Wind turbine propeller modeling in heliciel

Wind turbine power and wind speed: from the site resource to profitability

The power of a wind turbine grows with the cube of the wind speed: P = 0.5 × air density × swept area × speed³. A wind that is 20% stronger therefore gives 72% more power, and doubling the wind speed multiplies power by 8. This page shows how to assess the wind at a site, deduce the energy produced and the profitability of a wind turbine, then choose the nominal wind speed that sets the operating point of the propeller.

Wind speed determines the profitability of the project, but also the shape and size of the propeller. We therefore look for two speeds: the average speed measured at the site, which gives the average power and the profitability, and the operating speed (nominal speed), chosen as the reference for building the wind turbine.

In this page:

What factors make wind speed vary?
Average wind speed at the site and energy produced
Wind power formula
Why the average wind speed underestimates power
Calculating the profitability of a wind turbine project
Choosing the nominal wind speed of the wind turbine

World map of wind speed at 80 m, 5 km resolution: the wind resource varies from 3 to 9 m/s depending on the region

Wind speed map at 80 m height: from 3 m/s (blue) to 9 m/s (red), the wind resource varies greatly from one region to another.

1: What factors make wind speed vary?

Wind is not a regular, stable flow of energy. It varies with the seasons, the time of day, the height at which it is captured and the configuration of the terrain.

Instantaneous phenomena: gusts. They are difficult to assess without weather station records.

Daily phenomena: thermal breezes. By the sea, the breeze blows from sea to land by day and the other way round at night:

Thermal wind by the sea during the day: sea breeze from sea to land Thermal wind by the sea at night: land breeze from land to sea

Thermal breezes by the sea, by day (above) and by night (below).

In the mountains, valley and mountain breezes alternate in the same way:

Thermal wind in the mountains: rising valley breeze by day, descending mountain breeze at night

Valley breeze by day (top), mountain breeze at night (bottom).

Monthly and seasonal phenomena. In temperate zones, the most frequent winds are westerly and south-westerly, except in the far south-east, where south-easterly winds dominate. Each season has a general wind direction. For the northern hemisphere:

in winter, winds are south-westerly in the north and north-westerly in the south;
in spring, the most frequent winds are northerly and westerly;
in summer, the patterns are those of spring, but better established;
in autumn, the patterns are those of winter, but more regular.

Climate zones of the globe: cold, temperate and hot zones, which steer the wind regime

Climate zones: the prevailing wind regime depends on the zone where the site lies.

Annual phenomena. Annual data are gathered as wind maps. The Heliciel propeller and wing calculation software includes a series of wind maps by region and continent:

The wind map window in Heliciel: here the United States, with wind power and speed at 10, 30 and 50 m for the selected zone.

Vegetation on the ground also reveals the prevailing wind regime. The Griggs-Putnam index of deformity gives the wind strength from the deformation of trees:

Griggs-Putnam index of deformity: tree deformation by the prevailing wind, from index 0 (no deformity) to index VII (carpeting), with wind speeds

The eight levels of the Griggs-Putnam index, from an undeformed tree (0) to a tree flattened on the ground (VII), with the corresponding wind speeds.

Variations caused by altitude. Wind maps give the wind speed at heights from 10 metres. The variation of wind speed with height depends mainly on the nature of the terrain, and a simple law represents it:

Law of wind speed variation with height: V1/V2 = (h1/h2) to the power alpha

V1 and V2 are the wind speeds at heights h1 and h2, and the exponent α depends on the nature of the terrain:

Nature of the ground	Surface irregularities (cm)	Exponent α
Flat: sea, ice, snow, short grass	0 to 20	0.08 to 0.12
Small irregularities: pasture, fields, crops	20 to 200	0.13 to 0.16
Irregular: woods, sparsely populated areas	1,000 to 1,500	0.2 to 0.23
Very irregular: cities	1,000 to 4,000	0.25 to 0.4

Example: a two-bladed wind turbine of 20 m diameter is mounted on a 30 m tower. With the blades vertical, their tips lie at 40 m and 20 m. The speed ratio between the upper and the lower tip is then V2/V1 = (40/20)^α = 2^α, that is 1.07 for α = 0.1 and 1.32 for α = 0.4.

Variation of wind speed between the top and bottom of a two-bladed wind turbine, for two extreme values of the exponent alpha

Original diagram of the example, for α = 0.1 and α = 0.4. The ratios it shows (1.17 and 2) do not match the geometry described above; the ratios computed above are the reference.

