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Chapter 4—Wind Turbine Power 4–18 Figure 14: Generator efficiency for three generator sizes. If a two stage, 30-kW transmission with an efficiency curve such as Fig. 13 is selected, the rated output power is (0.96)(30) = 28.8 kW. Generators are always rated in terms of output power, so a 25-kW generator with efficiency of 0.9 has a rated input power 25/0.9 = 27.8 kW. The next size up, say 30 kW, would have a rated input power of 30/0.9 = 33.3 kW. Should we select the generator that is slightly undersized, or should we choose the next larger unit? In this particular situation, a good case can be made for choosing the smaller generator. There will be some slight savings in cost and weight, and some increase in average system efficiency because the generator will be operated at a higher fraction of its rating. The slight overload is acceptable because it is not present all the time. Any generator can supply 10 or 20 percent greater power than its rating for periods up to an hour if it is allowed to cool after that period. The wind is variable enough that periods of slight overload will be compensated by other periods of lighter load, so the average power delivered in a period of perhaps one hour would not be above the rated power. It should be remembered that the heat conduction away from the generator is greater in higher winds, and that the generator rating is determined for indoor or calm conditions. This effect may increase the practical rating of the generator by 5 percent or so. These factors of variable power operation and increased air cooling make it permissible to size the generator Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001
Chapter 4—Wind Turbine Power 4–19 by as much as 10 percent under the predicted steady state requirement. If we choose the 25-kW generator and connect it to the turbine whose shaft power is shown in Fig. 10, and if a two-stage transmission is assumed, the electrical power output as a function of wind speed will be as shown in Fig. 15. The shaft power input is also shown for comparison purposes. It is seen that both the shaft power and the electrical power output increase nearly linearly with wind speed up to their maximum values. This may seem somewhat surprising since the power in the wind increases as the cube of the wind speed. It is correct, however, since the low efficiencies at low wind speeds are responsible for linearizing the power output curve. We note in Fig. 15 that the electrical power output rises above zero at a wind speed of about 5 m/s. This wind speed at which electrical power production starts is called the cut-in speed u c . The turbine will develop enough mechanical power to rotate itself at slightly lower speeds, but this wind speed will actually supply all the generator and transmission losses so useful electrical power can be produced. Fig. 15 has been developed from actual turbine data and from reasonably complete models of the transmission and generator. Other turbines, transmissions, and generators will produce somewhat different curves with approximately the same shape. It is convenient to define a model for P e thatcanbeusedindiscussinganywindsystem. The simplest model would use a straight line to describe the variation in output power between cut-in and rated wind speeds. A straight line describes the output of the Sandia 17-m Darrieus rather well. We must remember, of course, that other monotonic functions will fit the observed data nearly as good as a straight line, or perhaps even better for some machines, and may yield more accurate energy estimates or more convenient analytic results. It will be seen later that a closed form expression for energy production can be obtained if P e is assumed to vary as u k between cut-in and rated wind speeds, where k is the Weibull shape parameter. Numerical integration is required if P e is assumed to vary as u, or in a linear fashion. Therefore, our choice of a somewhat complicated model will make later computations easier, and perhaps more accurate, than the choice of the simplest possible model. We therefore define the following equations for the electrical power output of a model wind turbine[9]: P e = 0 (u
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