Fast Linearized Models
Moderators: Bonnie.Jonkman, Jason.Jonkman

 Posts: 10
 Joined: Wed Jan 27, 2010 1:46 am
Fast Linearized Models
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 foreaft bending mode DOF (internal DOF index = DOF_TFA1)
Row/column 2 = 1st tower sidetoside bending mode DOF (internal DOF index = DOF_TSS1)
Row/column 3 = Variable speed generator DOF (internal DOF index = DOF_GeAz)
Row/column 4 = Drivetrain rotationalflexibility DOF (internal DOF index = DOF_DrTr)
Row/column 5 = 1st flapwise bendingmode DOF of blade 1 (internal DOF index = DOF_BF(1,1))
Row/column 6 = 1st flapwise bendingmode DOF of blade 2 (internal DOF index = DOF_BF(2,1))
Row/column 7 = 1st flapwise bendingmode DOF of blade 3 (internal DOF index = DOF_BF(3,1))
Row/column 8 = 1st edgewise bendingmode DOF of blade 1 (internal DOF index = DOF_BE(1,1))
Row/column 9 = 1st edgewise bendingmode DOF of blade 2 (internal DOF index = DOF_BE(2,1))
Row/column 10 = 1st edgewise bendingmode 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) 1.98206E01 op
Column 3 = individual pitch of blade 1 (rad) 1.98206E01 op
Column 4 = individual pitch of blade 2 (rad) 1.98206E01 op
Column 5 = individual pitch of blade 3 (rad) 1.98206E01 op
Order of Input Wind Disturbances in Linearized State Matrices:
Column 1 = horizontal hubheight 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 () 2.00000E01 op
Column 6 = linear vertical shear parameter () 0.00000E+00 op
Column 7 = horizontal hubheight 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/TP50038060: Definition of a 5MW 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 soclever MBC approach:
User’s Guide to MBC3 (Multiblade Coordinate Transformation Utility for 3Bladed Wind Turbines).
So, comparing the results of the FAST simultations with the MBCLinearized 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 1pcyclic 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 nondesired 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.
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 foreaft bending mode DOF (internal DOF index = DOF_TFA1)
Row/column 2 = 1st tower sidetoside bending mode DOF (internal DOF index = DOF_TSS1)
Row/column 3 = Variable speed generator DOF (internal DOF index = DOF_GeAz)
Row/column 4 = Drivetrain rotationalflexibility DOF (internal DOF index = DOF_DrTr)
Row/column 5 = 1st flapwise bendingmode DOF of blade 1 (internal DOF index = DOF_BF(1,1))
Row/column 6 = 1st flapwise bendingmode DOF of blade 2 (internal DOF index = DOF_BF(2,1))
Row/column 7 = 1st flapwise bendingmode DOF of blade 3 (internal DOF index = DOF_BF(3,1))
Row/column 8 = 1st edgewise bendingmode DOF of blade 1 (internal DOF index = DOF_BE(1,1))
Row/column 9 = 1st edgewise bendingmode DOF of blade 2 (internal DOF index = DOF_BE(2,1))
Row/column 10 = 1st edgewise bendingmode 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) 1.98206E01 op
Column 3 = individual pitch of blade 1 (rad) 1.98206E01 op
Column 4 = individual pitch of blade 2 (rad) 1.98206E01 op
Column 5 = individual pitch of blade 3 (rad) 1.98206E01 op
Order of Input Wind Disturbances in Linearized State Matrices:
Column 1 = horizontal hubheight 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 () 2.00000E01 op
Column 6 = linear vertical shear parameter () 0.00000E+00 op
Column 7 = horizontal hubheight 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/TP50038060: Definition of a 5MW 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 soclever MBC approach:
User’s Guide to MBC3 (Multiblade Coordinate Transformation Utility for 3Bladed Wind Turbines).
So, comparing the results of the FAST simultations with the MBCLinearized 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 1pcyclic 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 nondesired 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.
Javier Gil Soto
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Javier,
Before answering your questions, I need to better understand what you've done. My understand is:
*You've used the linearization feature of FAST to obtain linear periodic models of the landbased NREL 5MW turbine about several operating points.
*You've applied MBC3 to the linear periodic model, then azimuthaveraged the resulting matrices to obtain an LTI model.
*You've wrapped the LTI model inside a timedomain integrator in order to compare with responses obtained from FAST.
*For the closedloop cases with the LTI model, you've mimicked the pitch controller of the NREL 5MW in your timedomain integrator.
