By Niall McMahon

The Engineers Journal

May 2nd, 2013

Note: this article was amended on 26/06/2015 to correct the expression for average annual power output - there ought to be no *C _{p}*. The results ought to be unaffected.

(Essay based on a talk delivered to the Irish Mining and Quarrying Society at the Geological Survey of Ireland on Tuesday, April 16th 2013. A version of this note is also published at The Engineers Journal website. Some small typographical errors have been corrected.)

It is important for this discussion to be sure about some important fundamental quantities. Energy is the ability to do work; it is measured in joules (J). Power is the rate at which we use energy and is measured in watts (W). 1 W = 1 joule per second (J/s or Js^{-1}). It is usually convenient to measure energy in kilowatt-hours (kWh); a kWh is a *unit of electricity*. 1 kWh = 1000 x 3600 J = 3.6 MJ. A typical European home uses, averaged over a year, about 500 W continuously, or 4,380 kWh of energy in 1 year. A typical American home uses about 1000 W continuously, or 8,760 kWh of energy in 1 year. David MacKay's Sustainable Energy: Without the Hot Air is very useful for more commonsense discussion. It is the *energy* that we can capture over a typical year at a site that is of most use to engineers and financiers. Sometimes, however, when and how fast we get the energy, i.e. power, becomes important.

Wind turbines convert energy: the rotating blade assembly, the *rotor*, transforms wind energy into rotational kinetic energy; the generator converts this kinetic energy into electrical energy. Wind turbines are sometimes called wind turbine generators (WTGs); this is to emphasise the electrical aspect of their operation.

The instantaneous wind *power*, i.e. the rate of wind energy extraction, associated with a wind turbine is given by, *P = ½ ρ C _{p} A U^{3}*, where

There are many kinds of wind turbine but these can be grouped, generally, into two prototypes: (i) horizontal axis wind turbines (HAWTs) and (ii) vertical axis wind turbines (VAWTs). There are two principal types of VAWT, Savonius and Darrieus. Savonius turbines are *drag* machines while Darrieus turbines are *lift* machines. The HAWT is the most *efficient* design and, for this reason, is the most commonly deployed configuration. VAWTs will always be a part of the solution; at the same time, their niche seems to be for applications in which visual aesthetics matter most.

The current consensus is that a *small wind turbine* is a machine with a swept area of up to 200 m^{2}. This corresponds to a rotor diameter of about 16 m and a nominal *rated* power of 50 kW. There is no standard yet associated with "medium wind", but it is likely that such machines will be defined as larger than small wind turbines and with a maximum swept area of 1000 m^{2}, or a rotor diameter of about 32 m, and a nominal rated output of 500 kW. Wind turbines are getting larger. What was considered to be a large wind turbine in 1980 would be described as a medium-sized wind turbine today.

From the power formula given, it's easy to see that the rotor size is the most important way to describe a wind turbine. Other ways to classify a wind turbine include by: (i) its rotor orientation relative to the tower, i.e. upwind or downwind; (ii) The number of blades (two or three, most commonly); (iii) its pitch system, i.e. fixed/passive/dynamic pitch; (iv) its generator type (asynchronous, also called induction, or synchronous, including permanent magnet).

In technical specifications, the following quantities are often given: rated wind speed is this is the wind speed for which the power output of the machine is quoted; *rated power* is the maximum continuous power of a large wind turbine at its rated wind speed; *hub height* is the height of the hub centre-line above ground level; *solidity* is a measure of the ratio of the area of the blades to the swept area; *tip speed ratio* is the ratio of the speed of the blade tips to the free-stream wind speed; *cut-in wind speed* (often about 3 m/s) is the wind speed at which the turbine starts to generate; *cut-out wind speed* (often about 25 m/s) is the wind speed at which the turbine shuts down.

When deciding whether or not a wind turbine is suitable for your site, the *technical* things critical to success is securing good information about: (i) the wind resource; (ii) the wind turbine's performance; (iii) your own electrical load requirements.

