RUL prediction for aero-engines based on Copula similarity
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摘要:
针对航空发动机性能退化特征众多,以及特征相互影响等问题,考虑退化特征间的非线性相关关系,提出了基于Copula相似性的航空发动机RUL(Remaining Useful Life)预测方法。通过K-means聚类将航空发动机的工作状态分类,建立退化模型选取退化性能趋势最明显的3组传感器。基于Copula函数对选取的3组传感器进行相关性建模分析,构建发动机传感器之间的Copula结构。基于Copula相似性实现对航空发动机的剩余寿命预测。结果表明:基于Copula相似性的航空发动机RUL预测方法相较传统方法,在发动机运行周期的50%、70%、90%预测误差分别减少13.053%、31.328%、74.602%,预测精度得到提高。
Abstract:In view of many degradation features of aero-engine performance and their mutual influence, the RUL prediction method of aero-engine based on Copula similarity was proposed considering the nonlinear correlations of the degradation features. The working state of the aero-engine was classified through K-means clustering, and a degradation model was established to select three sets of sensors with the most obvious degradation performance trend. Based on the Copula function, the correlation modeling and analysis of the selected three sets of sensors were carried out to build the Copula structure between engine sensors. The prediction of the remaining life of aero-engine was realized based on Copula similarity. The results showed that, compared with the traditional method, the Copula similarity-based aero-engine RUL prediction method reduces the prediction errors by 13.053%, 31.328%, and 74.602% at 50%, 70%, and 90% of the engine operating cycle, respectively, and the prediction accuracy is improved.
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Key words:
- prediction and health management /
- performance degradation /
- remaining useful life /
- Copula /
- nonlinearity /
- similarity
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表 1 二元Copula函数简介
Table 1. Introduction to binary copula functions
函数
类型分布函数表达式 特性 Clayton
Copula$ C_\delta ^{\text{Cl}} = {\left( {{\mu ^{ - \delta }} + {v^{ - \delta }} - 1} \right)^{\tfrac{{ - 1}}{\delta }}} $ 上尾
相关Gumbel
Copula$ C_\theta ^{\text{Gu}} = \exp \left\{ {-\Bigg\{\bigg[(-{\text{ln}}\;\mu)^{\theta }+(-{\text{ln}}\;v)^{\theta }\bigg]\Bigg\}^{\tfrac{1}{\theta }}} \right\} $ 下尾
相关Frank
Copula$ C_\lambda ^{\rm{Fr}} = - \dfrac{1}{\lambda }\ln \left\{ {1 - \left[{\dfrac{{\left( {1 - {{\text{e}}^{ - \lambda \mu }}} \right)\left( {1 - {{\text{e}}^{ - \lambda v}}} \right)}}{{\left( {1 - {{\text{e}}^{ - \lambda }}} \right)}}} \right]} \right\} $ 不反映尾部
相关性序号 符号 描述 1 T2 风扇入口总温 3 T30 高压压气机出口总温 4 T50 低压涡轮出口总温 5 P2 风扇入口压力 6 P15 外涵总压 7 P30 高压压气机出口总压 8
9Nf
Ne风扇物理转速
核心机物理转速10 epr 发动机压比 11 Ps30 高压压气机出口静压 12 Phi 燃油流量与高压压气机出口总压比值 13 NRf 风扇换算转速 14 Nre 核心机换算转速 15 BPR 涵道比 16 farB 燃烧室燃气比 17 htBleed 引气焓值 18 Nf_dmd 设定风扇转速 19 PCNfR_dmd 设定核心机换算转速 20 W31 高压涡轮冷却引气流量 21 W32 低压涡轮冷却引气流量 注:发动机压比为高压压气机出口总压和风扇入口压力之比。 表 3 某发动机运行数据
Table 3. Operating data of an engine
$ t $ $ {y_1} (t) $ $\cdots$ $ x_1^l (t) $ $ x_2^l (t) $ $\cdots$ $ x_{_{21}}^l (t) $ 1 10.0047 489.05 604.13 17.1735 2 0.0015 518.67 642.13 23.3619 3 34.9986 449.44 555.42 8.8555 $ \vdots$ $ \vdots$ $ \vdots$ $ \vdots$ $ \vdots$ 223 34.9992 $\cdots$ 449.44 556.6 $\cdots$ 8.6695 表 4 聚类操作设置参数表
Table 4. Clustering parameters table of operation setting
类别 $ {Y_1} (t) $ $ {Y_2} (t) $ $ {Y_3} (t) $ 1 10.003 0.251 20 2 0.0012 0.001 100 3 25.003 0.621 80 4 42.003 0.841 40 5 20.003 0.701 0 6 35.003 0.841 60 表 5 历史样本Copula参数值
Table 5. Copula parameter values of historical sample
