# Numerical Simulation and Experimental Verification of Wind Field Reconstruction Based on PCA and QR Pivoting

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reconstruction Method Based on PCA and Sparse Sensing

_{k}.

_{k}and reconstruction coefficient α

_{k}can be obtained by Singular Value Decomposition (SVD). Given a snapshot matrix containing multiple state vectors $\mathrm{X}=\left[{x}_{1},{x}_{2},\dots ,{x}_{n}\right]$, $\mathrm{X}\in {\mathbb{R}}^{m\times n}$, the singular value decomposition of matrix X yields:

**v**as linear combinations of φ

_{k}that define a low-dimensional embedding space.

**y**($y\in {\mathbb{R}}^{r}$).

**v**. Point measurements require that the measurement matrix M can be structured in the following way:

_{j}are the canonical basis vectors for ${\mathbb{R}}^{n}$ with a unit entry at index j and zeros elsewhere. The form of M can be shown as follows:

**v**into the determination of the reconstruction coefficient a with the Moore–Penrose pseudoinverse. Subsequently, the reconstruction of

**v**is obtained using:

## 3. Preliminaries

#### 3.1. Construction of the Simulated Wind Tunnel Model

#### 3.2. Construction of an Experimental Wind Tunnel Model

#### 3.3. Construction of Snapshot Matrix

_{r}is the known wind speed at a reference height z

_{r}. The exponent (α) is an empirical coefficient, approximately 1/7, or 0.143 [27]. Furthermore, z

_{r}in the construction of the database is taken at 0.8 m. The wind velocities at z

_{r}change from 1 to 31 m/s with an interval of 5 m/s, and the incident angles are 30°, 70°, 110° and 150°. The turbulence intensity and viscosity ratio are set to 5% and 10%, respectively.

^{−5}.

## 4. Numerical Results of the Proposed Reconstruction Method

#### 4.1. Presentation of Reconstruction Results of Velocity Field and Pressure Field

#### 4.2. Influence of Sensor Number and Placement on Reconstruction Performance

**a**, and the number and location of sensors determine M. Thus the effect of the number and placement of sensors on the accuracy of reconstruction will be further discussed in this section.

_{λ}for a given measurement matrix M, which can be used to represent the stability of a linear equation:

_{λ}), the variation of both metrics is studied in detail with 1 to 20 sensors. Given our random sampling strategy, the results that follow will be statistical in nature, computing averages and variances for batches of randomly selected sampling.

_{λ}, log[Φ(K

_{λ})]. From Figure 17 and Figure 18, three conclusions can be drawn: First, the relative error and condition number are largest and the most unstable when the number of sensors is less than the number of basis vectors. Second, the variance of the 1000 trails, depicted by the red bars is also quite large, suggesting that the reconstruction performance for fewer sensors is highly sensitive to their placement. Third, the error and condition number both perform better as sensors increase, and the trends of both are similar, which supports the hypothesis that the condition number can be used to evaluate the performance of the sparse measurements.

_{λ}) and RE for six sensors in the velocity field and eight sensors in the pressure field. Specifically, the relative errors and condition number can change by orders of magnitude with the same sensors, but simply placed in different locations. From both figures, it can be seen that fewer sensors can produce both exceptional results and extremely poor performance depending upon the sensor locations, which illustrates that extremely high variability can be generated in reconstruction using random and sparse measurements.

#### 4.3. Reconstruction Results Considering Wind Shear Effect

_{r}), inlet velocity at z

_{r}and incident angle are not in the prepared database, i.e., the inlet boundary condition is completely unknown. The reconstruction calculation under this condition is of great significance to verify the feasibility of applying this method to the complex airflow around the real wind field. The following conclusions can be drawn from the analysis of errors in the table: (1) Compared with the uniform entrance boundary, the reconstruction error under the boundary condition considering the wind shear effect is significantly larger. (2) The reconstruction errors under the second and third boundary conditions indicate that relatively satisfactory results can still be obtained with limited sensors under complex and unknown inlet boundary conditions. (3) Interestingly, the reconstruction error of the pressure field is smaller than that of uniform wind speed, probably because the pressure field distribution database built considering the wind shear effect is closer to the actual pressure field distribution, which also indicates the importance of database construction in the reconstruction calculation.

## 5. Optimal Sensor Placement and Validation Results Based on QR Pivoting

#### 5.1. Criterions for Optimal Sensor Placement

_{λ}can be maximized by ${\mathrm{K}}_{\lambda}^{-1}$.

