(Solved): Problem 1.3: (a) Consider the analytic expression for the coefficient of power of a wind turbine pr ...

Problem 1.3: (a) Consider the analytic expression for the coefficient of power of a wind turbine proposed in [18], or $C_P(\lambda )=\left(\frac{c_1}{{\lambda }_1}-c_2\right)e^{-\frac{c_3}{{\lambda }_1}},{\lambda }_1=\frac{\lambda }{1-c_4\lambda }.$ Find analytic expressions for ${\lambda }_{OPT}$, the TSR for which $C_P(\lambda )$ is maximum, and for ${\lambda }_{MAX}$, the TSR for which $C_P(\lambda )$ is 0 . (b) Using the results of part (a), compute ${\lambda }_{OPT},C_{OPT}$, and ${\lambda }_{MAX}$ for $c_1=58$, $c_2=2.5,c_3=21$, and $c_4=0.035$. Plot $C_P(\lambda )$ for $\lambda$ ranging from 1 to 14 and check that the values that you computed are consistent with the graph. (c) Assuming a turbine with $R=9\mathrm{m}$ and a wind speed $v_W=10\mathrm{m}/\mathrm{s}$, plot $P_T$, the mechanical power available from the wind turbine as a function of the turbine speed. Label the axes in $\mathrm{kW}$ and rpm. What speed would the turbine reach if rotating freely (i.e., if $P_F=P_G=0$ )? (d) Consider a grid-tied squirrel-cage induction generator (SCIG) connected to the turbine through a gear, as shown in Fig. 1.22. The generating torque is given by ${\tau }_G=-\frac{k_{1}S}{1+{\left(k_{2}S\right)}^2},$ where $S={\omega }_S-n_P\omega$ is the so-called slip frequency in rad $/\mathrm{s},{\omega }_S=120\pi \mathrm{rad}/\mathrm{s}$ is the angular electrical frequency of the grid voltages (at $60\mathrm{Hz}$ ), $n_P=2$ is the number of pole pairs, and $\omega$ is the speed of the generator (in rad/s). The model represents the SCIG using a steady-state approximation. Assume that there are no friction losses, so that ${\tau }_{PM}={\tau }_G$ and $P_{PM}=P_G$ in (1.22)). Let $k_1=50,k_2=0.032$, and assume that the generator is connected to the wind turbine of parts (a) to (c) using a gear with ratio $G=29$. Plot on the same graph $P_T$ and $P_G={\tau }_G\omega$ (in $\mathrm{kW}$ ) as functions of the turbine speed ${\omega }_T$ in rpm. The curves should be similar to Fig. 1.8, but with ${\omega }_T$ on the xaxis. The values of ${\omega }_T^{\ast }$ where $P_T=P_G$ are the possible operating points of the turbine/generator set. Using the graph, estimate the speed and the power generated at this condition.