (S)
EVT also useful for wave climate modeling
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Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
EVT versus basic models
A wave energy example
A wave energy example
"340 patents between 1855 and the 1973 oilcrisis"
Several configurations of wave energy converters have been designed –
cf. early flying machines
Some ideas have survived – even at sea!
Lab testing of design needs realistic wave models
Early ideas – deterministic cosines
Now – irregular, random, Gaussian, wave models
Why randomness – control, efficiency, reliability, durability
Lindgren
Aspects on model choice
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Introductionwhere N represents the number of turns for the stator coils, and s the pole pitch. For a load im
pedance ZL, the output voltage (V) is related to the generator voltage E(t) as
The Lagrange wave model
EVT versus basic models
Lagrange waves in linear filter
(6
VðtÞ ¼ i ZL ¼ EðtÞ LCOIL diðtÞ=dt ri ðtÞ;
A wave energy
example
Monte Carlo experiments
where
L
COIL represents the inductances of the stator coils and r is the stator coil resistance.
Conclusions
III. RESULTS AND DISCUSSION
Two modern models of linear wave energy converters
The above model was used to calculate the buoy motion and couple it with the linear ge
erator to obtain the output voltages and currents. For concreteness, the sea state was taken to
characterized by the parameters reported off of the Oregon coast.35 For example, in Eq. (4) f
the summer time, the values were taken as Hs ¼ 1.5 m and xm ¼ 1.047 rad/s. The buoy
assumed to be cylindrical in shape and floating vertically in the water with the draft equal
Oregon model and Scandinavian model:
FIG. 1. Schematic of a buoy based linear generator.
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Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
EVT versus basic models
A wave energy example
FIG. 5. The time series for the wave and each buoy listed in Table I.
Theoretical calculations with different wave models
This amplification arises from the RAO gain. The irregular wave regime, however, comprised
several amplitude components and cyclic wave frequencies. This causes the buoy’s total heave
response to be determined by the combined effect of each wave amplitude component, cyclic
frequency, and the corresponding heave transfer function. Thus in this scenario, the average
output power increases between the buoys are due to the amplification to a greater degree of
the amplitude components near resonance, as compared to the suppression of the higher frequency components. As the buoy dimensions continue to increase, although the amplitude components within the resonance bandwidth are magnified to a greater degree, the bandwidth narrows causing the higher frequency components to be suppressed. This results in a lower
increase in the heave response for the buoy as compared to that of the regular wave regime.
Since the generated electrical power is proportional to the magnitude of the buoys’ heave, the
average output power is suppressed as well. In any case, Figure 6 clearly identifies buoy 4 as
the ideal choice from those listed in Table I since it generates the largest output power under
both wave regimes.
Monte Carlo simulation with synthetic waves gives theoretical figures for
energy production. The theoretical effect of a wave energy converter
depends on the wave model! Compare deterministic (sine) waves with
Gaussian waves for four buoy sizes:
FIG. 6. Calculated electrical output power from the liner generator for each buoy listed in Table I.
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
EVT versus basic models
A wave energy example
Why is this important?
Design, control, reliability:
What if waves are asymmetric?
Efficiency of different designs size, dimensions
Performance of control
mechanism - adjust period,
control ascending and
descending speed
Safety analysis - protect ceiling
and floor
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
EVT versus basic models
A wave energy example
Motivating questions
Is the Gaussian wave model good enough in order to describe the
extreme movements in a wave energy converter
or should one use the more complicated wave model that permits
statistically asymmetric waves
R
For example a Laplace model: X (t) = K (t − u) dΛ(u) with a
non-Gaussian spectral measure Λ(u) . . .
or the Lagrange wave model – physical motivation exists – . . .
or some other non-Gaussian process ?
Or perhaps rely only on measurements and EVT
Lindgren
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Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
The Gaussian wave model (1952)
The height W (t, s) of the water surface at location s at time t is a
Gaussian stationary (homogeneous) random process, expressed as a sum
X
W (t, s) =
Ak cos(κk s − ωk t + φk )
k
of moving cosines with
random amplitudes Ak and random phases φk
fixed frequencies ωk (1/wave period)
fixed wave numbers κk (1/wave length).
