Wind energy calculation

Wind energy calculation is the analysis of the conversion of the wind kinetic energy into electric energy via a wind turbine. In combination with solar energy and hydroelectric energy, wind energy is one of the main renewable energy sources available.

wind turbines located on a mountain
Wind turbines assembled together forming a wind farm

Wind energy calculation equation

To calculate the overall wind power of an idealised wind turbine the following equation is used:

\begin{gather} W_{e}=\frac{1}{2}\rho\,A\,P_{c}\,W_{s}^{3}\,N_{g} \end{gather}

Where \(\rho\) is the air density (this is almost always assumed to be constant and equal to 1.2 \({Kg}/{m^{3}}\), A is the frontal area of the wind turbine, \(P_{c}\) is the coefficient of performance (a non-dimensional number which gives an estimate of the overall wind turbine performance), \(W_{s}\) is the wind speed, \(N_{g}\) is the generator efficiency. The wind energy calculation equation above highlights that while increasing the efficiency of the generator and of the wind turbine (increasing \(N_{g}\) ) is important and to a certain extent new wind turbines designs are already progressively improving that parameter the most sensitive parameters which come into play are A which is obviously proportional to the square of the blade length and the wind speed. Increasing the size of a wind turbine is always the most direct way to increase its power output because not only the area increases but also naturally a bigger wind turbine is also higher and therefore is invested by higher speed winds as highlighted above. Since as the equation above is highlighting the wind energy is proportional to the cube of the wind speed choosing the right wind turbine location is absolutely critical, as in specific places within a wind farm site wind speed can be considerably higher than the site average.

Wind energy calculation with different wind turbine locations

Taking into account the wind energy calculation equation above choosing a geographic location which offers even a mild increase of 25% of wind speed doubles the overall power output. Very small variation of the wind farm site topology, like the presence of small valleys, surrounding hills can lead to significant variations in the local wind speed. In particular the best tool to correctly assess this effect is a Computational Fluid Dynamics (CFD) model which starts from the average wind speed typical of the geographic location of choice from historical data and gives as a final product the overall flow field in the location highlighting which are the specific areas where the flow accelerates and therefore the most suitable for the wind turbines location. For more details about CFD modelling see this related article. Interestingly the example below (and covered in more detail in the section wind power applications) highlights that in certain areas (the ones highlighted in green in the image below) the wind speed is 2.5 times the average:

wind turbine site assessment with areas with high velocity

Wind speed close to the ground level of a potential wind farm site with the red regions showing higher wind speeds

wind turbine site assessment showing velocity next to the ground

The highlighted area is where the local wind speed is high. In the dark blue area the wind speed is more than 60% higher than the average wind speed indicating the ideal location for the wind turbines

This means that placing a wind turbine in the area highlighted can offer an output which is 15 times higher than an equivalent wind turbine placed in the same location but in a different spot. It is important to choose the right wind turbine when such higher power outputs are predicted from the preliminary wind energy calculation equation above, as the blade and the overall wind turbine design has to be optimised for such power outputs. As wind energy equates to profit when investing in a wind farm optimising the wind turbine location is one of the most effective ways of increasing the ROI (Return Of Investment) of the profit can be increased with almost no impact on expenses as the wind turbine size is the biggest drive of wind turbine costs and for an identical wind turbine size a much increased wind power can be achieved.

Wind energy advantages

It is renewable:   it is a source of energy always available and does not rely on a fossil fuel which is in limited supply as coal, natural gas or petrol. As a renewable energy source does not produce any direct pollution either. It does nevertheless produce indirect pollution as a consequence of the industrial production of the wind turbines components.

It is cost efficient: due to the engineering progress of the wind turbines design a wind turbine can produce energy for many years, also the fact that wind turbines are becoming more and more common has naturally reduced the purchasing prices and both these effects are improving the ROI (Return Of Investment). Today wind energy cost is comparable and in some cases lower than traditional not renewable energy sources.

It can have high energy density: in certain geographic locations due to the presence of hills, valleys or of surrounding buildings in a city the wind flow can be accelerated to fairly high speeds, since the wind energy is proportional to the cube of the speed (see the section wind energy equation) in specific locations very high energy densities are possible.

It can be scaled: more wind turbines can be mounted together in a wind farm or alternatively bigger wind turbines can be used to increase the overall energy production. Using wind turbines with longer blades increases the effective frontal area which increases with the square of the blade length increasing consequently its energy production. A further effect of using bigger wind turbines is the increase in the height of the turbine centre and as wind speed increases with height this is a further factor contributing to increasing the overall power output.  

wind turbine blade being transported with a truck
Wind turbine blades dimensions have progressively increased in recent years

It is available in remote areas: wind energy can be used as the main energy source in remote locations which are not attached to the main electric grid, or which would be not economical to supply. Buildings or villages in remote locations are currently supplied with fuel electric generators or are connected to the grid by very long electric cable connections which are very costly and lead to high energy losses. In such a scenario clearly wind energy and renewable energies in general already show big advantages.

Wind energy disadvantages

Despite being renewable and economic wind energy is showing some disadvantages which are currently limiting its diffusion as a readily available alternative to non-renewable energy sources:

Wind turbines affect the landscape: wind turbines have become bigger to increase their efficiency (see previous points), this means that their impact on natural landscape has progressively increased. The moving blades are also badly interacting with the wildlife, especially with birds and bats. To overcome this limitation the most recent trend in designing wind farms is to place them offshore where they do not impact people and have a limited impact on wildlife.

