Floating offshore wind turbines - Aalborg · PDF fileWave forces on oating structures ... Though the fundamental laws of the mechanics of the simplest ... Introduction Wave forces

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 ([email protected]) Norwegian University of Science and Technology

Floating offshore wind turbines Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers,August 6–8, 2014

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Introduction Wave forces Dynamics Wind turbine loads References

Part I

Hydrodynamics and design of floating

offshore wind turbines



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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



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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.



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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)



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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



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Frequency considerations

• Example: Hywind (Statoil)

• 1P = 17 rpm/60min = 0.283Hz



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Wave forces



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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.



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Linear wave theory

(Faltinsen 1990)M. Muskulus ([email protected]) Norwegian University of Science and Technology


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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



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Long-term sea state

(Faltinsen 1990)



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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.



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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



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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



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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



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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)



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Morison formula – Corrections

Four-term Morison formula (Sarpkaya 1981):



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Morison formula – Typical values



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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〉



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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.



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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



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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.



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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)

)



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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.



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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)



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Example: Added mass – rough estimates

(Barltrop 1998)



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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.



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Model tests for the DeepCwind semisubmersible

(Coulling et al. 2013)

• Wave-only response improves signficantly when includingsecond-order wave forces



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Significance

(Coulling et al. 2013)

• In an operational case with turbulent wind, a difference of only 3percent occurs without second-order wave forces



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Heave plates

(Tao & Cai 2004)

Heave plates introduce additional viscous damping into the heave degreeof freedom (which is often critical)



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Heave plates

Additional viscous damping from heave plate:

(Tao & Cai 2004)

• Linear with KC number• Strongly depends on disk/cylinder diameter ratio



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Dynamics



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Equation of motion — General form

Mq + Bq + Cq = F

M InertiaB Damping Wave radiationC Stiffness Hydrostatic restoring, MooringF External Forces Wave excitation, Viscous forces



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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)



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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



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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



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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.



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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.



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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.



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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



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Wind turbine loads



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Mean rotor thrust

(Zaaijer 2007)



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Turbulence

• Stationary Gaussian process

• Simulated from a spectrum (e.g. Kaimal)

• Typically on a relatively coarse grid



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Turbulence

• Stationary Gaussian process

• Simulated from a spectrum (e.g. Kaimal)

• Typically on a relatively coarse grid



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Variability due to stochastic environment

(Zwick & Muskulus 2014)

• Wind turbines are strongly forced systems

• Significant variability due to fluctuating environment



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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



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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



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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



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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



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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



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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



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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



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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



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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



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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



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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



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References

N.D.P. Barltrop: Floating structures — A guide for design and analysis. CMPT(1998).

A.J. Coulling, A.J. Goupee, A.N. Robertson, J.M. Jonkman: Importance ofsecond-order difference-frequency wave-diffraction forces in the validation of aFast semi-submersible floating wind turbine model. Technical ReportNREL/CP-5000-57697, National Renewable Energy Laboratory (2013).

O.M. Faltinsen: Sea loads on ships and offshore structures. Cambridge UniversityPress (1990).

R.S. Langley: On the time domain simulation of second order wave forces andinduced responses. Applied Ocean Research 8 (1986), 134–143.

PJ Murtagh, B Basu, BM Broderick. Along-wind response of a wind turbinetower with blade coupling subjected to rotationally sampled wind loading.Engineering Structures 27 (2005), 1209–1219.

PH Madsen, S Frandsen: Wind-induced failure of wind turbines. EngineeringStructures 6 (1984), 281–287.

M.E. McCormick: Ocean engineering mechanics — with applications. CambridgeUniversity Press (2010).



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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.



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Floating offshore wind turbines - Aalborg · PDF fileWave forces on oating structures ... Though the fundamental laws of the mechanics of the simplest ... Introduction Wave forces