Dear Forum Members,

First, I would like to thank the NREL team for their hard work in the development of these Software Tools.

Following, I will post the problem that I have with the linearisation.

I intend to linearise the NREL 5MW turbine model in region 3 of operation, in order to design control strategies. I set up the linearisation with all the changes as describe in NREL user manual pages 39-44.

For the liniarisation, I use a wind profile hitting the turbine from front with a wind velocity of 14 m/s.

The fst file for linearisation is as follows

```
--------------------------------------------------------------------------------
------- FAST INPUT FILE --------------------------------------------------------
NREL 5.0 MW Baseline Wind Turbine for Use in Offshore Analysis.
Properties from Dutch Offshore Wind Energy Converter (DOWEC) 6MW Pre-Design (10046_009.pdf) and REpower 5M 5MW (5m_uk.pdf); Compatible with FAST v7.02.
---------------------- SIMULATION CONTROL --------------------------------------
False Echo - Echo input data to "echo.out" (flag)
1 ADAMSPrep - ADAMS preprocessor mode {1: Run FAST, 2: use FAST as a preprocessor to create an ADAMS model, 3: do both} (switch)
2 AnalMode - Analysis mode {1: Run a time-marching simulation, 2: create a periodic linearized model} (switch)
3 NumBl - Number of blades (-)
999.0 TMax - Total run time (s)
0.0125 DT - Integration time step (s)
---------------------- TURBINE CONTROL -----------------------------------------
0 YCMode - Yaw control mode {0: none, 1: user-defined from routine UserYawCont, 2: user-defined from Simulink/Labview} (switch)
9999.9 TYCOn - Time to enable active yaw control (s) [unused when YCMode=0]
0 PCMode - Pitch control mode {0: none, 1: user-defined from routine PitchCntrl, 2: user-defined from Simulink/Labview} (switch)
0.0 TPCOn - Time to enable active pitch control (s) [unused when PCMode=0]
1 VSContrl - Variable-speed control mode {0: none, 1: simple VS, 2: user-defined from routine UserVSCont, 3: user-defined
9999.9E-9 VS_RtGnSp - Rated generator speed for simple variable-speed generator control (HSS side) (rpm) [used only when VSContrl=1]
43093.6 VS_RtTq - Rated generator torque/constant generator torque in Region 3 for simple variable-speed generator control (HSS side) (N-m)
9999.9e-9 VS_Rgn2K - Generator torque constant in Region 2 for simple variable-speed generator control (HSS side) (N-m/rpm^2)
9999.9e-9 VS_SlPc - Rated generator slip percentage in Region 2 1/2 for simple variable-speed generator control (%)
1 GenModel - Generator model {1: simple, 2: Thevenin, 3: user-defined from routine UserGen} (switch) [used only when VSContrl=0]
True GenTiStr - Method to start the generator {T: timed using TimGenOn, F: generator speed using SpdGenOn} (flag)
True GenTiStp - Method to stop the generator {T: timed using TimGenOf, F: when generator power = 0} (flag)
9999.9 SpdGenOn - Generator speed to turn on the generator for a startup (HSS speed) (rpm) [used only when GenTiStr=False]
0.0 TimGenOn - Time to turn on the generator for a startup (s) [used only when GenTiStr=True]
9999.9 TimGenOf - Time to turn off the generator (s) [used only when GenTiStp=True]
1 HSSBrMode - HSS brake model {1: simple, 2: user-defined from routine UserHSSBr, 3: user-defined from Labview} (switch)
9999.9 THSSBrDp - Time to initiate deployment of the HSS brake (s)
9999.9 TiDynBrk - Time to initiate deployment of the dynamic generator brake [CURRENTLY IGNORED] (s)
9999.9 TTpBrDp(1) - Time to initiate deployment of tip brake 1 (s)
9999.9 TTpBrDp(2) - Time to initiate deployment of tip brake 2 (s)
9999.9 TTpBrDp(3) - Time to initiate deployment of tip brake 3 (s) [unused for 2 blades]
9999.9 TBDepISp(1) - Deployment-initiation speed for the tip brake on blade 1 (rpm)
9999.9 TBDepISp(2) - Deployment-initiation speed for the tip brake on blade 2 (rpm)
9999.9 TBDepISp(3) - Deployment-initiation speed for the tip brake on blade 3 (rpm) [unused for 2 blades]
9999.9 TYawManS - Time to start override yaw maneuver and end standard yaw control (s)
9999.9 TYawManE - Time at which override yaw maneuver reaches final yaw angle (s)
