Wind Energy

How Can I Calculate the Amount of Power Available at a Given Wind Speed?

Contributed By Eric Eggleston, 5 February 1998

Because air has mass and it moves to form wind, it has kinetic energy. You may remember from science class that:

kinetic energy (joules) = 0.5 x m x V²

where:
m = mass (kg) (1 kg = 2.2 pounds)
V = velocity (meters/second) (meter = 3.281 feet = 39.37 inches)

Usually, we're more interested in power (which changes moment to moment) than energy. Since energy = power x time and density is a more convenient way to express the mass of flowing air, the kinetic energy equation can be converted into a flow equation:

Power in the area swept by the wind turbine rotor:

P = 0.5 x rho x A x V³

where:
P = power in watts (746 watts = 1 hp) (1,000 watts = 1 kilowatt)
rho = air density (about 1.225 kg/m³ at sea level, less higher up)
A = rotor swept area, exposed to the wind (m²)
V = wind speed in meters/sec (20 mph = 9 m/s) (mph/2.24 = m/s)

This yields the power in a free flowing stream of wind. Of course, it is impossible to extract all the power from the wind because some flow must be maintained through the rotor (otherwise a brick wall would be a 100% efficient wind power extractor). So, we need to include some additional terms to get a practical equation for a wind turbine.

Wind Turbine Power:

P = 0.5 x rho x A x Cp x V³ x Ng x Nb

where:
P = power in watts (746 watts = 1 hp) (1,000 watts = 1 kilowatt)
rho = air density (about 1.225 kg/m³ at sea level, less higher up)
A = rotor swept area, exposed to the wind (m²)
Cp = Coefficient of performance (.59 {Betz limit} is the maximum thoretically possible, .35 for a good design)
V = wind speed in meters/sec (20 mph = 9 m/s)
Ng = generator efficiency (50% for car alternator, 80% or possibly more for a permanent magnet generator or grid-connected induction generator)
Nb = gearbox/bearings efficiency (depends, could be as high as 95% if good)

If there is any single equation that the beginning wind enthusiast should memorize, this is it.

Basic Aerodynamic Operating Principles of Wind Turbines

The Figure illustrates the basic aerodynamic operating principles of a horizontal axis wind turbine.

The wind passes over both surfaces of the airfoil shaped blade. It passes more rapidly over the longer (upper) side of the airfoil, creating a lower- pressure area above the airfoil. The pressure differential between top and bottom surfaces results in a force, called aerodynamic lift. In an aircraft wing, this forces causes the airfoil to "rise," lifting the aircraft off the ground. Since the blades of a wind turbine are constrained to move in a plane with the hub as its center, the lift force causes rotation about the hub. In addition to lift force, a "drag" force perpendicular to the lift force impedes rotor rotation. A prime objective in wind turbine design is for the blade to have a relatively high lift-to-drag ratio. This ratio can be varied along the length of the blade to optimize the turbine’s energy output at various wind speeds.

Is Low Frequency Noise a Problem for
Wind Turbines?

Provided by Neil Kelley
U.S. National Renewable Energy Laboratory

1) Because of the low rotational rates of the turbine blades, the peak acoustic energy radiated by large wind turbines is in the infrasonic range with a peak in the 8-12 Hz range. For smaller machines, this peak can extend into the low-frequency "audible" (20-20KHz) range because of high rotational speeds and multiple blades.

2) The levels of infrasound radiated by the largest wind turbines are very low in comparison to other sources of acoustic energy in this frequency range such as sonic booms, shock waves from explosions, etc. The danger of hearing damage from wind turbine low-frequency emissions is remote to non-existent. Generally there little acoustic energy much above about 4000 Hz so ultrasound is not a problem. <100>

3) Impulsive noise generation is generally confined to turbines whose rotors operate downwind of the support tower (downwind machine). In this case, impulses are generated by the interaction of the aerodynamic lift created on the rotor blades and the wake vortices being shed from the tower elements. Turbines that have their rotors upstream of the tower, except in very rare circumstances, do not generate impulses since there is nothing blocking the flow upwind of the rotor. The low-frequency noise generated from an upwind turbine is primarily the result of the interaction of the aerodynamic lift on the blades and the atmospheric turbulence in the wind. Because atmospheric turbulence is a random phenomenon, the radiated low-frequency noise also exhibits a random or non-coherent characteristic. Impulsive noise generated by the tower wake/rotor interaction, on the other hand, tends to be much less random or coherent and therefore much more detectable when it interacts with an intervening resonant structure.

4) In my paper published in the AWEA WindPower '87 conference proceedings ("A Proposed Metric for Assessing the Potential of Community Annoyance from Wind Turbine Low-Frequency Noise Emissions," SERI/TP-217-3261, Nov 1997 or Proceedings of WindPower '87, San Francisco), I discuss the development of a criteria for assessing potential annoyance. It is interesting to note that low-frequency sounds with a random characteristic are much less likely to induce human annoyance than those which are impulsive. Typically the random low-frequency noise induces annoyance only at much higher acoustic energy levels than impulsive sounds.