*For both the open and closesloop cases with the LTI model, you've replaced the torque controller of the NREL 5MW with the simple variablespeed control option (VSContrl = 1) available in FAST.
Can you confirm my understanding of your approach?
Now here are my questions:
*Was the time integration of your LTI model and the control sytem implemented within MATLAB/Simulink?
*Have you added back in the periodic steady state values to the outputs (y vector) in order to compare to the outputs of FAST? (That is, y = yop + deltay.)
*Have you applied the inverse MBC transform to the outputs (y vector) in order to compare to the outputs of FAST? (Of course, this is only needed for the outputs effected by MBC, i.e. the blade motions and loads, such as RootFx1.)
Best regards,
Before answering your questions, I need to better understand what you've done. My understand is:
*You've used the linearization feature of FAST to obtain linear periodic models of the landbased NREL 5MW turbine about several operating points.
*You've applied MBC3 to the linear periodic model, then azimuthaveraged the resulting matrices to obtain an LTI model.
*You've wrapped the LTI model inside a timedomain integrator in order to compare with responses obtained from FAST.
*For the closedloop cases with the LTI model, you've mimicked the pitch controller of the NREL 5MW in your timedomain integrator.
*For both the open and closesloop cases with the LTI model, you've replaced the torque controller of the NREL 5MW with the simple variablespeed control option (VSContrl = 1) available in FAST.
Can you confirm my understanding of your approach?
Now here are my questions:
*Was the time integration of your LTI model and the control sytem implemented within MATLAB/Simulink?
*Have you added back in the periodic steady state values to the outputs (y vector) in order to compare to the outputs of FAST? (That is, y = yop + deltay.)
*Have you applied the inverse MBC transform to the outputs (y vector) in order to compare to the outputs of FAST? (Of course, this is only needed for the outputs effected by MBC, i.e. the blade motions and loads, such as RootFx1.)
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
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

 Posts: 10
 Joined: Wed Jan 27, 2010 1:46 am
Re: Fast Linearized Models
Dear Jason,
I Apologize if I went too FAST in my first post. I´ll try to answer online to your questions:
*You've used the linearization feature of FAST to obtain linear periodic models of the landbased NREL 5MW turbine about several operating points.
The model is correct: landbased NREL 5MW turbine. As my aim was to control in region 3 of VS_CONT, I linearized the model for several horizontal hubheight wind speeds:
12:2:24 (m/s). In this case, plottings correspond to the 16 (m/s) model.
*You've applied MBC3 to the linear periodic model, then azimuthaveraged the resulting matrices to obtain an LTI model.
In this case, I used the mbc3 routines to obtain the MBCaveraged matrices. As mbc3 only computes the averaged state matrix, I added some lines to the script:
in order to obtain the rest of the averaged matrices. I wonder if there is a more efficient errorfree averaging procedure for all these matrices.
I use:
in my lin_options.dat file, but I think this point is not of relevance in this case.
*You've wrapped the LTI model inside a timedomain integrator in order to compare with responses obtained from FAST.
I prepared a simple wrapper in Simulink in order to integrate the MBCaveraged statespace representation of the nonlinear model. I used a rk4 fixedstep for integration.
*For the closedloop cases with the LTI model, you've mimicked the pitch controller of the NREL 5MW in your timedomain integrator.
As proposed in NREL/TP50038060, I used a gain scheduled PI controller. In the case of my last post, for 16 m/s wind speed an 11.36 Deg nominal pitch, I selected the corresponding scheduled gains.
I realize that some differences may arise due to gain changes in the nonlinear model, but I dont think this has something to do with the offset problems I documented earlier.
*For both the open and closesloop cases with the LTI model, you've replaced the torque controller of the NREL 5MW with the simple variablespeed control option (VSContrl = 1) available in FAST.
For both LTI and FAST models, I used the same appoach: simple VS_CONT model:
In the case of the simulink model, I wrote a embedded matlab function that emulates this behaviour. I realize that this model introduces a kind of nonlinearity, and is maybe a part of the problem.
For VS_CONT controlling purposes, I use the derivative of DOF_GeAz, which I´m not sure is the best variable.
Can you confirm my understanding of your approach?
Now here are my questions:
*Was the time integration of your LTI model and the control sytem implemented within MATLAB/Simulink?
As said before, I build a simple simulink wrapper for the averaged MBC statespace representation of the model.
*Have you added back in the periodic steady state values to the outputs (y vector) in order to compare to the outputs of FAST? (That is, y = yop + deltay.)
