# An Adversarial Single-Domain Generalization Network for Fault Diagnosis of Wind Turbine Gearboxes

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Proposed Method

#### 3.1. Problem Definition

#### 3.2. Structure

^{R}= E(X), used for data reconstruction. Let ${x}^{+}\in {S}^{+}$ and $x\in S$. Here, S and S

^{+}denote the source domain and the expanded domain, respectively.

^{+}, with substantial domain shift [32,33]. Subsequently, we learn domain-agnostic features from the source domain and these augmented domains to train a domain-generalized model, which is

_{classifier}can be computed using the cross-entropy function.

_{i}denoting the ith dimension of ŷ. y stands for the one-hot representation of the model’s true label, and y

_{i}indicates the ith dimension of y. K signifies the number of classes.

_{classifier}represents the classification loss and can be expressed as Equation (2), L

_{contract}signifies the semantic consistency constraint and its specific implementation will be detailed in Section 3.3, L

_{expand}induces domain shift and its specific implementation will be detailed in Section 3.4. The parameters α and β are hyperparameters that balance the training. Through adversarial training between VAE and F, the objective is to generate expanded domains with the maximum domain divergence from the source domain, thereby enhancing the model’s generalization capabilities in unknown domains.

^{+}is generated in the expanded domain S

^{+}using gradient ascent:

#### 3.3. Feature Extractor

^{+}, thereby ensuring semantic consistency.

^{+}∈ Z; z = F(x) and z

^{+}= F(x

^{+}). $\left|\right|.\left|\right|$ represents the Wasserstein distance between the source domain and the augmented domain; ensures that the data originates from the same fault type. L

_{contract}constrains the conditional distribution difference between the source and augmented domains, preventing the generation of ineffective samples in the augmented domain. However, this also limits the domain shift distance. Therefore, L

_{expand}is introduced to amplify the domain shift, and its specific definition will be elaborated in Section 3.4.

#### 3.4. Variational Autoencoder

_{expand}aims to expand the augmented domain, while L

_{contract}attempts to contract it. Through adversarial training of the two, we acquire a generalized dataset to train a robust generalization model.

#### 3.5. Model Optimization

Algorithm 1. ASDGN |

#Pre-train VAE |

Input: Source dataset $S={\{{x}_{i},{y}_{i}\}}_{i=1}^{n}$; VAE model $E(\theta ;X)$; pre-train epoch E1 |

for i = 1 to E1 do: |

Randomly sample from S |

Forward propagation and calculation Equation (7) |

Backward propagation to update θ by Equation (9) |

end |

Return: pre-trained VAE model |

#Domain Augmentation |

Input: Source dataset $S={\{{x}_{i},{y}_{i}\}}_{i=1}^{n}$; pre-trained VAE model $E(\theta ;X)$; pre-trained feature extractor F; classifier C; number of augmentation domains K; Adversarial train epoch E2 |

for i = 1 to K do: |

for i = 1 to E2 do: |

Randomly sample m data $X={\left\{{x}_{i}\right\}}_{i=1}^{m}$ from S; ${X}^{+}==X.clone()$ |

Forward propagation and calculation Equations (2), (5) and (8) |

Calculation Equation (3) |

Backward propagation to update ${X}^{+}$ by Equation (9) end |

Create ${S}^{+}={\left\{{x}_{i}^{+}\right\}}_{i=1}^{n}$ |

end |

Return: $D=\{S,{S}_{1}^{+},\cdot \cdot \cdot {S}_{K}^{+}\}$ |

#Domain Augmentation |

Input: Dataset $D=\{S,{S}_{1}^{+},\cdot \cdot \cdot {S}_{K}^{+}\}$; feature extractor F; classifier C; Task model train epoch E3. |

for i = 1 to E3 do: |

Randomly sample data from S |

Forward propagation and calculation Equation (2) |

Backward propagation to update $\omega $ by Equation (11) |

end |

Return: Task model |

## 4. Experiments

#### 4.1. Dataset Description

- Dataset 1: This dataset originates from a wind turbine gearbox fault simulation test rig, and the rig’s structure is depicted in Figure 5. The dataset encompasses four bearing health conditions under four loads: 0, 2, 4, and 8. The faults include Normal (N), Inner Race Fault (IR), Ball Fault (B), and Outer Race Fault (OR). The data sampling frequency is 20 kHz. This dataset provides an effective means to verify the robustness of the model.

