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8.3.2 Control of variable-speed, pitch-regulated turbines
A variable-speed generator is decoupled from the grid frequency by a power converter, which can control the load torque at the generator directly, so that the speed of the turbine rotor can be allowed to vary between certain limits. An often-quoted advantage of variable-speed operation is that below rated wind speed, the rotor speed can be adjusted in proportion to the wind speed so that the optimum tip speed ratio is maintained. At this tip speed ratio the power coefficient, C_{p} , is a maximum, which means that the aerodynamic power captured by the rotor is maximised. This is often used to suggest that a variable-speed turbine can capture much more energy than a fixed-speed turbine of the same diameter. In practice it may not be possible to realise all of this gain, partly because of losses in the power converter and partly because it is not possible to track optimum C_{p} perfectly.
变速发电机通过功率转换器与电网频率解耦，该转换器可以直接控制发电机上的负载转矩，从而允许涡轮机转子的转速在一定范围内变化。变速运行的一个常被引用的优点是，在额定风速以下，转子转速可以按比例调整以适应风速，从而保持最佳叶尖速度比。在该叶尖速度比下，功率系数 C_{p} 达到最大值，这意味着涡轮机叶片捕获的空气动力学功率被最大化。这常被用来表明，与相同直径的定速涡轮机相比，变速涡轮机可以捕获更多的能量。然而，在实践中，由于功率转换器中的损耗以及无法完美跟踪最佳 $C_{p}$，可能无法实现全部的增益。
Maximum aerodynamic efficiency is achieved at the optimum tip speed ratio \lambda=\lambda_{\mathrm{opt}} , at which the power coefficient C_{p} has its maximum value C_{p(\mathrm{max})} . Because the rotor speed \varOmega is then proportional to wind speed U. , the power increases with U^{3} and \varOmega^{3} , and the torque with U^{2} and \varOmega^{2} . The aerodynamic torque is given by
最大空气动力学效率在最佳叶尖速度比 \lambda=\lambda_{\mathrm{opt}} 时实现，此时功率系数 C_{p} 达到其最大值 C_{p(\mathrm{max})} 。由于转子转速 \varOmega 此时与风速 U 成正比，因此功率随 U^{3} 和 \varOmega^{3} 增大，而扭矩随 U^{2} 和 \varOmega^{2} 增大。空气动力学扭矩的表达式为
Q_{a}=\frac{1}{2}\rho A C_{q}U^{2}R=\frac{1}{2}\rho\pi R^{3}\frac{C_{p}}{\lambda}U^{2}
Since U\!=\!\varOmega R/\lambda we have
Q_{a}=\frac{1}{2}\rho\pi R^{5}\frac{C_{p}}{\lambda^{3}}\varOmega^{2}
In the steady state therefore, the optimum tip speed ratio can be maintained by setting the load torque at the generator, Q_{g} , to balance the aerodynamic torque, that is,
稳态条件下，可以通过在发电机处设置负载转矩 $Q_{g}$，以平衡气动转矩，从而维持最佳叶片尖速比，即：
{\it Q}_{g}=\frac{1}{2}\frac{\pi\rho R^{5}C_{p}}{\lambda^{3}G^{3}}\omega_{g}^{2}-{\it Q}_{L}
Here Q_{L} represents the mechanical torque loss in the drive train (which may itself be a function of rotational speed and torque), referred to the high-speed shaft. The generator speed is \omega_{\mathrm{g}}=G2 , where G is the gearbox ratio.
这里，Q_{L} 表示传递到高速轴上的驱动系机械扭矩损耗（本身可能旋转速度和扭矩的函数）。发电机转速为 $\omega_{\mathrm{g}}=G\varOmega$，其中 G 为齿轮箱传动比。
This torque-speed relationship is shown schematically in Figure 8.3 as the curve B1–C1. Although it represents the steady-state solution for optimum C_{p} , it can also be used dynamically to control generator torque demand as a function of measured generator speed. In many cases, this is a very benign and satisfactory way of controlling generator torque below rated wind speed.
图 8.3 示意性地显示了转矩-转速关系曲线 B1–C1。虽然它代表了最佳 C_{p} 的稳态解，但也可以动态地用于控制与测得的发电机转速相关的发电机转矩需求。在许多情况下，这是一种在额定风速以下控制发电机转矩的温和且令人满意的方案。
For tracking peak C_{p} below rated in a variable-speed turbine, the quadratic algorithm of Eq. (8.4) works well and gives smooth, stable control. However, in turbulent winds, the large rotor inertia prevents it from changing speed fast enough to follow the wind, so rather than staying on the peak of the C_{p} curve it will constantly fall off either side, resulting in a lower mean C_{p} . This problem is clearly worse for heavy rotors, and also if the C_{p}-\lambda curve has a sharp peak. Thus, in optimising a blade design for variable-speed operation, it is not only important to try to maximise the peak C_{p} , but also to ensure that the C_{p}-\lambda curve is reasonably flat-topped.
