FAST linearization V7

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,

Dear Jason,

Thank you so much for your quick email response and answer. Actually, I have applied inverse MBC to the linear model. The figure is after inverse MBC transformation. Does this mean the linear model is not so exact and have some accuracy problem?

Thank you again and all the best to you.

Yanhua

Dear Yanhua,

Have you applied the inverse MBC3 transform to the outputs i.e. have you taken the collective, cosine, and sine flapwise bending moments and converted them to individual blade bending moments? I would expect this to yield oscillations in the individual blade bending moment if the cosine and/or sine terms are nonzero; are they?

Best regards,

Dear Jason,

Sorry for the delay. Yep, I have applied the inverse MBC to the individual blade bending moments and also the cosine and/or sine terms are nonzero. The blade bending moments did not show any periodicity in 1P after inverse MBC. I tried another linearization by adding tower DOFs. But the results are similar and still fail to show the periodicity in 1P. So do you think what the possible problems are? Thanks a lot. I really feel desperate about this.

Happy Easter! Enjoy the holiday. And all the best to you.

Yanhua

Dear Yanhua,

I would guess that the problem is related to how you are applying the inverse MBC. I guess you wrote a script that that takes the collective, cosine, and sign terms and converts them individual blade coordinates i.e. q_b = q_0 + q_cCOS(azimuth_b) + q_sSIN(azimuth_b)–is that correct? Then, if q_c and/or q_s are nonzero, then q_b should naturally be periodic with the blade azimuth angle.

Best regards,

Dear Jason,

Nope, I think that the inverse MBC is right. I found something is wrong with my linearization results. I linearized the NREL 5MW wind model at 18m/s with two DOFs like Generator DOF and 1st blade flapwise bending DOF (using Test18.fst). The inputs used are three individual pitch angles and the horizontal wind speed is set as the disturbance input.

Can you ask a question about the set of VSContrl in “SERVODYN.dat”? Should we set this as 0 or 1? If I set as 1 (I tried), are the related parameters in “Simple variable-speed torque control” right?

I found the original 9 inputs including “Blade 1/2/3 pitch command” and “Extended input: collective blade-pitch command”. Because i want to design individual pitch control, I chose “Blade 1/2/3 pitch command” as the inputs. However, I have a confusion about the difference between “Blade 1/2/3 pitch command” and “Extended input: collective blade-pitch command”. If I chose the before as the inputs, do I need the latter one as the inputs? Thanks a lot.

By the way, I attached my linearization files. Can you help me to check and make sure everything is right?

Thanks a lot. All the best.

Yanhua

Dear Yanhua,

Yes, if you set VSContrl = 1 then the related parameters in the “simple variable-speed torque control” section are used. If you set VSContrl = 0, then you should select either GenModel = 1 or GenModel = 2 during a linearization analysis.

The “collective blade-pitch command” was added as an input to show the influence of consistent blade-pitch commands across all blades. You can always set the inputs you don’t need to zero.

Your input files look fine to me.

Best regards,