Lecture 11 Energy Technologies: Wind

Gang He

October 5, 2023

Sun, the source

Wind

\(P=\frac{1}{2}\rho \pi r^2 v^3\)

Where,
\(\rho\) = Air Density (\(kg/m^3\))
\(A\) = Swept Area (m2) = \(\pi r^2\)
\(v\) = Wind Speed (m/s)
\(P\) = Power (W)

Betz’s law: 59.3%

Average power

Rayleigh (a special type of Weibull) distribution

\(f(v)=\frac{2v}{c^2}\exp [-(\frac{v}{c})^2]\)

\(\bar{P}=\frac{6}{\pi}\cdot \frac{1}{2}\rho \pi r^2 (\bar{v})^3=1.91P\)

Use average power when dealing with average wind speed

Power curve

Important corrections

• Temperature: \(\rho = \frac{P\times M.W. \times 10^{-3}}{RT}=\frac{1 atm\times 28.97 g/mol \times 10^{-3}kg/g}{8.2056\times 10^{-5}m^3\cdot atm/(K\cdot mol)\times(273.15+T)K}\)
  
• Altitude: \(P=P_0 e^{-1.185\times 10^{-4}H}\) (H is elevation in meters)
  
• Tower height: \(\frac{v}{v_0}=(\frac{H}{H_0})^\alpha\) (\(\alpha\) is the friction coefficient)

Class of wind resources

Class	10 m (33 ft)		50 m (164 ft)
	Wind power density (W/m2)	Speed m/s (mph)	Wind power density (W/m2)	Speed m/s (mph)
1	0 - 100	0 - 4.4 (0 - 9.8)	0 - 200	0 - 5.6 (0 - 12.5)
2	100 - 150	4.4 - 5.1 (9.8 - 11.5)	200 - 300	5.6 - 6.4 (12.5 - 14.3)
3	150 - 200	5.1 - 5.6 (11.5 - 12.5)	300 - 400	6.4 - 7.0 (14.3 - 15.7)
4	200 - 250	5.6 - 6.0 (12.5 - 13.4)	400 - 500	7.0 - 7.5 (15.7 - 16.8)
5	250 - 300	6.0 - 6.4 (13.4 - 14.3)	500 - 600	7.5 - 8.0 (16.8 - 17.9)
6	300 - 400	6.4 - 7.0 (14.3 - 15.7)	600 - 800	8.0 - 8.8 (17.9 - 19.7)
7	400 - 1000	7.0 - 9.4 (15.7 - 21.1)	800 - 2000	8.8 - 11.9 (19.7 - 26.6)

Global wind power density map

Higher and bigger

Offshore wind

Challenges

Low speed wind

Smaller generator. \(\rightarrow\) Decreased generator weight and cost.

Operating at higher capacity in lower wind speeds. \(\rightarrow\) Greater generator efficiency.

Decreased tower head mass. \(\rightarrow\) Decreased foundation and tower costs.

Decreased PE system rating. \(\rightarrow\) Decreased PE system costs

Pros and Cons

Pros	Cons
Renewables	Variable & integration
Low emissions	Land use/NIMBY
Low costs	Distribution

Unexpected benefits

Hybrid power systems

References

Masters, Gilbert M. 2013. Renewable and Efficient Electric Power Systems. John Wiley & Sons.