#### 5.2. QR Pivoting for Sparse Sensor Placement

**Q**, an upper-triangular matrix

**R**and a column permutation matrix

**C**, i.e., $A{C}^{T}=QR$. QR factorization provides an approximate greedy method consistent with Equation (21), which can achieve the submatrix volume maximization to maximize the determinant. QR factorization increases the volume of the submatrix by choosing a new pivot column with the largest 2-norm and then subtracting its orthogonal projection onto the pivot column.

#### 5.3. Simulated Reconstruction Results of the Proposed Optimal Sensor Placement

_{λ}. In this section, the number of optimized sensors for velocity field reconstruction and pressure field reconstruction is respectively set as 6, 10 and 8, 15 with reference to Figure 12 and Figure 13. In addition, the noise level is 10%; the boundary condition with 90°, 15 m/s and boundary condition 3 in Table 2 are selected for discussion. For ease of representation, the two boundary conditions are named Boundary Condition 1 and Boundary Condition 2, respectively. After optimizing calculation, it is found that the optimal locations of the sensors obtained by QR pivoting are unique for the determined database, which means that only the optimal locations of 10 and 15 sensors need to be determined, so only the location of 10 sensors in the velocity field and the location of 15 sensors in the pressure field need to be shown in this section.

#### 5.4. Experimental Results of the Proposed Optimal Sensor Placement

- (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.

