The Physics and Economics of Wind Turbines

By Charles Norz, Electrical Engineer, Milwaukee, Wisc.

A big issue in generating power from any renewable energy source is the cost of generation versus that for conventional hydrocarbon fuel sources. Wind energy is estimated to have the lowest cost of all renewable options. Governments and private businesses have been investing in research in this technology and results are paying off. For example, it is estimated that the cost per kilowatt-hour (cents/kWh) from wind energy has been reduced by 80% over the last two decades. Recent high efficiency wind turbines develop electricity for about 11 to 13 cents/kWh depending on turbine design and location. However, the lowest cost of hydrocarbon fuel sources is coal, generating electricity for about 6 cents/kWh. Still, there are many opportunities to further reduce the cost of wind energy.

The basics
A few basic calculations provide good insight to the issues of wind turbine design. Wind is an air mass moving from a high-pressure area to one of low pressure. To calculate the energy in wind, consider a segment of air shaped like a horizontal cylinder. The energy in it depends on the volume of air, density, and wind speed. The mass per unit time for a slice of the cylinder is:

M = ρAV
where
M = mass
ρ= density
A = area
V = wind speed

The function of a wind turbine is to transform the wind’s kinetic energy into electricity. Therefore, we must start with a calculation of kinetic energy or E_k, where:

E_k = ½ MV²

Substituting the mass of the air cylinder (ρAV = M) gives

E_k = ½ ρAV³

Thus, the amount of energy in the wind depends on the density of the air, area (in this case, the area swept by the wind-turbine rotor) and the cube of the wind velocity. The equation underscores the point that selecting an area of strong winds is advantageous because the power in the wind increases with the cube of its speed.
The equation looks impressive, but wind turbines are not 100% efficient. If a turbine was completely efficient it would transform all kinetic energy from the wind into electricity. This would mean the wind velocity would drop to zero behind the blade. We know that is not the case. In fact, Albert Betz published a book in 1926 that showed it is only possible to extract 16/27 or 59% of the energy from a wind turbine. This is Betz’ law. Therefore the theoretical energy model for a wind turbine is:

E_{k max} = 16/27 (½ρAV³)

In practice, however, the amount of extractable energy ranges from only 40 to 47%.

Reducing costs

A brief recap is that we can extract less than half the energy from the wind and that depends on air density and wind speed. So the next question is: How can we further reduce the cost of producing electricity from the wind? Three main considerations are site selection, swept surface or rotor diameter, and reducing a turbine’s cost for capital, installation, and maintenance.

Site selection

It is obvious from the wind-power equation that it is best to place wind turbines in areas of strong sustainable winds. Low wind speeds have no significant extractable energy when compared to areas of even moderate wind speeds.
Site selection requires extensive study of an area’s topology, and annual wind speeds and directions. Wind analysts study meteorological trends and generate tools such as a wind rose that show annual distributions of wind speeds and their direction frequencies. Wind-farm investors are then presented with cost justifications based on farm locations. Turbine engineers can select a best design based on the type of winds at the location. One trend is to place wind turbines offshore to take advantage of the unobstructed winds over water.

Swept surface

The amount of energy extracted from the wind is directly proportional to the swept surface area. Large wind turbines leverage economies of scale with an increased blade diameter. The industry has seen a continual increase in diameters from 40 meters (131 ft) and 20 to 60-kW outputs in the 1970s to modern 90 m (295 ft) 3-MW designs. The largest wind turbine today is a 7+ MW 126 m (413 ft) three bladed design engineered by German based Enercon for a research and development project.  Wind-energy-the-facts.org estimates that with improved manufacture methods we could see 250-m (820 ft) rotors on 20 MW machines by 2020.

Swept surfaces usually leads to confusion regarding a best wind-turbine design. An important factor in rotor design is the tip-speed ratio (R_ts). This refers to the ratio between the wind speed and the blade-tip speed:
R_ts = V_{blade tip} /V_wind
where
V_{blade tip} = speed of the blade tip

Why is this important? Imagine a wind turbine spinning slowly, say 1 rpm. Most of the wind (and energy) would pass through the space between the blades, thus “wasting” the energy and reducing the efficiency of the turbine.

