Figure $1$: Reduced nonlinear model.

Exercise $1\mathrm{.}3$ Simulation of a Wind Turbine

A $5$ MW wind turbine is operating below rated wind conditions. With a reduced model $($see Figure $1$

based on $[2]),$ the motion of the rotor speed $$ can be described by:

$J{}^{}=M_a-M_G{r}_{GB}$

The gearbox with the ratio $r_{GB}=97$ is assumed loss$-$free. Further, $J=44\mathrm{.}5{10}^{6}kg{m}^2$ is the sum of

the moments of inertia about the rotation axis. The turbine operates at low wind speeds where the

torque controller adjust the generator torque $M_G$ based on the generator speed ${}_G$

${}_G=r_{GB}$

with $k=0\mathrm{.}0256N\frac{m}{(rpm{)}^2}$ based on $[3]$ by

$M_G=k{}_G^2$

to maintain the turbine in its aerodynamic optimum.

The turbine is operating at constant wind $8\frac{m}{s}$ and a rotor speed ${}_0=9\mathrm{.}196rpm.$ At $t_0=0s,$ the

wind changes to $9\frac{m}{s}$ and we assume that the aerodynamic torque changes instantaneous to the

corresponding $M_{a0}=2\mathrm{.}5{10}^{6}Nm.$ Here we neglect that in reality the aerodynamic torque depends

on the rotor speed as well.

a$)$ State and solve the initial value problem in $y=.$

b$)$ Which rotor speed will the turbine approximate?

c$)$ When does the rotor speed reach $10$ rpm $?$

d$)$ Plot the rotor speed over time.