文章快速检索 高级检索

Dynamic fault tree analysis using sequential binary decision diagrams
LI Peichang , YUAN Hongjie , LAN Jie , CHENG Ming
School of Reliability and Systems Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2016-01-11; Accepted: 2016-02-29; Published online: 2016-04-01 17:07
Foundation item: National Defense Basic Scientific Research Program of China (61325102)
Corresponding author. E-mail:yuanhongjie@buaa.edu.cn
Abstract: In order to solve the problem of the existing dynamic fault tree analysis method, such as state space explosion, low computational efficiency and limited application range, a method for dynamic fault tree analysis based on sequential binary decision diagram is proposed. First, dynamic logic gates are transformed into logic gates with sequential events. Next, sequential binary decision diagram model and Boolean operation with sequential events are presented. Then, failure paths of dynamic fault tree are obtained by sequential binary decision diagram and extensional Boolean operation. Finally, probability calculations for sequential events with multi-unit are deduced. With a certain ammunition as an example, considering the imperfect coverage problem, the dynamic fault tree is analyzed under the situations of exponential and non-exponential distribution. The results show that this method has the advantages of high efficiency, high accuracy and wide applicability, which provides a theoretical basis for the reliability analysis of complex dynamic systems.
Key words: dynamic fault tree     sequential binary decision diagrams     Boolean operation     reliability analysis     imperfect coverage

﻿故障树分析作为一种重要的系统可靠性分析方法，已成功应用于核工业、航空航天、电子通信等众多领域，为提高产品的安全性与可靠性发挥了巨大作用[1-2]。然而，传统的故障树分析方法把系统失效看作部件失效的组合，而忽视了部件间的优先关系、顺序关系、功能相关性等动态失效特性，因此，在分析具有动态失效特征的系统可靠性时存在较大误差。为解决系统的动态行为建模问题，Dugan 等[3-4]提出了动态故障树分析方法，定义一组动态逻辑门来描述部件的动态特征，弥补了传统故障树分析方法的不足。早期的动态故障树分析方法采用马尔可夫模型[5-6]，该模型只适用于部件寿命服从指数分布的情形，同时面临着状态空间爆炸的风险。1997年，Gulati和Dugan[7]提出模块化方法进行动态故障树分析，该方法只是在一定程度上缓解了马尔可夫模型面临的状态爆炸问题而未能从根本上解决。马尔可夫模型和模块化方法存在的缺陷限制了动态故障树定量分析的发展，众多研究者开始探索动态故障树分析的新方法。2003年,Amari等[8]提出一种基于复合梯形积分的数值方法，该方法求解动态故障树时不需要转化为马尔可夫模型，然而难以处理含重复单元的故障树。2005年，Boudali和Dugan[9]将动态故障树转化为贝叶斯网络进行分析，避免了状态爆炸问题的发生，但在处理部件寿命为非指数分布时，计算精度较低。2009年，Rao等[10]提出一种基于蒙特卡罗仿真方法，该方法适用于部件寿命服从任意分布类型，具备较高的计算精度，但需要大量的计算时间。

1 动态故障树基础 1.1 动态逻辑门

1) 优先与门

 图 1 两单元优先与门 Fig. 1 Priority AND gate with two units

2) 备份门

 图 2 3种备份门图形符号 Fig. 2 Graphic symbols for three types of spare gates

1.2 二元决策图

 (1)

 (2)

 (3)

1.3 顺序二元决策图

2 基于顺序二元决策图的分析方法 2.1 动态故障树的转化

1) 优先与门的转化

 图 3 输入为基本事件的优先与门转化 Fig. 3 Conversion of priority AND gates with basic events input

 图 4 输入含逻辑门的优先与门转化 Fig. 4 Conversion of priority AND gates with logic gate input

2) 温备份门的转化

 图 5 考虑不完全覆盖的温备份门转化 Fig. 5 Conversion of warm spare gates considering imperfect coverage
2.2 顺序二元决策图模型的建立

 图 6 与门的顺序二元决策图 Fig. 6 Sequential binary decision diagrams of AND gates

 图 7 或门的顺序二元决策图 Fig. 7 Sequential binary decision diagrams of OR gates

2.3 顺序二元决策图的分析评估

2.3.1 失效路径化简规则

 (4)

1)CiCj (∩表示逻辑关系“与”)

① 当CiCj=∅时，CiCj的发生概率：

 (5)

② 当CiCj≠∅时，令CiCj=DijDi/j=Ci/Dij，其意义是Ci中元素除去Dij中元素剩下的集合。此时Ci∩${{{\hat{C}}}_{j}}$的发生概率：

 (6)

2)CiCj

① 当CiCj=∅时，CiCj的发生概率：

 (7)

② 当CiCj≠∅时，CiCj的发生概率：

 (8)

3) ${{{\hat{C}}}_{i}}\cap {{{\hat{C}}}_{j}}$

① 当CiCj=∅时，CiCj的发生概率：

 (9)

