Post on 06-Mar-2018
Floating offshore wind turbines
Michael Muskulus
Department of Civil and Transport EngineeringNorwegian University of Science and Technology
7491 Trondheim, Norway
Stochastic Dynamics of Wind Turbines and Wave Energy AbsorbersAugust 6–8, 2014
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Part I
Hydrodynamics and design of floating
offshore wind turbines
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Learning objectives
• Design criteria
• Wave forces on floating structures
• Basic equation of motion
• Wind turbine loads
• Mooring systems
• Floating wind turbine concepts
• Basic dynamic instability
• Recent work
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Design challenges
• Highly dynamic and complex environment
• Nonlinearities
• Uncertainties in hydrodynamic loads and behavior
• Specialized analysis and software needs
Similar to challenges for fixed bottom wind turbines (cf. Schafhirt &Muskulus 2014), but with a few additional issues.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Design criteria
• Hydrodynamic stability
• Frequency considerations
• Platform survival (storm conditions)• No water on deck• Air gap (no slamming loads)• Mooring line fairlead breaking strength• Tendons / mooring lines no-slack• Sloshing in tanks
• Fatigue assessment
• System limits and performance• Power-cable limit on excursions• Maximum nacelle accelerations• Design static / dynamic platform pitch angle (vs. power production)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Design criteria
• Hydrodynamic stability
• Frequency considerations
• Platform survival (storm conditions)• No water on deck• Air gap (no slamming loads)• Mooring line fairlead breaking strength• Tendons / mooring lines no-slack• Sloshing in tanks
• Fatigue assessment
• System limits and performance• Power-cable limit on excursions• Maximum nacelle accelerations• Design static / dynamic platform pitch angle (vs. power production)
VIDEOSemisubmersible FOWT in extreme conditions
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Frequency considerations
• Example: Hywind (Statoil)
• 1P = 17 rpm/60min = 0.283Hz
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Wave forces
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Intro
William Thomson (Lord Kelvin, 1857)
• Now I think hydrodynamics is to be the root of all physical science,and it is at present second to none in the beauty of its mathematics.
Sir Geoffrey Ingram Taylor (1974)
• Though the fundamental laws of the mechanics of the simplestfluids, which possess Newtonian viscosity, are known andunderstood, to apply them to give a complete description of anyindustrially significant process is often far beyond our power.
Turgut Sarpkaya (2010)
• The emerging fact is that the current body of analytical,experimental, and operational knowledge is still inadequate todescribe the complex realities of fluid loading and dynamic responseof offshore structures as evidenced by often tragic and highlyexpensive failures.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Linear wave theory
(Faltinsen 1990)M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Wave spectra
(Faltinsen 1990)
• Spectral representation of short-term seastate(e.g. Pierson-Moskowitz / JONSWAP spectrum)
• Assumption: stationary Gaussian
• Typically wind sea and swell need to be considered
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Long-term sea state
(Faltinsen 1990)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Implications of linearity
(Faltinsen 1990)
Hydrodynamic problem dealt as two subproblems:
• Forces and moments on the body when the structure is restrainedfrom motion and there are incident waves: wave excitation loads(Froude-Krylov / diffraction forces and moments)
• Forces and moments on the body when the structure is forced tooscillate with a certain frequency in any rigid-body mode, and thereare no incident waves: added mass, damping and restoring terms.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Dimensional analysis
Wave forces depend on eight quantities:
f = Ψ(t,T ,D, λ, u0, u0, ρ, ν)
T Wave periodD Structural dimension (relevant)λ Wave lengthu0 Water particle velocity (maximum)u0 Water particle acceleration u0 = ωu0
In an M-L-T system, this leaves five dimensionless quantities(Buckingham Pi theorem):
f
ρu20D
= Φ
(t
T,u0D
ν,u0T
D,πD
λ
)t/T Dimensionless timeu0D/ν Reynolds numberu0T/D Keulegan-Carpenter (KC) numberπD/λ = kD/2 = ka Diffraction parameter
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Three basic approaches to wave excitation loads
Total wave force consists of components due to drag, inertia andscattering
• Morison formulaAccurate description if drag force issignificant, i.e., if KC > 5Inertia term can be used (CM = 2) forsmall structures (λ/D > 5)
• Froude-Krylov approximationPressure due to incident waves is usedon the surface of the structureApplicable for relatively smallstructures (λ/D > 5)
• Diffraction theoryNecessary for relatively largestructures, i.e., if λ/D < 5Viscous effects not represented
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Morison formula – Basic idea
F = CMAI u + CDAD |u|u
where AI = ρπ4 D2 and AD = 1
2ρD.
