*5.3. Feature Extractions and Fault Classifications Based under Scenario II*

*Data Set for Scenario II:* In this data set, it is composed of 'FF' samples and two types of '1AF + 4SFs' samples. The detailed information is shown in Figure 6—Scenario II. In order to evaluate the effectiveness of the proposed algorithm by comparison, four types of datasets are constructed, which is *XMPCA* II <sup>∈</sup> *<sup>R</sup>*{440,000×4×3000} , *XUMPCA* II <sup>∈</sup> *<sup>R</sup>*{22,000×80×3000} , *XFFT*+*MPCA* II <sup>∈</sup> *<sup>R</sup>*{550×800×4×3000} , and *XFFT*+*UMPCA* II <sup>∈</sup> *<sup>R</sup>*{100×220×80×3000} , respectively. All the detailed information can be found in Tables 5 and 6.

For *XMPCA* II <sup>∈</sup> *<sup>R</sup>*{440,000×4×3000} : '440,000' represents the dimensionality of the feature subspace, '4' stands for the dimensionality of parameter subspace, and '3000' illustrates the dimensionality of the sample subspace. For *XUMPCA* II <sup>∈</sup> *<sup>R</sup>*{22,000×80×3000} : '22,000' represents the dimensionality of the feature subspace, '80' stands for the dimensionality of the parameter subspace, and '3000' illustrates the dimensionality of the sample subspace.

For *XFFT*+*MPCA* II <sup>∈</sup> *<sup>R</sup>*{550×800×4×3000} : The original data set *X*II ∈ *R*{440,000×12,000} is projected into a frequency-domain subspace and reshaped into a tensor dataset *XFFT*+*MPCA* II <sup>∈</sup> *<sup>R</sup>*{550×800×4×3000} for the FFT + MPCA algorithm. For *XFFT*+*UMPCA* II <sup>∈</sup> *<sup>R</sup>*{100×220×80×3000} : The original data set *X*II ∈ *R*{440,000×12,000} is projected into a frequency-domain subspace and reshaped into a tensor dataset *XFFT*+*UMPCA* II <sup>∈</sup> *R*{100×220×80×3000} for the FFT + UMPCA algorithm..

In this section, Figures 12 and 13 illustrate the three-dimensional space visualization performance for fault classification for wind turbine systems subjected to an actuator fault and four sensors faults simultaneously under AWGN noises, respectively using MPCA, UMPCA, FFT + MPCA, and FFT + UMPCA. From the simulated result observation, all types of faulty condition can only be successfully classified by using the FFT + MPCA and FFT + UMPCA methodologies. Therefore, FFT has a positive impact on the improvement of the performance of the dimensionality reduction and feature extraction.

Specifically, from Figure 12a,b, the data sets cluster in a distributive way, although the UMPCA performs a bit better in classification. Encouragingly, from Figure 13a,b, the data sets cluster in three clear groups, indicating a clear fault classification and diagnosis for the three faulty/healthy conditions concerned. From Figure 13b, it is interesting to observe the data in the same group shape distinguishably. It is noted that faulty data in this study includes seven types of faults, such as effectiveness loss, sinusoidal faults, and random number disturbances and so forth, and the fault-free data are subjected to stochastic noises. Therefore, the classification by using the FFT + MPCA can recognize the difference between the data in the same large group. In other words, Figure 13b can also reflect the intrinsic properties of the original samples of the 4.8 MW wind turbine benchmark system.

(**b**) Classification using UMPCA

**Figure 12.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to single actuator and four sensor faults under AWGN noises, using (**a**) MPCA and (**b**) UMPCA, respectively.

(**b**) Classification using FFT + UMPCA

**Figure 13.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to the single actuator and four sensor faults under AWGN noises, using (**a**) FFT + MPCA and (**b**) FFT + UMPCA, respectively.

#### *5.4. Feature Extractions and Fault Classifications under Scenario III*

*Data Set for Scenario III:* In this data set, it includes 'FF' samples and six types of '2AFs + 2SFs' samples. The detailed information is shown in Figure 7—Scenario III. In order to validate the effectiveness of the proposed algorithm by comparison, four types of datasets are addressed, which is *XMPCA* III <sup>∈</sup> *<sup>R</sup>*{440,000×4×7000} , *XUMPCA* III <sup>∈</sup> *<sup>R</sup>*{22,000×80×7000} , *XFFT*+*MPCA* III <sup>∈</sup> *<sup>R</sup>*{550×800×4×7000} , and *XFFT*+*UMPCA* III <sup>∈</sup> *<sup>R</sup>*{100×220×80×7000} , respectively. All the detailed information can be found in Tables 5 and 6.

