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| Fig. 1: Power County Wind Farm, Idaho. (Source: Wikimedia Commons) |
A wind turbine is any device that converts kinetic energy from wind patterns to electrical energy through the use of a generator. By far the most popular type of commercial wind turbine is that which rotates vertically about a horizontal axis; such devices are called HAWTs (horizontal axis wind turbines). [1] These are the quintessential turbines that one pictures in his or her mind when one thinks of wind farms. A well-known example is the Power County Wind Farm, located southeast of American Falls, Idaho, shown in Fig. 1.
All commercially-available wind turbines function in a relatively similar way. When wind flows across the airfoil blade, it splits. Some flows over the top and the rest flows under the bottom. Because the distance the air must travel over the top of the blade is greater, the air going over the top travels faster, creating a pressure differential on either side of the blade. [1] This phenomenon is known as Bernoulli's Principle: as the velocity of a fluid flow increases, its pressure decreases. [2] The aforementioned pressure differential causes a lifting force that rotates the blade. The rotor then connects directly or through a gearbox to a generator, which converts the kinetic energy of the rotor/gearbox to electricity.
While it is easy to intuitively understand the basic physical principles behind wind turbine electricity generation, it is significantly more difficult to estimate and contextualize the amount of energy a turbine actually produces. Conceptually, we can formulate a simplification of the problem in order to better understand it. When wind blows at speed V (measured in m/s) it carries with it energy flux:
| Φ | = | 1 2 |
ρ | V3 |
where ρ is the mass density of air (1.225 kg/m3). The turbine spins, and delivers a power P (measured in watts) to its load. If the wind is assumed to blow perpendicularly to the turbine, the effective cross-section A of the turbine is therefore:
| A | = | P Φ |
In actuality, the problem is significantly more complex than this, and accurate modeling of turbine power generation requires calculations and simulations beyond the scope of this exploration.
A useful rule of thumb is that the effective cross section of a turbine is about half the physical cross section of the blade disc at the optimal wind speed (design point). Assuming from Table 1 and Fig. 2 that this is 10 m sec-1 and the power delivered is 6.0 MW, then we have
| Φ = | 1 2 |
× 1.225 kg m-3 × (10.0 m sec-1)3 = 613 W m-2 |
| A = | 3.0 × 106 W 613 W m-2 |
= 4894 m2 |
| R = ( | 2 A π |
)1/2 = ( | 2.0 × 4894 m2 π |
)1/2 = 55.8 m |
This is a reasonable estimate of the correct answer of 126 m / 2 = 63 m.
The first key factor to understand in turbine power production is windspeed. Calculating turbine power output would be simple if wind speeds were assumed to be constant. However, it has been empirically determined over time that specific probability density functions are able to, with reasonable accuracy, model wind speed distributions. The standard empirically based probability density function used for studies of surface wind speeds is the Weibull distribution. This distribution is characterized by a particular relationship between statistical moments: the skewness is a decreasing function of the ratio of the mean to the standard deviation. [3] The mean of a particular Weibull distribution is therefore the average windspeed experienced by a turbine over the course of a year.
Nameplate capacity is the amount of energy the turbine would produce if it ran 100% of the time at optimal wind speeds, and is given by the turbine's manufacturer. [4] Capacity factor (CF) is the actual energy produced over a specified period of time, divided by the nameplate capacity. [4] It thus quantifies the real-world performance of the turbine, as compared to its maximum possible energy production. Once we know the capacity factor of a given turbine, we know how much energy it is actually producing, and vice versa. We now have the tools necessary to understand and contextualize the amount of energy produced by any geared wind turbine.
Cited below is the actual performance of a 5,000 kW Senvion turbine with a blade diameter of 126 meters. The wind speed probabilities are given by a Rayleigh distribution with a mean wind speed of 7.5 m/s. The sum of all probabilities does not exactly equal 1.0 and the sum of hours per year does not exactly equal 8,760 because of the coarse resolution of the wind speed intervals. The annual average power production (1,682 kW) is the total energy output divided by 8,760 hours per year. The capacity factor is then 1,682 kW/5000 kW = 33.6%. [1]
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| Table 1: Senvion turbine performance at various conditions. [1] |
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| Fig. 2: Power Curve of Senvion 5,000 kW (126 m) Turbine, from Table 1. (Source: O. Friedman) |
Furthermore, I have graphed in Fig. 2 the power curve of the turbine; that is, the turbine power output at various windspeeds. The information used to create the graph is simply taken from the table cited above.
Notably, the power output of the turbine goes to zero when windspeed is 32 m/s. With increasing wind speed above the rated wind speed, the power output remains roughly constant until the cut-out wind speed is reached, at which point the power output is reduced to zero. The shutoff is accomplished through pitch control or active stall control and is needed to prevent damage to the turbine. Most turbines, even when shut off, are designed to survive wind speeds up to a destruction wind speed of 50 m/s for up to 10 minutes. [1] A new class of turbines is being designed to withstand sustained 10-minute wind speeds of up to 57.5 m/s. These typhoon-class wind turbines will be placed primarily offshore in locations prone to hurricanes or typhoons. For comparison, sustained 1-minute wind speeds in a Category 4 hurricane are 58.6 to 69.3 m/s. [1]
© Oliver Friedman. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] M. Z. Jacobson, 100% Clean, Renewable Energy and Storage for Everything (Cambridge University Press, 2020).
[2] R. A. Serway and J.W. Jewett, Physics for Scientists and Engineers, Fourth Edition, Vol. 1 (Harcourt College Publishing, 1995).
[3] A. M. Monahan et al., "The Probability Distribution of Land Surface Wind Speeds," J. Climate 224, 3892 (2011).
[4] "New York Wind Energy Guidebook," New York State Energy Research and Development Authority, September 2020.
