$7.(70$pts$)$ The power curve for the Vestas V$82$ wind turbine has been provided, and the relevant data is tabulated in the table. The wind turbine's rated velocity and power are $\backslash (13\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ and $1650$ kW $,$ respectively. This turbine features active stall control, which maintains the output power at $1650$ kW until it reaches its cut$-$out velocity of $\backslash (20\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ The cut$-$in velocity for this turbine is $\backslash (3\mathrm{.}5\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ and the rotor diameter is $82$ m $.$ At its rated velocity of $\backslash (13\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ the rotor speed is $14\mathrm{.}4$ RPM$.$ We plan to install this turbine at a site with Rayleigh wind statistics, where the average wind speed at hub height is $\backslash (10\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ density of air is $\backslash (1\mathrm{.}225\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$

a$)(15$pts$)$ Calculate the annual wind energy generated, assuming no losses in the gearbox, generator, inverter, etc.

b$)(15$pts$)$ Determine the overall average efficiency of the wind turbine.

c$)(10$pts$)$ Calculate the capacity factor of this wind turbine.

d$)(15$pts$)$ Imagine if the turbine were designed to work in its maximum Cp $(0\mathrm{.}46)$ between cutin wind speed and rated wind speed. Then what would be the new power curve, rotor average efficiency, and capacity factor of this new wind turbine $($Note: using new $\backslash (\backslash$mathrm$\{$cp$\}=0\mathrm{.}46\backslash ),$ the rated power of $1650$ kW would achieve at lower rated velocity.$)$

e$)(15$pts$)$ If the turbine $($in its original configuration with the given input data$)$ is installed at a location with Rayleigh wind statistics and an average wind speed of $\backslash (6\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ what would be the new capacity factor and the average efficiency of the turbine? $\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multicolumn$\{3\}\{|$l$|\}\{$Table: Vestas V$82-1\mathrm{.}65$MW wind turbine data$\}\backslash \backslash$

$\backslash$hline Wind Speed $[$m$/$s$]$ & Power $[$kW$]$ & Cp $\backslash \backslash$

$\backslash$hline $3$ & $0$ & $0\backslash \backslash$

$\backslash$hline $4$ & $28$ & $0\mathrm{.}135\backslash \backslash$

$\backslash$hline $5$ & $144$ & $0\mathrm{.}356\backslash \backslash$

$\backslash$hline $6$ & $309$ & $0\mathrm{.}442\backslash \backslash$

$\backslash$hline $7$ & $511$ & $0\mathrm{.}461\backslash \backslash$

$\backslash$hline $8$ & $758$ & $0\mathrm{.}458\backslash \backslash$

$\backslash$hline $9$ & $1017$ & $0\mathrm{.}431\backslash \backslash$

$\backslash$hline $10$ & $1285$ & $0\mathrm{.}397\backslash \backslash$

$\backslash$hline $11$ & $1504$ & $0\mathrm{.}349\backslash \backslash$

$\backslash$hline $12$ & $1637$ & $0\mathrm{.}293\backslash \backslash$

$\backslash$hline $13$ & $1650$ & $0\mathrm{.}232\backslash \backslash$

$\backslash$hline $14$ & $1650$ & $0\mathrm{.}186\backslash \backslash$

$\backslash$hline $15$ & $1650$ & $0\mathrm{.}151\backslash \backslash$

$\backslash$hline $16$ & $1650$ & $0\mathrm{.}125\backslash \backslash$

$\backslash$hline $17$ & $1650$ & $0\mathrm{.}104\backslash \backslash$

$\backslash$hline $18$ & $1650$ & $0\mathrm{.}087\backslash \backslash$

$\backslash$hline $19$ & $1650$ & $0\mathrm{.}074\backslash \backslash$

$\backslash$hline $20$ & $1650$ & $0\mathrm{.}064\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$