Wind Turbine performance, wind resource evaluation (probability distributions), capacity factor (5.1-5.3)

How was the equation for maximum power available in a turbine derived?

We took the mass of a thin slice of stream-tube thickness (dm=rho x change in volume = rho A dx), and used that in the equation for change in energy (dKE = (1/2)dmV^2). Then divided that by change in time to get power [as power is energy / time] (dKE /dt = (1/2)rhoAV^2 dx/dt)

Max power available in turbine equation

Pavail = (1/2)rhoAV^3

Force balance on a wind turbine

-Fsupportingcolumn = Fwind

What happens to the horizontal velocity component through the WT plane?

It remains the same

What happens to wind speed after passing through a turbine?

It decreases

True or false, eventually the power in the wake is restored to the power of the upstream

True

Relationship between velocity of the turbine plane to velocities of the upstream and wake planes

Vt = (1/2)(vinf + vwake)

what are at and aw?

Slowdown factors

Relationship between the slowdown factors

aw = 2at

Cp

Coefficient of performance for the wind turbine

Cp equation

Power generated by turbine / Power in wind; Prated / (1/2)rhoAtVinf^3

Betz Limit

the theoretical maximum fraction of the wind's energy that a wind turbine can extract; equal to 16/27, or about 59.3%

Why can't a turbine capture 100% of wind energy?

Air must keep moving past the turbine

Probability density function (pdf)

An equation used to compute probabilities of continuous random variables. The higher the curve the more likely that variable happens

Why are probability distributions used in wind energy analysis?

Because wind speeds vary over time

Mode

the most frequently occurring score(s) in a distribution

median

the middle score in a distribution; half the scores are above it and half are below it

Rayleigh PDF, aka f(V)

Used to commonly model wind speeds, special case of the Weibull PDF when the shape factor K = 2

If the f(V) curve is tall and thin, what does that mean?

Probability is concentrated over a small amount of variables

If f(V) curve is wide and short, what does it mean?

Probability is spread out over a large amount of variables

Vbar

mean velocity

c for Rayleigh PDF

c = 2Vbar / sqrt(pi)

Cumulative distribution function

Probability the wind speed will be between 0 and a particular value V

Median wind velocity

Put Rayleigh Cumulative distribution function equal to 0.5

Most probable wind velocity (mode)

Peak of f(V), Vhat = sqrt()2/pi)Vbar

Why must wind power calculations use (V^3)bar instead of (Vbar)^3

Because wind power depends on the cube of velocity, not the average velocity cubed. Average velocity cubed is affected by outliers

Why do high wind speeds contribute disproportionately to energy production?

Because power increases with (V^3)

Average wind power density equation

P/A = (3/pi)rhVbar^3

capacity factor

Actual Energy Generated / Max possible generation

What does capacity factor depend on?

How much wind actually blows, as it does not blow all the time. If it did you would get your mas possible generation