0.0 NacYawF - Final yaw angle for yaw maneuvers (degrees)
9999.9 TPitManS(1) - Time to start override pitch maneuver for blade 1 and end standard pitch control (s)
9999.9 TPitManS(2) - Time to start override pitch maneuver for blade 2 and end standard pitch control (s)
9999.9 TPitManS(3) - Time to start override pitch maneuver for blade 3 and end standard pitch control (s) [unused for 2 blades]
9999.9 TPitManE(1) - Time at which override pitch maneuver for blade 1 reaches final pitch (s)
9999.9 TPitManE(2) - Time at which override pitch maneuver for blade 2 reaches final pitch (s)
9999.9 TPitManE(3) - Time at which override pitch maneuver for blade 3 reaches final pitch (s) [unused for 2 blades]
8.70 BlPitch(1) - Blade 1 initial pitch (degrees)
8.70 BlPitch(2) - Blade 2 initial pitch (degrees)
8.70 BlPitch(3) - Blade 3 initial pitch (degrees) [unused for 2 blades]
8.70 BlPitchF(1) - Blade 1 final pitch for pitch maneuvers (degrees)
8.70 BlPitchF(2) - Blade 2 final pitch for pitch maneuvers (degrees)
8.70 BlPitchF(3) - Blade 3 final pitch for pitch maneuvers (degrees) [unused for 2 blades]
---------------------- ENVIRONMENTAL CONDITIONS --------------------------------
9.80665 Gravity - Gravitational acceleration (m/s^2)
---------------------- FEATURE FLAGS -------------------------------------------
True FlapDOF1 - First flapwise blade mode DOF (flag)
False FlapDOF2 - Second flapwise blade mode DOF (flag)
True EdgeDOF - First edgewise blade mode DOF (flag)
False TeetDOF - Rotor-teeter DOF (flag) [unused for 3 blades]
True DrTrDOF - Drivetrain rotational-flexibility DOF (flag)
True GenDOF - Generator DOF (flag)
False YawDOF - Yaw DOF (flag)
True TwFADOF1 - First fore-aft tower bending-mode DOF (flag)
True TwFADOF2 - Second fore-aft tower bending-mode DOF (flag)
True TwSSDOF1 - First side-to-side tower bending-mode DOF (flag)
True TwSSDOF2 - Second side-to-side tower bending-mode DOF (flag)
True CompAero - Compute aerodynamic forces (flag)
False CompNoise - Compute aerodynamic noise (flag)
---------------------- INITIAL CONDITIONS --------------------------------------
0.0 OoPDefl - Initial out-of-plane blade-tip displacement (meters)
0.0 IPDefl - Initial in-plane blade-tip deflection (meters)
0.0 TeetDefl - Initial or fixed teeter angle (degrees) [unused for 3 blades]
0.0 Azimuth - Initial azimuth angle for blade 1 (degrees)
12.1 RotSpeed - Initial or fixed rotor speed (rpm)
0.0 NacYaw - Initial or fixed nacelle-yaw angle (degrees)
0.0 TTDspFA - Initial fore-aft tower-top displacement (meters)
0.0 TTDspSS - Initial side-to-side tower-top displacement (meters)
---------------------- TURBINE CONFIGURATION -----------------------------------
63.0 TipRad - The distance from the rotor apex to the blade tip (meters)
1.5 HubRad - The distance from the rotor apex to the blade root (meters)
1 PSpnElN - Number of the innermost blade element which is still part of the pitchable portion of the blade for partial-span pitch control
0.0 UndSling - Undersling length [distance from teeter pin to the rotor apex] (meters) [unused for 3 blades]
0.0 HubCM - Distance from rotor apex to hub mass [positive downwind] (meters)
-5.01910 OverHang - Distance from yaw axis to rotor apex [3 blades] or teeter pin [2 blades] (meters)
1.9 NacCMxn - Downwind distance from the tower-top to the nacelle CM (meters)
0.0 NacCMyn - Lateral distance from the tower-top to the nacelle CM (meters)
1.75 NacCMzn - Vertical distance from the tower-top to the nacelle CM (meters)
87.6 TowerHt - Height of tower above ground level [onshore] or MSL [offshore] (meters)
1.96256 Twr2Shft - Vertical distance from the tower-top to the rotor shaft (meters)
0.0 TwrRBHt - Tower rigid base height (meters)
-5.0 ShftTilt - Rotor shaft tilt angle (degrees)
0.0 Delta3 - Delta-3 angle for teetering rotors (degrees) [unused for 3 blades]
-2.5 PreCone(1) - Blade 1 cone angle (degrees)
-2.5 PreCone(2) - Blade 2 cone angle (degrees)
-2.5 PreCone(3) - Blade 3 cone angle (degrees) [unused for 2 blades]
0.0 AzimB1Up - Azimuth value to use for I/O when blade 1 points up (degrees)
---------------------- MASS AND INERTIA ----------------------------------------
