# Linearisation degree of freedom units and locations

Hello

I’ve been trying to get my head around the results of linearisation analyses that I’ve been undertaking.

I’d like to confirm the units used for the state vector, x, (of xdot = Ax + Bu, y = Cx + Du) because this helps me confirm what my x and hence my y is. I think I’ve been making a mistake because I always think of a generator as having a speed, not a position, but would like to confirm.

I suspect that:
DOF - Description - Units

FlapDOF1 - First flapwise blade mode DOF (flag) - m
FlapDOF2 - Second flapwise blade mode DOF (flag) - m
EdgeDOF - First edgewise blade mode DOF (flag) - m
DrTrDOF - Drivetrain rotational-flexibility DOF (flag) - rad
GenDOF - Generator DOF (flag) - rad
YawDOF - Yaw DOF (flag) - rad
TwFADOF1 - First fore-aft tower bending-mode DOF (flag) - m
TwFADOF2 - Second fore-aft tower bending-mode DOF (flag) - m
TwSSDOF1 - First side-to-side tower bending-mode DOF (flag) - m
TwSSDOF2 - Second side-to-side tower bending-mode DOF (flag) - m
CompAero - Compute aerodynamic forces (flag) - N/A
CompNoise- Compute aerodynamic noise (flag) - N/A

Lastly, I did a linearisation with only GenDOF and CompAero TRUE. My only output defined in my .fst file is GenSpeed. My C matrix is [0 926.3]. 926.3 is equal to 97 / 0.1047, and my Gearbox ratio is 97 and a rad/s to rpm conversion is 1/0.1047.

So, if my degree of freedom is a generator one, and my output is generator speed why does the gearbox ratio appear in my C matrix? Is the generator DOF more like a rotor DOF?

Thanks for you help.

Alec

Also…

What is the sign convention for the torque in the linearisation results?

Is it positive torque causes an increase in rotor position, like all good physics would tell you, or is it positive torque slows the rotor and extracts energy, like all good wind turbine engineers think it should, and the rest of FAST considers that it does?

This would help explain why my B matrix is [ 0; -2.215x10^-6], when B is often [0; 1/mass].

Thanks

Alec

Another also…

What are the input units, for the generator torque (kNm or Nm) and the pitch (deg or rad)?

Thanks

Dear Alec,

the information you need is this PDF, technical and user’s manual for the fast_ad advanced dynamics code.

Hi Eneko

I’ve had a quick look through the document and couldn’t find anything to explain why a gearbox ratio is involved when you have GenDOF, generator torque as an input and generator rpm as an output in your linearisation results.

Also, not much on the units used for the values in the state space matrices. Maybe you could point out where in the document these are explained?

I have had a break through though. It turns out a lot of my problems were caused because negative gains are required for the controller gain for the torque control loop. (see my earlier query about the sign convention for torque) If you have the sign wrong, your MATLAB open loop bode plot still looks sensible, but the closed loop one does not.

Another problem I was having was with the MATLAB function bode(sys,w). I was using the w bit since my bode plots went down to 10^-20 rad/s, which, as it turns out, is a period of 5x10^13 years and significantly longer than the universe has been in existence (13.8x10^9 years)… Anyway… the MATLAB help describes w as

Which I took to mean you should do something like

``w = [0.01 100];``

and it would plot all the points needed between 0.01rad/s and 100rad/s. But it kept giving me daft results. Turns out it was only plotting the point at 0.01 and 100 and nothing in between. Something like this is needed:

```a = -2:0.1:2; b = 10*ones(1,size(a,2)); w = power(b,a);``` I write all the above in the hope that someone else who is having the same issue will stumble across this info and find it useful.

Still, any explanation about the units in the state space matrices and the GenDOF/gearbox ratio would be useful.

Thanks

Dear Alec,

Your interpretation of the units of the internal FAST DOFs in the FAST linearization results in your Apr 11 post is correct.

The internal generator DOF in FAST is the azimith angle (position) of the low-speed shaft (not high-speed shaft) at the entrance to the gearbox. The high-speed shaft rotates faster than the internal generator DOF via the specification of the gearbox ratio. The azimuth angle of the hub/rotor equals the value of the generator DOF plus the twist of the shaft via the drivetrain rotational-flexibility DOF.

In the FAST linearization results, it is only the internal states (x) that follow the internal FAST DOF convention. The control inputs (u) in the FAST linearization results have the same units/conventions as the associated inputs in the FAST control subroutines. For example, the generator torque control input has the units of Nm and is defined such that a positive value reflects power extracted from the wind and a negative value represents a motoring-up situation (power input). For the pitch-control input, the pitch angle is specified in rad. The system outputs (y) in the FAST linearization output are in the units/conventions described in the FAST user’s guide and associated output documentation. For example, the generator torque output is in kNm and the pitch angle output is in deg. You’ll see these unit conversions show up in the state-space matrices (A,B,C,D).

I hope that helps.

Best regards,

Thanks Jason

Yes, that helps.

Alec