On the other hand if the turbine spins “too” fast, two problems arise. The first is that the fast spinning blades acts like a wall to the wind. This reduces wind velocity in front of the blade much as the wind slows in front of a large building. This is a negative condition because the power of the wind is proportional to the cube of the wind speed. The second problem is that each blade of the turbine creates some turbulence in the air. When the blades spin too fast, each “slices” into the turbulence of the proceeding blade, again reducing the turbine efficiency.

A best tip-speed ratio depends on the number of blades in the rotor. The fewer blades, the faster the wind turbine spins to extract maximum power from the wind. Early experiments showed that a two-blade rotor has an optimum tip-speed ratio of about 6, a three-blade design about 5, and four blades, about 3. However, more recent highly efficient aerofoil designs have increased the numbers by 25 to 30%, which allows increasing rpm and therefore generating more power.

Reducing capital and maintenance costs
Manufacturers of wind turbines have been improving designs to reduce the system cost. Wind turbines are complex machines and so have many areas where costs can trimmed without a loss in performance.   Berkeley National Labs data base has shown that the costs of wind turbine had been declining but have recently seen some increases in costs of the past few years.

One consideration leverages the advantages of a two-blade turbine. The obvious is the reduced cost of one blade. As the trend of larger rotor diameters continues, material use and blade cost will also increase. Other advantages of a two-blade design include savings on smaller mechanical equipment due to the lower torque of faster rotor speeds. A lower turbine weight then allows reducing the size or eliminating yaw controls. Lower installation costs come from only one top-lift. And less equipment means lower maintenance costs. Also as more turbines are installed offshore, two bladed designs offer the advantage of less weight which can directly reduce the cost of the tower platforms.

But a cost comparison between three and two-blade designs is not as simple as eliminating the cost of one blade. The three-blade turbine is a proven design and its rotor solves some mechanical loading challenges. Two-blade designs need additional equipment in the rotor hub to compensate for the loading, and thus, may increase the cost of the rotor compared with a three-blade hub. However, a total cost savings from a two blade design would have to include all of the savings described above.

With all the benefits of a two blade design, why are three blade designs in such wide use? And what mechanical loading challenges do three-blade designs solve? Gyroscopic tendencies are one issue.

Spinning rotors act like gyroscopes in their plane of rotation. That is, they are content to rotate in a plane but offer great resistance when changing directions (yawing) out of that plane. This is problematic when wind direction changes and the wind turbine yaws into the wind to maximize power generation and equalize blade loads.

Two and three-blade rotors both generate gyroscopic forces. However, the advantage of a three blade design is that at least two of the blades are always out the vertical plane at one time, thus reducing shaft and gearbox stresses when the turbine yaws.
When the wind changes direction and the blades of a two-bladed design are in the vertical, 6 o’clock position, there is a minimal amount of force on the hub because the loading of the blades are relatively equal.  However when the blades begin to move to the horizontal position it generates unequal loading that adds stress to the hub and gearbox.

To solve the problems, most modern two-blade designs use a hub that is not rigidly fixed to the turbine shaft, so it “teeters” a few degrees to reduce stresses. As manufacturers build larger and heavier wind turbines, gyroscopic forces will increase requiring larger and stronger two and three-blade hubs and thus increasing costs.

There is, however, an idea that with proper research and investment could eliminate the large gyroscopic forces on two-blade windmills, thereby making them viable for 5 to 20MW machines. The idea is cyclical feathering.

It can be used for keeping a wind turbine facing into the wind without hub and gearbox stresses. The idea is to control pitch of individual blades, thereby decreasing gyroscopic forces on the rotor when yawing. This would take advantage of the wind’s kinetic energy on the blade to assist in turning the turbine into the wind. Such a control feature cyclically alters blade pitch as the wind direction changes so as to present different angles of attack between the blades and wind. Another plus for the design: it eliminates the need for yaw drive motors.

Experiments with cyclical feathering on wind turbines have been conducted on a small scale and show great potential. A similar control used on helicopter tail rotors has been reliable and effective. Continued research and investments are needed before this technology reaches large-scale wind turbines.

Discuss the physics and economics of wind turbines at www.EngineeringExchange.com

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