# Shaft RPM vs Time

{Edit: I’m new to this forum, had a tough time typing the formula, so I’m adding them now}

In the technical report of the 5-MW reference wind turbine “Definition of a 5-MW Reference wind turbine for offshore system development by J.Jonkman, S.Butterfield, W.Musial and G.Scott” , it is derived that the rotor speed error will behave as a second order system .

{J+(-∂P/ ∂θ)* Kd*N/ω}φ2 +{(-∂P/ ∂θ)* NKp1/ω - Po/ω*ω}φ1 +{(-∂P/∂θ)N Ki*1/ω}φ=0

where

``````     [b][u]∂P/  ∂θ[/u][/b] is the sensitivity of aerodynamic power to the rotor blade pitch angle
φ2,φ1 are the second, first order partial derivative of φ w.r.t time. φ is such that ∂φ/∂t=ΔΩ.
Ω is the rotor speed,ω is the rated rotor speed(constant)
J is the drive train moment of inertia
Kd, Ki, Kp are the derivative, integral and proportional gains respectively
Po is the rated power
N is the gear ratio
``````

This equation above is derived for region 3 of operation where the wind speeds are above the rated values.
How does the rotor rpm (or the rotor rpm speed) behave in region 2 (and 2&1/2) where windspeeds are less than the rated windspeed-from the graph generated in the labview simulation of the sample VI that comes along with the latest version of FAST, the graph generated resembles damped oscillations.

1.)I assumed the above equation to be correct in region 2 with a few changes in the coefficients, because I’m guessing that the rotor rpm exhibits damped periodic oscillations in this region too . Is this correct?

2.) In this equation, I believe that the only term that is dependent on wind speed is “∂P/ ∂θ”.Is this the term that is responsible for the change in rpm with change in windspeed? If yes, then even in region 2,is the variation of HSSrpm or LSSrpm with windspeed only approximately linear?

Also, I noticed that in the sample labview code for the same reference turbine(which comes along with the new FAST) the kp and ki values have been interchanged in the simple control block (as compared to the ones arrived in the report).

3.)((Region 2)) When I generate the graph of RPM (vs. time), it is indeed of the form A*exp(-bt)*sin(wt+c) +d. But w as calculated from the graph is different from the w as obtained using the values from the report.(using natural frequency = 0.6rad/s and damping ratio=0.7). The w from graph is almost 2.02 1/s whereas, theoretically (assuming that the equation is valid in region 2) it’s clearly less than 0.6 1/s. Am I going wrong somewhere? {Note: I used the kp and ki values given in the report and used the labview sample to generate the graph}

4.)Finally, I find that in the labview Simple control block, the torque is maintained constant at the rated torque (=Maxtorque). However according to this report, in region 3, it’s supposed to be inversely proportional to the generator speed.

These may be trivial doubts, if any one of you viewing this post knows the answer to any question in part or full, please post them.

Dear Harimanjunathan Sankaranarayanan,

Regarding questions 1-3, the equation you state (from the NREL 5-MW specifications report: nrel.gov/docs/fy09osti/38060.pdf) applies to above-rated conditions (Region 3) with active blade-pitch control for rotor speed regulation. The equation does not apply to below-rated conditions (Region 2), which typically involves fixed blade pitch and active torque control. For instance, in Region 2, one does not typically vary blade pitch, so ∂P/∂θ is not very useful, but ∂P/∂Ω or ∂P/∂V or ∂P/∂TSR may be. The rotor speed response characteristics in Region 2 will depend on the nature of the torque controller.

Regarding question 4, setting torque inversely proportional to speed (to ensure constant power) in Region 3 is common. Some wind turbines also have constant torque in Region 3, which leads to improved rotor-speed regulation at the expense of small variations in power.

I can’t comment on the LabView models provided with the latest version of FAST because I have not used these, but I’ll ask the developer to check the forum.

Best regards,

Hi,

Just wanted to follow-up on the questions related to Labview. You’re right that the code differs from the report values. Basically, the intention of the labview example was not to provide a complete controller, but just provide an example such that someone could see how labview and FAST are coupled and then implement any controller within labView.

In terms of the torque setting, constant torque was selected just for simplicity, however it should be direct to implement constant power. For kp and ki, this is a mistake. I’ll see if I can get this corrected, and also add a little more documentation that the example VI represents a simplified controller and is not the complete NREL 5MW controller.

Thanks for letting me know!

Paul

Dear Jason and Paul,
Thank you for both of your answers. I still have a few questions though. I’ve edited the labview code so that the Kp and Ki values are correct. After this, upon execution, even in region 3, I found that the Labview generated graph of rpm vs time almost always has a time period of about 3.1 s corresponding to a frequency of 0.3225 1/s . However according to the theory we calculated the Kp and Ki values based on an aim to achieve a rotor speed response of natural angular frequency 0.6 rad/s and damping ratio of 0.7 which gives an "angular frequency " (damped) of 0.428 rad/s which corresponds to a (damped) time period of 14.68 s and a frequency of 0.0681 1/s.

Thus the rpm signal’s damped frequency obtained theoretically doesn’t match with the damped freq. obtained from the graph of the corresponding output.Where am I going wrong in the calculations. I am pretty sure about the time period that I get from the output.

My second question is I noticed that the only difference between the pitch controller that comes with the 5MW turbine (Discon.f90) and the labview controller is that the rpm is filtered in the former. There is no other difference in the logic of both the implementations. Am I right?