For *XMPCA* III <sup>∈</sup> *<sup>R</sup>*{440,000×4×7000} : '440,000' represents the dimensionality of the feature subspace, '4' stands for the dimensionality of the parameter subspace, and '7000' illustrates the dimensionality of the sample subspace. For *XUMPCA* III <sup>∈</sup> *<sup>R</sup>*{22,000×80×7000} : '22,000' represents the dimensionality of the feature subspace, '80' stands for the dimensionality of parameter subspace, and '7000' illustrates the dimensionality of the sample subspace.

For *XFFT*+*MPCA* III <sup>∈</sup> *<sup>R</sup>*{550×800×4×7000} : The original data set *X*III ∈ *R*{440,000×28,000} is projected into a frequency-domain subspace and reshaped into a tensor dataset *XFFT*+*MPCA* III <sup>∈</sup> *<sup>R</sup>*{550×800×4×7000} for using the FFT + MPCA algorithm. For *XFFT*+*UMPCA* III <sup>∈</sup> *<sup>R</sup>*{100×220×80×7000} : The original data set *X*III ∈ *R*{440,000×28,000} is projected into a frequency-domain subspace and reshaped into a tensor dataset *XFFT*+*UMPCA* III <sup>∈</sup> *<sup>R</sup>*{100×220×80×7000} for the FFT <sup>+</sup> UMPCA algorithm.

In this section, Figures 14 and 15 exhibit the three-dimensional space visualization performance for fault classification for wind turbine systems subjected to two actuator faults and two sensor faults simultaneously under AWGN noise corruption, respectively by using MPCA, UMPCA, FFT + MPCA, and FFT + UMPCA.

**Figure 14.** *Cont.*

**Figure 14.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to two actuator and two sensor faults under AWGN noises, using (**a**) MPCA and (**b**) UMPCA, respectively.

(**a**) Classification using FFT + MPCA

**Figure 15.** *Cont.*

(**b**) Classification using FFT + UMPCA

**Figure 15.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to two actuator and two sensor faults under AWGN noises, using (**a**) FFT + MPCA and (**b**) FFT + UMPCA, respectively.

From Figure 14a, it is shown that two large groups are formed in the corresponding three-dimensional space based on the MPCA method. It is observed that the overlapping occurs between '{(A1 + A2) & (S1 + S3)}' and '{(A1 + A2) & (S3 + S4)}', and another overlapping happens among 'Fault Free', '{(A1 + A2) & (S1 + S2)}', '{(A1 + A2) & (S1 + S4)}', '{(A1 + A2) & (S2 + S3)}', and '{(A1 + A2) & (S2 + S4)}'. From Figure 14b based on the UMPCA technique, the visualization performance, with more formed data groups, is slightly better than that using the MPCA but is far from acceptable for classification.

Seven classes of faulty/healthy situations were successfully classified respectively by using the FFT + MPCA shown in Figure 15a and FFT + UMPCA exhibited by Figure 15b. More interesting, Figure 15b can clearly reflect the intrinsic properties of the original samples of the wind turbines, which indicates the FFT + UMPCA approach can also sense different types of faults in every single faulty situation.

### *5.5. Feature Extractions and Fault Classifications Based under Scenario IV*

*Data Set for Scenario IV:* In this data set, it consists in 'FF' samples and four types of '2AFs + 3SFs' samples. The detailed information is shown in Figure 8—Scenario IV. In order to evaluate the effectiveness of the proposed algorithm by comparison, four types of datasets are determined: *XMPCA* IV <sup>∈</sup> *<sup>R</sup>*{440,000×4×5000} , *XUMPCA* IV <sup>∈</sup> *<sup>R</sup>*{22,000×80×5000} , *XFFT*+*MPCA* IV <sup>∈</sup> *<sup>R</sup>*{550×800×4×5000} , and *XFFT*+*UMPCA* IV <sup>∈</sup> *<sup>R</sup>*{100×220×80×5000} , respectively. All the detailed information can be found in Tables 5 and 6.

For *XMPCA* IV <sup>∈</sup> *<sup>R</sup>*{440,000×4×5000} : '440,000' represents the dimensionality of the feature subspace, '4' stands for the dimensionality of the parameter subspace, and '5000' illustrates the dimensionality of the sample subspace. For *XUMPCA* IV <sup>∈</sup> *<sup>R</sup>*{22,000×80×5000} : '22,000' represents the dimensionality of the feature subspace, '80' stands for the dimensionality of the parameter subspace, and '5000' illustrates the dimensionality of the sample subspace.

