Economic Feasibility of a Wind Farm:

Laboratory Assignment #5
Wind Energy Production

Economic Feasibility of a Wind Farm:
If a wind farm is to be built, three fundamental questions must be answered: How much energy can be obtained from the wind, what price can that energy be sold, and how much does it cost to obtain that energy?
We will start with the first question. The more kinetic energy a wind turbine pulls out of the wind, the more the wind will be slowed down. If we tried to extract all the energy from the wind, the air would move away with no velocity and the air could not leave the turbine. This would prevent any air from entering the rotor of the turbine and would stop the turbine. If the wind passes though without being hindered at all, it likewise would not extract any energy from the wind. A wind turbine must lie between these two extremes to convert the energy in the wind to useful mechanical energy.
As it turns out, an ideal wind turbine slows down the wind by 2/3 of its original speed. To understand why, we have to use the fundamental physical law for the aerodynamics of wind turbines: Betz’ Law. Betz’ Law says that you cannot convert more than 16/27 (or 59%) of the kinetic energy in the wind to mechanical energy using a wind turbine. Betz’ law was first formulated by the German Physicist Albert Betz in 1919. His book Wind-Energie published in 1926 gives a good account of the knowledge of wind energy and wind turbines at that moment. It is quite surprising that one can make such a sweeping, general statement that applies to any wind turbine with a disc-like rotor. To prove the theorem requires a bit of math and physics. We won’t worry about that here.
You have seen in this course that the power potential varies in proportion to the cube of the wind speed. We can then use Betz’ Law, the Weibull distribution, the wind rose, and the power distribution of the wind to determine how much energy can be extracted from the wind. The graph below shows the power output for aparticular wind turbine after adjusting for Betz’s Law.
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The area under the gray curve gives the amount of wind power per square meter wind flow we may expect at this particular site. Notice the Weibull-like distribution. (Remember that the bulk of available wind energy will be found at wind speeds above the average wind speed at the site.) The area under the blue curve indicates the wind power that can theoretically be converted to mechanical power according to Betz’ law. Remember that this is 16/27 of the total power in the wind. The total area under the red curve indicates the electrical power a particular wind turbine will produce at this site. To determine this, the power curve for the turbine must be known. However, we note that the overall graph above is somewhat inaccurate because it ignores the “cut-in” and “cut-out” speed. Since the turbine only “cuts-in” or “starts to function” when the wind reaches some preset value, say 5 m/s, all of the wind energy below that will be lost. Usually, wind turbines are designed to start running at wind speeds somewhere around 3 to 5 meters per second. Likewise, the wind turbine will be programmed to stop at high wind speeds, for example, above about 25 meters per second, in order to avoid damaging the turbine. The stop wind speed is called the “cut-out” wind speed.
The power curve of a wind turbine is a graph that indicates how efficiently the electrical power output will be for the turbine at different wind speeds. For example, the graph shows a power curve for a typical 750 kW wind turbine. Power curves are generated from measurements made on actual working turbines. An anemometer is placed on a mast near the wind turbine (not on the turbine itself or too close to it, since the turbine rotor may create turbulence, and make wind speed measurements unreliable). If the wind speed is not fluctuating too rapidly, then the wind speed measurements from the anemometer may be used and the electrical power output from the wind turbine can be measured and the two values plotted together. The result is the power curve. Local turbulence and complex terrain may result in wind gusts hitting the rotor from varying directions. It may therefore be difficult to reproduce the power curve exactly in any given location.