## 6. Conclusions

- (1)
- In the simulation, the reconstruction errors of the uniform inlet are 0.21% and 6.46%, respectively, while the maximum reconstruction errors including wind shear effect are 1.21% and 6.41%, respectively, which indicates that the reconstruction algorithm based on PCA and sparse sensing can accurately and quickly obtain the distribution characteristics of the velocity and pressure of a 3D wind field.
- (2)
- The effects of the number of basis vectors, measurement noise, number of sensors and placement on the reconstruction results were systematically investigated. The results show that an excessive increase of the number of basis vectors will result in reconstruction errors being more sensitive to noise level.
- (3)
- Reconstruction accuracy can be significantly influenced by the arrangement of sensors when sensor costs are restricted (e.g., less than 20), especially for the pressure field, where the difference between the maximum and minimum reconstruction errors is even 20 times.
- (4)
- Matrix QR pivoting was integrated into the reconstruction algorithm to determine the optimal sensor placement, and its performance was validated by the simulation. The results indicate that QR pivoting-based sensor placement can achieve better reconstruction performance than random measurements, which reduces costs associated with the purchase, placement and maintenance of sensors.
- (5)
- Optimized sensor placement characteristics indicate that better reconstruction results can be obtained by placing the sensor in areas with large gradients of velocity and pressure, where reconstruction errors are also typically larger.
- (6)
- A wind tunnel experiment of velocity field reconstruction was performed to verify the practicability of the optimized reconstruction method based on QR pivoting, and the results indicate that a reasonably high accuracy 3D wind field can be obtained with only 10 sensors (the error of most points is less than 5% and the minimum error is only 0.74%).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Liu, Y.; Wang, Y.; Li, L.; Han, S.; Infield, D. Numerical weather prediction wind correction methods and its impact on computational fluid dynamics based wind power forecasting. J. Renew. Sustain. Energy
**2016**, 8, 33302. [Google Scholar] [CrossRef] [Green Version] - Zhao, J.; Guo, Y.; Xiao, X.; Wang, J.; Chi, D.; Guo, Z. Multi-step wind speed and power forecasts based on a WRF simulation and an optimized association method. Appl. Energy
**2017**, 197, 183–202. [Google Scholar] [CrossRef] - Jung, J.; Broadwater, R.P. Current status and future advances for wind speed and power forecasting. Renew. Sustain. Energy Rev.
**2014**, 31, 762–777. [Google Scholar] [CrossRef] - Reja, R.K.; Amin, R.; Tasneem, Z.; Ali, M.F.; Islam, M.R.; Saha, D.K.; Badal, F.R.; Ahamed, M.H.; Moyeen, S.I.; Das, S.K. A review of the evaluation of urban wind resources: Challenges and perspectives. Energy Build.
**2022**, 257, 111781. [Google Scholar] [CrossRef] - Liu, Z.; Jiang, P.; Zhang, L.; Niu, X. A combined forecasting model for time series: Application to short-term wind speed forecasting. Appl. Energy
**2020**, 259, 114137. [Google Scholar] [CrossRef] - Stathopoulos, C.; Kaperoni, A.; Galanis, G.; Kallos, G. Wind power prediction based on numerical and statistical models. J. Wind Eng. Ind. Aerodyn.
**2013**, 112, 25–38. [Google Scholar] [CrossRef] - Jiang, P.; Yang, H.; Heng, J. A hybrid forecasting system based on fuzzy time series and multi-objective optimization for wind speed forecasting. Appl. Energy
**2019**, 235, 786–801. [Google Scholar] [CrossRef] - Al-Yahyai, S.; Charabi, Y.; Gastli, A. Review of the use of Numerical Weather Prediction (NWP) Models for wind energy assessment. Renew. Sustain. Energy Rev.
**2010**, 14, 3192–3198. [Google Scholar] [CrossRef] - Al-Yahyai, S.; Charabi, Y.; Al-Badi, A.; Gastli, A. Nested ensemble NWP approach for wind energy assessment. Renew. Energy
**2012**, 37, 150–160. [Google Scholar] [CrossRef] - Zajaczkowski, F.J.; Haupt, S.E.; Schmehl, K.J. A preliminary study of assimilating numerical weather prediction data into computational fluid dynamics models for wind prediction. J. Wind Eng. Ind. Aerodyn.
**2011**, 99, 320–329. [Google Scholar] [CrossRef] - Zhang, X.; Weerasuriya, A.U.; Tse, K.T. CFD simulation of natural ventilation of a generic building in various incident wind directions: Comparison of turbulence modelling, evaluation methods, and ventilation mechanisms. Energy Build.
**2020**, 229, 110516. [Google Scholar] [CrossRef] - Wang, H.; Lei, Z.; Zhang, X.; Zhou, B.; Peng, J. A review of deep learning for renewable energy forecasting. Energy Convers. Manag.
**2019**, 198, 111799. [Google Scholar] [CrossRef] - Qureshi, A.S.; Khan, A.; Zameer, A.; Usman, A. Wind power prediction using deep neural network based meta regression and transfer learning. Appl. Soft Comput. J.
**2017**, 58, 742–755. [Google Scholar] [CrossRef] - Astrid, P.; Weiland, S.; Willcox, K.; Backx, T. Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Automat. Contr.
**2008**, 53, 2237–2251. [Google Scholar] [CrossRef] [Green Version] - Qamar, A.; Sanghi, S. Steady supersonic flow-field predictions using proper orthogonal decomposition technique. Comput. Fluids
**2009**, 38, 1218–1231. [Google Scholar] [CrossRef] - Willcox, K. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput. Fluids
**2006**, 35, 208–226. [Google Scholar] [CrossRef] [Green Version] - Jiang, C.; Soh, Y.C.; Li, H. Sensor and CFD data fusion for airflow field estimation. Appl. Therm. Eng.
**2016**, 92, 149–161. [Google Scholar] [CrossRef] - Liu, S.; Vempaty, A.; Fardad, M.; Masazade, E.; Varshney, P.K. Energy-aware sensor selection in field reconstruction. IEEE Signal Process. Lett.
**2014**, 21, 1476–1480. [Google Scholar] [CrossRef] - Chen, K.K.; Rowley, C.W. H2 optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system. J. Fluid Mech.
**2011**, 681, 241–260. [Google Scholar] [CrossRef] [Green Version] - Ranieri, J.; Chebira, A.; Vetterli, M. Near-optimal sensor placement for linear inverse problems. IEEE Trans. Signal Process.
**2014**, 62, 1135–1146. [Google Scholar] [CrossRef] [Green Version] - Lee, A.J.; Diwekar, U.M. Optimal sensor placement in integrated gasification combined cycle power systems. Appl. Energy
**2012**, 99, 255–264. [Google Scholar] [CrossRef] - Manohar, K.; Brunton, B.W.; Kutz, J.N.; Brunton, S.L. Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns. IEEE Control Syst.
**2018**, 38, 63–86. [Google Scholar] [CrossRef] [Green Version] - Zhang, D.; Zhou, Z.H. (2D)2 PCA: Two-directional two-dimensional PCA for efficient face representation and recognition. Neurocomputing
**2005**, 69, 224–231. [Google Scholar] [CrossRef] - Willcox, K.; Perairet, J. Balanced model reduction via the proper orthogonal decomposition. AIAA J.
**2001**, 40, 2323–2330. [Google Scholar] [CrossRef] - Kirby, M.; Sirovich, L. Application of the Karhunen-Loéve Procedure for the Characterization of Human Faces. IEEE Trans. Pattern Anal. Mach. Intell.
**1990**, 12, 103–108. [Google Scholar] [CrossRef] [Green Version] - Eriksson, P.; Jiménez, C.; Bühler, S.; Murtagh, D. A hotelling transformation approach for rapid inversion of atmospheric spectra. J. Quant. Spectrosc. Radiat. Transf.
**2002**, 73, 529–543. [Google Scholar] [CrossRef] [Green Version] - Akay, B.; Ragni, D.; Ferreira, C.S.; Van Bussel, G.J.W. Experimental investigation of the root flow in a horizontal axis wind turbine. Wind Energy
**2014**, 17, 1093–1109. [Google Scholar] [CrossRef] - Chaturantabut, S.; Sorensen, D.C. Nonlinear Model Reduction via Discrete Empirical Interpolation. Soc. Ind. Appl. Math.
**2010**. [Google Scholar] [CrossRef] - Barrault, M.; Maday, Y.; Nguyen, N.C.; Patera, A.T. Une méthode d’«intepolation empirique»: Application à la discrétisation efficace par base réduite d’equations aux dériveés partielles. Comptes Rendus Math.
**2004**, 339, 667–672. [Google Scholar] [CrossRef] - Drmac, Z.; Gugercin, S. A new selection operator for the discrete empirical interpolation method-improved a priori error bound and extensions. SIAM J. Sci. Comput.
**2016**, 38, A631–A648. [Google Scholar] [CrossRef] [Green Version] - Businger, P.; Golub, G.H. Linear least squares solutions by householder transformations. Numer. Math.
**1965**, 7, 269–276. [Google Scholar] [CrossRef] - Seshadri, P.; Narayan, A.; Mahadevan, S. Effectively subsampled quadratures for least squares polynomial approximations. SIAM-ASA J. Uncertain. Quantif.
**2017**, 5, 1003–1023. [Google Scholar] [CrossRef] [Green Version] - Heck, L.P.; Olkin, J.A.; Naghshineh, K. Transducer placement for broadband active vibration control using a novel multidimensional qr factorization. J. Vib. Acoust. Trans. ASME
**1998**, 120, 663–670. [Google Scholar] [CrossRef] - Sommariva, A.; Vianello, M. Computing approximate Fekete points by QR factorizations of Vandermonde matrices. Comput. Math. Appl.
**2009**, 57, 1324–1336. [Google Scholar] [CrossRef] [Green Version]

