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Chapter 4—Wind Turbine Power 4–44 u as a parameter. We shall use the Sandia 17-m Darrieus turbine for discussion purposes. The shaft power P m for this turbine as a function of shaft rotational speed n is shown in Fig. 26. P m is seen to rise to a maximum for each wind speed for a particular value of rotational speed. Higher wind speeds have more power in the wind, and the change in tip speed ratio with increasing wind speed causes the maximum to shift to a higher rotational speed. Maximum power is reached at 38 r/min in a 6 m/s wind and at 76 r/min in a 12 m/s wind. The maximum possible shaft power in a 12 m/s wind is eight times that in a 6 m/s wind, as would be expected from the cubic variation of power with wind speed. Figure 26: Shaft power output of Sandia 17-m Darrieus in variable-speed operation. Also shown in Fig. 26 are two vertical dashed lines at 42 and 52.5 r/min. We see that at 42 r/min we are close to the maximum shaft power for u = 6 m/s but are below the maximum for higher wind speeds. In fact, no additional shaft power can be obtained for winds above about 12 m/s because curves for higher wind speeds all follow the same line at this rotational speed. At 52.5 r/min we are close to the maximum shaft power at u = 8 m/s, and are able to Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001
Chapter 4—Wind Turbine Power 4–45 extract significantly more power from the wind for wind speeds above 8 m/s than was possible at 42 r/min. It would appear that if fixed speed operation is required, that 52.5 r/min is a good choice for sites with a high percentage of winds between 6 and 10 m/s since the turbine can deliver nearly the maximum possible shaft power over this wind speed range. A higher fixed shaft speed would be justified only if there were a significant fraction of wind speeds above 10 m/s. Increasing the shaft speed above 52.5 r/min would decrease the power output for wind speeds below 8 m/s, and this would be a significant penalty at many sites. Suppose now that the load can accept shaft power according to the curve marked Load in Fig. 26 and that the turbine is free to operate at any speed. The turbine will then operate at maximum power for any wind speed, so the energy output will be maximized. If this maximum energy output can be obtained without increasing losses or costs, then we have developed a system which extracts more energy from the wind at a lower cost per unit of energy. Variable speed operation requires a load which has a suitable curve of power input versus rotational speed. The optimum load will have a cubic variation of input power versus rotational speed. In examining various types of loads, we notice that the input power to pumps and fans often has a cubic variation with rotational speed. The power input to an electrical generator connected to a fixed resistance will vary as the square of the rotational speed. The power input to a generator used to charge batteries will vary even more rapidly than cubic, as we shall see in Chapter 6. Load curves like the one shown in Fig. 26 are therefore quite possible, so variable speed operation is possible with several applications. In addition to the variation of shaft power with rotational speed, we must also examine the variation of torque with rotational speed. If we take the shaft power curves in Fig. 26 and divide by the angular velocity at each point, we obtain the torque curves shown in Fig. 27. One important feature of these curves is that the torque is zero for n = 0, which means that the Darrieus turbine can not be expected to start without help. The friction in the transmission and generator produces an opposing torque, and the torque T m must exceed this opposing or tare torque before the turbine can be accelerated by the power in the wind. The turbine must be accelerated to some minimum rotational speed by an external power source so that T m will reach a sufficient value to accelerate the turbine up to operating speed. This particular machine would need to be accelerated up to perhaps 10 r/min before the wind would be able to accelerate it to higher speeds. This external power source is required for reliable starting of any Darrieus turbine. However, there have been cases where a combination of very low tare torques and strong, gusty winds allowed a Darrieus turbine to start on its own, sometimes with disastrous consequences. A Darrieus turbine should have a brake engaged when repairs are being made or at other times when rotation is not desirable. The torque rises to a maximum at a particular rotational speed for each wind speed, in the same manner as the shaft power. The peak torque is reached at a lower rotational speed than the peak shaft power, as can be seen by a careful comparison of Figs. 26 and 27. From Eq. 35 and the fact that maximum shaft power varies as the cube of the rotational speed, Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001
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