## Linearization of 5MW with Individual Pitch angles

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Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Linearization of 5MW with Individual Pitch angles

Dear all,

I've a question regarding some linearization results I've got, which I couldn't understand. So I guess your help is needed

I'm trying to linearized the 5MW standard wind turbine with the following features:

Enabled DOF: GenDOF
Enabled Control Inputs: individual pitch of blade 1,2,3

Also As recommended by FAST User's guide I've set the following

Code: Select all

`TrimCase = 3 (Collective Pitch)YCMode = 0PCMode = 0VSContrl = 1VS_RtGnSp = 9999.9E-9VS_RtTq = 43093.55 [N.m] (rated torque)VS_Rgn2K = 9999.9E-9VS_SlPc = 9999.9E-9BlPitch(i) = 12.0534 [deg]  (i=1,2,3) RotSpeed = 12.1 [rpm]`

Also didnot include any platform effects.

Code: Select all

`PtfmModel = 0 `

The wind input is a 16 m/s constant wind without any turbulence or shear components.
Sample of wind input file

Code: Select all

`   Time  HorSpd  WndDir  VerSpd  HorShr  VerShr  LnVShr  GstSpd!  (sec)   (m/s)   (deg)   (m/s)     (-)     (-)     (-)   (m/s)  0.000  16.000   0.000   0.000   0.000   0.000   0.000   0.000  0.025  16.000   0.000   0.000   0.000   0.000   0.000   0.000  0.050  16.000   0.000   0.000   0.000   0.000   0.000   0.000  0.075  16.000   0.000   0.000   0.000   0.000   0.000   0.000`

The linearized equation of motion is of the form

M*q''+C*q' + K*q = a1*beta1 + a2*beta2 + a3*beta3

q is the GenDOF
beta1,beta2,beta3 are the individual blade pitch angles

Taking a look at the linearized equations of motions before doing azimuth averaging, and plotting the gains a1, a2, a3 as functions of the azimuth angle will give the following

Aerodynamic_Gains.jpg (102.96 KiB) Viewed 3327 times

as far as I know is that the gains a1, a2, a3 are the aerodynamic gains which I expected, in such simplified model, to be equal to each other and constant (not changing with azimuth angle).

Any ideas how to explain the dependency on azimuth angle.

Edited to correct a mistake:
Actually the plotted gains F1, F2, F3 are the same gains a1, a2, and a3 mentioned in the equation.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization of 5MW with Individual Pitch angles

Dear Rannam,

Even without shear in the wind input, the NREL 5-MW turbine has a shaft tilt that will induce shear. A similar effect would be caused by a nacelle-yaw error (if there was one). Asymmetry in the flow will cause a 1P oscillation in the rotating frame (represented by the individual colors in your plot), which leads to a 3P oscillation in the fixed frame (represented by all colors in your plot combined).

I hope that helps.

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov

Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Re: Linearization of 5MW with Individual Pitch angles

Thanks Jason,
Your note helped me a lot, I totally forgot about the shaft tilt

By the way, I'm having some problems in accessing wind.nrel.gov today, I always get "The specified URL cannot be found".
I hope the problem will be solved soon.

Thanks again and best regards.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization of 5MW with Individual Pitch angles

Dear Rannam,

We've had some power outage issues recently (due to a storm), but the servers should be back up now.

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov

Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Re: Linearization of 5MW with Individual Pitch angles

Dear Jason,

I've another question, which needs your help to clear it out.
I'm trying to build a simple linearized model for a floating wind turbine with the following specifications:
1- Enabled DOF: Platform pitch tilt rotation DOF PtfmPDOF = true, and GenDOF = true.
2- Enabled Control Inputs: Generator Torque, collective pitch angle (2 inputs).
3- YCMode = 0
4- PCMode = 0
5- VSControl = 1; VS_RtTq=43093.55 and VS_RtGnSp=VS_Rgn2K=VS_SlPc=9999.9E-9 (as recommended in FAST-linearization Manual)
6- Steady wind with mean wind speed 18m/s and vertical shear power component 0.14.
7- PtfmModel = 3, Platform_ITIBarge4