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Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
Gaussian characteristics
A Gaussian sea is statistically symmetric:
the sea surface can be turned upside down
a wave movie can be run backwards
and you don’t see any difference
It needs to be combined with physics/hydrodynamics – gives the Lagrange
model
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
Gaussian generator and the orbital spectrum
In the Gaussian model the vertical height W (t, x) of a particle at the free
surface at time t and location x is an integral of harmonics with random
phases and amplitudes:
Z ∞
W (t, x) =
e i(κx−ωt) dζ(ω)
ω=−∞
ω 2 = g κ tanh κh
with S(ω) = the “orbital spectrum” and ζ(ω) is a Gaussian complex
“spectral process”.
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Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
The stochastic Lagrange model –
Describes horizontal and vertical movements of individual surface water
particles. Use
Z
W (t, u) = e i(κu−ωt) dζ(ω)
for the vertical movement of a particle with (intitial) reference coordinate u
and write X (t, u) for its horizontal location at time t
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Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
– with horizontal Gaussian movements
Use the same (vertical) Gaussian spectral process as in W (t, x) to generate
also the horizontal variation as
Z
X (t, u) = u + H(κ) e i(κu−ωt) dζ(ω)
where the filter function H depends on water depth h and on an
asymmetry parameter α:
H(κ) = i
α
cosh κh
+
= ρ(ω) e iθ(ω)
sinh κh
(−iω)2
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
The stochastic Lagrange model
The 2D stochastic first order Lagrange wave model is the pair of Gaussian
processes
(W (t, u), X (t, u))
All covariance functions and auto-spectral and cross-spectral density
functions for Σ(t, s) follow from the orbital spectrum S(ω, θ) and the filter
equation.
Space wave : keep time coordinate fixed
Time wave : keep space coordinate – X (t, u) – fixed
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
2D Lagrange waves
Asymmetric Lagrange 2D time waves (top) and space wave (bottom)
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
Explicit definitions – I
The model
(W (t, u), X (t, u))
is an implicitly defined model. The space and time models can be made
explicit by
u = X −1 (t, x)
equal to the reference point that is mapped to the observation point x.
Note: There may be many solutions to X (t, u) = x.
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Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
The Gaussian basis
The stochastic 2D Lagrange model
Explicit definitions
Explicit definitions – II
Space wave L(t0 , x) =
W (t0 , X −1 (t0 , x))
= photo of the surface
5
0
Time wave L(t, x0 ) is the parametric
curve:
t 7→ W (t, X −1 (t, x0 ))
= measured at a wave pole
Lindgren
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Introduction
(6c)
VðtÞ ¼ i ZL ¼ EðtÞ LCOIL diðtÞ=dt ri ðtÞ;
The Lagrange wave model
A second order linear filter
Lagrange waves in linear filterwhere LCOIL represents the inductances of the stator coils and r is the stator coil resistance.
Filtering by the Fourier method
Monte Carlo experiments
Conclusions
III. RESULTS AND DISCUSSION
The above model was used to calculate the buoy motion and couple it with the linear generator to obtain the output voltages and currents. For concreteness, the sea state was taken to be
characterized by the parameters reported off of the Oregon coast.35 For example, in Eq. (4) for
the summer time, the values were taken as Hs ¼ 1.5 m and xm ¼ 1.047 rad/s. The buoy is
assumed to be cylindrical in shape and floating vertically in the water with the draft equal to
The linear wave power extractor as linear filter
The linear wave power extractor of the
Scandinavian model is a linear filter
mY 00 (t) + zY 0 (t) + kY (t) = L(t)
where m is buoy+piston mass, z is
the damping from the magnet/coil,
and k depends on the buoy shape and
the anchoring spring. L(t) is the the
sea surface height and Y (t) is the
buoy/piston displacement
FIG. 1. Schematic of a buoy based linear generator.
