﻿ 一种考虑共因失效的PMS可靠性建模分析方法<sup>*</sup>
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A reliability modeling and analysis method for PMS considering common cause failure
WU Huan, JIAO Jian, ZHAO Tingdi
School of Reliability and Systems Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2017-06-07; Accepted: 2017-07-13; Published online: 2017-09-22 15:07
Corresponding author. JIAO Jian, E-mail: jiaojian@buaa.edu.cn
Abstract: Common cause failures (CCFs) in a system destroy the hypothesis that the failures are independent, which may significantly impact the reliability evaluation of the system, especially the phased-mission system (PMS). Aimed at the impact of probabilistic common cause failure (PCCF) on reliability of mission in PMS, this paper discussed the relationship between common cause events and extended the probabilistic model of common cause events using Bayesian theory to make the model fit for different statistical relations including mutually exclusive, s-independent and s-dependent. Moreover, a module-based modeling and analysis method using binary decision diagram (BDD) and Markov model was proposed. First, the fault tree of each phase was constructed. Then, considering CCF, BDD and Markov model were used to deal with the static and dynamic module in PMS respectively. Third, mission reliability was evaluated using total probability law. Finally, a case study of satellite for its orbit transfer was supplied to verify the effectiveness of the method. In addition, the result of this paper was compared with the existing case to analyze the influence of CCFs on mission reliability.
Key words: phased-mission system (PMS)     probabilistic common cause failure (PCCF)     binary decision diagram (BDD)     Markov model     dynamic

PMS可靠性评估中常常假设各个单元的失效是相互独立的，这种假设能够为具体求解带来很大方便，但是不能完全真实地反映实际情况。PMS中相似单元在同一阶段以及同一单元在多个阶段具有一定的相关性，若简单地在系统各单元失效相互独立的假设下进行系统可靠性分析与计算，常常会导致过大误差。PMS中往往存在由于某种共同的原因，简称共因(Common Cause，CC)，造成多个组件失效，从而导致系统失效，即共因失效(Common Cause Failure，CCF)。

1 基本假设条件

1) 组件是不可修复的。

2) 系统存在随机共因失效，且不同共因之间存在相关性。

3) 系统不同组件之间的失效概率都是相互独立的。

4) 组件失效服从指数分布。

5) Cij表示第i个阶段出现的第j个共因，其出现在阶段i开始时刻且在i阶段结束时结束，即该共因对组件影响时间为阶段持续时间。

2 共因事件模型

 (1)

P(En)En的发生概率，有，且EiEj=∅(ij)。

1) 当共因之间为互斥关系时，有

 (2)

2) 当共因之间相互独立时，有

 (3)

3) 当共因之间统计相关时，若

 (4)
3 考虑共因的PMS可靠性评估方法

3.1 基础方法理论

1) BDD模型

2) Markov模型

3.2 引入共因失效的基本思路

3.3 PMS可靠性评估步骤

1) 首先，将每个阶段用故障树表示，再根据多阶段任务系统的特点得到系统级的故障树；其次，根据Rauzy方法[18]将系统级故障树进行整合得到系统级相互独立的模块，再将每个独立模块作为底事件组成系统级故障树；最后，将系统故障树模型转化并化简为系统级最简BDD模型。

2) 根据其逻辑结构特点将上述的独立模块进行静态模块和动态模块归类。使用与、或、表决等静态逻辑门的为静态模块；包含至少一个功能相关、冷备份等动态逻辑门的为动态模块。

 (5)

 (6)

 (7)
4 案例分析 4.1 分析对象说明

 分系统 单机 符号 简介 姿轨控 姿态控制计算机 A 含Aa和Ab共2台，冷备份 陀螺 B 含Ba、Bb和Bc共3台，3取2 数字太阳敏感器 C 含Ca、Cb、Cc和Cd共4台，其中仅Ca和Cb热备份 红外地球敏感器 D 含Da和Db共2台，热备份 星敏感器 E 含Ea、Eb和Ec共3台，3取2 推进 490 N发动机 F 1台F 10 N推力器 G 含Ga和Gb共2套，热备份

1) 系统中存在3个共因：C41C51C52。对应失效概率分别为PC41=0.7，PC51=0.6，PC52=0.4。C41分别与C51C52统计独立；C51C52统计相关。其中：