0.0 YawBrMass - Yaw bearing mass (kg)
240.00E3 NacMass - Nacelle mass (kg)
56.78E3 HubMass - Hub mass (kg)
0.0 TipMass(1) - Tip-brake mass, blade 1 (kg)
0.0 TipMass(2) - Tip-brake mass, blade 2 (kg)
0.0 TipMass(3) - Tip-brake mass, blade 3 (kg) [unused for 2 blades]
2607.89E3 NacYIner - Nacelle inertia about yaw axis (kg m^2)
534.116 GenIner - Generator inertia about HSS (kg m^2)
115.926E3 HubIner - Hub inertia about rotor axis [3 blades] or teeter axis [2 blades] (kg m^2)
---------------------- DRIVETRAIN ----------------------------------------------
100.0 GBoxEff - Gearbox efficiency (%)
94.4 GenEff - Generator efficiency [ignored by the Thevenin and user-defined generator models] (%)
97.0 GBRatio - Gearbox ratio (-)
False GBRevers - Gearbox reversal {T: if rotor and generator rotate in opposite directions} (flag)
28.1162E3 HSSBrTqF - Fully deployed HSS-brake torque (N-m)
0.6 HSSBrDT - Time for HSS-brake to reach full deployment once initiated (sec) [used only when HSSBrMode=1]
"Dummy" DynBrkFi - File containing a mech-gen-torque vs HSS-speed curve for a dynamic brake [CURRENTLY IGNORED] (quoted string)
867.637E6 DTTorSpr - Drivetrain torsional spring (N-m/rad)
6.215E6 DTTorDmp - Drivetrain torsional damper (N-m/(rad/s))
---------------------- SIMPLE INDUCTION GENERATOR ------------------------------
9999.9 SIG_SlPc - Rated generator slip percentage (%) [used only when VSContrl=0 and GenModel=1]
9999.9 SIG_SySp - Synchronous (zero-torque) generator speed (rpm) [used only when VSContrl=0 and GenModel=1]
9999.9 SIG_RtTq - Rated torque (N-m) [used only when VSContrl=0 and GenModel=1]
9999.9 SIG_PORt - Pull-out ratio (Tpullout/Trated) (-) [used only when VSContrl=0 and GenModel=1]
---------------------- THEVENIN-EQUIVALENT INDUCTION GENERATOR -----------------
9999.9 TEC_Freq - Line frequency [50 or 60] (Hz) [used only when VSContrl=0 and GenModel=2]
9998 TEC_NPol - Number of poles [even integer > 0] (-) [used only when VSContrl=0 and GenModel=2]
9999.9 TEC_SRes - Stator resistance (ohms) [used only when VSContrl=0 and GenModel=2]
9999.9 TEC_RRes - Rotor resistance (ohms) [used only when VSContrl=0 and GenModel=2]
9999.9 TEC_VLL - Line-to-line RMS voltage (volts) [used only when VSContrl=0 and GenModel=2]
9999.9 TEC_SLR - Stator leakage reactance (ohms) [used only when VSContrl=0 and GenModel=2]
9999.9 TEC_RLR - Rotor leakage reactance (ohms) [used only when VSContrl=0 and GenModel=2]
9999.9 TEC_MR - Magnetizing reactance (ohms) [used only when VSContrl=0 and GenModel=2]
---------------------- PLATFORM ------------------------------------------------
0 PtfmModel - Platform model {0: none, 1: onshore, 2: fixed bottom offshore, 3: floating offshore} (switch)
"Dummy" PtfmFile - Name of file containing platform properties (quoted string) [unused when PtfmModel=0]
---------------------- TOWER ---------------------------------------------------
20 TwrNodes - Number of tower nodes used for analysis (-)
"NRELOffshrBsline5MW_Tower_Onshore.dat" TwrFile - Name of file containing tower properties (quoted string)
---------------------- NACELLE-YAW ---------------------------------------------
9028.32E6 YawSpr - Nacelle-yaw spring constant (N-m/rad)
19.16E6 YawDamp - Nacelle-yaw damping constant (N-m/(rad/s))
0.0 YawNeut - Neutral yaw position--yaw spring force is zero at this yaw (degrees)
---------------------- FURLING -------------------------------------------------
False Furling - Read in additional model properties for furling turbine (flag)
"Dummy" FurlFile - Name of file containing furling properties (quoted string) [unused when Furling=False]
---------------------- ROTOR-TEETER --------------------------------------------
0 TeetMod - Rotor-teeter spring/damper model {0: none, 1: standard, 2: user-defined from routine UserTeet} (switch) [unused for 3 blades]
0.0 TeetDmpP - Rotor-teeter damper position (degrees) [used only for 2 blades and when TeetMod=1]
0.0 TeetDmp - Rotor-teeter damping constant (N-m/(rad/s)) [used only for 2 blades and when TeetMod=1]
0.0 TeetCDmp - Rotor-teeter rate-independent Coulomb-damping moment (N-m) [used only for 2 blades and when TeetMod=1]