The power coefficient tells how efficiently a turbine converts the energy of the wind to electricity. To measure how technically efficient a wind turbine is, the electrical power output is divided by the incident wind energy input. Dividing the power curve value by the area of the rotor gives the power output per square meter of rotor area.
The average efficiency for a 700 kW wind turbine often less than 10%. The mechanical efficiency of this particular turbine is greatest (about 44%) at a wind speed around 9 m/s. It is a deliberate choice by the engineers who designed a turbine to maximize efficiency at the wind speeds of maximum available power. At low wind speeds efficiency is not so important, because there is not much energy to harvest. High wind speeds are too infrequent and the turbine must waste any excess energy above that for which the generator was designed. Efficiency therefore matters most in the region of wind speeds where most of the energy is to be found.
When thinking about the cost of a wind farm, it is not necessary to have the highest technical efficiency of the wind turbine. What matters is the cost of pulling kilowatt-hours out of the winds during the life of the turbine. Rememberthatthe fuel is free. The optimal turbine therefore is not necessarily the turbine with the highest energy output per year, but the one that generates electricity at the lowest possible cost. Of course, it is necessary to harvest whatever energy you can as long as costs per kilowatt-hour are kept low.
Another way of examining the annual energy output from a wind turbine is to look at the capacity factor for the turbine in a particular location, that is, how much of the turbine’s potential is being realized. By capacity factor we mean the actual annual energy output divided by the theoretical maximum output, if the machine were running at its rated (maximum) power during all of the 8766 hours of the year. Example: If a 700 kW turbine produces half a million kW-hr in a year, its capacity factor is = 500,000/(365.25*24*700) = 500,000/6,136,200 = 0.0815 or 8.15%.
Capacity factors may theoretically vary from 0 to 100 percent, but in practice they will usually range from 5 to 20% depending on the generator. They are mostly around 10%. Although a large capacity factor is preferred, it may not always be an economic advantage. In a very windy location, for instance, it may be advantageous to use a larger generator with a relatively small rotor diameter. This lowers the capacity factor (using less of the capacity of a relatively large generator), but it may mean a substantially larger annual energy production. Whether it is worthwhile to go for a lower capacity factor with a relatively larger generator depends both on wind conditions, and on the price of the different turbine models.
Questions and Exercises:
Submit both a Word Document and an Excel Spreadsheet for this lab that answers the following:

1. Estimate the annual total power output (in W/m2) for a single 750 kW turbine at your wind farm site. To do this you must combine your Weibull distribution with the power coefficient curve shown above.
Make a table in Excel. The table should contain 7 columns. The first column should contain all of the measured wind speeds (in miles per hour) at your wind farm site starting at the lowest speed and continuing up at 1-mph intervals up to the fastest speed. In the second column, convert the wind speeds from mph to m/s. The conversion factor is .447 m/s to 1 mph. The third column will contain the number of days that average wind speed was observed in 2014. The fourth column will be the probability of that wind speed (days divided by total days). In the fifth column, put the power/m2 of the wind at that speed. To get the power/m2 of the wind at a particular speed, you will use the P = 1/2*?v3 equation (assume air density to be 1.2 kg/ m3).Remember, air density changes with the temperature, so we are using a conservative estimate. The power will be in watts per square meters (W/m2) when the density of air is in kg/m3 and the velocity is in m/s. Remember that this is the power for each square meter of rotor diameter! In the sixth column, put the power coefficient (estimated from the curve above) at that particular wind speed. To obtain the power coefficient at a particular wind speed, estimate the value from the curve above knowing that the maximum value is 0.44 and the minimum value is zero. And finally in the seventh column calculate the power output per square meter that can be obtained from the turbine at that wind speed. To calculate the power output square meter at each wind speed, multiply the probability of that wind speed by the power/m2 of the wind at that speed and the power coefficient at that wind speed. This will give the turbine power output per square meter for each wind speed. Now, sum of all of the power outputs to get the average power output per square meter (in W/m2).
2. Estimate the annual total energy output (in W-hr/m2) for the same single 750 kW turbine at your wind farm site. If we multiply the total power output by the number of hours in a year (365.25 * 24) the total annual energy output (in W-hr) per square meters of rotor area is obtained for an average year.
3. On your Excel Spreadsheet, calculate the annual total energy output (in kW-hr) for your entire turbineat your site. Using the power coefficient of our 750 kW wind turbine, an air density of 1.2 kg/m3, and your Weibull distribution, you have calculated the annual energy output in W-hr per square meter of rotor area for different average wind speeds at the turbine hub height. Our 750 kW wind turbine has a 43 m rotor diameter. Knowing this you can calculate the rotor area. From the annual total energy output in W-hr/m2 and the rotor area,theannual total energy output in kW-hr can be calculated.
4. Calculate the capacity factor for your wind turbine. What value must increase in order for the capacity factor to increase? Comment on how you think you could increase this value.
5. To determine the cost of establishing a new wind farm, we must estimate using data from another location. Find the details of the Sleeping Bear wind farm from www.thewindpower.net (the specific page is easily accessible from Google). Assuming the Sleeping Bear wind farm cost $100 million to install, what was the cost per megawatt? Using this as the basis for your calculation, compute the cost setting up a 750 kW turbine at your site.Hint: assume that the cost per MW from the Sleeping Bear Wind Farm is the same as that for building your wind farm. Based on the cost of electricity in Tulsa ($0.17/kW-hr), how long would it take to recover just the cost of buying and installing a750 kW turbine at your site? Comment how economically feasible you think installing the 750 kW turbine described above is at your wind farm sight.
6. During a discussion about energy, someone says, “A wind turbine converts wind energy into electrical power.” What is wrong with this statement? What could you say to clarify and strengthen the explanation?

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