$ {L_{\rm{ru}}} $ $ Q_1^1 $ $ Q_2^1 $ $ Q_3^1 $ $ \tau _1^1 $ $ \tau _2^1 $ $ \tau _3^1 $ 223 Fra Cla Cla 0.479 0.454 0.495 181 Fra Joe Joe 0.531 0.460 0.524 209 Joe Joe Joe 0.472 0.332 0.333 ┇ ┇ ┇ ┇ ┇ ┇ ┇ 220 Cla Cla Cla 0.420 0.390 0.418 255 Fra Fra Fra 0.512 0.492 0.536 156 Joe Fra Fra 0.513 0.450 0.514 注:表中Fra表示Frank Copula,Cla表示Clayton Copula,Joe表示Joe Copula。 表 6 现役发动机参数值
Table 6. parameter values of in-service engine
传感器组合 Copula $ \tau $ (4,11) Clayton 0.474 (4,15) Frank 0.458 (11,15) Frank 0.519 表 7 现役发动机寿命预测值
Table 7. Life prediction of in-service engine
相似样本 $ {L_{\rm{ru}}} $ 50%Cycle 70%Cycle 90%Cycle 178 255 127.500 76.500 25.500 60 300 150 90 30 151 317 158.500 95.100 31.700 ┇ ┇ ┇ ┇ ┇ 108 230 115 69 23 35 188 94 56.4 18.8 16 172 86 51.600 17.200 预测结果 217.86 108.930 65.358 21.786 真实寿命 210 105 63 21 预测误差 7.860 3.930 2.358 0.786 表 8 方法对比误差表
Table 8. Error comparison table of two method
方法 估计误差 50%Cycle 70%Cycle 90%Cycle 传统方法[25] 26.346 20.014 18.037 基于Copula
相似性22.907 13.744 4.581 -
[1] XIONG Minglan,WANG Huawei,FU Qiang,et al. Digital twin-driven aero-engine intelligent predictive maintenance[J]. The International Journal of Advanced Manufacturing Technology,2021,114(11/12): 3751-3761. [2] XU Jiuping,WANG Yusheng,XU Lei. PHM-oriented integrated fusion prognostics for aircraft engines based on sensor data[J]. IEEE Sensors Journal,2014,14(4): 1124-1132. [3] LIAO Linxia,KÖTTIG F. Review of hybrid prognostics approaches for remaining useful life prediction of engineered systems,and an application to battery life prediction[J]. IEEE Transactions on Reliability,2014,63(1): 191-207. [4] LI Yiguang,NILKITSARANONT P. Gas turbine performance prognostic for condition-based maintenance[J]. Applied Energy,2009,86(10): 2152-2161. [5] 王华伟,吴海桥. 基于信息融合的航空发动机剩余寿命预测[J]. 航空动力学报,2012,27(12): 2749-2755. WANG Huawei,WU Haiqiao. Residual useful life prediction for aircraft engine based on information fusion[J]. Journal of Aerospace Power,2012,27(12): 2749-2755. (in Chinese doi: 10.13224/j.cnki.jasp.2012.12.018 WANG Huawei, WU Haiqiao . Residual useful life prediction for aircraft engine based on information fusion[J]. Journal of Aerospace Power,2012 ,27 (12 ):2749 -2755 . (in Chinese) doi: 10.13224/j.cnki.jasp.2012.12.018[6] SON K L,FOULADIRAD M,BARROS A,et al. Remaining useful life estimation based on stochastic deterioration models: a comparative study[J]. Reliability Engineering & System Safety,2013,112: 165-175. [7] GIANTOMASSI A,FERRACUTI F,BENINI A,et al. Hidden Markov model for health estimation and prognosis of turbofan engines[C]// International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Washington: ASME,2011: 681-689. [8] XIANG Sheng,QIN Yi,LUO Jun,et al. Multicellular LSTM-based deep learning model for aero-engine remaining useful life prediction[J]. Reliability Engineering & System Safety,2021,216: 107927. [9] MALINOWSKI S,CHEBEL-MORELLO B,ZERHOUNI N. Remaining useful life estimation based on discriminating shapelet extraction[J]. Reliability Engineering & System Safety,2015,142: 279-288. [10] LAM J,SANKARARAMAN S,STEWART B. Enhanced trajectory based similarity prediction with uncertainty quantification[C]//Annual Conference of the PHM Society. Texas: IJPHM,2014: 2513.1-2513.12. [11] KHELIF R,MALINOWSKI S,CHEBEL-MORELLO B,et al. RUL prediction based on a new similarity-instance based approach[C]//2014 IEEE 23rd International Symposium on Industrial Electronics. Piscataway,US: IEEE,2014: 2463-2468. [12] LIU Yingchao,HU Xiaofeng,ZHANG Wenjuan. Remaining useful life prediction based on health index similarity[J]. Reliability Engineering & System Safety,2019,185: 502-510. [13] YU Wennian,KIM I Y,MECHEFSKE C. An improved similarity-based prognostic algorithm for RUL estimation using an RNN autoencoder scheme[J]. Reliability Engineering & System Safety,2020,199: 106926. [14] PAN Zhengqiang,BALAKRISHNAN N. Reliability modeling of degradation of products with multiple performance characteristics based on gamma processes[J]. Reliability Engineering & System Safety,2011,96(8): 949-957. [15] ZHENG Jianfei,SI Xiaosheng,HU Changhua,et al. A nonlinear prognostic model for degrading systems with three-source variability[J]. IEEE Transactions on Reliability,2016,65(2): 736-750. doi: 10.1109/TR.2015.2513044 [16] RASMEKOMEN N,PARLIKAD A K. Condition-based maintenance of multi-component systems with degradation state-rate interactions[J]. Reliability Engineering & System Safety,2016,148: 1-10. [17] SUN Fuqiang,WANG Ning,LI Xiaoyang,et al. Remaining useful life prediction for a machine with multiple dependent features based on Bayesian dynamic linear model and copulas[J]. IEEE Access,2017,5: 16277-16287. doi: 10.1109/ACCESS.2017.2735966 [18] YANG Zhiyuan,ZHAO Jianmin,CHENG Zhonghua,et al. Reliability modeling of two-component system with degradation interaction based on copulas[C]//2018 Prognostics and System Health Management Conference. Piscataway,US: IEEE,2019: 138-143. [19] XI Zhimin,WANG Pingfeng. A Copula based sampling method for residual life prediction of engineering systems under uncertainty[C]//2012 IEEE Conference on Prognostics and Health Management. Piscataway,US: IEEE,2012: 1-9. [20] 宋仁旺,张岩,石慧. 基于Copula函数的齿轮箱剩余寿命预测方法[J]. 系统工程理论与实践,2020,40(9): 2466-2474. SONG Renwang,ZHANG Yan,SHI Hui. Prediction method for the remaining useful life of gearbox based on copula function[J]. Systems Engineering-Theory & Practice,2020,40(9): 2466-2474. (in Chinese doi: 10.12011/1000-6788-2019-0307-09 SONG Renwang, ZHANG Yan, SHI Hui . Prediction method for the remaining useful life of gearbox based on copula function[J]. Systems Engineering-Theory & Practice,2020 ,40 (9 ):2466 -2474 . (in Chinese) doi: 10.12011/1000-6788-2019-0307-09[21] SUKHANOVA E M. A test for independence of two multivariate samples[J]. Mathematical Methods of Statistics,2008,17(1): 74-86. doi: 10.3103/S1066530708010067 [22] 蔡菲,严正,赵静波,等. 基于Copula理论的风电场间风速及输出功率相依结构建模[J]. 电力系统自动化,2013,37(17): 9-16. CAI Fei,YAN Zheng,ZHAO Jingbo,et al. Dependence structure models for wind speed and wind power among different wind farms based on copula theory[J]. Automation of Electric Power Systems,2013,37(17): 9-16. (in Chinese doi: 10.7500/AEPS201207293 CAI Fei, YAN Zheng, ZHAO Jingbo, et al . Dependence structure models for wind speed and wind power among different wind farms based on copula theory[J]. Automation of Electric Power Systems,2013 ,37 (17 ):9 -16 . (in Chinese) doi: 10.7500/AEPS201207293[23] ATIQUE F,ATTOH-OKINE N. Using copula method for pipe data analysis[J]. Construction and Building Materials,2016,106: 140-148. doi: 10.1016/j.conbuildmat.2015.12.027 [24] SAXENA A,GOEBEL K,SIMON D,et al. Damage propagation modeling for aircraft engine Run-to-failure simulation[C]//2008 International Conference on Prognostics and Health Management. Piscataway,US: IEEE,2008: 1-9. [25] WANG Tianyi,YU Jianbo,SIEGEL D,et al. A similarity-based prognostics approach for Remaining Useful Life estimation of engineered systems[C]//2008 International Conference on Prognostics and Health Management. Piscataway,US: IEEE,2008: 1-6. -