Question

A fixed speed wind turbine can be characterized by the following equation

A fixed speed wind turbine can be characterized by the following equation rpm and R is the length of the blade -37 m. The wind speed is a random variable with the following CDF and PDF The characteristic wind speed is -30 m/s. Determine the following: (a) Find the PDF of G, fG(g), using the solution in section 1.3.1 of the notes (b) Find the equation for the CDF FG(g)P(G<g) from Fv(v) (c) Plot the PDF of G either by hand or using software. (d) What is the probability that g will exceed 0.15?

Answer 1

$(b)~CDF~of~G~is\\ F_G(g)=P(G\leq g)=P\left(V\leq wR g \right )=0;~g<0\\ =1-\exp\left(-\frac{wRg}{\theta} \right )\\ =1-\exp\left(-123.33g \right );~g\geq 0~when~w=100,~R=37,~\theta=30\\ (a)~The~PDF~of~G~is\\ f_G(g)=\frac{\mathrm{d}F_G(g) }{\mathrm{d} g}=123.33\exp\left(-123.33g \right );~g\geq 0\\ =0~otherwise\\ (c)$

R code:

t=seq(0,0.1,by=0.0001)
f=123.33*exp(-123.33*t)
plot(t,f,lwd=2,col=1,type="l",ylab=expression(f[G](g)),xlab="g")

$(d)~P(G>0.15)=1-F_G(0.15)=\exp\left(-123.33*0.15 \right )=0.000000009$