• Inertia coefficient CM : represents changes in the fluid due to thepresence of the cylinder under non-viscous potential flow.In a uniformly accelerated fluid CM = 2.0
• Drag coefficient CD : represents viscous effects due to turbulencewake region behind the cylinder — difficult to predict.NB: different values of CD apply for steady flow past the cylinder asopposed to oscillatory flow
• Simple superposition of both effects assumed
• Interaction effects become important for separation < 2D betweenmembers
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Morison formula – Alternative formulations
Relative velocity model
F = CMAI (u − x) + CDAD |u − x |(u − x)
Split form
F = CMAI u − CAAI x) + CDAD |u − x |(u − x)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Morison formula – Corrections
Four-term Morison formula (Sarpkaya 1981):
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Morison formula – Typical values
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Borgman linearization
For linear frequency calculations, e.g., the nonlinear drag term is modifiedand the Morison formula reads:
F ′ = CMAI u + CDADu
The right term is selected such that the sum of squares is minimized forall points in the timeseries, i.e., such that
∂〈(F − F1)2〉∂CD
AD = −2〈CDu2|u| − CDu
2〉 = 0
which leads to
CD = CD〈u2|u|〉〈u2〉
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Borgman linearization
CD = CD〈u2|u|〉〈u2〉
Assuming that wave surface elevation is zero-mean Gaussian and linearwave theory applies, one finds that
CD = CD(8π)σ3
u
σ2u
= CD
√(8/π)σu
and
FD(t) ∼ CDAD
√8
πσuu(t)
where σ2u =
∫∞0
Su(ω)dω.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Froude-Krylov force – General approach
Dynamic wave pressure
p = ρgH
2
cosh kz
cosh kdcos(kx − ωt)
Resulting horizontal force component
Fx = CH
∫S
pnx dS
NB: force coefficient CH corrects for changes in the fluid due to presenceof the structure — values not obtained by Froude-Krylov theoryAssumptions: small structure (D/λ < 5), wave height small
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Froude-Krylov force for the vertical cylinder
Cylinder of radius a, where x = a cos θ is the horizontal coordinate.Cylinder center at z , length of the cylinder l .
Fx = CHρgHa
2 cosh kd
∫ z+l/2
z−l/2
cosh kz dz
∫ 2π
0
cos(ka cos θ − ωt) cos θ dθ
= CHρV2J1(ka)
ka
sinh(kl/2)
(kl/2)˙u(z)
in terms of the horizontal water particle acceleration at the cylindercenter, with the Bessel function of the first kind (order one) J1.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
McCamy-Fuchs theory
• Analytical solution to the linear diffraction problem (wave excitationforce)
• Fixed vertical cylinder, surface piercing
Net force per unit axial length
fx =2ρgH
k
cosh kz
cosh kd
1√A1(ka)
cos(ωt − α)
where
A1(ka) = J′21 (ka) + Y
′21 (ka), α = tan−1
(J ′1(ka)
Y ′1(ka)
)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Relation to Morison approach
fx = CMρπa2uα
in which
CM =4
π(ka)2√A1
and uα is the water-particle acceleration at elevation z from the bottomat a phase lag of α.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
General diffraction theory
• Boundary-value problem with perturbative solution
• Linearization: Added mass, (wave radiation) damping and restoringfor the j-th motion mode qj :
Fk = −Akjd2qjdt2− Bkj
dqjdt− Ckjq
• NB: Added mass can be negative (e.g. catamarans)
• NB: Dependent on currents (quadratic correction)
• Added mass and damping terms experimentally accessible• Free-decay tests• Resonance testing (absorbed wave power)
• Determined through strip theory (two-dimensional approximation)• Numerical (Green’s function method)• Analytical (e.g. Lewis form technique)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Example: Added mass – rough estimates
(Barltrop 1998)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Second-order wave forces
Following Langley (1986) we write for the surface elevation
η(t) = <∑n
aneiωnt , with E[|an|2] = 2Gηη(ωn)dω
The low-frequency second order force is then
F = <∑n
∑m
ana∗mHnme
i(ωn−ωm)t
with a complex transfer function Hnm. This can be written as:
F =∑n
|an|2Hnn +∑k
Xkeiωk t
whereXk = 2
∑m
Hm+k,ma∗mam+k
The first term above represents a mean drift force whereas the secondterm is slowly-varying.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Model tests for the DeepCwind semisubmersible
(Coulling et al. 2013)