0.0 TeetSStP - Rotor-teeter soft-stop position (degrees) [used only for 2 blades and when TeetMod=1]
0.0 TeetHStP - Rotor-teeter hard-stop position (degrees) [used only for 2 blades and when TeetMod=1]
0.0 TeetSSSp - Rotor-teeter soft-stop linear-spring constant (N-m/rad) [used only for 2 blades and when TeetMod=1]
0.0 TeetHSSp - Rotor-teeter hard-stop linear-spring constant (N-m/rad) [used only for 2 blades and when TeetMod=1]
---------------------- TIP-BRAKE -----------------------------------------------
0.0 TBDrConN - Tip-brake drag constant during normal operation, Cd*Area (m^2)
0.0 TBDrConD - Tip-brake drag constant during fully-deployed operation, Cd*Area (m^2)
0.0 TpBrDT - Time for tip-brake to reach full deployment once released (sec)
---------------------- BLADE ---------------------------------------------------
"NRELOffshrBsline5MW_Blade.dat" BldFile(1) - Name of file containing properties for blade 1 (quoted string)
"NRELOffshrBsline5MW_Blade.dat" BldFile(2) - Name of file containing properties for blade 2 (quoted string)
"NRELOffshrBsline5MW_Blade.dat" BldFile(3) - Name of file containing properties for blade 3 (quoted string) [unused for 2 blades]
---------------------- AERODYN -------------------------------------------------
"NRELOffshrBsline5MW_AeroDyn.ipt" ADFile - Name of file containing AeroDyn input parameters (quoted string)
---------------------- NOISE ---------------------------------------------------
"Dummy" NoiseFile - Name of file containing aerodynamic noise input parameters (quoted string) [used only when CompNoise=True]
---------------------- ADAMS ---------------------------------------------------
"NRELOffshrBsline5MW_ADAMSSpecific.dat" ADAMSFile - Name of file containing ADAMS-specific input parameters (quoted string)
---------------------- LINEARIZATION CONTROL -----------------------------------
"NRELOffshrBsline5MW_Linear.dat" LinFile - Name of file containing FAST linearization parameters (quoted string)
---------------------- OUTPUT --------------------------------------------------
True SumPrint - Print summary data to "<RootName>.fsm" (flag)
3 OutFileFmt - Format for tabular (time-marching) output file(s) (1: text file [<RootName>.out], 2: binary file [<RootName>.outb], 3: both)
True TabDelim - Use tab delimiters in text tabular output file? (flag)
"ES10.3E2" OutFmt - Format used for text tabular output (except time). Resulting field should be 10 characters. (quoted string)
0.0 TStart - Time to begin tabular output (s)
1 DecFact - Decimation factor for tabular output {1: output every time step} (-)
1.0 SttsTime - Amount of time between screen status messages (sec)
-3.09528 NcIMUxn - Downwind distance from the tower-top to the nacelle IMU (meters)
0.0 NcIMUyn - Lateral distance from the tower-top to the nacelle IMU (meters)
2.23336 NcIMUzn - Vertical distance from the tower-top to the nacelle IMU (meters)
1.912 ShftGagL - Distance from rotor apex [3 blades] or teeter pin [2 blades] to shaft strain gages [positive for upwind rotors] (meters)
0 NTwGages - Number of tower nodes that have strain gages for output [0 to 9] (-)
TwrGagNd - List of tower nodes that have strain gages [1 to TwrNodes] (-) [unused if NTwGages=0]
3 NBlGages - Number of blade nodes that have strain gages for output [0 to 9] (-)
5,9,13 BldGagNd - List of blade nodes that have strain gages [1 to BldNodes] (-) [unused if NBlGages=0]
OutList - The next line(s) contains a list of output parameters. See OutList.txt for a listing of available output channels, (-)
"WindVxi" - Longitudinal, lateral, and vertical wind speeds
"GenPwr , GenTq" - Electrical generator power and torque
"BldPitch1" - Blade 1 pitch angle
"Azimuth" - Blade 1 azimuth angle
"RotSpeed , GenSpeed" - Low-speed shaft and high-speed shaft speeds
"OoPDefl1 , IPDefl1 , TwstDefl1" - Blade 1 out-of-plane and in-plane deflections and tip twist
"TTDspFA , TTDspSS , TTDspTwst" - Tower fore-aft and side-to-side displacments and top twist
END of FAST input file (the word "END" must appear in the first 3 columns of this last line).
--------------------------------------------------------------------------------
```

