Final answer:
To calculate the shut down hours per year due to high-speed winds, we can use the Rayleigh distribution and its cumulative distribution function. We can calculate the probability of wind speeds exceeding the furling wind speed and multiply it by the number of hours in a year.
Step-by-step explanation:
To determine the number of hours per year the wind turbine will be shut down due to excessively high-speed winds, we need to calculate the probability of wind speeds exceeding the furling wind speed. The Rayleigh distribution is used to describe wind speeds, and the cumulative distribution function (CDF) is given by:
CDF = 1 - exp(-(v^2)/(2σ^2)), where v is the wind speed and σ is the scale parameter (related to the average wind speed).
To calculate the shut down hours, we integrate the CDF from the furling wind speed (25 m/s) to infinity, and multiply the result by the number of hours in a year. Here is the step-by-step calculation:
- Calculate the scale parameter σ: σ = average wind speed / sqrt(2) = 9 / sqrt(2) m/s
- Calculate the CDF for the furling wind speed: CDF_furling = 1 - exp(-(25^2)/(2*(9/sqrt(2))^2))
- Calculate the probability of shut down: P_shut_down = CDF_furling
- Calculate the shut down hours per year: Shut_down_hours = P_shut_down * 365 * 24
Plug in the values and calculate the result.