Wind Energy#

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
import pandas as pd

Wind Power Density#

Consider air with density \(\rho\) moving through an area \(A\) at speed \(U\) (in the direction perpendicular to the area \(A\)). The mass flow rate is

\[ \frac{dm}{dt} = \rho A U \]

and has units M T-1. The kinetic energy of mass \(m\) moving at speed \(U\) is given by

\[ KE = \frac{1}{2} m U^2 \]

Wind power is the rate of kinetic energy flow of air through the given area and is given by

\[ P = \frac{1}{2} \frac{dm}{dt} U^2 = \frac{ \rho A U^3 }{2} \]

Wind power generated by a turbine depends on the swept area of the wind turbine blades and so we usually normalize the power by dividing by the area to assess to the wind power density:

\[ WPD = \frac{P}{A} = \frac{ \rho U^3 }{2} \]

Turbine Power Curve#

A wind turbine only captures a fraction of the total wind power available and a turbine usually operates only within a range of wind speeds from the cut-in wind speed up to the cut-out wind speed. Each turbine has a power curve which specifies the power output as a function of wind speed. The curve is usually an \(S\)-shaped curve from 0kW up to a maximum rated power. For example, the curve below shows data for a General Electric 2.5-200 wind turbine with cut-in speed 3m/s and cut-out speed 25m/s. The data was taken from wind-turbine-models.com.

U = np.arange(0,25.5,0.5)
P = np.hstack([[0,0,0,0,0,0,
                25,89,171,269,389,533,704,906,1136,1400,1674,1934,2160,2316,2416,2477,2514,2528],
                2530*np.ones(27)])
plt.figure(figsize=(12,4))
plt.plot(U,P,'.-'), plt.grid(True)
plt.xlabel('Wind Speed (m/s)'), plt.ylabel('Power (kW)')
plt.title('General Electric 2.5-120 Wind Turbine Power Curve')
plt.show()
../_images/719c7936ff1dba0c84c2d0d7223ecd533585ba82715cb62acbd945de93d92f6f.png

See also

Check out wind-turbine-models.com to see design specifications for different kinds of turbines.

Estimating Wind Power Generation#

Consider a wind turbine with a power curve given by a function \(P(U)\) of the wind speed \(U\). Wind speed \(U\) changes over time \(t\) and so the total energy generated from time \(t_0\) to \(t_f\) is given by

\[ E = \int_{t_0}^{t_f} P(U(t)) \, dt \]

If we have wind speed measurements \(U_0,\dots,U_N\) at intervals \(\Delta t = t_{i+1} - t_i\) (where \(U_i\) is wind speed at time \(t_i\)) then the total energy is

\[ E \approx \sum_{i=0}^{N-1} \left( \frac{P(U_{i+1}) + P(U_i)}{2} \right) \, \Delta t \]

Note that the approximation is given by the trapezoid rule.