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Wind Speed Distribution Weibull

 In order to calculate the likely power output from a given wind turbine it is necessary to understand the wind in the planned turbine location. It is very easy to find the average wind speed in a location

Wind speeds in most of the world can be modeled using the Weibull Distribution. This statistical tool tells us how often winds of different speeds will be seen at a location with a certain average (mean) wind speed. Knowing this helps us to choose a wind turbine with the optimal cut-in speed (the wind speed at which the turbine starts to generate usable power), and the cut-out speed (the speed at which the turbine hits the limit of its alternator  and can no longer put out increased power output with further increases in wind speed).

 Weibull Distribution and Wind Speeds Pictured here is an example of the Weibull Distribution of Wind Speeds for a site with an average (mean) wind speed of 7 metres per second (from Danish Wind Industry Association). It demonstrates visually how low and moderate winds are very common, and that strong gales are relatively rare. The line at 6.6 metres per second marks the median wind speed. 50% of the time the wind is lower than the median and 50% of the time it is stronger than the median. approximately 2. Standard performance figures provided by wind turbine manufacturers typically use a Shape value of 2 making this distribution a Rayleigh Distribution.

 The higher the value of Shape (from 1 to 3) the higher the median wind speed - i.e. locations with lots of low wind speeds as well as some very strong winds would have a value of shape of below 2, locations with fairly consistent wind speeds around the median would have a shape value of 3.

 Using the Weibull Distribution Since the Weibull Distribution can be used to calculate the probability of a particular wind speed at a particular location, it can be used to work out the number of hours per year that certain wind speeds are likely to recorded and therefore the likely total power output of a wind turbine per year. 

Example Electricity Generation Calculation

We can calculate the power generated by a wind turbine at different wind speeds as long as we know the diameter of the turbine rotors, and the overall efficiency of the turbine generator (below the Betz Limit of 59%). Then we simply multiply the number of hours at each wind speed by the power generated at that wind speed to give us the number of Watt Hours of power generated - or divide by 1,000 to give us the number of kwh of power generated per year (1 year = 8,772 hours).

 Looking at the Windpower 1000 - this domestic wind turbine has a rotor diameter of 1.75m, a cut-in speed of 5m/s, and a cut-out speed of 14m/s. Assuming an efficiency of around 35% and an average wind speed of 5 m/s (not actually realistic on most UK roofs see our article on roof mounting wind turbines ), the total electricity output is given by the Weibull Distribution at around 990 kWh per year.

Calculate the power of the wind hitting your wind turbine generator

There are many complicated calculations and equations involved in understanding and constructing wind turbine generators however the layman need not worry about most of these and should instead ensure they remember the following vital information:

 1) The power output of a wind generator is proportional to the area swept by the rotor - i.e. double the swept area and the power output will also double.

2) The power output of a wind generator is proportional to the cube of the wind speed - i.e. double the wind speed and the power output will increase by a factor of eight (2 x 2 x 2)! 

If you are not mathematically minded you can quit now, however it is well worth trying to understand what is going on here. 

The Power of Wind

Wind is made up of moving air molecules which have mass - though not a lot. Any moving object with mass carries kinetic energy in an amount which is given by the equation: 

Kinetic Energy = 0.5 x Mass x Velocity2

where the mass is measured in kg, the velocity in m/s, and the energy is given in joules. 

Air has a known density (around 1.23 kg/m3 at sea level), so the mass of air hitting our wind turbine (which sweeps a known area) each second is given by the following equation: 

Mass/sec (kg/s) = Velocity (m/s) x Area (m2) x Density (kg/m3) 

And therefore, the power (i.e. energy per second) in the wind hitting a wind turbine with a certain swept area is given by simply inserting the mass per second calculation into the standard kinetic energy equation given above resulting in the following vital equation: 

Power = 0.5 x Swept Area x Air Density x Velocity3 

where Power is given in Watts (i.e. joules/second), the Swept area in square metres, the Air density in kilograms per cubic metre, and the Velocity in metres per second. 

Read World Wind Power Calculation. 

The world's largest wind turbine generator has a rotor blade diameter of 126 metres and so the rotors sweep an area of PI x (diameter/2)2 = 12470 m2! As this is an offshore wind turbine, we know it is situated at sea-level and so we know the air density is 1.23 kg/m3. 

The turbine is rated at 5MW in 30mph (14m/s) winds, and so putting in the known values 

we get: Wind Power = 0.5 x 12,470 x 1.23 x (14 x 14 x 14) 

...which gives us a wind power of around 21,000,000 Watts. Why is the power of the wind (21MW) so much larger than the rated power of the turbine generator (5MW)? Because of the Betz Limit and inefficiencies in the system.

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