1 © Alexis Kwasinski, 2012 Low-power wind generation Power output of each generation unit in the order of a few kW. Power profile is predominately stochastic. Originally they were used for nautical and rural applications with dc generators. Cost is relatively low. More modern systems use permanent-magnet generators. Air-X W Rotor diameter: 1.15 m SW Windpower Whisper kW Rotor diameter: 2.7 m LNP kW Rotor diameter: 6.4 m

2 © Alexis Kwasinski, 2012 Low-power wind generation Bergey Excel 7.5 kW Rotor diameter: 6.4 m Solerner 3 kW YM-CZ3kW 3 kW SW Windpower Whisper kW Rotor diameter: 4.5 m Wind generators In Tokyo

3 © Alexis Kwasinski, 2012 Average wind power in the US

4 © Alexis Kwasinski, 2012 Average wind power in Europe resources/europe/Wind/Wind%20Map%20of%20Western%20Europe_files/euromap.gif

5 © Alexis Kwasinski, 2012 Generators: Synchronous machine Output: ac. Electric frequency depends on the rotor angular speed. Requires a dc input. Ideally P mec,in = P elect,out

6 © Alexis Kwasinski, 2012 Generators: Dynamos (Brushed dc generator) Output: ac + dc. AC component electric frequency depends on the rotor angular speed. Important maintenance and reliability issues caused by the brushes Ideally P mec,in = P elect,out

7 © Alexis Kwasinski, 2012 Brushless/Permanent magnet generators Output: ac. Electric frequency depends on the rotor angular speed. No issues with brushes Ideally P mec,in = P elect,out

8 © Alexis Kwasinski, 2012 Wind generators model The output in all types of generators have an ac component. The frequency of the ac component depends on the angular speed of the wind turbine, which does not necessarily matches the required speed to obtain an output electric frequency equal to that of the grid. For this reason, the output of the generator is always rectified. The rectification stage can also be used to regulate the output voltage. If ac power at a given frequency is needed, an inverter must be also added. There are 2 dynamic effects in the model: the generator dynamics and the wind dynamics.

9 © Alexis Kwasinski, 2012 Wind power Consider a mass m of air moving at a speed v. The kinetic energy is Then power is The last expression assumes an static wind behavior (i.e. v is constant) The mass flow rate dm/dt is Thus,

10 © Alexis Kwasinski, 2012 Typical Power-speed characteristics SW Windpower Whisper kW Rotor diameter: 2.7 m SW Windpower Whisper kW Rotor diameter: 4.5 m

11 © Alexis Kwasinski, 2012 Conversion efficiency In the previous slide, power does not varies with the cube of the wind speed. Why? Because not all the wind power is transmitted through the blades into the generator. Consider the next figure: vbvb vuvu vdvd Downwind Upwind Rotor area A

12 © Alexis Kwasinski, 2012 Conversion efficiency The wind energy “absorbed” by the wind turbine rotor equals the kinetic energy lost by the wind as it pass through the blades. Hence, the energy transmitted by the wind to the rotor blades is the difference between the upwind and the downwind kinetic energies: In the last equation it is assumed that there is no turbulence and the air passes through the rotor as a steady rate. If it is assumed that v b is the average between v u and v d, then the mass flow rate is If we define the ratio

13 © Alexis Kwasinski, 2012 Conversion efficiency Then The rotor efficiency is maximum when λ is 1/3. For this value, C p is Still, we still need to know how much of the “absorbed” power by the blades is transmitted to the generator. This conversion stage is characterized based on the tip-speed ration (TSR): Power in the wind Fraction extracted Rotor efficiency C p

14 © Alexis Kwasinski, 2012 Conversion efficiency From the course’s recommended book

15 © Alexis Kwasinski, 2012 Variable rotor speeds The maximum power point changes as the rotor speed changes. From the course’s recommended book

16 © Alexis Kwasinski, 2012 Wind stochastic nature Wind speed probability (then generated power, too) is an stochastic function. Wind speed probability can be represented using a Rayleigh distribution, which is a special case of a Weibull distribution. The Rayleigh distribution appears when a 2-dimentional vector has characteristics that: are normally distributed are uncorrelated have equal variance. A typical probability density distribution for wind speed is shown next. Rayleigh distributions approximates these curves.

17 © Alexis Kwasinski, 2012 Wind stochastic nature The Rayleigh probability density function is given by where c is a parameter. The average value of the random variable (wind speed v ) is The average power is If Then