For *XFFT*+*MPCA* IV <sup>∈</sup> *<sup>R</sup>*{550×800×4×5000} : The original data set *X*IV ∈ *R*{440,000×20,000} is projected into a frequency-domain subspace and reshaped into a tensor representation *XFFT*+*MPCA* IV <sup>∈</sup> *<sup>R</sup>*{550×800×4×5000} for the use of the FFT + MPCA algorithm. For *XFFT*+*UMPCA* IV <sup>∈</sup> *<sup>R</sup>*{100×220×80×5000} : The original data set *X*IV ∈ *R*{440,000×20,000} is projected into a frequency-domain subspace and reshaped into a tensor representation *XFFT*+*UMPCA* IV <sup>∈</sup> *<sup>R</sup>*{100×220×80×5000} for the implementation of the FFT <sup>+</sup> UMPCA technique.

In this subsection, Figures 16 and 17 exhibit the three-dimensional space visualization performance for fault classification for wind turbine systems subjected to two simultaneous actuator faults and three simultaneous sensors under AWGN noise corruptions, by using the MPCA, UMPCA, FFT + MPCA, and FFT + UMPCA algorithms, respectively.

(**b**) Classification using UMPCA

**Figure 16.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to two actuator and three sensor faults under AWGN noises, by using (**a**) MPCA and (**b**) UMPCA, respectively.

(**b**) Classification using FFT + UMPCA

**Figure 17.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to two actuator and three sensor faults under AWGN noises, (**a**) FFT + MPCA and (**b**) FFT + UMPCA, respectively.

From Figure 16a based on the MPCA, the data are clustering around three large sets, while in Figure 16b using the UMPCA, the data are clustering in a more distributed way. Both of the visualized results in Figure 16a,b fail to classify and diagnose the faults.

From Figure 17a,b, faulty conditions can be successfully classified by both FFT + MPCA and FFT + UMPCA algorithms. Specifically, it is worthy to point out that the corresponding three-dimensional space visualization behaviours in Figure 17a,b shape differently in comparison with Figure 16a,b, respectively. From what is exhibited in Figure 17a, one can see that the FFT + MPCA methods outperform the MPCA. The reason behind this is that the intrinsic structures of the obtained experimental data sets were reconstructed by using Fourier transform bases. Additionally, these samples are mapped into the multi-dimensional frequency-domain subspace, which means the visualized performance/behaviour are different from the MPCA-based circumstances. Furthermore, the performance of the FFT + UMPCA in Figure 17b is much better than that based on the UMPCA in Figure 16b. Consequently, the FFT has a positive impact on the improvement of the performance of the fault classification and diagnosis. From Figure 17b, one can see that the FFT + UMPCA approach can also recognize the differences of data in the same group.

#### *5.6. Feature Extractions and Fault Classifications Based under Scenario V*

*Data Set for Scenario V:* In this data set, it is combined by 'FF' and '2AFs + 4SFs' samples. The detailed information is shown in Figure 8—Scenario V. In order to validate the effectiveness of the proposed algorithm by comparison, four types of datasets are built: *XMPCA* <sup>V</sup> <sup>∈</sup> *<sup>R</sup>*{440,000×4×2000} , *XUMPCA* <sup>V</sup> ∈ *R*{22,000×80×2000} , *XFFT*+*MPCA* <sup>V</sup> <sup>∈</sup> *<sup>R</sup>*{550×800×4×2000} , and *XFFT*+*UMPCA* <sup>V</sup> <sup>∈</sup> *<sup>R</sup>*{100×220×80×2000} , respectively. All the detailed information can be found in Tables 5 and 6.

For *XMPCA* <sup>V</sup> <sup>∈</sup> *<sup>R</sup>*{440,000×4×2000} : '440,000' represents the dimensionality of the feature subspace, '4' stands for the dimensionality of the parameter subspace, and '2000' indicates the dimensionality of the sample subspace. For *XUMPCA* <sup>V</sup> <sup>∈</sup> *<sup>R</sup>*{22,000×80×2000} : '22,000' represents the dimensionality of the feature subspace, '80' stands for the dimensionality of the parameter subspace, and '2000' represents the dimensionality of the sample subspace.

For *XFFT*+*MPCA* <sup>V</sup> <sup>∈</sup> *<sup>R</sup>*{550×800×4×2000} : The original data set *X*<sup>V</sup> ∈ *R*{440,000×8000} is projected into a frequency-domain subspace and reshaped into a tensor dataset *XFFT*+*MPCA* <sup>V</sup> <sup>∈</sup> *<sup>R</sup>*{550×800×4×2000} for the use of the FFT + MPCA algorithm. The original data set *X*<sup>V</sup> ∈ *R*{440,000×8000} is projected into a frequency-domain subspace and reshaped into a tensor representation *XFFT*+*UMPCA* <sup>V</sup> <sup>∈</sup> *R*{100×220×80×2000} for the FFT + UMPCA technique.

