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 400400 W

Rotor diameter: 1.15 m

SW WindpowerWhisper 200

1 kWRotor diameter: 2.7 m

LNP 6.4-50005 kW

Rotor diameter: 6.4 m

2 © Alexis Kwasinski, 2012

Low-power wind generation

Bergey Excel7.5 kW

Rotor diameter: 6.4 mSolerner

3 kW

YM-CZ3kW3 kW

SW WindpowerWhisper 500

3 kWRotor diameter: 4.5 m

Wind generatorsIn Tokyo

3 © Alexis Kwasinski, 2012

Average wind power in the US

http://rredc.nrel.gov/wind/pubs/atlas/maps.html

4 © Alexis Kwasinski, 2012

Average wind power in Europe

http://www.geni.org/globalenergy/library/renewable-energy-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 Pmec,in = Pelect,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 Pmec,in = Pelect,out

7 © Alexis Kwasinski, 2012

Brushless/Permanent magnet generators

• Output: ac. Electric frequency depends on the rotor angular speed.

•No issues with brushes

• Ideally Pmec,in = Pelect,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,

21

2KW mv

21

2KdW dm

P vdt dt

31

2P Av

dmAv

dt

10 © Alexis Kwasinski, 2012

Typical Power-speed characteristics

SW WindpowerWhisper 200

1 kWRotor diameter: 2.7 m

SW WindpowerWhisper 500

3 kWRotor 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:

vb

vu

vd

Downwind

Upwind Rotor areaA

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 vb is the average between vu and vd, then the mass flow rate is

• If we define the ratio

2 2( ) 1( )

2u d

b u d

d E E dmP v v

dt dt

2u dv vdm

Adt

d

u

v

v

13 © Alexis Kwasinski, 2012

Conversion efficiency

• Then

• The rotor efficiency is maximum when λ is 1/3. For this value, Cp is 0.593.

• 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):

2 2 2 3 21 1 1( ) (1 )(1 )

2 2 2 2u u

b u d u

v vP A v v Av

Power in the wind

Fraction extracted

Rotor efficiencyCp

rotor tip speed( )

wind speed 60 b

rpm DTSR

v

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. Rayleighdistributions 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

0. ( )

2v v f v dv c

2

2

2( )

v

cvf v e

c

3 3 3 3

0

3 6( ) ( )

4avgv v f v dv c v

36 1

2avgP Av

31( )

2avg avgP A v