Linearisation file .dat

```
--------------------------------------------------------------------------------
---------------------- FAST LINEARIZATION CONTROL FILE -------------------------
NREL 5.0 MW offshore baseline linearization input properties.
---------------------- PERIODIC STEADY STATE SOLUTION --------------------------
True CalcStdy - Calculate periodic steady state condition {False: linearize about initial conditions} (flag)
3 TrimCase - Trim case {1: find nacelle yaw, 2: find generator torque, 3: find collective blade pitch} (switch)
0.0001 DispTol - Convergence tolerance for the 2-norm of displacements in the periodic steady state calculation (rad )
0.0001 VelTol - Convergence tolerance for the 2-norm of velocities in the periodic steady state calculation (rad/s)
---------------------- MODEL LINEARIZATION -------------------------------------
36 NAzimStep - Number of equally-spaced azimuth steps in periodic linearized model (-)
1 MdlOrder - Order of output linearized model {1: 1st order A, B, Bd, C, D, Dd; 2: 2nd order M, C, K, F, Fd, VelC, DspC, D, Dd} (switch)
---------------------- INPUTS AND DISTURBANCES ---------------------------------
2 NInputs - Number of control inputs [0 (none) or 1 to 4+NumBl] (-)
3,4 CntrlInpt - List of control inputs [1 to NInputs] { 3: generator torque, 4: collective blade pitch }
1 NDisturbs - Number of wind disturbances [0 (none) or 1 to 7] (-)
1 Disturbnc - List of input wind disturbances [1 to NDisturbs] {1: horizontal hub-height wind speed} (-)
```