The linearized model has the form: Mx"+Cx'+Kx = F + Fd
x = [xi, psi]' where xi is the platform pitch angle and psi is the rotor azimuth angle. x'= [xi', Omega_r] and x"=[xi", Omega_r'].
My question is regarding the K Matrix. Through my theoretical development, the obtained K matrix has K(1,1) as non zero element, the other elements in the matrix are 0. However the obtained one from FAST is as following: K= [1.641e+9 -456.28; 2.4336e+6 4.0222]. The element K(2,1) = 2.4336e+6 is too big, and indicates that the platform pitching angle affects the equation of motion of the rotor which is simply modeled as

J_eq*Omega_r' = Q_aero - N*Q_gen

where:
- J_eq is the equivalent moment of inertia of the rotor and generator referenced to the law speed shaft side,
- Omega_r' is the rotor angular acceleration,
- Q_aero is the aerodynamic torque,
- Q_gen is the generator torque,
- N is the gearbox ratio.

Expanding the aerodynamic torque at the operating point we get:

Q_aero = Q_aero(op) + a1*D_beta + a2*D_Omega_r + a3*D_V - a4*D_xi'

where:
- a1,...a4 are the partial derivatives of the aerodynamics torque for the corresponding variable evaluated at the operating point.
- D_beta is the perturbation of collective blade pitch angle.
- D_Omega_r is the perturbation of rotor speed.
- D_V is the perturbation of wind speed.
- D_xi' is the perturbation of platform pitching velocity.
I can't see in this simple model any term related to platform pitching angle, the only term related to platform motion is it's velocity.

The same problem I've noticed when I used an onshore turbine, (PtfmModel = 0), and enabled the tower fore-aft first mode.

The only reason for such dependency I can think about is as following: the pitching angle of the platform (or fore-aft binding of the tower) changes the actual area of the rotor that is perpendicular to wind direction, when the platform angle is 0 this area is a circle and equals to the rotor disc, when the platform pitching angle deviates from 0 the rotor disc should be projected to the vertical plane to get the real area (elliptical) that is facing the wind. However this will introduce a term related to cos(xi) or to cos(tower fore-aft binding angle).

I'm wondering if you could direct me to get an answer to this question.

Best Regards.
Rannam Chaaban.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization of 5MW with Individual Pitch angles

Dear Rannam,

I suspect the coupling between platform pitch angle and rotor azimuth angle that you are seeing in your FAST linearization results is coming through what is known as the "aerodynamic stiffness" -- that is, the displacement of the turbine can impact the airfoil angle of attack and resulting aerodynamic loads, leading to an effective stiffness.

I'm assuming the stiffness matrix you reported is only for a given azimuth angle, and not azimuth averaged over many azimuth angles. You may wish to azimuth average the stiffness matrix over many azimuth angles.

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov

Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Re: Linearization of 5MW with Individual Pitch angles

Dear Jason,
The stiffness matrix reported in my post is the azimuth average one, the following figure shows the change of the stiffness matrix with the azimuth angle
StffMat.jpeg (114.55 KiB) Viewed 3239 times

The change in elements K>(1,2) and K(2,2) is around zero, so averaging will give small values. However, this does not apply for element K(2,1) which is still big compared to K(1,1) that I get in my theoretical model.
I'm wondering if you could guide me to some literature that talks about such topic, and that would enable me to extract some theoretical model that can explain the obtained results.

Best Regards
Rannam Chaaban.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization of 5MW with Individual Pitch angles

Dear Rannam,

I can't think of a good reference that explains "aerodynamic stiffness."

It is a bit peculiar to me that the stiffness matrix is not symmetric. This may be a result of the way the quasi-steady induction is recalculated with every perturbation in the linaerization process (and may not be realistic). In some cases, it is more accurate to assume a frozen wake during linearization--see, for example, the forum topic found here: viewtopic.php?f=4&t=394. I suggest you see if your linearization results are drastically different with frozen wake. If so, I would use the results with frozen wake.

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov

Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Re: Linearization of 5MW with Individual Pitch angles

Dear Jason,

Here are the result I've obtained using the frozen talk assumption.