**Figure 9.**Joint effect of the number of basis vectors and the noise level (%) on the reconstruction error (velocity field).

**Figure 10.**Joint effect of the number of basis vectors and the noise level (%) on the reconstruction error (pressure field).

**Figure 16.**Relative reconstruction error (%) for different number of sensors with four distributions.

**Figure 17.**The average and variance (red bars) over 1000 trials of logarithm of the condition number of K

_{λ}and relative errors (velocity field).

**Figure 18.**The average and variance (red bars) over 1000 trials of logarithm of the condition number of K

_{λ}and relative errors (pressure field).

**Figure 23.**Reconstructed velocity nephogram (boundary condition 1) of the first plane with ten sensor positions (black points).

**Figure 24.**Reconstructed pressure nephogram (boundary condition 1) of the first plane with fifteen sensor positions (black points).

**Figure 25.**Reconstructed velocity nephogram (boundary condition 2) of the first plane with ten sensor positions (black points).

**Figure 26.**Reconstructed pressure nephogram (boundary condition 2) of the first plane with fifteen sensor positions (black points).

Mesh Number | Relative Error (%) |
---|---|

986,523 | 0.12 |

415,243 | 0.22 |

195,489 | 0.58 |

Inlet Boundary Conditions | Wind Speed (m/s) at z_{r} | Incident Angle | z_{r} (m) | Exponent (α) |
---|---|---|---|---|

1 | 5 | 50° | 0.8 | 0.143 |

2 | 10 | 90° | 0.8 | 0.2 |

3 | 15 | 130° | 0.7 | 0.2 |

**Table 3.**Reconstruction errors (%) of velocity and pressure fields under three different wind shear effects.

Inlet Boundary Conditions | Reconstruction Error of Velocity Field | Reconstruction Error of Pressure Field |
---|---|---|

1 | 0.51 | 5.48 |

2 | 1.14 | 6.28 |

3 | 1.21 | 6.41 |

**Table 4.**Logarithm of condition number of K

_{λ}and relative reconstruction errors (%) of random placement and QR pivoting in velocity field (Boundary Condition 1).