• Wave-only response improves signficantly when includingsecond-order wave forces
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Significance
(Coulling et al. 2013)
• In an operational case with turbulent wind, a difference of only 3percent occurs without second-order wave forces
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Heave plates
(Tao & Cai 2004)
Heave plates introduce additional viscous damping into the heave degreeof freedom (which is often critical)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Heave plates
Additional viscous damping from heave plate:
(Tao & Cai 2004)
• Linear with KC number• Strongly depends on disk/cylinder diameter ratio
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Dynamics
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Equation of motion — General form
Mq + Bq + Cq = F
M InertiaB Damping Wave radiationC Stiffness Hydrostatic restoring, MooringF External Forces Wave excitation, Viscous forces
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Response in irregular seas
Wave elevation:
ζ =N∑j=1
Aj sin(ωj t − kjx + εj)
Steady-state response to j-th component:
Aj |H(ωj)| sin(ωj t + δ(ωj) + εj)
determines RAO (Response Amplitude Operator) — response amplitudeper unit wave amplitude (transfer function)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Added mass
Presence of a body in a fluid causes complex changes in the flow.May theoretetically be calculated using diffraction theory.As a first approximation, can be treated as an additional mass of fluid ma
that is trapped by the body.The hydrodynamic force on a body in an accelerating fluid is (cf. inertiaterm in Morison formula):
F = FFroude−Krylov + maxfluid
The force required to accelerate a submerged structure is:
F = (mstructure + ma)xstructure
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Equation of motion — Frequency domain
(M + A(ω))q + B(ω)q + Cq = F (ω)
M InertiaA(ω) Addedd mass Frequency-dependentB(ω) Damping Frequency-dependent wave radiationC Stiffness Hydrostatic restoring, MooringF (ω) External force Wave excitation
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Natural periods
Ti = 2π
(Mii + Aii
Cii
) 12
• For an unmoored structure there are no resonance periods in surge,sway and yaw.
• Typical periods for a moored structure: T > 60s in these DOF.
• Standard design criterion: T > 20s in heave, pitch and roll.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Natural periods — Semi-submersible
Heave degree of freedom:
T3 = 2π
(M + A33
ρgAw
) 12
,
where Aw is waterplane area.
• Possible to achieve T3 > 20s in a semi-submersible by large waterplane area.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Natural periods — TLP
Heave degree of freedom:
T3 = 2π
(M + A33
EA/l
) 12
,
where E ,A, l are modulus of elasticity, cross-section and length of thetendons.
• Lower than wave excitation spectrum, i.e., T3 < 5 s.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Equation of motion — Time domain
Cummins equation:
(M + A∞)q +
∫ t
0
K (t − τ)q(t)dτ + Cq = F
with
K (t) = − 2
π
∫ ∞0
ω [A(ω)− A∞] sin(ωt)dω
=2
π
∫ ∞0
B(ω) cos(ωt)dω
A∞ = A(ω) +1
ω
∫ ∞0
K (t) sin(ωt)dt = limω→∞
A(ω)
B∞ = 0
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Wind turbine loads
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Mean rotor thrust
(Zaaijer 2007)
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Turbulence
• Stationary Gaussian process
• Simulated from a spectrum (e.g. Kaimal)
• Typically on a relatively coarse grid
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Turbulence
• Stationary Gaussian process
• Simulated from a spectrum (e.g. Kaimal)
• Typically on a relatively coarse grid
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Variability due to stochastic environment
(Zwick & Muskulus 2014)
• Wind turbines are strongly forced systems
• Significant variability due to fluctuating environment
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Variability due to stochastic environment
(Zwick & Muskulus 2014)
• Fatigue analysis shows a bias for < 60 min
• With probability 5 percent a difference in DEL of ±10 percentbetween estimate and expectation for 60 min analysis
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
The rotor: Basic model
• Loads based on lift/drag component in direction of horizontal axis;Rotor thrust load modelled as
Ta =1
2ρπR2ct(λ, θ)v2
rel
where λ = ωRvrel
is the tip-speed-ratio, and θ is pitch angle.