The most interesting sites are therefore those with the least rugged terrain, where the exponent α is low. Besides the loss of wind near the ground, which forces us to build expensive towers, the speed difference between the upper and lower tips unbalances the wind turbine and generates vibrations and extra loads.

Variations caused by obstacles and the shape of the site. Without a precise map of the available wind resource, we can still locate sites that deserve attention and make local measurements there, more specific and shorter. Some regions are naturally blessed by the god Aeolus: coastal areas, high plateaus and mountains, islands of course, or some open plains. The corridors of large rivers, where the wind can funnel, are also favoured: following the river bed, the wind takes a relatively constant direction.

More generally, the most remarkable sites are those with the lowest roughness, where the wind meets few obstacles. We therefore prefer to install wind turbines in places that are slightly elevated and open. Inland, hills can hide particularly interesting sites, especially if their slope is gentle and gradual. Not only is roughness low there, but the airflow is compressed by the obstacle and speeds up. Its speed can sometimes almost double: this is the "hill effect".

Hill effect on wind turbine wind: speed profiles over a gentle hill (A), a cliff-top plateau (B) and a rocky peak (C)

Wind speed profiles over three reliefs. A: the gentle hill accelerates the wind without disturbing it. B: the cliff creates eddies. C: the rocky peak disturbs the flow in front of and behind it.

A: hill with gentle slopes and a rounded summit: very favourable site.
B: steep hill or cliff top: site causing strong turbulence, to be avoided.
C: rocky peak, trees, buildings, houses: we avoid installing close by, because the wind is disturbed upwind and downwind of the obstacle.

2: Average wind speed at the site and energy produced

We have gathered the wind data for the site and consulted the maps to obtain an average speed, increased to account for height. What energy will we produce? To find out, we determine the power of the wind at the site.

3: Wind power formula

The kinetic energy of an object is calculated from its mass and its speed:

kinetic energy = 0.5 × mass × speed²

Take the second as the reference interval. For a wind moving at constant speed, the mass is that of the air crossing the surface in one second, that is its volume multiplied by the density. The volume equals the speed (m/s) multiplied by the area S (m²), and the mass equals this volume multiplied by the air density (kg/m³). The kinetic energy per second becomes:

0.5 × mass × speed² = 0.5 × (speed × S × density) × speed²

We obtain the power of the wind crossing the area S:

P = 0.5 × density × S × speed³

P in watts, density in kg/m³, S in m², speed in m/s.

This formula implies two essential things.

Air density enters the power. It varies with altitude and temperature. In Heliciel, you change the characteristics of the ambient fluid: density varies with the altitude and temperature you enter, and viscosity also varies with the chosen temperature. At the same wind speed, a site at altitude therefore has less wind power than one at sea level.
A small change in speed is enough to cause a large change in power. Take a site where the wind is 10 m/s and another where it is 12 m/s, with a density of 1.2 kg/m³ and an area of 1 m². The speed difference between the two sites is 20% (12/10 = 1.2).
- P site 1 = 0.5 × 1.2 × 1 × 10³ = 600 W
- P site 2 = 0.5 × 1.2 × 1 × 12³ = 1,036.8 W
- Power ratio: 1,036.8 / 600 = 1.72, that is 72% more power.

Doubling the speed multiplies the power by 8.

4: Why the average wind speed underestimates power

Back to the average speed measured at the site. Suppose, for simplicity, an average calculated over two months with two wind speeds, for a density of 1.2 kg/m³ and an area of 1 m²:

speed in March = 10 m/s and speed in April = 20 m/s;
average speed = (10 + 20) / 2 = 15 m/s;
power at the average speed = 0.5 × 1.2 × 1 × 15³ = 2,025 W.

Now calculate the power in March and in April, then average them:

power in March = 0.5 × 1.2 × 1 × 10³ = 600 W;
power in April = 0.5 × 1.2 × 1 × 20³ = 4,800 W;
average of the powers = (600 + 4,800) / 2 = 2,700 W.

The average of the powers is larger than the power of the average. Forecasting power from an average speed therefore underestimates it, and the more the data are averaged, the less accurate we are. To forecast the power and energy of the site properly, it is better to sum the powers generated at the different speeds recorded, over the shortest possible periods. Weather data are valuable, and monthly records are useful. Even with weather data, values can vary by 25% from one year to the next: we therefore put the accuracy of forecasts in perspective and treat them with caution. A wind farm has good years and bad years.

Mathematical methods nevertheless make it possible to assess the power from the average speed, such as the Weibull distribution or the Rayleigh distribution.