The linearisation goes well, it finds steady state solution in 50 steps. The result is as follows:

```
This linearized model file was generated by FAST (v7.02.00d-bjj, 20-Feb-2013) on 05-Feb-2015 at 08:53:18.
The aerodynamic calculations were made by AeroDyn (v13.00.02a-bjj, 20-Feb-2013).
NREL 5.0 MW Baseline Wind Turbine for Use in Offshore Analysis.
Some Useful Information:
Type of steady state solution found Trimmed collective blade pitch (TrimCase = 3)
Azimuth-average rotor speed, RotSpeed (rad/s) 1.26711E+00
Period of steady state solution (sec) 4.95868E+00
Iterations needed to find steady state solution 60
Displacement 2-norm of steady state solution (rad) 6.29953E-05
Velocity 2-norm of steady state solution (rad/s) 4.66512E-05
Number of equally-speced azimuth steps, NAzimStep 36
Order of linearized model, MdlOrder 1
Number of active (enabled) DOFs 12 (24 states)
Number of control inputs, NInputs 2
Number of input wind disturbances, NDisturbs 1
Number of output measurements 13
Order of States in Linearized State Matrices:
Row/column 1 = 1st tower fore-aft bending mode DOF (internal DOF index = DOF_TFA1)
Row/column 2 = 1st tower side-to-side bending mode DOF (internal DOF index = DOF_TSS1)
Row/column 3 = 2nd tower fore-aft bending mode DOF (internal DOF index = DOF_TFA2)
Row/column 4 = 2nd tower side-to-side bending mode DOF (internal DOF index = DOF_TSS2)
Row/column 5 = Variable speed generator DOF (internal DOF index = DOF_GeAz)
Row/column 6 = Drivetrain rotational-flexibility DOF (internal DOF index = DOF_DrTr)
Row/column 7 = 1st flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,1))
Row/column 8 = 1st flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,1))
Row/column 9 = 1st flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,1))
Row/column 10 = 1st edgewise bending-mode DOF of blade 1 (internal DOF index = DOF_BE(1,1))
Row/column 11 = 1st edgewise bending-mode DOF of blade 2 (internal DOF index = DOF_BE(2,1))
Row/column 12 = 1st edgewise bending-mode DOF of blade 3 (internal DOF index = DOF_BE(3,1))
Row/column 13 to 24 = First derivatives of row/column 1 to 12.
Order of Control Inputs in Linearized State Matrices:
Column 1 = electrical generator torque (N·m) 4.30936E+04 op
Column 2 = rotor collective blade pitch (rad) 1.51366E-01 op
Order of Input Wind Disturbances in Linearized State Matrices:
Column 1 = horizontal hub-height wind speed (m/s) : See selected wind file for op
Order of Output Measurements in Linearized State Matrices:
Row 1 = WindVxi (m/s)
Row 2 = GenPwr (kW)
Row 3 = GenTq (kN·m)
Row 4 = BldPitch1 (deg)
Row 5 = Azimuth (deg)
Row 6 = RotSpeed (rpm)
Row 7 = GenSpeed (rpm)
Row 8 = OoPDefl1 (m)
Row 9 = IPDefl1 (m)
Row 10 = TwstDefl1 (deg)
Row 11 = TTDspFA (m)
Row 12 = TTDspSS (m)
Row 13 = TTDspTwst (deg)
```

In order to compare the results to the nonlinear model I have a second wind profile in which I set a step in the velocity from 14 m/s to 15m/s.

In Matlab, I simulate the averaged state space matrixes for the linearised model. In order to compare the results I use a wind step of one (corresponding to the 14m/s to 15m/s in the nonlinear case).

And I get the following result

In the figures, with black is represented nonlinear model output, and with blue the liniarised model output . Moreover, in the case of the liniarised model, the GenPwr and RotSpeed decrease, instead of getting to the same steady state as in the case of the nonlinear model. This effect is present when I set the FLAPDOF1 and EDGEDOF to be TRUE.

I cannot make any sense of this behavior, and I have tried several things, but ran out of ideas.

I hope that you can help me with a suggestion, on why is it happening, as maybe I am doing something wrong in the linearisation setup, or when I compare the nonlinear model to the liniarised model.

Thank you in advance!