Downloaded 20 Aug 2013 to 130.235.252.18. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jrse.aip.org/about/r
Lindgren
Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
A second order linear filter
Filtering by the Fourier method
Filter frequency function
The piston displacement Y (t) is obtained by Fourier transformation:
Y(ω) = L(ω) Hfilter (ω)
where the filter has frequency function
Hfilter (ω) =
m (iω)2
1
+ z (iω) + k
Inverse Fourier transformation gives Y (t) = F −1 Y(t)
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Quantities of interest
Lagrange experimental setup
Gaussian and Lagrange waves in linear filter
Quantities of interest
Compare asymmetry of Gaussian
and Lagrangian waves as input
to linear filter . . .
and asymmetry after passage
through a linear filter
Figure shows an experiment
from NTNU with Gaussian
waves – reduction in power by
25% compared to sinuoidal
waves
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Quantities of interest
Lagrange experimental setup
Gaussian and Lagrange waves in linear filter
Experimental setup for a Lagrange experiment
Jonswap orbital spectrum for W (t, u0 )
Water depth h = 25m (Lysekil)
Degree of asymmetry: α = 1 strong asymmetry
Damping z depends on the loading on the generator
Significant wave height: Hs = 7m, Hs = 1.41m (Lysekil)
Peak period: 11s, 6s (Lysekil)
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Quantities of interest
Lagrange experimental setup
Gaussian and Lagrange waves in linear filter
The linear filter removes part of the asymmetry . . .
Filtered Lagrange waves Lagrange waves
Big waves: Hs = 7m, Soft spring: z = 0.2, Strong wave asymmetry: α = 1
Waves
4
2
0
−2
−4
−5
Derivative
4
99.9%
99% 2
95%
75%
50% 0
25%
5% −2
1%
0.1%
−4
0 5
−5 0 5 1015
4
2
0
−2
−4
−10 0 10
4
99.9%
99% 2
95%
75%
50% 0
25%
5% −2
1%
0.1%
−4
−5 0 5
Lindgren
Second derivative
4
99.9%
99.9%
99% 2
99%
95%
95%
75%
75%
50% 0
50%
25%
25%
5% −2
5%
1%
1%
0.1%
0.1%
−4
−40−20 0 20
4
99.9%
99% 2
95%
75%
50% 0
25%
5% −2
1%
0.1%
−4
−5
99.9%
99%
95%
75%
50%
25%
5%
1%
0.1%
0
Aspects on model choice
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Quantities of interest
Lagrange experimental setup
Gaussian and Lagrange waves in linear filter
. . . but not in the extremes
0.5
0
0
5
x
10
0.5
5
x
Slopes above 5% quantile
1
0.5
0
4
6
8
10
8
10
x
1
0
0
F(x| X>=4.823)
All slopes
1
10
F(x| X>=4.2104)
Filtered Lagrange
Filtered Gaussian
Still asymmetry in slopes at level crossings
1
0.5
0
4
6
x
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Quantities of interest
Lagrange experimental setup
Gaussian and Lagrange waves in linear filter
Base data from the Lysekil station - resonant conditions
F(x| X>=1.5)
F(x| X>=1.5)
Filtered Gaussian at level crossings
1
0.5
0
1.5
2
2.5
x
Filtered Lagrange at level crossings
3
1
0.5
0
1.5
2
2.5
3
x
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Quantities of interest
Lagrange experimental setup
Gaussian and Lagrange waves in linear filter
Base data from the Lysekil station - non-resonant conditions
Filtered Gaussian at level crossings
F(x| X>=0.2)
0.5
F(x| X>=0.2)
1
0.5
0
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
x
Filtered Lagrange at level crossings
1
0
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
x
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Introduction
The Lagrange wave model
Lagrange waves in linear filter
Monte Carlo experiments
Conclusions
Conclusions
A Lagrange wave model gives realistic asymmetric slope distributions
Lagrange waves become more "Gaussian" after linear filtration like in
a wave power station
Still considerable asymmetry at high levels
Model simulations with irregular sea should take asymmetry into
account
Full scale experiment + EVT still needed
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