{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)* N*Kp*1/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.

Thank you for your time.