No, I dont. For plotting purposes, I remove the operational values from the FAST nonlinear solution. My aim was to compare the performance of the models in terms of the solution mean values.
I know that I can obtain better approximations when adding operational values (y = yop + deltay) to the LTI solution, but I dont know how to do it properly when having rotor speed variations. As far as I understand, I could transform the 36 nonaveraged yop values (each 10 Deg) into a lookup table and enter in this table with the azimut position. For MBC transformation purposes, I assume that azimut position can be approximated by:
which I'm not sure is the best approach (see figure). Is this the correct way to do it?
It would be maravillous if it could be done in realtime. That way, one should be able to develop more efficient control schemes.
*Have you applied the inverse MBC transform to the outputs (y vector) in order to compare to the outputs of FAST? (Of course, this is only needed for the outputs effected by MBC, i.e. the blade motions and loads, such as RootFx1.)
For plotting purposes, I applied back the MBC transformation as follows:
After checking RotTripletIndicesStates, RotTripletIndicesCntrlInpt, RotTripletIndicesOutput from mbc3.m, I assume that the MBC transformation is correct and the state space representation is in the nonrotating frame.
Therefore, I applied back the MBC transformation to the same RotTripletIndicesOutput.
Thank you in avance.
Kind regards,
Javier.
I Apologize if I went too FAST in my first post. I´ll try to answer online to your questions:
*You've used the linearization feature of FAST to obtain linear periodic models of the landbased NREL 5MW turbine about several operating points.
The model is correct: landbased NREL 5MW turbine. As my aim was to control in region 3 of VS_CONT, I linearized the model for several horizontal hubheight wind speeds:
12:2:24 (m/s). In this case, plottings correspond to the 16 (m/s) model.
*You've applied MBC3 to the linear periodic model, then azimuthaveraged the resulting matrices to obtain an LTI model.
In this case, I used the mbc3 routines to obtain the MBCaveraged matrices. As mbc3 only computes the averaged state matrix, I added some lines to the script:
Code: Select all
MBC_AvgB = MBC_AvgB + MBC_B(:,:,iaz);
if (exist('BMat'))
MBC_AvgB = MBC_AvgB/NAzSteps; % azimuthaverage
end
in order to obtain the rest of the averaged matrices. I wonder if there is a more efficient errorfree averaging procedure for all these matrices.
I use:
Code: Select all
2 MdlOrder
in my lin_options.dat file, but I think this point is not of relevance in this case.
*You've wrapped the LTI model inside a timedomain integrator in order to compare with responses obtained from FAST.
I prepared a simple wrapper in Simulink in order to integrate the MBCaveraged statespace representation of the nonlinear model. I used a rk4 fixedstep for integration.
*For the closedloop cases with the LTI model, you've mimicked the pitch controller of the NREL 5MW in your timedomain integrator.
As proposed in NREL/TP50038060, I used a gain scheduled PI controller. In the case of my last post, for 16 m/s wind speed an 11.36 Deg nominal pitch, I selected the corresponding scheduled gains.
I realize that some differences may arise due to gain changes in the nonlinear model, but I dont think this has something to do with the offset problems I documented earlier.
*For both the open and closesloop cases with the LTI model, you've replaced the torque controller of the NREL 5MW with the simple variablespeed control option (VSContrl = 1) available in FAST.
For both LTI and FAST models, I used the same appoach: simple VS_CONT model:
Code: Select all
1 VSContrl
1173.0 VS_RtGnSp
43093.0 VS_RtTq
3.1318E2 VS_Rgn2K
9999.9E9 VS_SlPc
In the case of the simulink model, I wrote a embedded matlab function that emulates this behaviour. I realize that this model introduces a kind of nonlinearity, and is maybe a part of the problem.
For VS_CONT controlling purposes, I use the derivative of DOF_GeAz, which I´m not sure is the best variable.
Can you confirm my understanding of your approach?
Now here are my questions:
*Was the time integration of your LTI model and the control sytem implemented within MATLAB/Simulink?
As said before, I build a simple simulink wrapper for the averaged MBC statespace representation of the model.
*Have you added back in the periodic steady state values to the outputs (y vector) in order to compare to the outputs of FAST? (That is, y = yop + deltay.)
No, I dont. For plotting purposes, I remove the operational values from the FAST nonlinear solution. My aim was to compare the performance of the models in terms of the solution mean values.