- 2.
- Dataset 2 [43]: The gear fault data is collected from the gearbox fault simulation test rig, as shown in Figure 6a. It includes two conditions of speed–load, 20–0 and 30–2. Under each condition, there are five types of gear fault states, health, chipped, miss, root, and surface, as illustrated in Figure 6b. The data sampling frequency is 5120 Hz. The two conditions in this dataset have significant differences, which effectively validates the generalization performance of the model.

#### 4.2. Comparison Experiment

#### 4.3. Experimental Results

#### 4.4. Visualization Analysis

#### 4.5. Ablation Experiments

_{contract}and L

_{expand}during the domain expansion model optimization process, we conducted ablation experiments to compare the effects on model classification accuracy and the training process when either

_{Lcontract}or L

_{expand}are omitted. The experimental results are shown in Figure 12 and Figure 13.

_{expand}drops significantly. This is because, although the expanded domain increases the number of samples, it does not cover the unknown domain, resulting in an inadequate generalization capability. The ASDGN without L

_{contract}has slightly lower accuracy than the standard ASDGN and shows poorer stability during iterations. This is attributed to the absence of L

_{contract}’s constraint, leading to the inclusion of some faulty samples in the generated samples. Additionally, it is worth noting that since the ASDGN pre-trains the feature extractor and classifier during the domain expansion phase, its initial accuracy when training the task model is higher than that of a CNN model trained from scratch.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The illustration depicts the schematics of DA, DG, and SDG. The distinct domains signify data originating from different operational conditions. (

**a**) DA: extracting transferable features from both the source and target domains to facilitate knowledge transfer. (

**b**) DG: extracting domain-invariant features from multiple source domains, aiming for direct application to an unseen target domain. (

**c**) SDG: harnessing domain-invariant features solely from a single-source domain, eliminating the multi-source domain constraints inherent in DG.

**Figure 6.**(

**a**) shows the structure diagram of the gearbox fault simulation testbench, and (

**b**) illustrates the five fault states. From left to right, they are surface, miss, chipped, health, and root.

**Figure 9.**Taking task G2 as an example, this figure contrasts the iterative curves of the baseline and proposed models. (

**a**) The graph depicts the iterative convergence curve of the CNN; (

**b**) the graph depicts the iterative convergence curve of the ASDGN.

**Figure 10.**Taking B3 as an example, the figure shows the feature visualization of the feature layer for each method.

**Figure 12.**Taking G1 as an example, we compare the iterative processes of the CNN and ASDGN without L

_{contract}, or L

_{expand}.

**Figure 13.**A comparison of the accuracy rates for the CNN and ASDGN without L

_{contract}or L

_{expand}on Dataset 1.

**Table 1.**Comparison of the research problem addressed in this study with previous research in terms of problem settings and the scope of application. Note: This table lists the training conditions and applicability scope of different methods. In training conditions, “✓” indicates that the data is required, and “✕” means the data is not needed. In target conditions, “✓” signifies that the model is applicable to that condition, and “✕” suggests it is not suitable for that condition.

Method | Training Conditions | Target Conditions | ||||||
---|---|---|---|---|---|---|---|---|

Source Domain Data | Source Domain Labels | Target Domain Data | Target Domain Labels | Multi-Source Domain for Training | Same Distribution between the Source and Target Domains | Different Distribution between the Source and Target Domains | Multi-Target Domains | |

