# FAST linearization V7

Hi all,

Thanks for your time. I linearized the 5MW FAST model at 36 azimuth angles (10 degrees increase) then obtained the first-order average-azimuthed linear model using MBC. I activated 4 DOFs (Generator DOF and First flapwise blade mode DOF). My linearization is done under wind speed 18m/s without any vertical shear. In this operating point, the steady pitch angle is 14.93 degrees. There are 3 inputs: individual pitch angle of blade i (i=3) and no disturbance.

The inputs for the linearized model (after MBC transformation) is [uo,uc,us] which means the zero component,cosine, sine components respectively.

I am trying to compare the simulation results between the nonlinear model and the obtained linearized model. I studied how the rotor speed and 3 flap-wise bending moments change if the pitch angle changes from 14.93 degrees to 14.93+1 degrees. The simulation model is like:

The results from nonlinear model and linearized model is shown as following

We can see that the rotor speed in linearized model is right but for the flapwise bending moments are unstable. Can anyone help me to find the possible problems? Thank you so much.

Yanhua

Dear Yanhua,

I would not call the increasing flapwise moment from the linearized model "unstable’, as the increase does not grow exponentially.

The problem I see in your comparison is that the rotor speed has been changed as a result of the pitch step, but this change in rotor speed doesn’t seem to be affecting the rotor-azimuth angle of your linearized model.

Best regards,

Dear Jason,

You are right. I changed the azimuth angle this time. I approximated the azimuth angle as the integral of rotor speed. But the problem still remained. I can’t figure out what the reason is. Actually I found my problem is same with the case 2 “- 2 Deg Pitch Increment in Open Loop Condition.” in previous post in [url]http://forums.nrel.gov/t/fast-linearized-models/249/1]. He said that the possible reason may be “the offset error in the DOF_GeAz derivative causes non stationary terms in some output variables.” But I can not understand it. Can you help me to fix this problem?

Thank you so so much.

All the best.

Yanhua

Dear Yanhua,

I’m not sure. To eliminate the variations in rotor speed and azimuth as a source of possible error, I suggest simplifying the model to 3 DOFs i.e. the first flapwise blade mode of each blade. Do you get the result you expect then?

Best regards,

Dear Jason,

Thanks for all your help. Actually, I still have the problem that the output of flapwise bending moment from the linearized model (after MBC) is increasing slowly with time. I tried using 3 DOFs only the first flapwise blade mode of each blade and the result is same. You know, the 3 outputs change from 3 flapwise bending moments xfl1, xfl2, xfl3 to xflo (the mean value), xflc (cosine component),xfls (sine component) after MBC transformation. The studied case is the pitch angle changes from 14.93 degrees to 14.93+1 degrees. The outputs of xflo, xflc, xfls (similar with xflc) are shown as:

You can see the output of xflo is right. But xflc and xfls is changing with time. I think xflc and xfls are accumulated variables but not the value at a time. They should be a constant value. Because with 1-degree pitch angle change, the values of xflc and xfls should stay at a constant value for example 50. But the actual outputs of xflc and xfls are keep increasing like 50+50+50… as time increases. So I add a deviation module for xflc and xfls each. And then after the inverse MBC, the output of flapwise bending moment is what I want.

But I can’t figure out why it is like this. Can you help me with this? Do you have any idea?

Thanks a lot.

All the best.
Yanhua

Dear Yanhua,

I’m sorry, but I still don’t know enough about what you are doing to understand what the problem is. My understanding is as follows:

• Your linearized model has 6 states (blade flapwise bending deflection and first time derivatives), 3 inputs (blade pitch), and 3 outputs (blade-root moments).
• You’ve applied the MBC-transformation and azimuth-averaged the linearized matrices so that A, B, C, and D are now LTI with the states, inputs, and outputs in terms of 0 (mean), C (cosine), and S (sine) components.
• You apply a step-change in the 0 (mean) input (without changing the C and S inputs), time integrate the LTI system, and obtain linearly increasing C and S outputs.

Is my understanding correct?

How are the states reacting to the step change in the 0 input? What happens if you eliminate the states, yielding only the D matrix?

Best regards,

Dear Jason,

Your understanding is correct except that there 4 DOFs (Generator DOF and First flapwise blade mode DOF) and 4 outputs (rotor speed and 3 flapwise bending moments). I am using FAST V7 for linearization. My linearization is done under wind speed 18m/s without any vertical shear. In this operating point, the steady pitch angle is 14.93 degrees. There are 3 inputs: individual pitch angle of blade i (i=3) and no disturbance. I give a step change in the 0 input, zero for cosine and sine inputs.The four states rotor azimuth angle, xfl0,xflc,xfls(similar with xflc) are like:

Is there any thing wrong with the states? Actually I think xflc and xfls should be stable like xfl0 but it is not. Because I do linearization on FAST V7, I don’t think it can eliminate the states, yielding only the D matrix. If it can, can you tell me how to do this? Thanks a lot.

All the best.

Yanhua

Dear Yanhua,

You have a state (rotor azimuth) this is linearly increasing; what happens when you eliminate the generator DOF such that the azimuth angle and rotor speed remain fixed?

