# A Novel Digital Twin Framework for Aeroengine Performance Diagnosis

^{*}

## Abstract

**:**

## 1. Introduction

- The proposed digital twin framework provides a new approach for effectively integrating mechanism models and data-driven models in engine performance diagnosis.
- The proposed digital twin framework incorporates the strengths of mechanism models and data-driven models, allowing for continuous updates to follow engine performance changes while ensuring high-precision monitoring. Compared to purely model-based or data-driven methods, the proposed digital twin framework exhibits a higher prediction accuracy and diagnostic precision.
- Serving as a more reliable and efficient tool for gas path diagnosis, the proposed digital twin framework enables more intuitive monitoring of the engine health and provides more effective technical support for establishing engine health management systems.

## 2. Methodology

#### 2.1. Component-Level Mechanism Model

_{1}” denotes the high-pressure spool, and “N

_{2}” represents the low-pressure spool. Table 1 presents the design point performance of this engine, while Table 2 provides the definitions of the station numbers.

#### 2.1.1. Thermodynamic Calculation of Gas Path Components

_{cor}represents the non-dimensional speed corresponding to the compressor, N represents the actual physical speed of the compressor, β

_{c}represents the auxiliary coordinate on the compressor characteristic map, W represents the mass flow rate at the compressor outlet, π

_{c}represents the pressure ratio of the compressor, and η

_{c}represents the efficiency of the compressor. The total temperature T

_{c}and total pressure P

_{c}at the compressor outlet section can be calculated as follows:

_{P}represents the air’s specific heat capacity at constant pressure.

_{2}), and the auxiliary coordinate (β

_{t}) on the HPT component characteristic map, the component characteristic parameters for the HPT can be obtained from the map. Therefore, the thermodynamic calculations of the turbine can be expressed as follows:

_{HPTcor}represents the non-dimensional speed of HPT, W

_{45}represents the mass flow rate at the outlet of HPT, π

_{HPT}represents the HPT pressure ratio, η

_{HPT}represents the HPT efficiency, and β

_{t}represents the HPT auxiliary coordinate on the HPT map. Therefore, the total temperature and total pressure at the outlet section of HPT can be calculated as follows:

_{g}represents the specific heat ratio of the gas. Therefore, the power generated per second by HPT can be calculated as follows:

#### 2.1.2. Steady-State and Dynamic Balance Calculation

_{f}as inputs. Based on the thermodynamic calculations of the key components in the aeroengine gas path as described earlier, the equilibrium equations for the steady-state and transient conditions are solved using the Newton–Raphson method [37]. In order to complete the overall simulation of the aeroengine gas path, the power balance and the gas mass flow continuity principle are considered. The equilibrium equations for the aeroengine under steady-state conditions can be expressed as follows:

_{L}represents the low-pressure shaft moment of inertia, and J

_{H}represents the high-pressure shaft moment of inertia.

#### 2.1.3. Mechanism Model Solving Process

#### 2.2. Data-Driven Model

#### 2.2.1. Extreme Gradient Boosting Machine (XGBoost)

_{i}is drawn from a dataset D = {(x

_{i}, y

_{i})} that contains n samples with m features. K represents the additive function, and F represents the space of regression trees. f

_{k}represents the function to the decision tree. The XGBoost model reduces the error of the ensemble of trees through the objective function, which is calculated as follows:

_{i}represents the actual value, y

^{*}

_{i}represents the predicted value, t represents the number of iterations to minimize the error, and Ω represents the penalty function.

#### 2.2.2. Data-Driven Model Framework Based on PSO-XGBoost

#### 2.3. Multimodal Fusion Model

#### 2.3.1. Multimodal Fusion Using Tensor Representations

_{m}represents the input representations. The input tensor Z is then transformed by a linear layer g to generate the vector representation, which is expressed as follows:

#### 2.3.2. Low-Rank Multimodal Fusion (LMF)

^{i}

_{m}

_{,k}}

^{M}

_{m}

_{=1}}

^{R}

_{i}

_{=1}. Therefore, the low-rank weight tensor can be modified as follows:

_{k}, and these vectors are recombined and concatenated to form the low-rank factors for each of the M modalities. Therefore, Equation (25) can be rewritten as follows:

#### 2.4. Fault Diagnosis Model

#### 2.4.1. Sparse Stacked Autoencoder Network (SSAE)

_{(1)}, s

_{(2)}, …, s

_{(l)}], the weight matrices are represented as w = [w

^{(1)}, w

^{(2)}, …, w

^{(l+1)}], and the bias matrices are represented as b = [b

^{(1)}, b

^{(2)}, …, b

^{(l+1)}]. The encoding process can be written as follows:

#### 2.4.2. Training Process of SSAE

^{(1)}= [w

^{(1,1)}, w

^{(1,2)}] and bias set b

^{(1)}= [b

^{(1,1)}, b

^{(1,2)}] are obtained. Then, the output values of the non-linear mapping function in the first layer are used as inputs for the next autoencoder to train the second autoencoder. The updated weight set w

^{(2)}= [w

^{(2,1)}, w

^{(2,2)}] and bias set b

^{(2)}= [b

^{(2,1)}, b

^{(2,2)}] for the second layer subnetwork are obtained. This process continues layer by layer until all subnetworks are trained, resulting in w = [w

^{(1,1)}, w

^{(2,1)}, …, w

^{(l,1)}, w

^{(l+1,1)}] and b = [b

^{(1,1)}, b

^{(2,1)}, …, b

^{(l,1)}, b

^{(l+1,1)}]. Finally, all the subnetworks are concatenated, and the global loss function is used to adjust the weights (w) and biases (b).

