Crest/trough and front/back asymmetric waves in wave ......ron (NdÐFeÐB) permanent magnets,...

IntroductionThe Lagrange wave model

Does it matter?Monte Carlo experiments

Conclusions and references

Crest/trough and front/back asymmetric wavesin wave energy systems

Georg Lindgren

Workshop on�Can Stochastic Geometry handleDynamics of Risk Management?�

Lund, April 18-20, 2018

Lindgren Energy systems 1 / 23




A wave energy example

The origin of the study

Two wave energy converters: Oregon model and Scandinavian model:

the piston-driven permanent magnet (attached to the buoy) to induce voltages as shown schemati-

cally in Fig. 1. The rotor could be constructed of an alternating assembly of Neodymium–Iron–Bo-

ron (Nd–Fe–B) permanent magnets, interspersed with soft iron pole pieces mounted on a threaded

aluminum shaft.21 The magnets would be stacked in pairs to force the opposing fluxes of the magne-

tomotive forces through the iron pole pieces and across the air gap to the stator. Regardless of the

dimensional details, the induced electromotive force E(t) for such a configuration would be

E(t)¼ dkf /dt, with kf being the flux linkage between the rotor and the stator coils. For the LG, with zdenoting the vertical displacement, and Up the peak flux, one has

kf ¼ NU:pcos½pz=s�; (6a)

and hence

EðtÞ ¼ �N Upðp=sÞ: sinðpz=sÞ: dz=dt; (6b)

where 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

VðtÞ ¼ i ZL ¼ EðtÞ � LCOIL diðtÞ=dt� ri ðtÞ; (6c)

where LCOIL represents the inductances of the stator coils and r is the stator coil resistance.

III. RESULTS AND DISCUSSION

The above model was used to calculate the buoy motion and couple it with the linear gen-

erator 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 Hs¼ 1.5 m and xm¼ 1.047 rad/s. The buoy is

assumed to be cylindrical in shape and floating vertically in the water with the draft equal to

FIG. 1. Schematic of a buoy based linear generator.

063137-4 M. A. Stelzer and R. P. Joshi J. Renewable Sustainable Energy 4, 063137 (2012)

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/rights_and_permissions






A standard simulation model

From WEC-Sim (Wave Energy Converter SIMulator), an open-source waveenergy converter simulation tool in Matlab.






Theoretical calculations with di�erent wave models

Monte Carlo simulation with synthetic waves gives theoretical numbers forenergy production. The theoretical e�ect of a wave energy converterdepends on the wave model! Compare deterministic (sine) waves withGaussian waves for four buoy sizes:

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 fre-

quency components. As the buoy dimensions continue to increase, although the amplitude com-

ponents within the resonance bandwidth are magnified to a greater degree, the bandwidth nar-

rows 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.

FIG. 5. The time series for the wave and each buoy listed in Table I.

FIG. 6. Calculated electrical output power from the liner generator for each buoy listed in Table I.








Motivating questions

Is the Gaussian wave model good enough in order to describe theexciting forces in the converter

or should one use the more complicated wave model that permitsasymmetric waves

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 ?





The stochastic 2D Lagrange modelParticle orbits

Gaussian generator for the Lagrange model

In the Gaussian model the vertical height W (t, x) of a particle at the freesurface at time t and location x is an integral of harmonics with randomphases and amplitudes:

W (t, x) =

∫ ∞ω=−∞

e i(κx−ωt) dζ(ω),

approximated by a real sum

W (t, x) =∑k

Ak cos(κkx − ωkt + φk)

with random Ak and φk .






The stochastic Lagrange model �

Describes joint horizontal and vertical movements of individual surfacewater particles. Use

W (t, u) =

∫e i(κu−ωt) dζ(ω)

for the vertical movement of a particle with (intitial) reference coordinate uand write X (t, u) for its horizontal location at time t ...






� with horizontal Gaussian movements

... and the same (vertical) Gaussian spectral process ζ(ω) as in W (t, u) togenerate also the horizontal variation.