What do the blade flapwise states look like?

Best regards,

Hi Jason,

I tried only using three DOF (1st flapwise blade mode), but the result is not so good. The states xfl0, xflc,xfls are all unstable like this:
xfl0.jpg
I attached my linearized file, can you help me check it, please? I feel really desperate about my result. I really appreciate your kind help. Thanks a lot.

All the best.

Yanhus

Dear Yanhua,

You’re FAST model and linearization output look reasonable to me.

After applying MBC3, you calculate the azimuth-averaged A, B, C, and D matrices, but applying a step input yields unstable states? How are you applying the step input and computing the time-domain solution?

Best regards,

Dear Jason,

Yes, I am really confused about this, but it is indeed unstable. This is what I used for the time-domain solution.Linearized model.jpg

I give “Pitch angle step” a step input then obtain the unstable results. Can you help me check it, please? Thank you so much.

All the best.

Yanhua

Dear Yanhua,

I’m not an expert in MATLAB/Simulink, but there seem some inconsistencies in your “linearized model” block:

• The state-derivatives seem to be calculated as x5-x7 in the “states” section, but are listed as x4-x6 in the “state vector” and “outputs” sections.
• The “goto” and “from” numbers are not consistent.

Best regards,

Dear Jason,

Yes, It indeed has the above problem. Sorry for the faults. I changed them. It seems reasonable now for 3 DOFs. But the problem still exists for 4 DOFs including the generator DOF. I checked my simulation and I don’t think something is wrong with it.[attachment=0]linearmodel_4DOF.jpg[/attachment]

But the state (rotor azimuth) is linearly increasing. How do you think of this? How does this happen? Thanks a lot.

All the best.

Yanhua

Dear Yanhua,

Once you apply MBC3 and azimuth average the A, B, C, and D matrices, I would expect the column of the A and C matrices associated with the generator-azimuth angle state to be effectively zero, which means that the generator azimuth angle can be eliminated as a state (resulting in 7 states instead of 8 for your case). Is this not what you seeing?

Best regards,

Dear Jason,

Actually no. I tried both FAST V7 and FAST V8. I found they both have this problem. I attached the GetMats.m, mbc3.m, my linearization result (4 DOFs: generator DOF, 1st flapwise) and other related procedures. Can you spare me some time? Can you help me to check it, please? I changed mbc3.m a little bit (only adding the related sentences of obtaining MBC_AvgB, MBC_AvgC etc. ). Thanks a lot for your kind help.

All the best.

Yanhua
FAST V7 linearizaiton.rar (33.8 KB)

Dear Yanhua,

I see that the (5,1) and (6,1) elements of MBC_AvgA are zero, but the (7,1) and (8,1) elements of MBC_AvgA are nonzero as a result of MBC3 and azimuth-averaging. Regardless, I still think you must eliminate this column (as well as the first column from MBC_AvgC) from your state-space model. This is because:

• AvgAMat clearly shows no influence from the generator-azimuth angle (the azimuth averaging eliminates the influence)
• The MBC_NaturalFrequency and MBC_DampingRatio clearly show a rigid-body mode associated with the generator-aximuth state i.e. there is no stiffness and the state will not return after being perturbed

Once you eliminate the generator-azimuth state (by eliminating the first column from MBC_AvgA and MBC_AvgC) you will not get a state that linearly increases after any perturbation, which is what you want.

I hope that helps.

Best regards,

Dear Jason,

Thanks a lot. When I eliminate the generator-azimuth state, it is exactly what I want. However, how to understand MBC_NaturalFrequency and MBC_DampingRatio? How to obtain the information of a rigid-body mode associated with the generator-aximuth state from MBC_NaturalFrequency and MBC_DampingRatio? I know little about this. Can you recommend some document for this? Thank you again!

All the best.

Yanhua

Dear Yanhua,

As described in this forum topic: http://forums.nrel.gov/t/learizing-baseline-5mw-wind-turbine-with-fast/494/1, rigid-body modes (i.e. modes without stiffness) show up in MBC3 as a pair of zero-valued (or near-zero-valued) frequencies with +/- inf damping (i.e., eigenvalues with real values only). That is, each rigid-body mode will introduce an additional mode beyond the number of enabled DOFs and the damping is unphysical.

Best regards,

Dear Jason,

Thanks for your time. Actually, I found something wrong about the linearization result. If we eliminate the generator-azimuth state in the linearized state-space model, the flapwise bending moment from linearization will lose the periodicity just as the red line shown in the Fig. 1 I attached. The blue one is the linearized model including the generator azimuth state while the yellow one is from nonlinear model. How can I keep the periodicity of flapwise bending moment from the linearization? Thanks a lot.

Dear Yanhua,

Your linear model is obtained by applying MBC3 and azimuth-averaging, correct? In this case, the states and outputs in the rotating frame have been transformed into the nonrotating frame. You will see the periodicity of the flapwise bending moment again if you transform the output of the linearized model back into the rotating frame (by applying the inverse MBC3).

Best regards,