Figures 18 and 19 show the three-dimensional space visualization performance for fault classification for wind turbine systems subjected to two simultaneous actuators faults and four simultaneous sensors faults corrupted by AWGN noisy signals, respectively using different algorithms, such as MPCA, UMPCA, FFT + MPCA, and FFT + UMPCA.

From Figure 18a based on the MPCA, the faulty-data cluster in 10 groups, and one of them is overlapped with the fault-free data, indicating unsuccessful fault classification. From Figure 18b based on the UMPCA technique, the visualization performance is relatively better than that using the MPCA, as there is no overlapping between the faulty-data and fault-free data. However, the performance in Figure 18b is still not satisfactory, since the distances between the faulty-data are too large and the distances between the fault-free data and some of the faulty-data are quite close.

From Figure 19a, one can see that FFT+ MPCA method has a much better classification performance than the MPCA, shown in Figure 18a, as the faulty data and fault-free data are clearly classified into two separated groups by using FFT + MPCA. Comparing Figure 19b by using FFT + UMPCA to Figure 18b via the UMPCA, the faulty data and fault-free data are separated into two large groups in Figure 19b, showing a clear classification between the faulty data and fault free data. As a result, it is evident that the FFT has a positive impact on the improvement of the performance of the fault classification and diagnosis. In addition, from Figure 19b, the data in the same group are not clustered so close compared with Figure 19a. As the faulty data is a combination of the data subjected to different types, such as effectiveness loss, sinusoidal fault signal, random number disturbances, and so forth, this means the FFT + UMPCA can sense the difference between these data.

(**b**) Classification using UMPCA

**Figure 18.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to two actuator and four sensor faults under AWGN noises, using (**a**) MPCA and (**b**) UMPCA, respectively.

(**b**) Classification using FFT + UMPCA

**Figure 19.** Three-dimensional space visualization performance for fault classification for wind turbines subjected to two actuator and four sensor faults under AWGN noises, using (**a**) FFT + MPCA and (**b**) FFT + UMPCA, respectively.

It is noted that the MPCA approach determines a tensor-to-tensor projection that captures most of the signal variation present in the original tensor representation, whereas, the UMPCA method uses the tensor-to-vector projection. For the MPCA technique, some of the correlations of the principal components among the projected directions are neglected to some extent. Compared with MPCA, UMPCA can exclude the possibilities of getting significant features with similar geometric structures, depending on the methodology of tensor-to-vector projection. The reason behind is that the UMPCA algorithm concentrates on extracting and determining the uncorrelated principal components rather than the conventional principal components in the MPCA technique. Moreover, the FFT preprocessing technique can enhance the data classification capability of the UMPCA. As a result, this is why the proposed FFT + UMPCA can effectively classify the fault under all five scenarios above.

#### **6. Conclusions**

In this paper, fast Fourier transform (FFT) and uncorrelated multi-linear principal component analysis (UMPCA) techniques were integrated for fault classification of the 4.8 MW benchmark wind turbine systems subjected to multiple actuator and sensor faults under five scenarios of actuator and sensor faults. The detailed comparison studies were carried out, and the effectiveness of the proposed algorithm was well demonstrated. It is worthy to point out, among all the used algorithms, the FFT has a positive impact on the improvement of the performance of the fault diagnosis and classification. The proposed FFT plus UMPCA algorithm can not only classify the various classes of faulty conditions but can also recognize the differences between the data within the same class.

In the future, it is of interest to investigate data-driven fault prognosis and remaining useful life prediction for wind turbine systems. It is also promising to enhance fault diagnosis and prognosis performance by using hybrid methods (by integrating various data-driven fault diagnosis/prognosis methods or even by combining model-based approaches and data-driven based methods).

**Author Contributions:** Conceptualization, Y.F., Z.G. and Y.L.; methodology, Y.F., Z.G. and Y.L.; software, Y.F. and Y.L.; validation, Y.F.; formal analysis, Y.F., Y.L., A.Z., X.Y. and Z.G.; writing—original draft preparation, Y.F.; writing—review and editing, Z.G.; supervision, Z.G., Y.L.; funding acquisition, Z.G. and A.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Nature Science Foundation of China (NNSFC) under grant 61673074, and the Alexander von Humboldt Foundation under grant GRO/1117303 STP.

**Acknowledgments:** The authors would like to thank the research support from theE&E faculty at University of Northumbria (UK), the National Nature Science Foundation of China (NNSFC) under grant 61673074, and the Alexander von Humboldt Foundation under grant GRO/1117303 STP.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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