I know that I can obtain better approximations when adding operational values (y = yop + deltay) to the LTI solution, but I dont know how to do it properly when having rotor speed variations. As far as I understand, I could transform the 36 nonaveraged yop values (each 10 Deg) into a lookup table and enter in this table with the azimut position. For MBC transformation purposes, I assume that azimut position can be approximated by:
Code: Select all
azimut = (RotSpeed_op+deltaRotSpeed)*t
which I'm not sure is the best approach (see figure). Is this the correct way to do it?
It would be maravillous if it could be done in realtime. That way, one should be able to develop more efficient control schemes.
*Have you applied the inverse MBC transform to the outputs (y vector) in order to compare to the outputs of FAST? (Of course, this is only needed for the outputs effected by MBC, i.e. the blade motions and loads, such as RootFx1.)
For plotting purposes, I applied back the MBC transformation as follows:
Code: Select all
function Ur = MBC_trans(Unr, rotpos)
% inputs:
% Unr [3x1] nonrotating MBC set.
% rotpos [1] azimut position. Calculated with the former formula.
% output:
% Ur [3x1] rotating MBC set.
az1 = 0 + rotpos;
az2 = az1 + 2*pi/3;
az3 = az2 + 2*pi/3;
% compute transformation matrix
c1 = cos(az1);
c2 = cos(az2);
c3 = cos(az3);
s1 = sin(az1);
s2 = sin(az2);
s3 = sin(az3);
cos_col = [c1;c2;c3];
sin_col = [s1;s2;s3];
tt = [ones(3,1), cos_col, sin_col];
Ur = tt*Unr;
After checking RotTripletIndicesStates, RotTripletIndicesCntrlInpt, RotTripletIndicesOutput from mbc3.m, I assume that the MBC transformation is correct and the state space representation is in the nonrotating frame.
Therefore, I applied back the MBC transformation to the same RotTripletIndicesOutput.
Thank you in avance.
Kind regards,
Javier.
Javier Gil Soto
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Javier,
Thanks for clarifying your approach.
It is difficult to answer your questions because you have supplied a lot of information and are using more than just NRELdeveloped codes and models. Neither do I have the time to assess all of all your steps in detail. So, instead of answering your direct questions, I'll try to give provide general information.
In general,the linear model should follow the nonlinear well for small perturbations about the linearization point. If your linear model is not tracking the nonlinear model well for small perturbations, I suspect there is a problem somewhere in the solution or comparison. It may help to simplify the model as much as possible in order to track down a problem (e.g., start by eliminating the generator azimuth DOF).
MBC is useful for rotors with 3 or more blades. MBC does not eliminate the periodicity, it simply transforms the rotating states of a model into the nonrotating frame (a linear model of mixed rotating and nonrotating states can then be represented in the nonrotating frame). For example, a 1pcyclic load in the rotating frame contributes to 0p and 3p loading in the nonrotating frame. Much of the periodicity of the mixed frame linear matrices is captured by the MBC transform. Further azimuthaveraging of the lightlyperiodic MBCtransormed matrices has been shown to be quite adequate, such that direct periodic techniques are likely not needed for 3bladed rotors. See, for example, this reference: Stol A, Moll HG, Bir G, and Namik H. "A Comparison of MultiBlade Coordinate Transformation and Direct Periodic Technicques for Wind Turbine Control Design." 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exhibition, 5–8 January 2009, Orlando, FL. AIAA2009479. January 2009.
MBC is very useful to the design of independent blade pitch controllers. Again, see the reference above or review the papers it references.
Perhaps someone with more experience with the application of linear models and/or MBC can provide further guidance.
Best regards,
Thanks for clarifying your approach.
It is difficult to answer your questions because you have supplied a lot of information and are using more than just NRELdeveloped codes and models. Neither do I have the time to assess all of all your steps in detail. So, instead of answering your direct questions, I'll try to give provide general information.
In general,the linear model should follow the nonlinear well for small perturbations about the linearization point. If your linear model is not tracking the nonlinear model well for small perturbations, I suspect there is a problem somewhere in the solution or comparison. It may help to simplify the model as much as possible in order to track down a problem (e.g., start by eliminating the generator azimuth DOF).
MBC is useful for rotors with 3 or more blades. MBC does not eliminate the periodicity, it simply transforms the rotating states of a model into the nonrotating frame (a linear model of mixed rotating and nonrotating states can then be represented in the nonrotating frame). For example, a 1pcyclic load in the rotating frame contributes to 0p and 3p loading in the nonrotating frame. Much of the periodicity of the mixed frame linear matrices is captured by the MBC transform. Further azimuthaveraging of the lightlyperiodic MBCtransormed matrices has been shown to be quite adequate, such that direct periodic techniques are likely not needed for 3bladed rotors. See, for example, this reference: Stol A, Moll HG, Bir G, and Namik H. "A Comparison of MultiBlade Coordinate Transformation and Direct Periodic Technicques for Wind Turbine Control Design." 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exhibition, 5–8 January 2009, Orlando, FL. AIAA2009479. January 2009.