Error Metric | 6 Sensors | 10 Sensors | ||
---|---|---|---|---|

QR | Random | QR | Random | |

log[Φ(K_{λ})] | 1.47 | 3.15 | 1.38 | 1.62 |

RE | 1.23 | 24.15 | 0.66 | 0.80 |

**Table 5.**Logarithm of condition number of K

_{λ}and relative reconstruction errors (%) of random placement and QR pivoting in pressure field (Boundary Condition 1).

Error Metric | 8 Sensors | 15 Sensors | ||
---|---|---|---|---|

QR | Random | QR | Random | |

log[Φ(K_{λ})] | 2.37 | 3.61 | 1.35 | 1.71 |

RE | 12.06 | 101.46 | 7.25 | 10.65 |

**Table 6.**Logarithm of condition number of K

_{λ}and relative reconstruction errors (%) of random placement and QR pivoting in velocity field (Boundary Condition 2).

Error Metric | 6 Sensors | 10 Sensors | ||
---|---|---|---|---|

QR | Random | QR | Random | |

log[Φ(K_{λ})] | 2.36 | 2.84 | 2.52 | 2.76 |

RE | 1.63 | 12.06 | 1.55 | 4.21 |

**Table 7.**Logarithm of condition number of K

_{λ}and relative reconstruction errors (%) of random placement and QR pivoting in pressure field (Boundary Condition 2).

Error Metric | 8 Sensors | 15 Sensors | ||
---|---|---|---|---|

QR | Random | QR | Random | |

log[Φ(K_{λ})] | 2.63 | 3.40 | 1.04 | 2.11 |

RE | 15.38 | 83.05 | 6.99 | 18.16 |

Sensor Distribution | Relative Reconstruction Error (%) |
---|---|

QR Pivoting | 5.04 |

10 random sensors | 7.89 |

15 random sensors | 6.83 |

**Table 9.**Measured value (m/s), CFD value (m/s), reconstructed value (m/s) and error (%) of each location.

Location | Verification Points | CFD | Error | QR Pivoting | Error | Random 10 | Error | Random 15 | Error |
---|---|---|---|---|---|---|---|---|---|

1 | 7.07 | 7.24 | 2.33 | 7.28 | 3.03 | 7.82 | 10.64 | 7.57 | 7.01 |

2 | 6.81 | 8.28 | 21.63 | 6.73 | 1.14 | 6.99 | 2.61 | 7.09 | 4.15 |

3 | 7.89 | 8.45 | 7.03 | 7.37 | 6.65 | 8.02 | 1.63 | 7.74 | 2.01 |

4 | 6.62 | 7.09 | 7.13 | 7.12 | 7.59 | 7.58 | 14.60 | 7.47 | 12.87 |

5 | 7.15 | 8.05 | 12.64 | 7.10 | 0.74 | 7.57 | 5.81 | 7.46 | 4.35 |

6 | 7.30 | 8.61 | 17.95 | 7.40 | 1.30 | 7.84 | 7.37 | 7.65 | 4.81 |

7 | 7.13 | 7.69 | 7.82 | 7.00 | 1.94 | 7.31 | 2.52 | 7.36 | 3.24 |

8 | 6.94 | 7.59 | 9.42 | 7.27 | 4.87 | 7.76 | 11.83 | 7.60 | 9.52 |

9 | 8.54 | 8.08 | 5.40 | 7.08 | 17.14 | 7.54 | 11.77 | 7.46 | 12.68 |

10 | 6.94 | 7.09 | 2.10 | 7.18 | 3.42 | 7.67 | 10.46 | 7.46 | 7.47 |

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**MDPI and ACS Style**

Liu, S.; Zhang, G.
Numerical Simulation and Experimental Verification of Wind Field Reconstruction Based on PCA and QR Pivoting. *Appl. Sci.* **2023**, *13*, 2927.
https://doi.org/10.3390/app13052927

**AMA Style**

Liu S, Zhang G.
Numerical Simulation and Experimental Verification of Wind Field Reconstruction Based on PCA and QR Pivoting. *Applied Sciences*. 2023; 13(5):2927.
https://doi.org/10.3390/app13052927

**Chicago/Turabian Style**

Liu, Shi, and Guangchao Zhang.
2023. "Numerical Simulation and Experimental Verification of Wind Field Reconstruction Based on PCA and QR Pivoting" *Applied Sciences* 13, no. 5: 2927.
https://doi.org/10.3390/app13052927