## 3. Digital Twin Framework Functions and Processes

#### 3.1. Mechanism Model Update for Performance Degradation

#### 3.2. Gas Path Fault Diagnosis

**×**ndicates the fault categories included in the case.

#### 3.3. Digital Twin Process

## 4. Case Study

#### 4.1. Gas Path Parameters Prediction Verification

_{f}), to make the study case more realistic, we simulated the typical flight trajectory of a commercial twin-spool turbofan engine, and the variations of the flight mission control data are shown in Figure 12. From Figure 12, it can be observed that except for the cruise phase, H keeps changing, while W

_{f}gradually increases during the climb phase. The mechanism model generates the corresponding flight data based on this flight control pattern, with one data point generated per second. The data-driven model utilizes the generated flight data as training samples, while the mechanism model updates the performance degradation by extracting the engine operating points from the flight data.

_{i}and y

^{*}

_{i}represent the actual value and predicted value of the i-th data sample, respectively.

#### 4.2. Gas Path Fault Diagnosis Verification

**+**digital twin framework can clearly distinguish between the nine fault types, with only a small number of fault samples causing confusion. However, in the three fault cases (Case 7, Case 8, and Case 9) involving multiple coupled gas path faults, there are relatively more instances of confusion. This is because extracting multiple fault features becomes more challenging. For the proposed SSAE

**+**digital twin framework in this paper, Case 7, Case 8, and Case 9 have seven, six, and eight misclassified fault samples, respectively. Specifically, in Case 7, three fault samples are misclassified as Case 8, and four fault samples are misclassified as Case 9. In Case 8, three fault samples are misclassified as Case 7, and three fault samples are misclassified as Case 9. In Case 9, five fault samples are misclassified as Case 7, and three fault samples are misclassified as Case 8. However, the misclassified fault samples constitute only a small portion of the overall samples, and compared to the other three methods, the number of misclassified fault samples is significantly reduced.

**+**SSAE method reaches 98.6%, significantly outperforming the three aforementioned methods.

**+**SSAE method achieves improvements of 10%, 8.5%, and 6.1% in the overall accuracy, respectively. This indicates a significant enhancement in the diagnostic performance of the SSAE through the fusion of multimodal data in the digital twin framework. The digital twin framework, unlike single-modal information-based gas path fault diagnosis methods, effectively extracts valuable features from the engine gas path data from different modalities through the LWF method. This improves data reliability and enhances the accuracy of the gas path fault diagnosis in engines by leveraging the fusion of multimodal information.

#### 4.3. Economic Evaluation of Gas Path Performance Degradation

_{fe}) generated by the flight mission due to degradation:

_{fd}represents the fuel flow rate of the engine after degradation, T

_{i}represents the total duration of the flight mission, and i represents the i-th flight phase among n flight segments. The calculation method for the additional fuel cost (C

_{f}) incurred by the entire airline fleet due to engine degradation is

_{up}represents the fuel price, and TNumE represents the total number of engines in the entire fleet.

_{f}) for the next engine in the flight cycle due to degradation is approximately USD 209.5. This significant increase in the fuel cost highlights the detrimental impact of aviation engine degradation on the economy of engine users. In conclusion, this study demonstrates that aviation engine degradation can have a substantial negative effect on operational costs for engine users, emphasizing the importance of monitoring and managing the engine health to mitigate these economic implications.

## 5. Conclusions

- The mechanism model updated with performance degradation achieves a significant improvement in the total MAPE, decreasing from 2.443% to 0.583%, and the overall MAPE of the data-driven model is 0.232%. The proposed digital twin framework achieves an overall MAPE of only 0.125%. Compared to the mechanism model and the data-driven model, the digital twin framework exhibits a significantly improved prediction accuracy for aerodynamic parameters across the entire flight mission, enabling continuous updates as the engine operates over the long term, and better reflecting the engine’s health condition.
- Compared to three deep learning models, DBN, CNN, and SSAE, the proposed digital twin framework demonstrates distinct advantages in diagnostic accuracy, achieving overall accuracy improvements of 10%, 8.5%, and 6.1%, respectively, with an overall diagnostic rate of 98.6%. In terms of the diagnostic time, the digital twin framework performs comparably to the other models.
- This study presents an estimation method for calculating the fuel cost incurred by engine degradation and assesses the resulting typical flight mission loss, which amounts to approximately USD 209.5 per engine after undergoing 3000 flight cycles. The proposed digital twin framework for aeroengines accurately tracks the engine’s degradation and enables gas path diagnosis under a flight mission, providing effective data support for engine health management. In conclusion, this framework contributes to improving maintenance decision making, ensuring the reliability and safety of aviation operations, and reducing engine operating costs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Parameter | Value | Unit |
---|---|---|