MBC is very useful to the design of independent blade pitch controllers. Again, see the reference above or review the papers it references.
Perhaps someone with more experience with the application of linear models and/or MBC can provide further guidance.
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
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

 Posts: 10
 Joined: Wed Jan 27, 2010 1:46 am
Re: Fast Linearized Models
Dear Jason,
I will take a close look to the references you mentioned. Thanks a lot for your attention.
Perhaps you are wright when you say I must start with a simplifyed model, but before doing that, I have to make sure that I am not misunderstanding the FAST linearization process.
Let say I perform a linearization at an operation point that is above rated wind speed (pitch control zone). What I suppose is that the linearized state matrix refers to the dynamics of the openloop wind turbine. Therefore, the contribution of selected inputs / disturbances are conveniently represented with [B] and [Bd] matrices.
Matrices [C], [D] and [Dd] allow us to build the openloop transfer functions from selected inputs/disturbances to selected outputs, let say From BldPitch to RotSpeed. In this case, the only assumption I have to do is that matrices [B], [Bd], [C], [D] and [Dd] can be azimutaveraged in the same manner I buid the averaged state matrix.
In the case I analyze by example, 16 m/s wind speed, the results of the linearization for selected transfer functions (From GenTq, BldPitch and HorWindV to RotSpeed) are shown in the following figure:
My aim was to use all these averaged transfer functions in a simulink LTI model so as to start playing with different control algoritms. The point is that the system that I am trying to control, y = H(s) [u; ud],
appears to be somehow controlled, so I wonder if is there anything wrong with this basic approach.
Thank you in advance.
Kind Regards,
I will take a close look to the references you mentioned. Thanks a lot for your attention.
Perhaps you are wright when you say I must start with a simplifyed model, but before doing that, I have to make sure that I am not misunderstanding the FAST linearization process.
Let say I perform a linearization at an operation point that is above rated wind speed (pitch control zone). What I suppose is that the linearized state matrix refers to the dynamics of the openloop wind turbine. Therefore, the contribution of selected inputs / disturbances are conveniently represented with [B] and [Bd] matrices.
Matrices [C], [D] and [Dd] allow us to build the openloop transfer functions from selected inputs/disturbances to selected outputs, let say From BldPitch to RotSpeed. In this case, the only assumption I have to do is that matrices [B], [Bd], [C], [D] and [Dd] can be azimutaveraged in the same manner I buid the averaged state matrix.
In the case I analyze by example, 16 m/s wind speed, the results of the linearization for selected transfer functions (From GenTq, BldPitch and HorWindV to RotSpeed) are shown in the following figure:
My aim was to use all these averaged transfer functions in a simulink LTI model so as to start playing with different control algoritms. The point is that the system that I am trying to control, y = H(s) [u; ud],
appears to be somehow controlled, so I wonder if is there anything wrong with this basic approach.
Thank you in advance.
Kind Regards,
Javier Gil Soto
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Javier,
Yes, the linearized state matrix refers to the dynamics of the openloop turbine (see exception below). And yes, the [C], [D] and [Dd] matrices refer to the openloop dynamics from inputs/disturbances to outputs. And yes, the [B], [Bd], [C], [D] and [Dd] can be azimuthaveraged in the same manner that you build the averaged state matrix, [A]. Perhaps someone with more experience with linear systems analysis can comment on your Bode plot results.
Exception: the influence of the generator torque can be included in your linear model if the torque is implemented as a simple function of the turbine states (such as generator speed), without introducing new states. For example, the influence of the generator torque can be included when VSContrl = 1 or VSContrl = 0 with GenModel = 1 or 2. What torque option do you have set in your model before linearizing? Could your Bode plot results be influenced by the generator torque?
Best regards,
Yes, the linearized state matrix refers to the dynamics of the openloop turbine (see exception below). And yes, the [C], [D] and [Dd] matrices refer to the openloop dynamics from inputs/disturbances to outputs. And yes, the [B], [Bd], [C], [D] and [Dd] can be azimuthaveraged in the same manner that you build the averaged state matrix, [A]. Perhaps someone with more experience with linear systems analysis can comment on your Bode plot results.