Wind turbines are noisy: to transform mechanical energy into electricity wind turbines use a gearbox and usually the mechanical friction between gears is producing noise which can impact people living close to the wind farm. Some progress has been made in evolving the mechanical components used in wind turbines gearboxes with the aim to reduce their noise and therefore their impact on people. But again moving to offshore wind farms offers the opportunity of limiting this effect as well.

It is cannot be turned on demand: as many other renewable energy sources wind energy cannot be turned on just when more electric energy is required by the grid as it is possible for instance with a gas turbine energy plant. The only renewable source of energy which has got this features is hydroelectric when used in combination with water reserves which can be regulated according to the instantaneous energy requirement. A recent way to overcome this limitation is to use the wind energy to pump water in water reserves located at high altitude when the grid requires less energy and use water turbines when the energy request of the grid is at its peak.

It is discontinuous: during the year wind speed does vary unpredictably with time, this means that this is a randomly varying energy source. This effect can be compensated as stated above by using water reserves as a potential energy storage system or more often wind energy is just one of the elements of a complete energy strategy and when the wind energy contribution is reduced other energy sources like hydroelectric or gar turbines are used.

How to overcome wind energy disadvantages - main solutions

Reduction of the wind turbines impact on landscape: a recent trend is to create offshore wind farms. In this case wind turbines are located offshore and far from the human view reducing their impact.

offshore wind farm with a boat
Offshore wind farm

The main disadvantage of this solution is that wind turbines are still to be anchored to the seabed with long pillars which are costly and therefore impact the ROI and can potentially impact sea life. The cost of wind turbines maintenance is higher and the maintenance operations themselves are more challenging due to the distance. Similarly the wind farm has to be connected to the land with a power cable connection which is both costly and leads to small additional energy losses.

A different way to reduce the visual impact of wind turbines, but also to enhance the energy production in local regions where the wind speed is increased by surrounding buildings or by buildings shapes mini wind turbines integrated in buildings are a viable solutions:

integrated wind turbines on top of the strata building in London
Integrated wind turbines in the Strata building in London [1]

If fully integrated with the building of choice these mini turbines have a very small visual impact on the urban landscape and can even become a part of the overall building architecture. This solution has the additional advantage that the energy produced is used directly in the same location or even in the same building considerably reducing the electricity transport losses. In combination with solar energy and with thermal building insulation mini wind turbines are one of the main tools available to achieve an energy independent building.

Wind energy intermittency:  electricity is overproduced when demand is low and wind speed pick up and is not enough when demand is at its peak. To solve this issue the electricity produced in excess is used to pump water from a low altitude water reservoir to one placed higher up:

high ground water reservoir with exiting flow from a dam
High ground reservoir and dam used to produce hydroelectric energy

This solution is very effective in terms of power losses as water turbines can reach levels of efficiency which are close to 95%. The main drawback of this solution is the added system complication which increases the costs and therefore is impacting the ROI and sometimes simply the geographic location of such a system does not allow to add the water reservoirs required by this system.

How to enhance wind energy advantages – innovative designs

As wind power is proportional to the cube of wind speed as shown in the wind energy calculation equation and wind speed increases with height many innovative designs are looking at using flying and therefore high altitude wind turbines. One of the classic solutions is to create flying balloons, usually filled with helium, with a Venturi tube built into them which is further accelerating the flow in a small wind turbine placed in the middle of the balloon:

flying balloon with integrated wind turbine in the centre
Flying balloon with integrated wind turbine in the centre [2]

The main advantage of a flying wind turbine is the increased wind speed which is the most sensitive parameter to increase a wind turbine energy production. Also the fixed installation costs are reduced as the flying balloon is typically movable.

An alternative flying wind turbine design is the one developed by Makani© in California where a flying glider is attached to a rope and made running in circles while being invested by wind. The glider is equipped with propellers which are attached to electric generators and are passively run by the glider going into descend, when the glider is climbing in its upwards trajectory the propellers are not active and the plane is purely relying on the thrust from the wind interaction to continue along its trajectory. 

tethered wind turbine working principle scheme
Work principle of flying wind turbines tethered to the ground [3]
wind turbines mounted on tethered glider
Wind turbines mounted on tethered glider [3]

This solution is very interesting because the glider sweeps along a very high portion of area while being sustained by the incoming wind, the combination of high altitude and high swept area are increasing the maximal wind power output as presented in the wind energy equation presented above.

Conclusions

Wind energy is nowadays the most cost effective renewable energy source which is already a valid and economically viable alternative to fossil fuel sources. It is more efficient in terms of ratio between profit and costs with respect to the solar energy (Photovoltaic) and provided that the correct wind turbine location is chosen a single wind turbine can have very high energy outputs. The impact on wind energy on the landscape though is considerably higher than solar energy for example, for this reason the best bet at the moment are the innovative solutions presented in the article above which will certainly become common in the near future. CFD is becoming an invaluable tool to assess in detail the effect of the wind turbine location within a defined wind farm site, and as presented in the article above a considerable increase in profit is possible for identical costs simply slightly relocating the wind turbine.  

Resources

1 – By Cmglee – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=20745152

2 – Altaeros energy – http://www.altaeros.com

3 – Makani Energy – https://makanipower.com/technology/

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