• Standard model used for control-system or simplified dynamicalstudies.
• Weakly non-Gaussian
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
The rotor: Rotational sampling
(Murtagh et al. 2004)
• Stochastic component: Numerically integrated from two-pointcovariance function
• Periodic components: At multiples 1P, 2P, 3P, . . . , of rotorfrequency?
• Amplitudes: Can now be studied numerically
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
The rotor: Rotational sampling
(Murtagh et al. 2004)
• Stochastic component: Numerically integrated from two-pointcovariance function
• Periodic components: At multiples 1P, 2P, 3P, . . . , of rotorfrequency?
• Amplitudes: Can now be studied numerically
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Rotor thrust modelling
• Goal• A simple model that is accurate enough for response analysis and
fatigue calculations
• Ambition• Significantly faster than time-domain• Similar accuracy
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Combined periodic and stochastic wide-band signal
• Classical treatment according to Madsen & Frandsen (1984):
Y (t) = Z (t) + X (t)
where X (t) is stationary Gaussian process characterised by spectraldensity SX (ω), and
Z (t) = <N∑
n=0
αn exp(in(ωRt + θ)).
• Spectral moments:
λk = 2
∫ ∞0
[ωkSX (ω) +
1
2
N∑α=1
(ωk |αn|2δ(ω − nωR)
)]dω
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Upcrossing intensity
• Crossing frequency conditional on random phase θ = θ0. Anupcrossing of the level ξ by Y (t) corresponds to an upcrossing ofthe time-dependent level ξ − Z (t) by X (t):
νY (ξ, t|θ0) =
∫ ∞−z(t)
(x + Z (t))fXX (ξ − Z (t), x)dx (1)
= 2πν0φ
(ξ − Z (t)
σX
)Ψ
(−Z (t)
ω0σX
)(2)
where Ψ(x) = φ(x)− xφ(−x).
• Crossing frequency by averaging Rice’s formula over a full periodT0 = 2π/ωR :
νY (ξ) =2π
T0
∫ T0/2
−T0/2
ν0φ
(ξ − Z (t)
σX
)Ψ
(−Z (t)
ω0σX
)dt
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Model fitting
• ARMA model plus periodic components (not SARIMA model)• Difficult to separate out the “seasonal” (deterministic) components
• Example: AR(0.9997) with σ = 1
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Model fitting
• ARMA model plus periodic components (not SARIMA model)• Difficult to separate out the “seasonal” (deterministic) components
• Example: AR(0.9997) with σ = 1
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Model fitting
• Harmonic components multiples of ω = 0.04 Hz (3P = 0.397 Hz)
• Z (t) = 2Zω(t) + 2Z2ω(t) + 0.4Z3ω(t) + 0.4Z4ω(t)
• X (t) = 0.9997X (t − 1) + N(0, 1)
• Y (t) = X (t) + Z (t) + mean
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
Model fitting
• Harmonic components multiples of ω = 0.04 Hz (3P = 0.397 Hz)
• Z (t) = 2Zω(t) + 2Z2ω(t) + 0.4Z3ω(t) + 0.4Z4ω(t)
• X (t) = 0.9997X (t − 1) + N(0, 1)
• Y (t) = X (t) + Z (t) + mean
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
References
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M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014
Introduction Wave forces Dynamics Wind turbine loads References
References
M Muskulus, S Schafhirt: Design optimization of wind turbine support structures— a review. ISOPE Journal of Ocean and Wind Energy 1 (2014), 12–22.
T. Sarpkaya: Morison equation and the wave forces on offshore structures.Technical Report CR82.008, US Navy Civil Engineering Laboratory (1981).
L. Tao, S. Cai: Heave motion suppression of a Spar with a heave plate. OceanEngineering 31 (2004), 669–692.
M. Zaaijer: Introduction to wind energy. Lecture notes Offshore Wind FarmDesign, TU Delft (2007).
D Zwick, M Muskulus: The simulation error caused by input loading variability inoffshore wind turbine structural analysis. Wind Energy, to appear.
M. Muskulus (michael.muskulus@ntnu.no) Norwegian University of Science and Technology
Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014