FAST 7 Linearization

Wind & Water Computer-Aided Engineering Software Tools
1

Dear Jason,
I found some particularity linearizing the 5 MW reference on-shore wind turbine (FAST v7). The A matrix of the state space model obtained after linearization at different wind speeds in region 3 (making a pitch trim) differs if the torque characteristic is used as explained in the FAST manual (making VSContrl=1, VS_RtGnSp=1E-9, VS_RtTq=Tn, VS_Rgn2=1E-9, VS_SlPc=1E-9) or leaving as a “standard curve” (i.e. VSContrl=1, VS_RtGnSp= 1173.7, VS_RtTq=Tn, VS_Rgn2= 0.0255764, VS_SlPc=10). The linearization points result the same (torque, rotational speed, pitch angle, etc), but the difference appears to be in two coefficients corresponding to the First derivatives of Variable speed generator DOF (internal DOF index = DOF_GeAz)and Drivetrain rotational-flexibility DOF (internal DOF index = DOF_DrTr) with respect to the state Variable speed generator DOF (internal DOF index = DOF_GeAz (rows 7 and 8, and column 7 in the pictures below).

For example:
---------------------- FEATURE FLAGS ------------------------------
False FlapDOF1 - First flapwise blade mode DOF (flag)
False FlapDOF2 - Second flapwise blade mode DOF (flag)
False EdgeDOF - First edgewise blade mode DOF (flag)
False TeetDOF - Rotor-teeter DOF (flag) [unused for 3
True DrTrDOF - Drivetrain rotational-flexibility DOF
True GenDOF - Generator DOF (flag)
False YawDOF - Yaw DOF (flag)
True TwFADOF1 - First fore-aft tower bending-mode DOF (flag)
False TwFADOF2 - Second fore-aft tower bending-mode DOF
True TwSSDOF1 - First side-to-side tower bending-mode DOF
False TwSSDOF2 - Second side-to-side tower bending-mode DOF
True CompAero - Compute aerodynamic forces (flag)
False CompNoise - Compute aerodynamic noise (flag)

Linearization point:
Wind speed= 23.568 m/s,
Gen speed= 1173.7 rpm,
Pitch angle= 21.497 °,
Torque= 44205

For “standard curve” values
std.png
For Fast manual values
man.png
The difference is for 1: Run FAST and for 3: do both.

Therefore, my question is: do you know what could be the reason and which one is correct?
Thank you very much in advance.

Best regards,
Bernabé Ibáñez

2

Dear Bernabe,

The difference between these two results is caused by the linearization of the torque-speed curve, which results in a derivative dT/domega, which is zero in the approach recommended in the manual and nonzero in the other approach. In the other approach, you are linearizing the torque-speed curve at the “bend” in the curve, where the derivative is not well defined. This is resulting in a nonzero dT/domega that is modifying the state matrix “A”. I would suggest using the approach recommended in the manual.

Best regards,

3

Dear All,

I have a question related to the damping ratios obtained from the linearization procedure in FASTv7. As I understand, at during linearization, at a point matrix A from the discrete-time state-space form of the EoM is obtained and then the natural frequencies and the related modal damping are determined via solving the eigenvalue problem. This matrix A includes the mass, stiffness and damping matrices of the system.

My question is as follows: In case of non-orthogonal damping matrix, how is this damping ratio (obtained from the linearization) linked to a specific mode? From my point of view, given that the damping matrix is non-orthogonal, we cannot decouple the system into SDoF and find the damping of each mode. I cannot understand the physical meaning of this damping ratio and how a non-orthogonal damping matrix is connected to these specific modal damping ratios.

Thank you very much for your time!

Best regards

Georgios

4

Dear Georgios,

I agree with your first paragraph except that the state-matrix A represents the linearized continuous-time state-space form (rather than the discrete-time).

If the underlying damping matrix was not orthogonal like the mass and stiffness matrices, like it is in Rayleigh damping, the equations of motion in modal coordinates are coupled through off-diagonal terms in the modal damping matrix, and consequently, the system possesses complex modes instead of real normal modes. Regardless, the damping ratio calculated by the MBC3 post-processor is calculated by finding eigenvalues with positive imaginary components and dividing the real part of this eigenvalue by the natural frequency.

Best regards,

5

Dear Jason,

Thank you very much for your help and your fast response!

Best regards,

Georgios