Wind power theory
Wind power theory
Table of Contents
Wind turbine types
Wind turbines currently used in utility-scale onshore and offshore wind farms are typically 3 bladed, horizonal axis wind turbines where the rotor is upwind of the nacelle and tower. Horizontal axis wind turbines with upwind rotors require yaw motors and controls to keep the rotor facing the wind direction.
Most offshore wind turbines are direct drive (no gearbox), while onshore wind turbine designs are split between direct drive and constant speed turbines with gearboxes.
Other possible turbine types are possible, but not in common use.
Wind kinetic energy through turbine
The kinetic energy K of a mass of air m travelling at a velocity v is
\\ \hspace{1cm}K = \frac{1}{2} mv^2. \\ \\
The mass of air of density \rho flowing through a turbine rotor area A each second is
\\ \hspace{1cm}m = A \rho v.\\\\
The wind power density P is the kinetic energy of the wind flowing through a wind turbine in one second is thus
\\ \hspace{1cm}P = \frac{1}{2} \rho A v^3.\\\\
Thus the total power available is proportional to the swept area of a turbine rotor, as you would expect, and to the cube of the wind speed, which is perhaps surprising.
The bad news is that even the best designed wind turbine cannot theoretically tap into all of that power. The issue is straightforward – if you managed to extract all the power then the wind behind the wind turbine would have to have zero kinetic energy, which means it would be stationary. There would thus be no room available for new wind to pass through the turbine rotor, so the process of extracting power from the wind would stop.
The Betz law
There is thus a coefficient which can be calculated which gives the absolute maximum possible fraction of the total kinetic energy in the wind which can be extracted by a wind turbine. The coefficient is known as the Betz coefficient C_p and is
\\ \hspace{1cm}C_p = \frac{16}{27} = 0.593 = 59.3\% \\ \\
derived here.
Utility scale turbines will typically extract 75% to 80% of the power from the wind allowed by the Betz law limit. However, no one ever mentions the efficiency of a wind turbine, relative to the Betz limit. Contrast that with the emphasis on the efficiency of solar panels, regarded as much more important. The lack of obvious interest in wind turbine efficiency may be due to the lower variation of efficiency between different wind turbines.
Wind turbine spacing
The Betz limit applies to a single wind turbine. However, within a multiple turbine wind farm, each turbine has a downwind wake, consisting of slower and more turbulent air. If the downwind wake from an upwind turbine forms a major part of the upwind air for a downwind turbine, the output from the downwind turbine will be less than the output of the upwind turbine. Not only that, but the downwind turbine will suffer more mechanical strains because of the turbulence added to the stream of air by the upwind turbine.
To avoid severe impacts from turbine wakes on downwind turbines, the spacing in any direction between wind turbines is generally a factor of six to ten times the rotor diameter of the turbines.
Wind does not blow in the same direction all the time, so the relationship between upstream and downstream turbines is dynamic, applying only at times when the wind is blowing in a particular direction.
Effect of turbine hub height
The wind is both stronger and more consistent higher up.
The US DoE estimates [page vi] that in the flat Great Plains region, increasing the hub height of a wind turbine from 80 to 110m increased the wind speed by 0.5 to 1.0 m/s. Moving from 80 to 160m hub height increases the wind speed by 1.0 to 1.5 m/s
Although an improvement of only 10% or so in wind speed may not sound like a lot, remember that wind power output depends on the cube of the wind speed. So a 10% increase in wind speed gives a 33% increase in power output from a turbine.
The DoE also estimates [also page vi] that capacity factors tend to rise with hub height. For a given turbine type, moving from 80 to 110m adds 2 to 4%. From 110 to 140m adds another 2 to 4%, and from 140 to 160m adds another 1%.
Capacity Factors
Although capacity factors have significance for a given maximum turbine output, to some extent capacity factors can be designed in. For instance, compare two wind turbines with the same rotor diameter and hub height in regions of similar wind. If one wind turbine has twice the generator maximum output of the other, then it will have a lower capacity factor. That is, the capacity factor is a function of the total energy which could be delivered to the generator by the wind passing through the rotor swept area, vs the maximum electrical output from the generator in the nacelle.
However, the consistency of the wind in the region at that hub height also affects the capacity factor.
When assessing the economics of a web farm, the important criterion tends to be total electricity generated versus project cost. Because the grid connection has significant costs, it may make financial sense to cap the useful total generator output by going with a lower grid connection capacity, which has the effect of increasing the capacity factor as measured at the grid connection.
Increasing the capacity of the connection to the grid will always provide more output and more revenue. However, at some value, the increase in the cost of a higher capacity grid connection is higher than the additional revenue which would be received. An offshore wind farm connection to the onshore grid might be 25% of the overall capital costs.
The inverters and grid connection of a utility scale solar PV farm are similarly expensive parts of the overall system. Thus the financially optimum DC power output from the solar panels is often 30 to 40% more than the AC capacity of the grid connection, giving an ILR (inverter (maximum) loading ratio of 1.3 to 1.4).