Exception: the influence of the generator torque can be included in your linear model if the torque is implemented as a simple function of the turbine states (such as generator speed), without introducing new states. For example, the influence of the generator torque can be included when VSContrl = 1 or VSContrl = 0 with GenModel = 1 or 2. What torque option do you have set in your model before linearizing? Could your Bode plot results be influenced by the generator torque?
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
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

 Posts: 10
 Joined: Wed Jan 27, 2010 1:46 am
Re: Fast Linearized Models
Dear Jason,
Thank you for your patience. As you suggested, I performed different simulations with different generator torque definitions (simple VSCONT and constant torque respectively), obtaining slightly different results. If I am wright, better results (also less damped) are obtained when using a constant torque definition, which is a normal assumtion when working in region 3. I also suppose that the differences are due to the nonlinear behaviour of the simple VSCONT generator in regions 2 1/2 and 3.
The rest of the problems I documented earlyer had to do with numerical residuals in the state matrices. Therefore, I can now obtain the results that I expected. Moreover, I took my time in analyzing the evolution of the cyclic terms in the state matrices for different azimuth positions, so I get some insight on the effect of the MBC transformation. Thank you very much for the references you mentioned in your second post.
If I may be permitted to make one more (basic) question, I have one regarding to the tower shadow effect. If I am wright, the tower shadow has 3p contribution in nonrotating variables (ie. tilt moment in the yaw bearing) and 1p contribution in rotating variables (ie. Fx and My at the blade root of each blade). In order to evaluate this effect, I performed different simulations using the following parameters in the aerodyn input file:
(model: 5Mw baseline onshore turbine)
After analyzing results, I realized that the tower shadow effect is not so strong as I expected. Maybe the problem has to do with a bad selection of the former parameters. Maybe this tower shadow effect is somehow neglected when performing a linearization. Maybe I am misunderstanding the effect of the tower shadow, which I am afraid is the most probable thing.
The point is that I would like to make sure about which terms should be always taken in account and which terms can be negected.
Thank you in advance.
Kind regards,
Thank you for your patience. As you suggested, I performed different simulations with different generator torque definitions (simple VSCONT and constant torque respectively), obtaining slightly different results. If I am wright, better results (also less damped) are obtained when using a constant torque definition, which is a normal assumtion when working in region 3. I also suppose that the differences are due to the nonlinear behaviour of the simple VSCONT generator in regions 2 1/2 and 3.
The rest of the problems I documented earlyer had to do with numerical residuals in the state matrices. Therefore, I can now obtain the results that I expected. Moreover, I took my time in analyzing the evolution of the cyclic terms in the state matrices for different azimuth positions, so I get some insight on the effect of the MBC transformation. Thank you very much for the references you mentioned in your second post.
If I may be permitted to make one more (basic) question, I have one regarding to the tower shadow effect. If I am wright, the tower shadow has 3p contribution in nonrotating variables (ie. tilt moment in the yaw bearing) and 1p contribution in rotating variables (ie. Fx and My at the blade root of each blade). In order to evaluate this effect, I performed different simulations using the following parameters in the aerodyn input file:
Code: Select all
90.000 HH
0.3 TwrShad
3.00 ShadHWid
7.75 T_Shad_Refpt
(model: 5Mw baseline onshore turbine)
After analyzing results, I realized that the tower shadow effect is not so strong as I expected. Maybe the problem has to do with a bad selection of the former parameters. Maybe this tower shadow effect is somehow neglected when performing a linearization. Maybe I am misunderstanding the effect of the tower shadow, which I am afraid is the most probable thing.
The point is that I would like to make sure about which terms should be always taken in account and which terms can be negected.
Thank you in advance.
Kind regards,
Javier Gil Soto
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)
Mechanical Engineering Department
http://www.imac.unavarra.es/
Public University of Navarre
http://www.unavarra.es/
Pamplona (Spain)

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Javier,
I'm glad your MBC results now make sense to you.
AeroDyn v12.58 and earlier have only one towerinfluence model. This is a towershadow model that only creates a wake deficit behind the tower, so it only influences the aerodynamics of a rotor operating downwind of the tower.
v13.00.00bjj has two towerinfluence models. The default model is the same towershadow model from v12.58, as described above. The second towerinfluence model includes a tower dam model for upwind influence, as described in the AeroDyn Theory Manual: http://www.nrel.gov/docs/fy05osti/36881.pdf.