DL | ✓ | ✓ | ✕ | ✕ | ✕ | ✓ | ✕ | ✕ |

DA | ✓ | ✓ | ✓ | ✕ | ✕ | ✓ | ✓ | ✕ |

DG | ✓ | ✓ | ✕ | ✕ | ✓ | ✓ | ✓ | ✓ |

SDG | ✓ | ✓ | ✕ | ✕ | ✕ | ✓ | ✓ | ✓ |

Task | Train Data | Text Data | ||
---|---|---|---|---|

Condition | Number | Condition | Number | |

B1 | 0 (Nm) | 1000 × 4 = 4000 | 0 Nm, 2 Nm, 4 Nm, 8 Nm | 300 × 4 × 4 = 4800 |

B2 | 2 (Nm) | 1000 × 4 = 4000 | ||

B3 | 4 (Nm) | 1000 × 4 = 4000 | ||

B4 | 8 (Nm) | 1000 × 4 = 4000 | ||

G1 | 20–0 (rpm-V) | 1000 × 5 = 5000 | 20 rpm–0 V, 30 rpm–2 V | 300 × 5 × 2 = 3000 |

G2 | 30–2 (rpm-V) | 1000 × 5 = 5000 |

Method | CNN | DT-DDG | L2A-OT | AMInet | RTDGN | Proposed |
---|---|---|---|---|---|---|

B1 | 80.25 ± 0.71 | 84.42 ± 0.49 | 85.84 ± 0.82 | 88.02 ± 0.73 | 81.10 ± 0.51 | 90.90 ± 0.73 |

B2 | 74.94 ± 0.96 | 84.91 ± 0.87 | 87.85 ± 0.95 | 86.91 ± 0.71 | 82.29 ± 0.55 | 88.99 ± 0.62 |

B3 | 79.28 ± 0.49 | 83.32 ± 0.91 | 83.61 ± 0.57 | 89.69 ± 0.32 | 79.70 ± 0.33 | 94.07 ± 0.59 |

B4 | 79.00 ± 0.43 | 84.83 ± 0.16 | 84.08 ± 1.00 | 90.64 ± 0.32 | 82.29 ± 1.08 | 93.83 ± 0.63 |

Average | 78.37 | 84.37 | 85.35 | 88.82 | 81.35 | 91.95 |

Method | CNN | DT-DDG | L2A-OT | AMInet | RTDGN | Proposed |
---|---|---|---|---|---|---|

G1 | 65.66 ± 0.29 | 79.06 ± 0.46 | 81.42 ± 1.07 | 86.86 ± 0.70 | 78.51 ± 0.56 | 88.43 ± 1.01 |

G2 | 66.67 ± 0.72 | 84.35 ± 0.95 | 82.46 ± 1.31 | 87.63 ± 0.74 | 82.11 ± 0.74 | 91.65 ± 1.08 |

Average | 66.17 | 81.71 | 81.94 | 87.25 | 80.31 | 90.04 |

Model | Training Time (s) |
---|---|

CNN | 219.37 |

DT-DDG | 311.54 |

L2A-OT | 392.78 |

AMInet | 413.49 |

RTDGN | 288.15 |

Proposed | 506.42 |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Wang, C.; Liu, H.; Zhang, C.; Fu, Z.; Ding, L.; Bai, C.; Zhang, H.; Wei, Y.
An Adversarial Single-Domain Generalization Network for Fault Diagnosis of Wind Turbine Gearboxes. *J. Mar. Sci. Eng.* **2023**, *11*, 2384.
https://doi.org/10.3390/jmse11122384

**AMA Style**

Wang X, Wang C, Liu H, Zhang C, Fu Z, Ding L, Bai C, Zhang H, Wei Y.
An Adversarial Single-Domain Generalization Network for Fault Diagnosis of Wind Turbine Gearboxes. *Journal of Marine Science and Engineering*. 2023; 11(12):2384.
https://doi.org/10.3390/jmse11122384

**Chicago/Turabian Style**

Wang, Xinran, Chenyong Wang, Hanlin Liu, Cunyou Zhang, Zhenqiang Fu, Lin Ding, Chenzhao Bai, Hongpeng Zhang, and Yi Wei.
2023. "An Adversarial Single-Domain Generalization Network for Fault Diagnosis of Wind Turbine Gearboxes" *Journal of Marine Science and Engineering* 11, no. 12: 2384.
https://doi.org/10.3390/jmse11122384