The first thing is the wind resource. Mick Sagrillo has a nice idea that wind consists of two components, *quantity* and *quality*, or *wind speed* and its *turbulence* content. Wind speed increases with height above the ground. As you might imagine, this effect is more pronounced in areas that have rough surfaces, e.g. urban areas, forest, hills. These places are also associated with high turbulence, i.e. rapid changes in flow speeds and directions at all scales. A wind turbine tower ought to be as high as possible. As a rule of thumb, the turbine should be placed upwind of obstacles, about 10 m above everything within a radius of 150 m. See Paul Gipe's Wind Energy Basics for a more detailed treatment.

Anemometers and wind vanes are used to measure wind speed and direction. Depending on the size of the installation, i.e. how much money there is to spend, it is best practice to measure these quantities at the site for a whole year, as outlined in IEC 61400-12. (There are cheaper, and less accurate, methods.) From this information, assuming that turbulence is sufficiently low and correcting for any errors, a *wind speed distribution* for the site can be determined. This describes the number of hours per year that the wind blew at various wind speeds, e.g. we might find out that the wind blew at speeds between 4.5 m/s and 5.5 m/s for 10% of the year. And so on. The average wind speed at the site can be easily calculated.

A second step is to understand the performance of the provisionally selected wind turbines. Each wind turbine has a *power curve* that describes its power output at regular intervals of wind speed. See the accompanying theoretical *estimate* for a Vestas V52. (This is our own rough calculation, not Vestas's.) Experimental, *independently measured*, power curves are essential.

*A theoretical power curve calculated by the author for a Vestas V52-850kW large wind turbine.*

Using the wind speed distribution at the site together with the wind turbine power curve, we can estimate an *annual average energy production* (AEP). This is done by combining information from the power curve and the wind speed distribution. Power times hours yields kWh, or energy. The actual methodology used is a little more sophisticated. Approximately, you can use the expression:

*P = ρ (⅔ D) ^{2} U_{av}^{3}*

(This expression is derived with the assumption that the wind speeds follow a *Rayleigh distribution*. See Wind Energy Explained, or another text book, for more.)

Where *P* is the average power output for the year, *ρ* is the local air density, *D* is the rotor diameter and *U _{av}* is the average wind speed at the site. Multiplying this figure by 8760, the number of hours in the year, will give an approximate AEP. This estimated AEP can be divided by the total energy that the machine would produce if operating at rated power for the year to produce a site-specific ratio called the

The last essential technical thing is to consider the site's load requirements. While the overall energy capture for the year will be, on average and depending on the quality of the resource and machine information, about right, the power will vary. The best use of power for a small site is for auto-production, i.e. to use everything produced on-site. This is possible for large round-the-clock operations. For others, there will be times that the wind is blowing and the site is quiet. At these times, the only economic option at present is to sell the power, usually via a broker, to the grid. Doing this with most of your energy is much less economically sensible than using it. Wind speeds tend to peak during the daytime hours, i.e. from before the working day to before midnight. Understanding your site's day-to-day load fluctuations then becomes important for the financing of the project.

Other, non-technical, considerations are perhaps most important. These include ensuring that the proposed turbine will be a good neighbour, arranging planning permission, permission to generate from the CER, if you are planning to export, and arranging a suitable grid connection with ESB Networks.

As an example, using the approach outlined, it's instructive to think about two extreme possibilities and a mid-point, i.e. : (i) where all the energy from a turbine is used on-site to offset grid power; (ii) where all the energy is sold to the grid; (iii) where half the energy is used on-site and half is sold to the grid.

We assume an average home, with plenty of land, and with an electricity power requirement of 500 W continuous over a year; this is 4380 kWh of energy over a year. Imagine that the wind blows at the site, about 15 m above ground, at an average speed of 5 m/s over the year. This is a very good site!

With these two conditions, the load requirement and the wind speed, we can estimate that a well-matched wind turbine, i.e. one that can provide 4830 kWh over a year at this site, will have a rotor diameter of around 4.5 metres and a nominal rating of about 3.5 kW. It will cost perhaps Euro 15,000 (at least) to have such a machine working. We assume that the machine has a life-span of 25 years.