The inputs you've used apply only to the default towershadow model, and so, will not impact the aerodynamic loads of the NREL 5MW baseline turbine, which has an upwind rotor.
Best regards,
I'm glad your MBC results now make sense to you.
AeroDyn v12.58 and earlier have only one towerinfluence model. This is a towershadow model that only creates a wake deficit behind the tower, so it only influences the aerodynamics of a rotor operating downwind of the tower.
v13.00.00bjj has two towerinfluence models. The default model is the same towershadow model from v12.58, as described above. The second towerinfluence model includes a tower dam model for upwind influence, as described in the AeroDyn Theory Manual: http://www.nrel.gov/docs/fy05osti/36881.pdf.
The inputs you've used apply only to the default towershadow model, and so, will not impact the aerodynamic loads of the NREL 5MW baseline turbine, which has an upwind rotor.
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
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

 Posts: 15
 Joined: Sat Nov 02, 2013 10:16 pm
 Organization: Graduate Unerversity of Advanced Tecnology
 Location: Iran
Re: Fast Linearized Models
Hello
As far as I know, FAST models the drivetrain of a wind turbine as follows:
2Ht d(ωt)/dt =TwKtg(δtδg)Dtg(ωtωg)
d(δtδg)/dt=d(δtg)/dt=ωtωg
2Hg d(ωg)/dt =Ktg(δtδg)+Dtg(ωtωg)Tg
where;
subscripts t and g indicate the rotor and generator, respectively;
ω: angular speed;
δtg: torsional angle;
Ktg: shaft stiffness;
Dtg: damping coefficient;
H: inertia constant;
Tg: electromagnetic torque;
thus we have three state variables, including Δωt, Δωg, and Δδtg.
In FAST linearization, I'd like to know which states model the mentioned variables.
Moreover, I'd like to know why GenDOF should be disabled when linearizing a constantspeed wind turbine. I think it leads to ignore variable Δωg.
regards
As far as I know, FAST models the drivetrain of a wind turbine as follows:
2Ht d(ωt)/dt =TwKtg(δtδg)Dtg(ωtωg)
d(δtδg)/dt=d(δtg)/dt=ωtωg
2Hg d(ωg)/dt =Ktg(δtδg)+Dtg(ωtωg)Tg
where;
subscripts t and g indicate the rotor and generator, respectively;
ω: angular speed;
δtg: torsional angle;
Ktg: shaft stiffness;
Dtg: damping coefficient;
H: inertia constant;
Tg: electromagnetic torque;
thus we have three state variables, including Δωt, Δωg, and Δδtg.
In FAST linearization, I'd like to know which states model the mentioned variables.
Moreover, I'd like to know why GenDOF should be disabled when linearizing a constantspeed wind turbine. I think it leads to ignore variable Δωg.
regards

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Hamid Reza,
Your equations are effectively what FAST does for a 2DOF generator + drivetrain model, except that I'm not sure where the "2"s in your equation come from.
For this 2DOF model, FAST linearization will produce a 4state model, including Δδg and Δωg corresponding to the generator DOF and Δδtg and Δωtg corresponding to the drivetraintorsion DOF. As discussed in my Sep 02, 2014 response in the forum topic found here, it is common to remove the azimuth angle perturbation (Δδg) as a state. Then, with a little manipulation, you can transform this 4state model into your desired 3state model.
Best regards,
Your equations are effectively what FAST does for a 2DOF generator + drivetrain model, except that I'm not sure where the "2"s in your equation come from.
For this 2DOF model, FAST linearization will produce a 4state model, including Δδg and Δωg corresponding to the generator DOF and Δδtg and Δωtg corresponding to the drivetraintorsion DOF. As discussed in my Sep 02, 2014 response in the forum topic found here, it is common to remove the azimuth angle perturbation (Δδg) as a state. Then, with a little manipulation, you can transform this 4state model into your desired 3state model.
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
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

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Hamid Reza,
Regarding your second question, the generator DOF need not be disabled when linearizing a constantspeed wind turbine; in this case, it is possible to linearize such a model, keeping the slip of the generator.
Best regards,
Regarding your second question, the generator DOF need not be disabled when linearizing a constantspeed wind turbine; in this case, it is possible to linearize such a model, keeping the slip of the generator.
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
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

 Posts: 15
 Joined: Sat Nov 02, 2013 10:16 pm
 Organization: Graduate Unerversity of Advanced Tecnology
 Location: Iran
Re: Fast Linearized Models
Dear Jason
Thank you very much for your time and quick response
Just 1 more question:
you wrote that with a little manipulation, I can transform this 4state model into my desired 3state model. I was wondering if it means that I should remove the row and column corresponding to the azimuth angle perturbation (Δδg) from the state matrix (A).
thanks again.
