Verification of Floating Offshore Wind Linearization Functionality in OpenFASTNicholas JohnsonJason Jonkman, Ph.D.Alan Wright, Ph.D.Greg HaymanAmy Robertson, Ph.D.

EERA DeepWind’201916-18 January, 2019Trondheim, Norway

2

Introduction: The OpenFAST Multi-Physics Engineering Tool

• OpenFAST is DOE/NREL’s premier open-source wind turbine multi-physics engineering tool

• FAST has undergone a majorrestructuring, w/ a newmodularization framework (v8)

• Framework originally designedw/ intent of enabling full-systemlinearization, but functionalityis being implemented in stages

InflowWind

ElastoDyn

ServoDyn

MAP++, MoorDyn,or FEAMooring

HydroDyn

AeroDyn

External Conditions

Applied Loads

Wind Turbine

Hydro-dynamics

Aero-dynamics

Waves & Currents

Wind-Inflow Power Generation

Rotor Dynamics

Platform Dynamics

Mooring Dynamics

Drivetrain Dynamics

Control System & Actuators

Nacelle Dynamics

Tower Dynamics Now calledOpenFAST

3

• OpenFAST primary used for nonlinear time-domainstandards-based load analysis (ultimate & fatigue)

• Linearization is about understanding:o Useful for eigenanalysis, controls design,

stability analysis, gradients for optimization,& development of reduced-order models

• Prior focus:o Structuring source code to enable linearizationo Developing general approach to linearizing mesh-mapping

w/n module-to-module coupling relationships, inc. rotationso Linearizing core (but not all) features of InflowWind, ServoDyn,

ElastoDyn, BeamDyn, & AeroDyn modules & their couplingo Verifying implementation

• Recent work (presented @ IOWTC 2018):o Linearizing HydroDyn, & MAP++, & couplingo State-space implementation of wave-excitation

& wave-radiation loads• This work – Verifying implementation for FOWT

Background: Why Linearize?

( )( )( )

x X x,z,u,tZ0 Z x,z,u,t with 0z

y Y x,z,u,t

=∂

= ≠∂

=

.op

u u u etc= + ∆

x A x B uy C x D u

∆ ∆ ∆∆ ∆ ∆

= += +

1

op

X X Z ZA etc.x z z x

− ∂ ∂ ∂ ∂ = − ∂ ∂ ∂ ∂

with

Modulex, z

X, Z, Yu y

4

• Wave-radiation “memory effect” accounted for in HydroDyn by direct time-domain (numerical) convolution

• Linear state-space (SS) approximation:o SS matrices derived from

SS_Fitting pre-processor using4 system-ID approaches

Background: State-Space-Based Wave Radiation

( ) ( )t

Ptfm Rdtn Rdtn Ptfm Rdtn0

Rdtn Rdtn Rdtn Rdtn PtfmPtfm Rdtn

Rdtn Rdtn Rdtn

q F K t q d F

x A x B qq F

F C x

τ τ τ= − −

= +

≅

∫

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• First-order wave-excitation loads accounted for in HydroDyn by inverse Fourier transform

• Linear SS approximation:o SS matrices derived from extension

to SS_Fitting pre-processor using system-ID approach

o Requires prediction of wave elevation time tc into future to address noncausality i.e.

Background: State-Space-Based Wave Excitation

-40 -30 -20 -10 0 10 20 30 40 50 60-8

-6

-4

-2

0

2

4

6

8

1010 5 Impulse Response Functions: original and causalized

12ct s≈

( )cExctnK t( )ExctnK t

( ) ( )

( ) ( )

j tExtn Exctn

Extn Extn Exctn

Extn Extn Extn Extn cc Exctn

Extn Extn Extn

1F X , e d F2

F K t d F

x A x BF

F C x

ωζ ω β ζ ω ωπ

ζ τ ζ τ τ

ζζ

∞

−∞

∞

−∞

=

= −

= +≅

∫

∫

( ) ( )c ct t tζ ζ= +

6

Full System

( )

( )

( )

( )

( )

( )

( )

( )

( )

ED

BD

HD

ED ED

1BD BDop

opHD

HD

A 0 0

A 0 A 0

0 0 A

0 0 00 0 0

0 0 B 0 0 0 0 C 0 0U0 0 0 B 0 0 0 G 0 C 0y

0 0 00 0 0 0 0 B 0

0 0 C0 0 0

etc.