为了跟踪额定转速以下的最大 C_{p} 值，公式(8.4)中的二次算法效果良好，能提供平稳、稳定的控制。然而，在湍流风况下，大型转子的惯性阻碍了其快速改变转速以跟踪风速，因此它无法始终保持在 C_{p} 曲线的峰值上，而是会不断偏离两侧，导致平均 C_{p} 值降低。这种问题对于重型转子尤其明显，并且当 C_{p}-\lambda 曲线具有尖锐峰值时也会加剧。因此，在优化用于可变速运行的叶片设计时，不仅要努力最大化峰值 $C_{p}$，还要确保 C_{p}-\lambda 曲线具有较为平坦的峰顶。
It is possible to manipulate the generator torque to cause the rotor speed to change faster when required, so staying closer to the peak of the C_{p} curve. One way to do this is to modify the torque demand by a term proportional to rotor acceleration (Bossanyi 1994):
Figure 8.3 Schematic torque-speed curve for a variable-speed pitch-regulated turbine
\mathcal{Q}_{g}=\frac{1}{2}\frac{\pi\rho R^{5}C_{p}}{\lambda^{3}G^{3}}\omega_{g}^{2}-Q_{L}-B\dot{\omega}_{g}
where B is a gain that determines the amount of inertia compensation. For a stiff drive train, and ignoring frequency converter dynamics, the torque balance gives
I\dot{\boldsymbol\Omega}=Q_{a}-G Q_{g}
where I is the total inertia (of rotor, drive train and generator, referred to the low-speed shaft) and \varOmega is the rotational speed of the rotor. Hence
(I-G^{2}B)\dot{\Omega}=Q_{a}-\frac{1}{2}\frac{\pi\rho R^{5}C_{p}}{\lambda^{3}G^{2}}\omega_{g}^{2}+G Q_{L}
Thus, the effective inertia is reduced from I to I-G^{2}B , allowing the rotor speed to respond more rapidly to changes in wind speed. The gain B should remain significantly smaller than I/G^{2} otherwise the effective inertia will approach zero, requiring huge power swings to force the rotor speed to track closely the changes in wind speed.
Another possible method is to use available measurements to make an estimate of the wind speed, calculate the rotor speed required for optimum C_{p} , and then use the generator torque to achieve that speed as rapidly as possible. The aerodynamic torque can be expressed as
Q_{a}=\frac{1}{2}\rho A C_{q}R U^{2}=\frac{1}{2}\rho\pi R^{5}\varOmega^{2}C_{q}/\lambda^{2}
where R is the turbine radius, \varOmega the rotational speed, and C_{q} the torque coefficient. If drive train torsional flexibility is ignored, a simple estimator for the aerodynamic torque is
{Q_{a}}^{*}=G Q_{g}+I\dot{\varOmega}=G Q_{g}+I\dot{\omega}_{g}/G
where I is the total inertia. A more sophisticated estimator could take into account drive train torsion, etc. From this it is possible to estimate the value of the function F(\lambda)=C_{q}(\lambda)/\lambda^{2} as
F^{*}(\lambda)=\frac{Q_{a}^{*}}{\frac{1}{2}\rho\pi R^{5}(\omega_{g}/G)^{2}}
Knowing the function F(\lambda) from steady state aerodynamic analysis, one can then deduce the current estimated tip speed ratio \lambda^{*} (see also Section 8.3.16 for a better estimation method). The desired generator speed for optimum tip speed ratio can then be calculated as
\omega_{d}=\omega_{g}\widehat{\lambda}/\lambda^{*}
where \widehat{\lambda} is the optimum tip speed ratio to be tracked. A simple PI controller can then be used, acting on the speed error \omega_{\mathrm{g}}-\omega_{\mathrm{d}} , to calculate a generator torque demand that will track \omega_{\mathrm{d}} . The higher the gain of PI controller, the better will be the C_{p} tracking, but at the expense of larger power variations. Simulations for a particular turbine showed that a below rated energy gain of almost 1\% could be achieved, with large but not unacceptable power variations.
Holley et al. (1999) demonstrated similar results with a more sophisticated scheme, and also showed that a perfect C_{p} tracker could capture 3\% more energy below rated, but only by demanding huge power swings of plus and minus three to four times rated power, which is totally unacceptable.
Because such large torque variations are required to achieve only a modest increase in power output, it is usual to use the simple quadratic law, possibly augmented by some inertia compensation as in Eq. (8.5) if the rotor inertia is large enough to justify it.
As turbine diameters increase in relation to the lateral and vertical length scales of turbulence, it becomes more difficult to achieve peak C_{p} anyway because of the non-uniformity of the wind speed over the rotor swept area. Thus if one part of a blade is at its optimum angle of attack at some instant, other parts will not be.
In most cases, it is actually not practical to maintain peak C_{p} from cut-in all of the way to rated wind speed. Although some variable-speed systems can operate all of the way down to zero rotational speed, this is not the case with limited range variable-speed systems based on the widely used doubly fed induction generators. These systems only need a power converter rated to handle a fraction of the turbine power, which is a major cost saving. This means that in low wind speeds, just above cut-in, it may be necessary to operate at an essentially constant rotational speed, with the tip speed ratio above the optimum value.
At the other end of the range, it is usual to limit the rotational speed to some level, usually determined by aerodynamic noise constraints or blade leading-edge erosion, which is reached at a wind speed that is still some way below rated. It is then cost-effective to increase to torque demand further, at essentially constant rotational speed, until rated power is reached. Figure 8.3 illustrates some typical torque-speed trajectories, which are explained in more detail below. Turbines designed for noise-insensitive sites may be designed to operate along the optimum C_{p} trajectory all of the way until rated power is reached. The higher rotational speed implies lower torque and in-plane loads, but higher out-of-plane loads, for the same rated power. This strategy might be of interest for offshore wind turbines.