Note in passing that calculating power for an area of 1 m² simplifies things a lot. Power per square metre is the power density of the site. It simplifies the equation, and it gives the power of the wind turbine directly: just multiply it by the area swept by the propeller.

5: Calculating the profitability of a wind turbine project

To assess the profitability of a wind turbine project, we compare the money invested and the money produced.

Money invested. We list the materials and labour, not forgetting maintenance.
Money produced. We multiply the energy produced by its selling price. The average power at the site must therefore be converted into energy produced:
- The power at the site is reduced by the efficiency of the wind turbine. According to the table below, the annual overall conversion efficiency ranges from about 14 to 20% for small wind turbines, and from 23 to 36% for medium-size ones, depending on wind speed (see also the page on wind energy). Example: if the average power at the site is 150 W per m² and our wind turbine has an efficiency of 20%, the power produced per m² of swept area is 150 × 0.2 = 30 W = 0.03 kW.
- What is sold is not power but energy. The selling unit of electrical energy is the kilowatt-hour (kWh): 1 kWh is produced by delivering a power of 1 kW for one hour. A year has 8,760 hours; with 0.03 kW of average power per m², the annual production is:

8,760 h × 0.03 kW = 262.8 kWh per m² per year

Worked example with a dated price. A kWh was charged 12.8 cents (average value in Los Angeles, 2013). The wind turbine then provides savings of 262.8 × 0.128 = 33.6 dollars per m² per year at this site. A wind turbine with 1 m blades sweeping 3.14 m² would save 3.14 × 33.6 = about 106 dollars per year. Replace this price with the one in your own contract to get your own estimate.

Table of estimated annual energy production of small and medium-size wind turbines: overall conversion efficiency by annual mean wind speed (French-language table)

Annual energy production as a function of mean wind speed, from 4 to 9 m/s (table in French). Small wind turbines: overall efficiency from 0.20 to 0.14. Medium-size wind turbines: from 0.36 to 0.23. In this table, overall efficiency decreases as mean wind speed increases.

Map of wind speeds in France at 50 m height over open plains, from under 4.5 m/s to 7.5 m/s (French-language map)

Wind speeds in France at 50 m above ground over open plains (map in French): from under 4.5 m/s inland to 7.5 m/s on some coasts and in the Rhône valley.

The table below gives the main purchase conditions for electricity produced from some renewable sources in France. These tariffs date from the 2006 and 2007 orders and change quickly: they are certainly not today's. For current values, consult the French ministry's website (developpement-durable.gouv.fr).

Sector	Order	Contract length	Example tariffs for new installations (euro cents)
Hydraulic	1 March 2007	20 years	6.07 c€/kWh + bonus between 0.5 and 2.5 for small installations + bonus between 0 and 1.68 c€/kWh in winter depending on the regularity of production
Biogas and methanisation	10 July 2006	15 years	between 7.5 and 9 c€/kWh depending on power, + energy efficiency bonus between 0 and 3 c€/kWh, + methanisation bonus of 2 c€/kWh
Wind energy	10 July 2006	15 years (onshore) 20 years (offshore)	- onshore wind: 8.2 c€/kWh for 10 years, then between 2.8 and 8.2 c€/kWh for 5 years depending on the site. - offshore wind: 13 c€/kWh for 10 years, then between 3 and 13 c€/kWh for 10 years depending on the site.
Photovoltaic energy	10 July 2006	20 years	- Mainland: 30 c€/kWh, + building-integration bonus of 25 c€/kWh - Corsica, overseas departments, Mayotte: 40 c€/kWh, + building-integration bonus of 15 c€/kWh
Geothermal	10 July 2006	15 years	- Mainland: 12 c€/kWh, + energy efficiency bonus between 0 and 3 c€/kWh - Overseas departments: 10 c€/kWh, + energy efficiency bonus between 0 and 3 c€/kWh

6: Choosing the nominal wind speed of the wind turbine

As we have seen, we read the average wind speed from maps or from measurements on the ground, paying attention to how often each wind occurs. We then choose the nominal wind speed and the rotation speed from three elements:

the characteristics of the generator;
the power captured;
the wind speed taken as the reference speed.

Once the blade is built, we can no longer change its twist. A pitch variation remains possible, but the twist must have been optimized for the chosen wind speed (the nominal speed) and rotation speed (the operating point).

To go further, we can link this nominal speed to the shape of the rotor with the page on the wind turbine rotor and tip speed ratio, then draw the blades in the wind turbine propeller design tutorial. The bibliography on propellers, wings and wind turbines lists the references used.