−

=

∂ − ∂

Glue Code

AeroDyn (AD)

ServoDyn (SrvD)

ElastoDyn (ED)InflowWind (IfW)

• D-matrices (included in G) impactall matrices of coupled system, highlighting important role of direct feedthrough

• While A(ED) contains mass, stiffness, & damping of ElastoDyn structural model only, full-system A contains mass, stiffness, & damping associated w/ full-system coupled aero-hydro-servo-elastics, including FOWT hydrostatics, radiation damping, drag, added mass, & mooring restoring

Background: Final Matrix Assembly

with

x A x B u

y C x D u

∆ ∆ ∆

∆ ∆ ∆

+

+

= +

= +

( ) ( ) ( )IfW IfW IfWy D u∆ ∆=

( ) ( ) ( )SrvD SrvD SrvDy D u∆ ∆=

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

ED ED ED ED ED

ED ED ED ED ED

x A x B u

y C x D u

∆ ∆ ∆

∆ ∆ ∆

= +

= +

( ) ( ) ( )AD AD ADy D u∆ ∆=

op op

U U0 u yu y

∆ ∆∂ ∂= +∂ ∂

BeamDyn (BD)( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

BD BD BD BD BD

BD BD BD BD BD

x A x B u

y C x D u

∆ ∆ ∆

∆ ∆ ∆

= +

= +

HydroDyn (HD)( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

HD HD HD HD HD

HD HD HD HD HD

x A x B u

y C x D u

∆ ∆ ∆

∆ ∆ ∆

= +

= +

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

IfW IfW

SrvD SrvD

ED ED ED

BD \BD \BD

HD AD AD

HD HD

MAP MAP

u y

u yx u y

x x u yu yx u y

u y

u y

∆ ∆

∆ ∆

∆ ∆ ∆

∆ ∆ ∆ ∆∆ ∆

∆ ∆ ∆

∆ ∆

∆ ∆

+

+

= = =

MAP++ (MAP)( ) ( ) ( )MAP MAP MAPy D u∆ ∆=

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FAST v7

Results: Campbell Diagram of NREL 5-MW Turbine Atop OC3-Hywind Spar

• Modules enabled: ElastoDyn, ServoDyn, HydroDyn, & MAP++ • Approach (for each rotor speed): Find periodic steady-state OP Linearize to find

A matrix MBC Azimuth-average Eigenanalysis Extract freq.s & damping

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• Modules enabled: ElastoDyn, ServoDyn, HydroDyn, MAP++, AeroDyn, & InflowWind• Approach (for each wind speed): Define torque & blade pitch Find periodic steady-

state OP Linearize to find A matrix MBC Azimuth-average EigenanalysisExtract freq.s & damping

Results: Campbell Diagram of NREL 5-MW Turbine Atop OC3-Hywind Spar – w/ Aero

FAST v7

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Results: Time Series Comparison of Nonlinear & Linear Models

• Modules enabled: ElastoDyn, ServoDyn, HydroDyn, & MAP++ • Nonlinear approach (for each sea state): Time-domain simulation w/ waves• Linear approach (for each sea state): Find steady-state OP Linearize to find A, B, C, D

matrices Integrate in time w/ wave-elevation input derived from nonlinear solution

Hs = 0.67 m, Tp = 4.8 sHs = 2.44 m, Tp = 8.1 sHs = 5.49 m, Tp = 11.3 s

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Conclusions & Future Work• Conclusions:

o Linearization of underlying nonlinear wind-system equations advantageous to:

– Understand system response– Exploit well-established methods/tools for analyzing linear systems

o Linearization functionality has been expanded to FOWT w/n OpenFASTo Verification results:

– Good agreement in natural frequencies between OpenFAST & FAST v7– Damping differences impacted by trim solution, frozen wake, perturbation

size on viscous damping, wave-radiation damping– Nonlinear versus linear response shows impact of structural nonlinearites

for more severe sea states• Future work:

o Improved OP through static-equilibrium, steady-state, or periodic steady-state determination, including trim

o Eigenmode automation & visualizationo Linearization functionality for:

– Other important features (e.g. unsteady aerodynamics of AeroDyn)– Other offshore functionality (SubDyn, etc.)– New features as they are developed

Carpe Ventum!

Jason Jonkman, Ph.D.+1 (303) 384 – 7026jason.jonkman@nrel.gov

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• A linear model of a nonlinear system is only valid in local vicinity of an operating point (OP)

• Current implementation allows OP to be set by given initial conditions (time zero) or a given times in nonlinear time-solution

• Note about rotations in 3D:o Rotations don’t reside in a linear spaceo FAST framework stores module

inputs/outputs for 3D rotationsusing 3×3 DCMs ( )

o Linearized rotationalparameters taken to be 3small-angle rotations aboutglobal X, Y, & Z ( )

Approach & Methods: Operating-Point Determination

Λ

opu u u for most variables= + ∆

op for rotationsΛ Λ ∆Λ=

X

Y

Z

∆θ∆θ ∆θ

∆θ

=

Z Y

Z X

Y X

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1

∆θ ∆θ∆Λ ∆θ ∆θ

∆θ ∆θ

− ≈ − −

with

( )( ) ( )