Running the frozen wake enabled FAST and AeroDyn to get linearized model of the floating turbine did not converge after 9999 seconds. So I was not able to get the linearized model of floating turbine. Instead I've removed the platform effect be setting PtfmModel = 0, and enabling the first mode for tower fore-aft motion. (in addition to GenDOF, so in total 2DOF system, Inputs GenTq + Collective Pitch, No Disturbance).

Linearizing the model using the enabled frozen wake assumption will give averaged K matrix as: avgK = [1.779e+6 1924; 46492 6175.7]; and the periodic change of the elements of K with azimuth are as following:
Frzn_StffMat_Tower_1st_Foreaft_Mode.jpeg (114.28 KiB) Viewed 3229 times

Now, running linearization without the frozen wake assumption will give an averaged K matrix as: avgK = [1.779e+6 -10.806; 39524 -123.17]; and the periodic change of K elements with azimuth angle as:
StffMat_Tower_1st_Foreaft_Mode.jpeg (124.92 KiB) Viewed 3229 times

Again, K matrix is not symmetric, and K(2,1) is still big, and the only explanation for it is as you mentioned: Aerodynamic Stiffness, which for the moment is still ambiguous for me.

Anyway, as you mentioned that the frozen wake assumption will give more close to real results, I've to ask your help to get a linearized model for the case when I've enabled pitching angle of floating platform, in addition to individual blade pitching.

Best regards.
Rannam Chaaban
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization of 5MW with Individual Pitch angles

Dear Rannam,

It looks like the assymmetry in the stiffness matrix persists with and without frozen wake. There is no reason why the matrix needs to be symmetric, but it is interesting when it isn't.

Regarding the inability of the model with platform-pitch DOF enabled to reach a steady-state condition, the error message FAST returns provides some suggestions on how to proceed. Have you plotted the Displacement and Velocity 2-Norms to see if they are converging? What features of the FAST model have you enabled? Have you tried increasing the sytsem damping values?

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov

Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Re: Linearization of 5MW with Individual Pitch angles

Dear Jason,

Following are the errors I get when I try to linearize with frozen wake assumption, and Barge platform.

Code: Select all

`  Running FAST (v6.01, 12-Aug-2005).   NREL 5.0 MW Baseline Wind Turbine for Use in Offshore Analysis.  Heading of the aerodyn.ipt file : NREL 5.0 MW offshore baseline aerodynamic input properties; Compatible with Aero  Detected hub-height wind file:    "..\..\WP_Baseline\laminare\Laminare_H90_18mps_VShr014.hh"   Aerodynamics loads calculated using AeroDyn(12.58, 28-Jun-2005)   NOTE: AToler changed from 0.005 to 1E-6 in order to reduce numerical errors  during FAST linearization.   Beginning iteration to find a steady state solution of type:   Trimmed collective blade pitch (TrimCase = 3)                Avg Rotor    Nacelle  Generator      Blade   Dsplcmnt   Velocity  Iter    Time     Speed Yaw Demand     Torque      Pitch     2-norm     2-norm  Nmbr   (sec)     (rpm)      (deg)     (kN-m)      (deg)      (rad)    (rad/s)  -----------------------------------------------------------------------------     1    4.96  11.54597    0.00000   43.09355   14.99196  0.2998987  0.1618203 WARNING:                                                               Small angle assumption violated in SUBROUTINE SmllRotTrans() due to   a large platform displacement.  The solution may be inaccurate.   Future warnings suppressed.  Simulation continuing...      2    9.92   8.72620    0.00000   43.09355   14.72962  1.7990638  0.4654096   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...     3   14.88   3.17486    0.00000    0.00001   13.13206  2.2133889  0.7662420   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VINDERR.  High VNB velocity encountered during induction factor calculation.   Blade number    2   Element number 17   VNW =             2.884152   VNB =            -100.3423   Program continues...   Program warning in Subroutine VINDERR.  High VNB velocity encountered during induction factor calculation.   Blade number    2   Element number 17   VNW =             2.875064   VNB =            -100.9163   Program continues...   Program warning in Subroutine VINDERR.  High VNB velocity encountered during induction factor calculation.   Blade number    2   Element number 17   VNW =              2.86593   VNB =            -101.4829   Program continues...   Program warning in Subroutine VINDERR.  High VNB velocity encountered during induction factor calculation.   Blade number    2   Element number 16   VNW =             2.851676   VNB =            -100.3411   Program continues...   Program warning in Subroutine VINDERR.  High VNB velocity encountered during induction factor calculation.   Blade number    2   Element number 17   VNW =             2.856744   VNB =            -102.0423   Program continues...   Program warning in Subroutine VINDERR.  Induced velocity warning written 5 times.   The message will not be repeated, though the condition may persist.   Program continues...     4   19.83  -4.25642    0.00000    0.00962    8.90583  3.8157122  1.0601093    5   24.79 -12.79717    0.00000    0.02342    1.16076  7.7903652  0.6011980    6   29.75  -5.69786    0.00000   43.09355  -10.62852  5.3169513  1.9201106    7   34.71  23.27443    0.00000   43.09355  -19.05614 12.9125834  4.6072226    8   39.67  60.70856    0.00000   43.09355  -13.76484  4.1766005  1.9580559    9   44.63  35.58517    0.00000   43.09355    9.25223  2.7398889  4.9526567   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations.   AeroDyn will continue using induction factors from previous successful time step.   Program continues...   Program warning in Subroutine VIND. Induction factor calculation did not converge after1000 iterations. `