Thank you very much for your time and quick response
Just 1 more question:
you wrote that with a little manipulation, I can transform this 4state model into my desired 3state model. I was wondering if it means that I should remove the row and column corresponding to the azimuth angle perturbation (Δδg) from the state matrix (A).
thanks again.

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Hamid Reza,
Yes, removing the azimuth angle perturbation (Δδg) as a state involves removing a row and column from the statespace model. Then, with a little manipulation you can convert the resulting 3state model from using Δωg, Δδtg, and Δωtg as states to using Δωt, Δωg, and Δδtg as states. That is: Δωt = Δωg + Δωtg.
Best regards,
Yes, removing the azimuth angle perturbation (Δδg) as a state involves removing a row and column from the statespace model. Then, with a little manipulation you can convert the resulting 3state model from using Δωg, Δδtg, and Δωtg as states to using Δωt, Δωg, and Δδtg as states. That is: Δωt = Δωg + Δωtg.
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
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

 Posts: 38
 Joined: Mon Nov 30, 2015 4:13 am
 Organization: IK4Ikerlan
 Location: Spain
Re: Fast Linearized Models
Dear All,
This is the most related post that I found it, but I still feeling confused.
Running GetMats.m function must I achieve wind turbine rotor blades dynamics expressed in rotating frames (AMat, BMat, ...), while running MBC3.m function the dynamics of individual blades were expressed in a fixed (nonrotating) frame (MBC_A, MBC_B, ...), right? And if I want to obtain more accurate model I should linearize the model at a number of different azimuth steps, right? Well, if I do that I need to take the average values to form my state space matrix (AvgAMat, AvgBMat,... in rotating reference frame), but:
 Why are not available this average value in non rotating reference frame (only I found MBC_AvgA)?
 Can I achieve the rest of this components (MBC_AvgB, MBC_AvgBd, MBC_AvgC, MBC_AvgD and MBC_AvgDd) as the same way that MBC_AvgA it is calculated?
MBC_AvgA = zeros(ns);
MBC_AvgA = MBC_AvgA + MBC_A(:,:,iaz);
MBC_AvgA = MBC_AvgA/NAzSteps;
Thank you in advance.
Best regards,
JOE
This is the most related post that I found it, but I still feeling confused.
Running GetMats.m function must I achieve wind turbine rotor blades dynamics expressed in rotating frames (AMat, BMat, ...), while running MBC3.m function the dynamics of individual blades were expressed in a fixed (nonrotating) frame (MBC_A, MBC_B, ...), right? And if I want to obtain more accurate model I should linearize the model at a number of different azimuth steps, right? Well, if I do that I need to take the average values to form my state space matrix (AvgAMat, AvgBMat,... in rotating reference frame), but:
 Why are not available this average value in non rotating reference frame (only I found MBC_AvgA)?
 Can I achieve the rest of this components (MBC_AvgB, MBC_AvgBd, MBC_AvgC, MBC_AvgD and MBC_AvgDd) as the same way that MBC_AvgA it is calculated?
MBC_AvgA = zeros(ns);
MBC_AvgA = MBC_AvgA + MBC_A(:,:,iaz);
MBC_AvgA = MBC_AvgA/NAzSteps;
Thank you in advance.
Best regards,
JOE
Joannes Olondriz
Engineer  IK4Ikerlan
Control and Monitoring area, IK4Ikerlan (Olandixo)
20500 ArrasateMondragon (Gipuzkoa). Spain
Paseo J.Mª Arizmendiarrieta, 2
(+34)943 712400  jolondriz@ikerlan.es
Engineer  IK4Ikerlan
Control and Monitoring area, IK4Ikerlan (Olandixo)
20500 ArrasateMondragon (Gipuzkoa). Spain
Paseo J.Mª Arizmendiarrieta, 2
(+34)943 712400  jolondriz@ikerlan.es

 Posts: 4933
 Joined: Thu Nov 03, 2005 4:38 pm
 Location: Boulder, CO
 Contact:
Re: Fast Linearized Models
Dear Joannes,
Yes, your understanding is correct. In general, I agree that AvgAMat etc. is less useful than MBC_AvgA etc.
Best regards,
Yes, your understanding is correct. In general, I agree that AvgAMat etc. is less useful than MBC_AvgA etc.
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
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
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