## 5 Optimal Sensor Placement and Validation Results Based on QR Pivoting
### Experimental Results of the Proposed Optimal Sensor Placement  
As described in Section 3.2, an experimental wind tunnel model is constructed to demonstrate the feasibility of the reconstruction algorithm and its optimal placement. Due to the limited experimental conditions and the importance of obtaining the velocity field in a wind farm for wind power forecasting, only the reconstruction results of the velocity field are validated in this section. The measurement is divided into four groups: 10 QR pivoting-based sensors; 10 randomly placed sensors; 15 randomly placed sensors; and 10 verification locations. Randomly placed sensors are used for comparison, and measurements at 10 verification locations are taken as reference values for error calculations. The steps of experimental data acquisition are as follows:  
1. Calibrate the anemometers and insert them in the appropriate locations. 2. Turn on the fan and adjust it to the appropriate operating frequency. 3. After the fan works for 5 min, collect the data continuously for 10 min. 4. Calculate the average wind speed of each anemometer. 5. Repeat the above steps until all data acquisition is completed. In this experiment, the number of basis vectors is 3 and the fan frequency is set to 25 Hz. The reconstructed nephograms and sensor placements are shown in Figure 27. The nephograms can accurately show the wind speed distribution characteristics, which verifies the reliability of the PCA-based reconstruction algorithm in practice. In addition, the relative reconstruction errors for the three sensor placements are shown in Table 8. It is obvious that the optimal placement based on QR pivoting can achieve a more accurate reconstruction than random placement, no matter whether the number of random sensors is 10 or 15. This validates that in practical applications, QR pivoting can effectively reduce the number of sensors required while ensuring the reconstruction accuracy. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Error Metric**} & \multicolumn{3}{c}{**6 Sensors**} & \multicolumn{3}{c}{**10 Sensors**} \\ \cline{2-5}  & **QR** & **Random** & **QR** & **Random** \\ \hline \(\log\{\Phi(\text{K}_{\lambda})\}\) & 2.36 & 2.84 & 2.52 & 2.76 \\ \(RE\) & 1.63 & 12.06 & 1.55 & 4.21 \\ \hline \end{tabular}
\end{table} 
Table 6: Logarithm of condition number of \(\text{K}_{\lambda}\) and relative reconstruction errors (%) of random placement and QR pivoting in velocity field (Boundary Condition 2). \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Error Metric**} & \multicolumn{3}{c}{**8 Sensors**} & \multicolumn{3}{c}{**15 Sensors**} \\ \cline{2-5}  & **QR** & **Random** & **QR** & **Random** \\ \hline \(\log\{\Phi(\text{K}_{\lambda})\}\) & 2.63 & 3.40 & 1.04 & 2.11 \\ \(RE\) & 15.38 & 83.05 & 6.99 & 18.16 \\ \hline \end{tabular}
\end{table} 
Table 7: Logarithm of condition number of \(\text{K}_{\lambda}\) and relative reconstruction errors (%) of random placement and QR pivoting in pressure field (Boundary Condition 2). Figure 27: Reconstructed velocity nephograms and sensor locations. Similar to the error analyses in Section 4.2 and 4.3, the experimental results will be discussed from three aspects: number of basis vectors; number of sensors; and errors of ten verification points. First, the effect of the basis vectors on the experiment is explored, which is also the validation of the conclusions in Section 4.2. Since the measurement error is generally large and unavoidable in the experiment, given the conclusion of Section 4.2, i.e., the reconstruction error with more basis vectors is sensitive to the noise, the number of basis vectors are determined as two, three, four and five, respectively. Figure 28 shows the reconstruction errors with a different number of basis vectors. Compared with 10 sensors, 15 sensors can significantly reduce errors, but it is still higher than the QR pivoting-based placement.