XM(t, u) = u +

∫HM(κ) e i(κu−ωt) dζ(ω)

where the �lter function HM depends on water depth h:

HM(κ) = icoshκh

sinhκh






The stochastic Lagrange model

The 2D stochastic �rst order free Lagrange wave model is the pair ofGaussian processes

(W (t, u),XM(t, u))

All covariance functions and auto-spectral and cross-spectral densityfunctions for Σ(t, s) follow from the orbital spectrum S(ω, θ) and the �lterequation.

Space wave : keep time coordinate �xed

Time wave : keep space coordinate � XM(t, u) � �xed






Front-back asymmetry

The model HM(κ) = i coshκhsinhκh gives front-back statistically symmetric

waves.

Adding a slope-dependent term gives asymmetric waves. For example,

∂2X (t, u)/∂t2 = ∂2XM(t, u)/∂t2 + αW (t, u),

H(κ) = icoshκh

sinhκh+

α

(−iω)2= ρ(ω) e iθ(ω)

Implies an extra phase shift (to the phase θ = π/2 in the free model).A general form for X (t, u) is

X (t, u) = u +∫e i(κu−ωt+θ(ω))ρ(ω) dζ(ω)






2D Lagrange waves

Asymmetric Lagrange 2D time waves (top) and space wave (bottom)

0 20 40 60 80 100 120 140 160 180 200−5

0

5

0 20 40 60 80 100 120 140 160 180 200−5

0

5






Is the Lagrange model useful?

The joint Gaussian character of the vertical, W (t, u), and horizontal,X (t, u), component makes it possible to compute exact statisticaldistributions of wave characteristics in 2D and 3D:

wave steepness

wave asymmetry in time and space

wave front velocity






Is the Lagrange model realistic

when it comes to:

Particle movements - particle orbits Unclear?

Wave geometry - front-back and crest-trough asymmetry Yes!

3D properties - horseshoe-like patterns In theory, yes!






Orbit shape and orientation depend on wave asymmetry

The coupled model with α = 1: Wave asymmetry has strong relation toorbit orientation.

−5 −4 −3 −2 −1 0 1 2 3 4 5−2

−1

0

1

2

time [s]

Positive skewness, α = 1

−5 −4 −3 −2 −1 0 1 2 3 4 5−2

−1

0

1

2

orbit [m]

Strongly up−tilting orbits

−5 −4 −3 −2 −1 0 1 2 3 4 5−2

−1

0

1

2

time [s]

Negative skewness, α = 1

−5 −4 −3 −2 −1 0 1 2 3 4 5−2

−1

0

1

2

orbit [m]

Irregularly up−tilting orbits





Lagrange waves in linear �lter

Back to wave energy

Does wave asymmetry matter in the wave energy example?





Lagrange waves in linear �lter

The linear wave power extractor as linear �lter

Simplistic model: The linear wave power ex-tractor as a linear �lter

mZ ′′(t) + zZ ′(t) + kZ (t) = L(t)

where m is total mass, z is total damping,and k depends on the buoy shape and theanchoring spring. L(t) is the sea surface vari-ation and Z (t) is the buoy/piston displace-ment from equilibrium.Neglects hydrodynamic forces!

the piston-driven permanent magnet (attached to the buoy) to induce voltages as shown schemati-

cally in Fig. 1. The rotor could be constructed of an alternating assembly of Neodymium–Iron–Bo-

ron (Nd–Fe–B) permanent magnets, interspersed with soft iron pole pieces mounted on a threaded

aluminum shaft.21 The magnets would be stacked in pairs to force the opposing fluxes of the magne-

tomotive forces through the iron pole pieces and across the air gap to the stator. Regardless of the

dimensional details, the induced electromotive force E(t) for such a configuration would be

E(t)¼ dkf /dt, with kf being the flux linkage between the rotor and the stator coils. For the LG, with zdenoting the vertical displacement, and Up the peak flux, one has

kf ¼ NU:pcos½pz=s�; (6a)

and hence

EðtÞ ¼ �N Upðp=sÞ: sinðpz=sÞ: dz=dt; (6b)

where 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

VðtÞ ¼ i ZL ¼ EðtÞ � LCOIL diðtÞ=dt� ri ðtÞ; (6c)

where LCOIL represents the inductances of the stator coils and r is the stator coil resistance.

