# Remaining Useful Life Estimation of Aircraft Engines Using a Modified Similarity and Supporting Vector Machine (SVM) Approach

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology for RUL Estimation

#### 2.1. The Scheme of the Modified Similarity Methodology Based on Run-to-Failure Data

#### 2.1.1. Determination of Time Range for Similarity Measurement

#### 2.1.2. Calculation of the Similarity Measure

#### 2.1.3. Definition of the Weight Function

#### 2.1.4. RUL Estimation of the Operating System

#### 2.1.5. Optimization of the Weight-Adjust Coefficient $\alpha $

#### 2.2. The Scheme of the Similarity and SVM Methodology Based on Deteriorated Data

## 3. Model Applications to an Aero Engine

#### 3.1. The Estimation of RUL for an Airplane Engine with Run-to-Failure Data Though the Modified Similarity Methodology

- (1)
- Build the failure space (two-dimensional space) and calculate the projection of the failure values in the failure space, as shown as the hollow dots in Figure 3;
- (2)
- Calculate the center of these projection dots, as shown as star dot in Figure 3;
- (3)
- Calculate the projection dot of the performance parameters on the failure space at a certain cycle;
- (4)
- Calculate the Euclidean distance between the projection dot of the performance parameters in the failure space at a certain cycle and the center of the projected dots in the failure space in an operating mode, which is shown in Figure 4.

#### 3.2. The Estimation of RUL for an Aero Engine with Deteriorated Data Though the Similarity and SVM Methodology

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Methodologies for RUL estimation. RUL: residual useful life; SVM: supporting vector machine; CM: condition maintenance; PHM: prognostics and health management; PCM: proportional covariate model; HMM: hidden Markov model.

Mode | Mode 1 | Mode 2 | Mode 3 | Mode 4 | Mode 5 | Mode 6 |
---|---|---|---|---|---|---|

PC1 | 0.6082 | 0.5892 | 0.7959 | 0.7185 | 0.6087 | 0.5363 |

PC2 | 0.3803 | 0.4005 | 0.1903 | 0.2641 | 0.3034 | 0.4329 |

Run Time | 156 Cycles | 161 Cycles | 166 Cycles | 171 Cycles | 176 Cycles |
---|---|---|---|---|---|

RUL | 2.5986 | 2.4292 | 2.3260 | 2.4667 | 2.3691 |

Ranking | Sample Number | Sampling Interval | Lifetime | RUL | Weight |
---|---|---|---|---|---|

1 | 38 | 224–228 | 287 | 59 | 0.192924 |

2 | 82 | 193–197 | 223 | 26 | 0.164397 |

3 | 115 | 211–215 | 260 | 45 | 0.162683 |

4 | 12 | 120–124 | 242 | 118 | 0.093304 |

5 | 29 | 164–168 | 228 | 60 | 0.082048 |

6 | 103 | 219–223 | 243 | 20 | 0.080489 |

7 | 64 | 122–126 | 154 | 28 | 0.060018 |

8 | 53 | 205–209 | 259 | 50 | 0.058619 |

9 | 78 | 176–180 | 228 | 48 | 0.055197 |

10 | 34 | 244–248 | 286 | 38 | 0.050321 |

Operating Time | Traditional Similarity Method | Modified Similarity Method | ||
---|---|---|---|---|

Predicted RUL | Error (%) | Predicted RUL | Error (%) | |

176 | 225.69 | 0.1358 | 214.50 | 5.0871 |

177 | 238.11 | 5.3600 | 219.63 | 2.8201 |

178 | 213.58 | 5.4953 | 221.18 | 2.1310 |

179 | 209.86 | 7.1432 | 219.50 | 2.8742 |

180 | 203.53 | 9.9426 | 221.56 | 1.9642 |

… | … | … | … | … |

196 | 197.23 | 12.7305 | 221.62 | 1.9388 |

197 | 200.43 | 11.3132 | 219.35 | 2.9427 |

198 | 205.42 | 9.1069 | 220.53 | 2.4208 |

199 | 200.55 | 11.2619 | 225.75 | 0.1121 |

200 | 194.23 | 14.0578 | 226.77 | 0.3401 |

… | … | … | … | … |

216 | 188.03 | 16.8012 | 226.80 | 0.3558 |

217 | 184.37 | 18.4209 | 226.17 | 0.0741 |

218 | 182.68 | 19.1695 | 225.31 | 0.3036 |

219 | 183.04 | 19.0067 | 224.09 | 0.8448 |

220 | 181.41 | 19.7290 | 224.15 | 0.8169 |

Sample Number | Lifetime | ||||
---|---|---|---|---|---|

$F{t}_{1}~F{t}_{5}$ | 231.12598 | 289.93651 | 214.7592 | 299.89567 | 372.86392 |

$F{t}_{6}~F{t}_{10}$ | 232.23493 | 174.22486 | 290.63039 | 183.51086 | 239.97266 |

$F{t}_{11}~F{t}_{15}$ | 214.47475 | 262.17492 | 215.14943 | 238.10682 | 297.89302 |

$F{t}_{15}~F{t}_{20}$ | 301.69223 | 236.68347 | 201.39653 | 243.82267 | 255.27005 |

Reference Samples | W_{11} | W_{13} | W_{18} | W_{17} | W_{3} |
---|---|---|---|---|---|

Weights | 0.2526 | 0.2258 | 0.1957 | 0.1678 | 0.1581 |

Work Time | Predicted Failure Time | Error (%) |
---|---|---|

121–125 | 201.4377 | 10.86 |

126–130 | 189.9478 | 15.95 |

131–135 | 188.2144 | 16.71 |

136–140 | 187.5556 | 17.01 |

141–145 | 188.0059 | 16.81 |

146–150 | 192.7102 | 14.73 |

151–155 | 203.6575 | 9.88 |

156–160 | 227.415 | 0.62 |

161–165 | 227.8171 | 0.80 |

166–170 | 230.5832 | 2.02 |

171–175 | 219.4822 | 2.88 |

176–180 | 217.1443 | 3.91 |

181–185 | 220.0715 | 2.62 |

186–190 | 224.6852 | 0.58 |

191–195 | 219.9376 | 2.68 |

196–200 | 232.6292 | 2.93 |

201–205 | 231.4903 | 2.42 |

206–210 | 234.244 | 3.64 |

211–215 | 233.7521 | 3.43 |

216–220 | 230.2518 | 1.88 |

221–225 | 228.2501 | 0.99 |

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**MDPI and ACS Style**

Chen, Z.; Cao, S.; Mao, Z.
Remaining Useful Life Estimation of Aircraft Engines Using a Modified Similarity and Supporting Vector Machine (SVM) Approach. *Energies* **2018**, *11*, 28.
https://doi.org/10.3390/en11010028

**AMA Style**

Chen Z, Cao S, Mao Z.
Remaining Useful Life Estimation of Aircraft Engines Using a Modified Similarity and Supporting Vector Machine (SVM) Approach. *Energies*. 2018; 11(1):28.
https://doi.org/10.3390/en11010028

**Chicago/Turabian Style**

Chen, Zhongzhe, Shuchen Cao, and Zijian Mao.
2018. "Remaining Useful Life Estimation of Aircraft Engines Using a Modified Similarity and Supporting Vector Machine (SVM) Approach" *Energies* 11, no. 1: 28.
https://doi.org/10.3390/en11010028