Wind roses
The above graphic shows the wind rose plot for La Guardia airport in 2008. The graphic image was produced by Breeze Software and licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
A wind rose is a concise chart showing the distribution of speeds and directions of the wind in a particular place. The length of the spokes represents the probability that the wind blows in that direction on average. The length (along the spoke) of each colour coded regions represents the probability that the wind is both in that direction and has a speed within the range indicated by the colour coding. The “bullseye” region in the centre (light blue here) contains the probability that there is no wind (3.6%). The spokes represent the direction the wind is coming from.
Wind roses were used mainly for airport planning – planes can have lower take off and landing speeds if travelling directly into the wind, so it is advantageous to orient runways in the direction that most wind will be experienced.
Wind roses form part of the data used to design the turbine layout of a wind farm, to minimise wake interference between the turbines and thus maximise output from the turbines on a particular lease area. One design technique [abstract paragraph 1] is to make the spacing between turbines longer in the direction of the prevailing winds and shorter in the direction perpendicular to that. Another way of looking at it is that the wind farm is less efficient when the wind is blowing along lines of closely-spaced turbines rather than perpendicular to them.
Wake steering
Normally the operational imperative is to keep the wind turbine rotor axis facing directly into the wind (so that the plane of rotation of the rotor is perpendicular to the wind). The rotor is invariably in front of the tower as there is little point in allowing the tower to shield the rotor from the wind.
However, there is the potential to steer the downwind wake from a upwind wind turbine by yawing the turbine rotor plane of rotation [abstract] so that it is not directly facing the wind. The intent is to steer the wake away from a downwind turbine, increasing the wind speed seen by that turbine and reduce the turbulence and strain from being directly in the wake of an upwind turbine. Such techniques are in their infancy, but modelling suggests that a small single digit percentage increase in the overall output from the wind farm can be achieved.
Large scale wind farm power density
Currently the largest onshore wind farms can generate average output up to 10 W/m^2 and offshore wind farms can generate an average of 4 W/m^2 of wind farm area respectively, in wind farms spanning hundreds or one thousand of square km. Smaller wind farms can achieve higher wind power density, but are generally less economic. Effectively the wind farms are removing momentum from the wind in the lowest atmospheric layer in which the wind turbines operate.
However, modelling of atmospheric layers shows that much larger wind farms, of 20,000 sq km or more would not allow the same average power density per square metre. In fact the available power is much lower – something like 0.3\;to\; 1.0 W/m^2. The issue is that such large wind farms would not allow sufficient wind momentum transfer from higher layers of the atmosphere to fully replenish the surface layer momentum, so the average power available is restricted.
The land around onshore wind farms generally presents a high surface roughness which generates more turbulence and encourages mixing of vertical layers. Nearby mountainous areas would be a good intuitive example But the offshore sea surface is all but flat. The wind can whip up waves metres high – not enough to promote a high degree of mixing between vertical layers of wind. Offshore, the vertical momentum transfer between layers is more due to natural friction. If the height of the wind turbines is, say, 200m then a wind farm of 1000 sq km or less can be replenished with wind momentum from only a few 200m layers. As wind farm size increases, the momentum must be replenished from higher and higher 200m layers, requiring a longer and longer distance to occur effectively. For a particular direction of the wind, and the number of turbines in that direction, if the wind cannot be replenished within the length of the wind farm, the maximum output from the wind farm is reduced, and only the leading, upwind turbines will have a decent output.
The implication is that the maximum output of large, individual offshore wind farms is naturally restricted. Given the costs of cabling, offshore wind turbines are no more than 7 to 19 rotor diameters apart.
The obvious solution for offshore wind is to cover only, say, 10% of the sea surface with offshore wind farms up to 1000 sq km in area. Surface layer wind momentum replenishment can then happen between separated wind farms, keeping turbine array cabling costs at a reasonable level, while ensuring turbine capacity factor and average output is maintained.
For onshore wind farms, the transfer of momentum to replenish the wind speeds at the surface level depends heavily on the topography of the terrain – clearly mountainous regions mix air from different heights more rapidly than undulating plains will. Fluid dynamic atmospheric simulations can give a good indication of how wind farms should be laid out in the terrain.
The good news is that these effects appear only with much larger wind farms than are currently planned. Generally there is plenty of space on the ocean to space out offshore wind farms – for instance the UK probably has enough economic exclusion ocean zone area surrounding the GB mainland to provide two or three times the energy required by the UK in 2050, even while spacing out offshore wind farms to cover no more than 10% of this zone.
Length scale of weather fronts
The length scale of weather fronts is around 1,000 km or 600 miles, called the synoptic scale.
The implication is that the weather and wind is likely to be similar in a region up to 1,000 km in length, which will move over time.
For the UK, 1,000 km is similar to the distance between the far north of Scotland and the south coast. Thus the wind speeds could all be high or low together over the UK – it is not a large enough area for the wind in the north and the wind in the south to be independent of each other all the time. By contrast the USA is much larger and the weather on the east and west coasts should generally have a much lower correlation.
The physical extent of a grid, relative to the synoptic scale, is important when assessing the likelihood of the whole of a grid being becalmed at the same time. It affects the likely duration of gaps in wind power requiring backup generation. Or put another way, all other things being equal, average onshore wind power output over the whole of the UK is likely to show more variation than average onshore wind power over the whole of the USA, though smaller regions of the USA may show similar variability to the UK.