2) 组件的失效率描述。组件的内部失效率分别为：λA=2.44, λB=1.22, λC=6.10, λD= 2.44, λF=1.72, λG=1.22(单位均为10-8 min-1)；组件由于共因所导致的失效率分别为：λ41(A)=2, λ41(C)=3, λ41(E)=6, λ41(G)=7, λ51(B)=1, λ51(D)=2, λ51(F)=3, λ52(A)=5, λ52(C)=6, λ52(E)=2, λ52(G)=4(单位均为10-4 min-1)。

3) 阶段持续时间。每个阶段的持续时间分别为：T1=45, T2=698, T3=35, T4=120, T5=57(单位均为min)。

4.2 可靠性建模与评估

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 图 1 系统BDD模型 Fig. 1 BDD model of system

 模块 组件 E1/10-5 E2 E3 E4/10-5 E5 E6 E7 E8 M24 Ba4 1.086 1.086×10-5 1.086×10-5 1.086 1.086×10-5 1.086×10-5 1.086×10-5 1.086×10-5 Bb4 1.068 1.068×10-5 1.068×10-5 1.068 1.068×10-5 1.068×10-5 1.068×10-5 1.068×10-5 Bc4 1.068 1.068×10-5 1.068×10-5 1.068 1.068×10-5 1.068×10-5 1.068×10-5 1.068×10-5 M63 Ca3 4.746 4.746×10-5 4.746×10-5 4.746 4.746×10-5 4.746×10-5 4.746×10-5 4.746×10-5 Cb3 4.746 4.746×10-5 4.746×10-5 4.746 4.746×10-5 4.746×10-5 4.746×10-5 4.746×10-5 M34 Cc4 5.478 0.035 4 5.478×10-5 5.478 0.035 4 0.035 4 5.478×10-5 0.035 4 M44 Cd4 5.478 0.035 4 5.478×10-5 5.478 0.035 4 0.035 4 5.478×10-5 0.035 4 M54 Ga4 1.068 0.080 6 1.068×10-5 1.068 0.080 6 0.080 6 1.068×10-5 0.080 6 Gb4 1.068 0.080 6 1.068×10-5 1.068 0.080 6 0.080 6 1.068×10-5 0.080 6 M73 Da3 1.891 1.891×10-5 1.891×10-5 1.891 1.891×10-5 1.891×10-5 1.891×10-5 1.891×10-5 Db3 1.891 1.891×10-5 1.891×10-5 1.891 1.891×10-5 1.891×10-5 1.891×10-5 1.891×10-5 M85 F5 1.643 1.643×10-5 0.0170 1.643 0.017 0 1.643×10-5 0.017 0 0.017 0

 (9)
 (10)
 (11)
 图 2 M1对应的的Markov模型 Fig. 2 Markov model of M1

 模块 E1 E2 E3 E4 E5 E6 E7 E8 M15 1.000×10-8 0.009 87 1.000×10-8 0.009 64 0.009 87 0.004 22 0.009 64 0.004 22 M24 3.422×10-10 3.422×10-10 3.422×10-10 3.422×10-10 3.422×10-10 3.422×10-10 3.422×10-10 3.422×10-10 M63 2.252×10-9 2.252×10-9 2.252×10-9 2.252×10-9 2.252×10-9 2.252×10-9 2.252×10-9 2.252×10-9 M34 5.478×10-5 0.0354 5.478×10-5 5.478×10-5 0.035 4 0.035 4 5.478×10-5 0.035 4 M44 5.478×10-5 0.035 4 5.478×10-5 5.478×10-5 0.035 4 0.035 4 5.478×10-5 0.035 4 M54 1.141×10-10 0.006 5 1.141×10-10 1.141×10-10 0.006 5 0.006 5 1.141×10-10 0.006 5 M73 3.576×10-10 3.576×10-10 3.576×10-10 3.576×10-10 3.576×10-10 3.576×10-10 3.576×10-10 3.576×10-10 M85 1.643×10-5 1.643×10-5 0.017 0 1.64 3×10-5 0.017 0 1.643×10-5 0.017 0 0.017 0

5 结论

1) 利用贝叶斯理论扩展了PMS中随机共因失效的概率模型，使其具有更大的适用范围。

2) 提出了将随机共因失效引入PMS可靠性评估的基本思路，并给出了详细的可靠性评估步骤。

3) 给出了分别利用BDD和Markov模型以及全概率公式进行求解的概率模型。

4) 通过案例分析，一方面验证了方法的可行性和有效性，另一方面确认了如果不考虑共因失效的影响，可靠性评估结论将过于乐观。

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#### 文章信息

WU Huan, JIAO Jian, ZHAO Tingdi

A reliability modeling and analysis method for PMS considering common cause failure

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(5): 1088-1094
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0386