**** MACHINE SPECIFICATION ****

Demand (W) D (m) Rated (W) Cost (Euro) 500 4.5 3534 14844

By using conservative feed-in tariff values, i.e. how much money you get for each kWh sold to the grid, and the approximate retail cost of electricity from the grid per kWh, simple pay-back on the original capital investment can be calculated for the three situations.

**** OFFSET ALL (ELECTRICAL) ENERGY **** Yr # Eur/kWh Eur/Yr Cumulative Benefit (Eur) 1 0.15 657 657 2 0.15 657 1314 3 0.15 657 1971 4 0.15 657 2628 ... 22 0.15 657 14454 23 0.15 657 15111 ... Initial cost break-even after 23 years (2 years inside machine lifespan). **** SELL ALL ENERGY **** Yr # Eur/kWh Eur/Yr Cumulative Benefit (Eur) 1 0.09 418 418 2 0.09 418 837 3 0.09 418 1256 4 0.09 418 1675 ... 35 0.09 418 14658 36 0.09 418 15077 With 0.09 cent FIT, ... Initial cost break-even after 36 years (11 years beyond machine lifespan). **** USE HALF ENERGY, SELL REST **** Yr # Eur/kWh Eur/Yr Cumulative Benefit (Eur) 1 550 550 2 550 1100 3 550 1650 4 550 2200 ... 26 550 14305 27 550 14855 ... Initial cost break-even after 27 years (2 years beyond machine lifespan).

These numbers are very approximate, of course, but the scale is about right. It is clear that for such a site, with a good wind resource of 5 m/s, and with the feed-in tariffs currently available in the Republic of Ireland, it is not economic in a simple financial sense to install a small wind turbine. Rather, installing a small wind turbine should be seen as something similar to buying a new car, although somewhat better for the environment.

These numbers change dramatically if: (i) the wind resource is better, even by a small amount; (ii) the wind turbine size is increased by a moderate amount; (iii) the feed-in tariffs rise; (iv) the cost of electricity from the grid rises.

For example, if the average wind speed at the site was 6 m/s rather than 5 m/s, for the same conditions and calculation method, the simple pay-back drops to 13 years, 18 years and 14 years, respectively. If the lower-band FIT rose to UK levels, say 0.40 Eur per kWh, the numbers for an average wind speed of 5 m/s come out as 23 years, 8 years and 12 years, respectively.

This analysis also assumes an unchanging cost of grid energy; in fact, this is likely to increase over the coming years. If the cost of electricity was to double overnight, then the "payback" period halves for a site that uses almost all the turbine's output on-site.

If a site has an average annual wind speed of, say, 6 m/s at 60 m above ground (this is not an exceptionally good site) and if the continuous load requirement at the site is, say, 170000 W or 170 kW, i.e. 1.5 GWh of energy in a year, then how do the figures look? Using the same method of calculation, and assuming a payment of 0.09 Eur per kWh sold to the grid and a purchase price of 0.15 Eur per kWh from the grid, the simple pay-back figures are 6 years, 11 years and 7 years, respectively. Evidently, scale matters and the simple economics improve dramatically with machine size.

For a large energy user, such as a mine, that can use all the energy generated on-site to displace grid power, for example, a wind turbine can relatively quickly re-coup the capital investment. Displacing on-site diesel generation can be a little trickier as the output from a turbine varies; smoothing its output with local generation would require frequent adjustment of the on-site diesel generators. Off-setting grid power is easier.

it's important to note that the numbers presented here are approximate and indicative only. Additional complexities not considered include the actual cost of grid or locally generated electricity (e.g. diesel/gas) to your organisation, the annual maintenance costs associated with the turbine (~ approx. Eur 15000 per annum for a large wind turbine), among other things. Do not use these numbers as absolute values for any serious purpose including making financial decisions - the usual cautions apply.

Finally, in case you're wondering, the calculation method is implemented as a Python script.

Most material © Niall McMahon. See legals and disambiguation for more detail. Don't forget that opinions expressed here are not necessarily shared by others, including my employers.