## 5 Lidar Scan Pattern Optimization
### Metrics for Assessing Measurement Quality  
To compare the effectiveness of different scan patterns, an appropriate metric representing lidar measurement quality must be defined. One simple metric that describes measurement accuracy is the MSE between the true rotor effective wind speed at the turbine \(u_{\mathit{eff}}\) and its estimated value based on the lidar measurement \(\hat{u}_{\mathit{eff}}\):  
\[\mathbf{E}\left[\left|u_{\mathit{eff}}-\hat{u}_{\mathit{eff}}\right|^{2}\right]. \tag{1}\]  
However, because lidar-based estimates of the rotor effective wind speed are not perfectly correlated with the wind at the rotor due to wind evolution and other sources of measurement error, it is beneficial to low-pass filter the measurements before they are used by the controller. As will be described in the next section, an optimal filter \(H_{opt}\) can be derived that minimizes the measurement MSE. Therefore, a useful metric for determining measurement accuracy is the MSE between the true rotor effective wind speed and its optimally-filtered lidar-based estimate:  
\[\mathbf{E}\left[\left|u_{\mathit{eff}}-H_{opt}\hat{u}_{\mathit{eff}}\right|^{ 2}\right]. \tag{2}\]  
Metrics describing measurement quality can be made more meaningful by including the response of the turbine to the wind inflow, instead of concentrating on the wind variables alone. For example, the impact of the wind inflow on a turbine variable of interest \(y_{WT}\), when the imperfect measurement \(\hat{u}_{\mathit{eff}}\) is used as input to the feedforward controller, can be used to judge measurement quality. Assuming the objective of the lidar-assisted controller is to minimize the magnitude of a wind turbine variable, the variance of \(y_{WT}\) is an appropriate measurement quality metric. Specifically, for the control objective of minimizing the mean square error between the generator speed \(\omega_{gen}\) and the rated generator speed \(\omega_{gen,0}\), this metric becomes:  
\[\mathbf{E}\left[\left|\omega_{gen}-\omega_{gen,0}\right|^{2}\right]. \tag{3}\]  
Finally, measurement quality can be judged by the highest frequency at which the measurement remains correlated with the true rotor effective wind speed. In this case, the objective of the scan pattern optimization is to maximize the correlation bandwidth, allowing the feedforward controller to mitigate the impact of as much of the frequency content of the wind as possible. The correlation between the rotor effective wind speed \(u_{\mathit{eff}}\) and its lidar-based estimate \(\hat{u}_{\mathit{eff}}\) as a function of frequency is described by the magnitude-squared coherence between the two signals: \(\gamma_{u_{\mathit{eff}}\hat{u}_{\mathit{eff}}}^{2}\left(f\right)\). The magnitude-squared coherence between two signals \(a\) and \(b\) is defined as  
\[\gamma_{ab}^{2}\left(f\right)=\frac{\left|S_{ab}\left(f\right)\right|^{2}}{S_ {aa}\left(f\right)S_{bb}\left(f\right)}, \tag{4}\]  
where \(S_{ab}\left(f\right)\) is the cross-power spectral density (CPSD) between signals \(a\) and \(b\), \(S_{aa}\left(f\right)\) is the power spectral density (PSD) of \(a\), and \(S_{bb}\left(f\right)\) is the PSD of \(b\). Values of coherence range from 0 to 1, where a value of 0 indicates that the two signals are completely uncorrelated while a value of 1 indicates perfect correlation. A reasonable value of coherence for representing the boundary between frequency components that are uncorrelated and correlated is 0.5 [51]. Therefore, a useful metric for indicating measurement quality is the frequency at which the measurement coherence crosses below 0.5:  
\[f:\ \gamma_{u_{\mathit{eff}}\hat{u}_{\mathit{eff}}}^{2}\left(f\right)=0.5. \tag{5}\]