## Linearization - Disturbance

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Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Linearization - Disturbance

Hallo,
I’ve a question regarding the linearized wind turbine Model using FAST. The output state space equation of the linearized system is given as
X' = A.X+B.U+Bd.W
Y=C.X+D.U+Dd.W
And what I’m trying to do is the use the Disturbance accommodating control method to compensate the vertical wind shear with the help of Individual Pitch control.
I’m concerned about the disturbance part W as it is always given in the non rotating frame. The available input wind disturbance includes
1- Horizontal hub-height wind speed
2- Horizontal wind direction
3- Vertical wind speed
4- Horizontal wind shear
5- Vertical power law wind shear
6- Linear vertical wind shear
7- Horizontal hub-height wind gust
And none of them include disturbance in the rotating frame. I’m wondering if it is possible in FAST to include disturbance input for each blade alone in the rotating frame similar to what is done by Wright, 2004, Modern Control Design for Flexible Wind Turbines, Ch7.
Also, how the vertical power law wind shear disturbance should be interpreted in the state space equation. Normally this is just a value given in TurbSim to generate the appropriate wind profile with the requested vertical wind shear.

Thanks in advance for your help
Rannam Chaaban.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Rannam.Chaaban
Posts: 30
Joined: Wed Jul 20, 2011 9:25 am
Organization: Siegen University, Siegen, Germany
Location: Germany

### Re: Linearization - Disturbance

Hallo,

As there was no answer for this post for sometime, I'll rephrase my question:

What is the difference between the "5-Vertical power law wind shear" and "6- Linear vertical wind shear" in the linearized state space equation, and how can we interpret these disturbance inputs.
Also, I'm wondering if there any work have been done using these disturbance inputs.

Thanks in advance for your help.
Rannam Chaaban.
MSc. Rannam Chaaban
Universität Siegen
Paul-Bonatz Strasse 9-11
57076 Siegen / Germany

Jason.Jonkman
Posts: 4952
Joined: Thu Nov 03, 2005 4:38 pm
Location: Boulder, CO
Contact:

### Re: Linearization - Disturbance

Dear Rannam,

I'll leave the controls-related question(s) to a control experts, but here is my answer regading your question on the difference between power-law and linear vertical shear perturbations. The wind speed at a given point in space and time based on AeroDyn's so-called "hub-height" wind file format is given by Equations 118 - 120 in the AeroDyn Theory Manual, found here: http://wind.nrel.gov/designcodes/simula ... Theory.pdf. The vertical power-law shear shear exponent is identified as "Vshr" and the linear vertical wind shear coefficient is identified as "Vshr_lin" in Equation 118. During a linearization analysis in FAST, wind speed disturbances "5" and "6" are equivalent to perturbations of "Vshr" and "Vshr_lin" about their nominal values, respectively (the nominal values of "Vshr" and "Vshr_lin" are the values specified in the "hub-height" wind file used during the FAST linearization analysis).

I hope that helps.

Best regards,
Jason Jonkman, Ph.D.
Senior Engineer | National Wind Technology Center (NWTC)

National Renewable Energy Laboratory (NREL)
15013 Denver West Parkway | Golden, CO 80401
+1 (303) 384 – 7026 | Fax: +1 (303) 384 – 6901
nwtc.nrel.gov

Jason.Laks
Posts: 1
Joined: Fri Jun 17, 2011 3:03 pm
Organization: University of Colorado
Location: Colorado

### Re: Linearization - Disturbance

Hi, Rannam-

From a control perspective, these "disturbance" inputs are most easily addressed when using multi-blade coordinates (MBC). However, in the linearized model, it is possible to take these inputs and convert them to equivalent blade local wind disturbances. In either the MBC or non-MBC cases, you would normally work with an average state-space model. See the article:

J. Laks, L. Y. Pao, A. D. Wright, N. Kelley, and B. Jonkman. "The Use of Preview Wind Measurements for Blade Pitch Control," IFAC J. Mechatronics, 21(4): 668-681, June 2011

It is also possible to work with non-averaged models (e.g., search on periodic control and wind turbines). A starting point for interpretation is to imagine the wind speeds at every point across the rotor plane changing simultaneously with a distribution equivalent to the shear profile defined in the FAST manual; then the magnitude of the effect felt by the turbine is determined by where the blades are placed within the rotor plane at each rotor position (i.e. azimuth). The linearized model describes how perturbations in each wind profile translate into perturbations in other turbine variables. A power law will normally produce a larger ratio between size of the perturbations at the bottom and top of the rotor, and one might expect more variation in the linearized models as a result.

Without getting into all the detail of linearized models, you can also view vertical (or horizontal) shear as a loading on the turbine blades that is periodic with rotor position. And, you might expect linear-vertical shear to produce mostly once-per-rev (1P) sinusoidal loading while a power-law would produce 1P and more harmonics thereof. Once you get FAST up and running, you can try out both linear and power law shears and take a look at the frequency content in the blade loads.

At any rate, once you have a model that describes how perturbations in wind speed (either shear/MBC or blade local) generate perturbations in other turbine variables, you can take/research any desired control approach to mitigate their effect.

Good Luck!
-Jason

P.S. To get more intuition, also take a look at where each blade experiences its peak load while simulating with different levels of vertical and horizontal shear.

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