III. RESULTS AND DISCUSSION

The above model was used to calculate the buoy motion and couple it with the linear gen-

erator 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 Hs¼ 1.5 m and xm¼ 1.047 rad/s. The buoy is

assumed to be cylindrical in shape and floating vertically in the water with the draft equal to

FIG. 1. Schematic of a buoy based linear generator.







Lagrange experimental setupGaussian and Lagrange waves in linear �lter

Experimental setup for a numerical Lagrange experiment

Pierson-Moskowitz orbital spectrum for W (t, u0)

Water depth h = 20m

Degree of asymmetry: α = 3

Damping z depends on the loading on the generator

Wave height: Hs = 2.5m, Peak period: Tp = 4s, 6s, 10s

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Frequency [rad/s]

S(w

) [m

2 s /

rad]

PM−spectral densities

w

p = 2π/4

wp = 2π/6

wp = 2π/10






Front-back asymmetry statistics

Solid lines: CDF's of full crest front (x) and back (o) periodsDashed curves: CDF's of half crest front (x) and back (o) periods

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

T [s]

F(T

)

Tp = 10

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

T [s]

F(T

)

Tp = 4






Relative e�ciency of converter: Lagrange vs Gaussian

The average power of the converter is

P = γE

((∂Z (t)

∂t

)2)

where Z (t) is the position of the magnet and γ a damping coe�cient

Figures show relative e�ciency PLagrange/PGauss < 1 as function of γ

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.9

0.95

1

1.05

1.1

1.15

Tp = 4[s], ω0 = 2π/6

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.9

0.95

1

Tp = 6[s], ω0 = 2π/6

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.85

0.9

0.95

1

Tp = 10[s], ω0 = 2π/6






Including hydrodynamic forces - Yingguang Wang, 2018

(M + A)Z ′′(t) +

∫ t

−∞K (t − τ)Z ′(τ) dτ + Phs

= Pwave(t) + Pext(t) + Pvisc(t)

Here, Z (t) is the height of the buoy as before, and the Pxx are all di�erenthydrodynamic forces acting on the buoy.This can be simulated by WEC-Sim.It takes the wave time function as input.






Wang uses second order Stokes waves

Second order Stokes waves are crest/trough asymmetric ...

... but front/back statistically symmetric

With full hydrodynamic equation, the Stokes wave model givesPStokes/PGauss > 1.

My question: Does front/back asymmetric Stokes/Lagrange waves +full equation give

PStokes/Lagrange

PGauss

{< 1

> 1

No answer yet!





Conclusions

A Lagrange wave model gives realistic asymmetric slope distributions

Gaussian input in simulations may give too optimistic/pessimistice�ciency

Model simulations with irregular sea should take asymmetry intoaccount

Full scale experiment needed





References

Pierson, W.J.(1961). Model of random seas based on the Lagrangian equations ofmotion. New York University

Lindgren, G. and Aberg, S. (2008). First order stochastic Lagrange models forfront-back asymmetric ocean waves. J. OMAE, 131

Lindgren, G. (2010). Slope distribution in front-back asymmetric stochasticLagrange time waves. Adv. Appl. Prob., 42

Lindgren, G. (2015). Asymmetric waves in wave energy systems analysed by thestochastic Gauss-Lagrange wave model. Proc. Estonian Acad. Sci., 64

Yingguang Wang (2018). A novel method for predicting the power outputs ofwave energy converters. Acta Mechanica Sinica,https://doi.org/10.1007/s10409-018�0755-2

WEC-Sim � Wave Energy Converter SIMulator (2015-2018).http://wec-sim.github.io/WEC-Sim/


Crest/trough and front/back asymmetric waves in wave ......ron (NdÐFeÐB) permanent magnets,...

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