5) While downwind wind turbines inherently are pre-disposed for producing impulsive noise because of the wakes from tower structural elements, careful designs can reduce such emissions to below detectable levels and therefore will cause no community annoyance. The overwhelming majority of wind turbines that have been installed in Europe are of the upwind rotor design and therefore low-frequency noise has not been considered a major issue there. By contrast, the primary concern there has been the much higher frequency broadband and discrete frequency noise associated with the unsteady aerodynamic forces on the blades, often in the blade tips.

Basic Principles of Wind Turbine
Power Production

The output of a wind turbine varies with the wind's speed through the rotor. This relationship is usually shown graphically in a power curve (Figure).

The "rated wind speed" is the wind speed at which the "rated power" is achieved and generally corresponds to the point at which the conversion efficiency is near its maximum. In many systems, the power output above the rated wind speed is mechanically or electrically maintained at a constant level, allowing more stable system control.

Note that at lower wind speeds, the power output drops off sharply. This can be explained by the cubic power law, which states that the power available in the wind increases eight times for every doubling of wind speed (and decreases eight times for every halving of the wind speed).

Using the power curve, it is possible to determine roughly how much power will be produced at the average or mean wind speed prevalent at a site. In the example above, the turbine would produce about 20% of its rated power at an average wind speed of 15 miles per hour (or 20 kilowatts if the turbine was rated at 100 kilowatts). This is somewhat lower than most modern wind turbines.

How Does A Wind Turbine's Energy Production Differ from Its Power Production?

While wind turbines are most commonly classified by their rated power at a certain rated wind speed, annual energy output is actually a more important measure for evaluating a wind turbine's value at a given site.

We know that . . .

Energy = Power x Time

This means that the amount of time a wind turbine produces a given power output is just as important as the level of power output itself. And wind turbine operators don't get paid for producing a large amount of power for a few minutes (except in rare circumstances.) They get paid by the number of kilowatt-hours (kWh) their turbines produce in a given time period.

The best crude indication of a wind turbine's energy production capabilities is its rotor diameter--which determines its swept area, also called the capture area. A wind turbine may have an impressive "rated power" of 100 kW, but if its rotor diameter is so small that it can't capture that power until the wind speed reaches 40 mph (18 m/s), the wind turbine won't rack up enough time at high power output to produce a reasonable annual energy output.

Expected energy output per year can be reliably calculated when the wind turbine's capacity factor at a given average annual wind speed is known. The capacity factor is simply the wind turbine's actual energy output for the year divided by the energy output if the machine operated at its rated power output for the entire year. A reasonable capacity factor would be 0.25 to 0.30. A very good capacity factor would be 0.40.

NOTE: Capacity factor is very sensitive to the average wind speed. When using the capacity factor to calculate estimated annual energy output, it is extremely important to know the capacity factor at the average wind speed of the intended site.

Lacking a calculated capacity factor, the machine's power curve can actually provide a crude indication of the annual energy output of any wind turbine. Using the power curve, one can find the predicted power output at the average wind speed at the wind turbine site. By calculating the percentage of the rated power (RP) produced at the average wind speed, one can arrive at a rough capacity factor (RCF) for the wind turbine at that site. And by multiplying the rated power output by the rough capacity factor by the number of hours in a year, (8,760), a very crude annual energy production can be estimated. For example, for a 100 kW turbine producing 20 kW at an average wind speed of 15 mph, the calculation would be:

100 kW (RP) x .20 (RCF) = 20 kW x 8760 hours = 175,200 kWh

Actually, because of the effect of the cubic power law, the annual energy output will probably be somewhat higher than this figure at most windy sites. This is determined by the wind power distribution, which shows the percentage of time the wind blows at various wind speeds over the course of an average year. Lacking precise data on a given site, there are two common wind distributions used to make energy calculations for wind turbines: the Weibull distribution and a variant of the Weibull called the Rayleigh distribution that is thought to be more accurate at sites with high average wind speeds.

Energy output is also greatly influenced by more subtle features of a wind turbine's design, including

• cut-in speed, or the wind speed at which it begins to produce power (if the turbine's cut-in speed is significantly below a site's average wind speed, problems are inevitable)
• the power it produces at moderate wind speeds, determined largely by blade airfoil shape and geometry
• the cut-out speed (the wind speed at which the turbine may be shut down to protect the rotor and drive train machinery from damage) or high wind stalling characteristics.
• operating characteristics such as low speed on-off cycling, shut-down behavior, and overall reliability, which together determine the turbine's availability to produce power when the wind speeds are in its operating range
• the efficiency of drive train components, such as the generator and gear box.

These more subtle features should not be underestimated when looking for ways to improve energy output. In recent years, the U.S. wind industry has begun using seemingly insignificant refinements in blade airfoil shapes to increase annual energy output from 10 to well over 25 percent. These increases have helped to dramatically lower the cost of wind-generated energy and increase the number of areas in the U.S. at which wind plants are feasible.