## 6 Frequency Domain Optimization
### Detailed Optimization Example: Circular Scan Pattern  
All measurement quality calculations are performed for a wind field with mean wind speed \(U\) = 13 m/s (1.6 m/s above rated wind speed for the NREL 5 MW reference turbine) and IEC Class B normal turbulence model turbulence intensity (Stein  
beams, the additional measurement ranges afforded by pulsed lidars improve measurement accuracy compared to a CW lidar. All lidar measurement scenarios improve upon a single point measurement of the wind speed at hub height. However, the measurement accuracy is much higher for the ideal rotor disk average wind speed measurements than for any realistic lidar scenario. Here, the only source of measurement error is wind evolution due to the 50 m preview distance between the measurement plane and the rotor disk. The optimal coherence bandwidth, measurement MSE, and generator speed MSE values shown in Figure 5 for the seven lidar measurement scenarios are listed in Table 1, along with the scan parameters required to achieve them. For the two pulsed lidar scenarios, the scan radius \(r\), preview distance \(d\), and focus distance \(F\) indicate the values corresponding to the farthest range gate measured. Table 1 reveals that the optimal scan parameters depend on the measurement quality metric that is being optimized. Note that the optimal scan parameters for maximizing coherence bandwidth and minimizing generator speed MSE are almost always the same. However, the optimal scan radius and preview distance values that minimize measurement MSE are typically greater than they are for the other two metrics. The reasons why the optimal scan parameters that maximize coherence bandwidth or minimize generator speed MSE are shorter than for minimizing measurement MSE are related to the frequencies that are weighted most heavily when calculating each metric (see Equations (10) and (11)). For example, due to the frequency response of the closed-loop transfer function \(T_{\omega_{\textit{pos}}u_{\textit{off}}}\), the generator speed MSE is most sensitive to measurement error in the frequency band near 0.05 Hz, while the measurement MSE is more sensitive to measurement error at lower frequencies, where more of the energy in the wind is concentrated. This explains why generator speed MSE is minimized (and coherence bandwidth is maximized) using shorter preview distances, which prevent the coherence at higher frequencies from decaying too much from wind evolution. When minimizing measurement MSE, on the other hand, it is more important to measure farther away, thereby reducing LOS errors, which affect the low frequencies as much as the high frequencies. Figure 5: Maximum achievable coherence bandwidth and minimum achievable rotor effective wind speed measurement MSE (normalized by rotor effective wind speed variance) and linearized generator speed MSE (normalized by generator speed MSE with baseline feedback control) corresponding to the optimized scan parameters for each scan pattern category. Measuring farther away from the turbine adds the benefit of more volume averaging along the lidar beam due to the wider range weighting function for CW lidars. Longer preview distances also permit larger scan radii to be achieved with smaller cone angles, allowing greater radial coverage of the rotor disk area while reducing LOS errors. The additional volume averaging and greater rotor disk coverage help approximate the rotor effective wind speed, further explaining why longer preview distances and larger scan radii are optimal for minimizing MSE, where additional wind evolution at high frequencies can be tolerated. It should be noted that the optimal scan radius and preview distance parameters listed in Table 1 are specific to a 126 m-rotor diameter turbine. However, when expressed in non-dimensional units of rotor diameters and rotor radii, the optimal scan parameters roughly translate to different rotor sizes.