( )( ) ( )

( )

2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 Z X Y Z X Y X Y Z Y X Y Z X Z X Y ZX X Y Z Y Z

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2X Y Z X Y Z X Y Z X Y Z X Y Z X

1 1 1 11

1 1 1

∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ

∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆

∆Λ

+ + + + + + − − + + + + + + −+ + + + +

+ + + + + + + + + + + + + +

=( ) ( )

( ) ( )( ) ( )

( )

2 2Y Z

2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2Z X Y Z X Y X Y Z X X Y Z Y Z X Y ZX Y X Y Z Z

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2X Y Z X Y Z X Y Z X Y Z X Y Z

1 1 1 11

1 1 1

θ ∆θ

∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ

∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆

+

− + + + + + + − + + + + + + −+ + + + +

+ + + + + + + + + + + + +

( ) ( )( )

( ) ( )( )

2 2 2X Y Z

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2Y X Y Z X Z X Y Z X X Y Z Y Z X Y Z X Y Z X Y Z

2 2 2 2 2 2 2 2 2 2 2 2 2 2X Y Z X Y Z X Y Z X Y Z X Y

1 1 1 1 1

1 1

θ ∆θ ∆θ

∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ

∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ ∆θ

+ +

+ + + + + + − − + + + + + + − + + + + +

+ + + + + + + + + + + +( )2 2 2 2Z X Y Z1 ∆θ ∆θ ∆θ

+ + +

X

Y

Z

x

y

z Z∆θ

X∆θ

Y∆θ

xyz

XYZ

Λ =

∆θ

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Approach & Methods: Module LinearizationModule Linear Features States (x, z) Inputs (u) Outputs (y) Jacobian Calc.

ElastoDyn(ED)

• Structural dynamics of:oBladesoDrivetrainoNacelleoToweroPlatform

• Structural degrees-of-freedom (DOFs) & their 1st time derivatives (continuous states)

• Applied loads along blades & tower

• Applied loads on hub, nacelle, & platform

• Blade-pitch-angle command

• Nacelle-yaw moment• Generator torque

• Motions along blades &tower

• Motions of hub, nacelle, & platform

• Nacelle-yaw angle & rate• Generator speed• User-selected structural

outputs (motions &/or loads)

• Numerical central-difference perturbation technique*

HydroDyn(HD)

• Wave excitation• Wave-radiation

added mass• Wave-radiation

damping• Hydrostatic

restoring• Viscous drag

• State-space-based wave-excitation (continuous states)

• State-space-based radiation (continuous states)

• Motions of platform• Wave-elevation

disturbance

• Hydrodynamic applied loads along platform

• User-selected hydrodynamic outputs

• Analytical for state equations

• Numericalcentral-difference perturbation technique* for output equations

MAP++ (MAP)

• Mooring restoring

• Mooring line tensions (constraint states)

• Positions of connect nodes (constraint states)

• Displacements of fairleads

• Tensions at fairleads• User-selected mooring

outputs

• Numericalcentral-difference perturbation technique*

*Numerical central-difference perturbation technique (see paper fortreatment of 3D rotations)

( ) ( )op op op op op op

op

X x x,u ,t X x x,u ,tX etc.x 2 x

∆ ∆

∆

+ − −∂=

∂

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StructuralDiscretization

HydrodynamicDiscretization

Mapping

• Module inputs & outputsresiding on spatial boundariesuse a mesh, consisting of:o Nodes & elements (nodal

connectivity)o Nodal reference locations

(position & orientation)o One or more nodal fields,

including motion, load, &/or scalar quantities

• Mesh-to-mesh mappings involve:o Mapping search – Nearest

neighbors are foundo Mapping transfer – Nodal fields

are transferred• Mapping transfers & other

module-to-module input-output coupling relationships have been linearized analytically

Approach & Methods: Glue-Code Linearization

op op

U U0 u yu y

∆ ∆∂ ∂= +∂ ∂

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

IfW

AD

IfW

ED ED ED EDSrvD

BD AD HD MAPED

BD BD\BD

BD ADopAD

AD

HDAD

MAP HD

HD

UI 0 0 0 0 0u

0 I 0 0 0 0 0uU U U Uu 0 0 Iu u u uu

U U Uu 0 0 0 0 0u u u uu

U0 0 0 0 0 0u uu U0 0 0 0 0 0

u0 0 0 0 0 0 I

∆

∆

∆

∆ ∆

∆

∆

∆

∂

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂= =

∂ ∂ ∂ ∂ ∂ ∂

∂

op

etc.

with

( )yuU ,0 =