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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= − −
= +
≅
∫
5
• 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∆ ∆=
7
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
8
• 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
9
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
10
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 – [email protected]
12
• 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
11
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
13
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
∆ ∆
∆
+ − −∂=
∂
14
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 =