This last error is repeated till time is 9999, then FAST will abort.

I've collected all the used files for this linearization for both with and without frozen wake assumption zipped file (http://www.rannam.com/downloads/2DOF_IT ... n_Walk.zip), Just run the Matlab file 'LinWT_Frzn.m' to see the full list of errors.
By the way, there is no problem is linearization without frozen wake assumption.

Another idea I would like to mention, is that this unexpected term K(2,1) = 2.2386e+06 (averages term from linearizing floating turbine) in the stiffness matrix does not make any big problem for the model when I move to state space form (which I'm using for my control), as this term is divided by the the term H*J_eq = 90*4.378e7 = 3.94e9 (Jeq = equivalent moment of inertia of the rotor and generator, H = Hub Height), this will give a value in state matrix A of -0.00056815 which is about 100 times smaller than the other values, and it does not affect the eigenvalues of state matrix A.

Thank you very much for you help, and Best regards
Rannam.
Last edited by Rannam.Chaaban on Thu Jun 13, 2013 2:47 am, edited 1 time in total.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization of 5MW with Individual Pitch angles

Dear Rannam,

I'm assuming that you grabbed the FAST executable supporting frozen wake during linearization that is available from here: http://wind.nrel.gov/public/jjonkman/FA ... zation.exe. This version was created quite a while ago and does not include the features of HydroDyn, such as waves, hydrodynamic/hydrostatic loads, and moorings. Thus, if you've enabled platform DOFs, the turbine will not be supported by buoyancy or moorings and will simply "fall" due to gravity -- hence all of the errors you are getting. Unfortunately, we have not created a more recent version of FAST that supports frozen wake during linearization.

Regardless, your results with the tower-bending mode DOF used in place of the platform-pitch DOF indicate that the main cause of the asymmetric stiffness matrix does not really depend on the use of frozen wake or not, so, I wouldn't worry about applying frozen wake. Also, it sounds like the asymmetry in the stiffness matrix is not causing a problem in your subsequent analysis, so, is not of too much concern.

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov

Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Re: Linearization of 5MW with Individual Pitch angles

Dear Jason,

The used FAST executable with frozen wake is the one you have mentioned.
As the frozen wake assumption does not affect the result, I assume that your first suggestion about the asymmetry in stiffness matrix is still due to Aerodynamic stiffness.

Thank you very much for your help.

Kind regards.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization of 5MW with Individual Pitch angles

Dear Rannam,

Yes, I assume that the asymmetric stiffness matrix is the result of "aerodynamic stiffness". When I led you to the version of FAST with frozen wake, I had forgotten how old this version was.

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov