Hi all,

First of all I would like to thank you for such a great and well documented project on Wind Turbine simulation.

I was doing a bit of research on Wind Turbine dynamics and control. Unfortunately, I´m not an expert on control issues, so apologize in advance for the obvious questions on that field.

Taking the model NRELOffshrBsline5MW_Onshore, I have performed a linearization at several operating points in region 3 (above rated Wind Speed).

A standard header of one of my linearized solutions:

Order of States in Linearized State Matrices:

Row/column 1 = 1st tower fore-aft bending mode DOF (internal DOF index = DOF_TFA1)

Row/column 2 = 1st tower side-to-side bending mode DOF (internal DOF index = DOF_TSS1)

Row/column 3 = Variable speed generator DOF (internal DOF index = DOF_GeAz)

Row/column 4 = Drivetrain rotational-flexibility DOF (internal DOF index = DOF_DrTr)

Row/column 5 = 1st flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,1))

Row/column 6 = 1st flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,1))

Row/column 7 = 1st flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,1))

Row/column 8 = 1st edgewise bending-mode DOF of blade 1 (internal DOF index = DOF_BE(1,1))

Row/column 9 = 1st edgewise bending-mode DOF of blade 2 (internal DOF index = DOF_BE(2,1))

Row/column 10 = 1st edgewise bending-mode DOF of blade 3 (internal DOF index = DOF_BE(3,1))

Order of Control Inputs in Linearized State Matrices:

Column 1 = electrical generator torque (N·m) 4.30930E+04 op

Column 2 = rotor collective blade pitch (rad) 1.98206E-01 op

Column 3 = individual pitch of blade 1 (rad) 1.98206E-01 op

Column 4 = individual pitch of blade 2 (rad) 1.98206E-01 op

Column 5 = individual pitch of blade 3 (rad) 1.98206E-01 op

Order of Input Wind Disturbances in Linearized State Matrices:

Column 1 = horizontal hub-height wind speed (m/s) 1.60000E+01 op

Column 2 = horizontal wind direction (rad) 0.00000E+00 op

Column 3 = vertical wind speed (m/s) 3.47900E+00 op

Column 4 = horizontal shear parameter (-) 0.00000E+00 op

Column 5 = vertical power law shear exponent (-) 2.00000E-01 op

Column 6 = linear vertical shear parameter (-) 0.00000E+00 op

Column 7 = horizontal hub-height wind gust (m/s) 0.00000E+00 op

Order of Output Measurements in Linearized State Matrices:

Row 1 = GenSpeed (rpm)

Row 2 = GenTq (kN·m)

Row 3 = GenPwr (kW)

Row 4 = RotSpeed (rpm)

Row 5 = RotThrust (kN)

Row 6 = RotTorq (kN·m)

Row 7 = RotPwr (kW)

Row 8 = RootFxc1 (kN)

Row 9 = RootFxc2 (kN)

Row 10 = RootFxc3 (kN)

Row 11 = RootFyc1 (kN)

Row 12 = RootFyc2 (kN)

Row 13 = RootFyc3 (kN)

Row 14 = RootFzc1 (kN)

Row 15 = RootFzc2 (kN)

Row 16 = RootFzc3 (kN)

Row 17 = RootMxc1 (kN·m)

Row 18 = RootMxc2 (kN·m)

Row 19 = RootMxc3 (kN·m)

Row 20 = RootMyc1 (kN·m)

Row 21 = RootMyc2 (kN·m)

Row 22 = RootMyc3 (kN·m)

Row 23 = RootMzc1 (kN·m)

Row 24 = RootMzc2 (kN·m)

Row 25 = RootMzc3 (kN·m)

Row 26 = TipDxc1 (m)

Row 27 = TipDxc2 (m)

Row 28 = TipDxc3 (m)

Row 29 = TipDyc1 (m)

Row 30 = TipDyc2 (m)

Row 31 = TipDyc3 (m)

Row 32 = TipDzc1 (m)

Row 33 = TipDzc2 (m)

Row 34 = TipDzc3 (m)

Row 35 = TwrBsFxt (kN)

Row 36 = TwrBsFyt (kN)

Row 37 = TwrBsFzt (kN)

Row 38 = TwrBsMxt (kN·m)

Row 39 = TwrBsMyt (kN·m)

Row 40 = TwrBsMzt (kN·m)

Row 41 = TTDspFA (m)

Row 42 = TTDspSS (m)

Row 43 = TTDspAx (m)

Row 44 = BlPitch1 (deg)

Row 45 = BlPitch2 (deg)

Row 46 = BlPitch3 (deg)

Row 47 = HorWindV (m/sec)

Row 48 = HorWndDir (deg)

Row 49 = TSR (-)

In order to compare the performance of the linearized model with the nonlinear model, I prepared three basic simulations:

1.- Extreme Operating Wind Gust (max 8 m/s) in Closed Loop Condition.

2.- 2 Deg Pitch Increment in Open Loop Condition.

3.- 4 m/s Wind Speed Increment in Closed Loop Condition.

For the linearized model, I reproduce (as far as I can) the pitch control documented in:

Technical Report NREL/TP-500-38060: Definition of a 5-MW Reference Wind Turbine for Offshore System Development (great document, by the way).

In order to simplify the Variable Speed Control definition, I took the standard VS_CONT model with constant torque in region 3.

My first attempt was to use eigenanalysis.m routine and average out the linearized models obtained for different azimuts,

but then I found the so-clever MBC approach:

User’s Guide to MBC3 (Multi-blade Coordinate Transformation Utility for 3-Bladed Wind Turbines).

So, comparing the results of the FAST simultations with the MBC-Linearized model, following questions arise:

1.- Extreme Operating Wind Gust in Closed Loop Condition.

The linearized model exhibits a good agreement with FAST simulation, but I wonder if is there any way to reproduce the 1p-cyclic effects in the linearized model.

I realize that the cyclic effects are filtered out when performing the linearization, but maybe there exist an alternative way to deal with this effects. Otherwise,

I understand I could simulate the whole nonlinear model of the Wind Turbine in the Simulink environment with the aid of FAST_SFunc.mexw32.

It would be maravillous if one could develop individual pitch control strategies with the aid of the linearized models. Any Ideas?

2.- 2 Deg Pitch Increment in Open Loop Condition.

I have a problem with the linearized model. My thougth is that the problem is related with the evolution of the state variables.

What I expected to obtain was a steady condition in an operating point with a different rotational speed, but the offset error in the DOF_GeAz derivative causes non stationary terms in some output variables.

Mainly those which has a 1p periodic behaviour.

I try to work arround this problem, but no way to do that. I think it could maybe have to do with the procedure used to obtain the averaged matrices, which in this case is a simple average of all the matrices, let say A, B, Bd, C, D and Dd MBC transformed matrices.

From control point of view, this behaviour is related to the presence of integral terms, but I would need more understanding on the problem to filter out this non-desired terms.

This graph represents the evolution of RootFxc1 in this loadcase. As mentioned, it would be nice to obtain a steady value in this output.

See Figure PitchDelta_RootFxc1_(kN).png

3.- 4 m/s Wind Speed Increment in Closed Loop Condition.

In this case, the pitch controller command an increment of the collaborative pitch in order to recover the rated speed (also the rated power).

What I would spect in this case is a steady condition in all the outputs, but I don´t.

If we take a look at a variable as the Rotor Thrust, it´s OK.

See Figure WindSpeedDelta_RotThrust_(kN).png

If we analyze the evolution of RootFxc1, what we see is that a periodic behaviour arises.

See Figure WindSpeedDelta_RootFxc1_(kN).png

For this variable, the FAST model propose a steady situation with lower mean and higher amplitude so what I would spect for my LTI model was a constant steady value, beter than a periodic one.

On the other hand, it would be nice if the LTI model would be able to reproduce this kind of effects. I suppose this is a kind of MBC transformation of the offset error in one of the state variables.

Any idea about this topic?

Thank you in advance.

Kind Regards,

Javier.