Thrust | 95.31 | kN |

Total flow rate | 322.65 | kg/s |

Total pressure ratio | 33.8 | |

SFC | 10.17 | g/(kN·s) |

Number | Significance |
---|---|

amb | Ambient |

13 | Bypass |

18 | Bypass nozzle outlet |

2 | FAN outlet |

25 | Low-pressure compressor outlet |

3 | High-pressure compressor outlet |

4 | Combustion chamber outlet |

45 | High-pressure turbine outlet |

5 | Low-pressure turbine outlet |

8 | Nozzle outlet |

Fault Type | Flow Capacity (A) | Isentropic Efficiency (B) | Ratio A:B | Range |
---|---|---|---|---|

Compressor fouling | ↓ | ↓ | ~3:1 | (0, −7.5%), (0, −2.5%) |

Turbine fouling | ↓ | ↓ | ~2:1 | (0, −4%), (0, −2%) |

Turbine erosion | ↑ | ↓ | ~2:1 | (0, 4%), (0, −2%) |

Case | FAN Fouling | LPC Fouling | HPC Fouling | LPC Erosion | HPT Fouling | LPT Fouling | LPT Erosion |
---|---|---|---|---|---|---|---|

1 | × | ||||||

2 | × | ||||||

3 | × | ||||||

4 | × | ||||||

5 | × | ||||||

6 | × | ||||||

7 | × | × | × | × | |||

8 | × | × | × | × | × | ||

9 | × | × | × | × | × |

Component | Health Parameter | Coefficient | Value |
---|---|---|---|

FAN | DF_{m} | a | 0.01 |

b | 0.14675 | ||

DF_{η} | a | 0.01 | |

b | 0.11878 | ||

LPC | DF_{m} | a | 0.01 |

b | 0.1571 | ||

DF_{η} | a | 0.01 | |

b | 0.1088 | ||

HPC | DF_{m} | a | 0.01 |

b | 0.25236 | ||

DF_{η} | a | 0.01 | |

b | 0.29621 | ||

HPT | DF_{m} | a | 0.01 |

b | 0.10705 | ||

DF_{η} | a | 0.01 | |

b | 0.1516 | ||

LPT | DF_{m} | a | 0.001 |

b | 0.16544 | ||

DF_{η} | a | 0.01 | |

b | 0.008 |

Parameters | Definition | Reference Noise Level |
---|---|---|

N_{1} | LP spool speed | 0.25% |

N_{2} | HP spool speed | 0.25% |

T_{2} | Fan outer exit temperature | 0.75% |

T_{25} | LPC exit temperature | 0.75% |

T_{3} | HPC inlet temperature | 0.75% |

T_{5} | LPT exit temperature | 0.75% |

Model | N_{1} | N_{2} | T_{2} | T_{25} | T_{3} | T_{5} | Average |
---|---|---|---|---|---|---|---|

mechanism model | 1.516% | 3.276% | 1.162% | 4.184% | 1.286% | 3.235% | 2.443% |

mechanism model update | 0.251% | 0.249% | 0.749% | 0.752% | 0.748% | 0.748% | 0.583% |

data-driven model | 0.122% | 0.137% | 0.223% | 0.152% | 0.278% | 0.481% | 0.232% |

digital twin | 0.076% | 0.081% | 0.011% | 0.061% | 0.171% | 0.352% | 0.125% |

Network | Network Structure |
---|---|

SSAE | 6 layers of modules |

digital twin + SSAE | 6 layers of modules |

DBN | 1 hidden layer and 1 visible layer |

CNN | 3 convolution layers and 2 fully connected layers |

Approach | Total Accuracy (%) | CPU Time (s) |
---|---|---|

DBN | 88.6% | 52.11 |

CNN | 90.1% | 69.39 |

SSAE | 92.5% | 47.25 |

SSAE + digital twin | 98.6% | 93.96 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Wang, Y.; Wang, X.; Yang, K.; Zhao, Y.
A Novel Digital Twin Framework for Aeroengine Performance Diagnosis. *Aerospace* **2023**, *10*, 789.
https://doi.org/10.3390/aerospace10090789

**AMA Style**

Wang Z, Wang Y, Wang X, Yang K, Zhao Y.
A Novel Digital Twin Framework for Aeroengine Performance Diagnosis. *Aerospace*. 2023; 10(9):789.
https://doi.org/10.3390/aerospace10090789

**Chicago/Turabian Style**

Wang, Zepeng, Ye Wang, Xizhen Wang, Kaiqiang Yang, and Yongjun Zhao.
2023. "A Novel Digital Twin Framework for Aeroengine Performance Diagnosis" *Aerospace* 10, no. 9: 789.
https://doi.org/10.3390/aerospace10090789