- Wind modelling analysis on a given location, through Rayleigh and Weibull probability density function determination
- Compare the calculated Rayleigh and Weibull plots to on-screen images (see below for sample graph)
Introduction and Theory:
In this era of huge energy demands and shortage of the existing conventional energy sources, wind energy is one of the most established and cost-competitive renewable energy sources. A wind turbine is an effective way to harness the energy contained in the wind. Basically, the energy contained within the wind is in the form of kinetic energy. A turbine transforms this kinetic energy into mechanical energy which is inturn transformed into electrical energy with the means of a generator. Energy produced by the turbine is proportional to the energy contained in the wind column.
The Kinetic Energy of a stream of wind is given by:
KE = ½ mv2 = ½ ρ (AΔx) v2,
where ρ is the air density, A is the cross section area of the stream, and Δx is in the direction of wind movement (mass = density * volume = density * area * delta length = ρAΔx).
Then the Wind Energy Density, Pw, is given by the derivative of KE with respect to time, divided by the Area
Pw = dKE/dt * (1/A) = ½ ρ A (Δx/dt) v2 *(1/A)
Pw = ½ ρ (dx/dt) v2 = ½ ρ v v2 = ½ ρ v3
Pw = ½ ρ v3 (watts/meter2)
Also, it’s important to note that the density of the air changes with temperature
ρ = P/(RT,) (kg/m3)
where P = Pressure, R = the Gas Constant, and T = Temperature. The air density, ρ, is frequently assumed to be constant at sea level and equal to 1.2929 kg/m3.
In order to install and to get the turbine working in good efficiency, there are many factors to be considered such as: availability of direct wind, height of the location, average wind speed and annual wind pattern. The studies about the various parameters are collectively called as “wind performance monitoring”. The size and the generator characteristics of the turbine can be decided only after the wind performance monitoring.
The energy contained in the wind varies exponentially in proportion with the velocity of the wind. Logically, the best location for wind energy production would be somewhere where there is a steady wind, from the same direction, and at a high enough speed for the turbine to work at optimum efficiency as determined by its aerodynamics
Normally the wind measurements are made for a longer period of time for example, one month, or two months. Because the data over a larger time period smoothens out the occasional irregularities in wind distribution due to the prevailing weather conditions of the location. The wind data is taken from an anemometer attached to the weather station installed at the site. The anemometer gives the wind direction and the wind speed values. The readings are fed into a data logger continuously for over a period of time.
The wind speed at which the turbine starts to spin from the rest is called “cut in” velocity and the velocity at which the turbine stops out from rotation is called “cut off” velocity. So the cut in and cut off velocities for the location can only be fixed after a detailed wind distribution statistical study over a longer period. A statistical density function of the wind velocity is needed. This will give someone investigating the feasibility of a wind production facility at a particular location a better idea of how well it will work out. A density function which shows most of its values falling in the optimum wind speed range will indicate a potential treasure trove of wind energy.
Weibull density function
There is various density functions used for wind speed monitoring. The most commonly used are Weibull density function and Rayleigh density function. Weibull density function is two variable density function namely the scale parameter(c) and shape parameter (k), while the Rayleigh density function a single variable density function -wind speed (u).Which makes Weibull somewhat versatile and Rayleigh somewhat simpler to use.
The wind speed u is described as Weibull distribution if its probability density function is
(k > 0,u > 0,c > 1) (1)
where k is shape parameter, cis scale parameter and u is the wind speed.
The Weibull cumulative distribution function is given by,
Determining the Weibull parameters
There are several methods available for determining the Weibull parmeters c and k. If the mean and variance of the wind speed are known, the following equations can be used to determine c and k.
where is the standard deviation, is the mean wind speed and is the gamma function
Example of Weibull probability density function, for various values of k, are given in the below figure. As shown, as the value of k increases, the curve has a sharper peak, indicating that there is less wind variation.
The Rayleigh probability density function is given by
The Rayleigh cumulative distribution function is
The figure shows a Rayleigh probability density function for different mean wind speeds.
Implementation of density functions
For testing, we measured actual wind values of the site using an anemometer and a data logger. The data logger measures and stores the wind speed values once every 10 seconds. i.e. 8640 values a day. The actual wind speed values of 50 days have been used to plot the density functions.
After collecting all the required values, the frequency of each wind speed is calculated. For ease of calculation only multiples of 0.5 up to 7 is taken. The rest of the values are rounded off to the nearest multiple of 0.5.The counted frequency is then used to calculate the probability density and cumulative density of each wind speed value
The Weibull and Rayleigh density functions can give the students a fairly accurate depiction of the wind conditions of this site. These values can be used later on to predict the power production capabilities of a turbine in the same site.