② 当Ci∩Cj≠∅时，令Ci∩Cj=DijDij={f1,f2,…,fk}，其中Dij的元素顺序与其在Ci内的顺序一致，Dij=〈f1,f2,…，fk〉。

Dij≠DjiCi∩Cj的发生概率：

 (10)

P0={Ci中从ei1f1的前一个元素}；

Q0={Cj中从ej1f1的前一个元素}；

Pl={Ci中从fl的后一个元素到fl+1的前一个元素} (l=1,2,…,k-1)；

Ql={Cj中从fl的后一个元素到fl+1的前一个元素} (l=1,2,…,k-1)；

Pk={Ci中从fk的后一个元素到ein}；

Qk={Cj中从fk的后一个元素到ejm}。

Pi∩Qi=Ei=Ei1∪Ei2∪…∪Eixi(i=1,2,…,k)，其中Ei表示所有符合PiQi元素顺序的排列，则Ci∩Cj=〈E0,f1,E1,…，Ek-1,fk,Ek 〉=Gij1Gij2∪…∪Gijx,其中Gijm(m=1,2,…,x)为Ci∩Cj的不交化序列。

 (11)

2.3.2 顺序事件发生概率计算

1) 对于两单元的顺序事件A →B (或B →A)

 (12)

 (13)

 (14)

 (15)

 (16)
 (17)

2) 对于n单元的顺序事件A1A2→…→An

A1A2,…,An中不包括温备份件，其失效密度函数分别为fi(t)(i=1,2,…,n),则在任务时间t内，顺序事件A1A2→…→An发生的概率为

 (18)

A1A2,…,An中包括温备份件AiAi工作状态下的失效密度函数为fi(t)，贮备状态下的失效密度函数为fi,α(t)、单元Ai所处备份结构的主件是Aj

i＜j时，顺序事件发生的概率：

 (19)

i＞j时，顺序事件发生的概率：

 (20)
3 实例分析

 图 8 系统的动态故障树 Fig. 8 Dynamic fault tree of the system

1) 动态故障树的转化

 图 9 转化后的故障树 Fig. 9 Fault tree after conversion

2) 顺序二元决策图模型的建立

 图 10 系统的顺序二元决策图 Fig. 10 Sequential binary decision diagrams of the system

3) 顺序二元决策图的分析评估

① A →B

A →B·(A→C)

A →B·A→C·D

A →B·A→C·D·B·(E→F)

A →B·A→C·D·B·E→F·(F→E)

A →B·A→C·D·B·E→F·F→E·

(G→E)

 (21)
3.1 指数分布情形

 图 11 部件指数分布情形系统可靠度随时间变化曲线 Fig. 11 System reliability curves changes over time with elements obeying exponential distribution

 t/h R(t) 本文方法 马尔可夫模型方法 5 000 0.951 0 0.951 0 10 000 0.903 7 0.903 7 30 000 0.729 6 0.729 6 50 000 0.577 8 0.577 8 100 000 0.321 6 0.321 6

3.2 非指数分布情形

 图 12 部件非指数分布情形系统可靠度随时间变化曲线 Fig. 12 System reliability curves changes over time with elements obeying non-exponential distribution

 t/h R(t) 相对误差/% 本文方法 蒙特卡罗仿真方法 5 000 0.963 4 0.963 4 0 10 000 0.927 2 0.927 2 0 30 000 0.784 3 0.784 4 0.01 50 000 0.647 2 0.647 4 0.03 100 000 0.357 2 0.357 5 0.08

 方法 本文方法 蒙特卡罗仿真方法 运算时间/s 24.894 0 48.093 7

4 结 论

1) 方法考虑备份结构的不完全覆盖问题，对含优先与门和温备份门的复杂动态故障树进行了建模分析。

2) 方法对比了完全覆盖和非完全覆盖情形的可靠度计算，说明了失效覆盖问题对于系统可靠性的影响。

3) 方法通过指数分布和非指数分布2种情形的实例分析，并与马尔可夫模型和蒙特卡罗仿真方法进行了对比，显示了其具备计算精度高、计算效率高、适用性广泛的优点。

 [1] LENG L, LIU Y. Fault tree reliability analysis for passive medium pressure safety injection system in nuclear power plant[J]. Energy and Power Engineering, 2013, 5 (4) : 264 –268. DOI:10.4236/epe.2013.54B051 [2] NYSTROM B,AUSTRIN L,ANKARBACK N,et al.Fault tree analysis of an aircraft electric power supply system to electrical actuators[C]//Probabilistic Methods Applied to Power Systems,2006.Piscataway,NJ:IEEE Press,2006:1-7. [3] DUGAN J B, BAVUSO S J, BOYD M A. Dynamic fault-tree for fault-tolerant computer systems[J]. IEEE Transactions on Reliability, 1992, 41 (3) : 363 –376. DOI:10.1109/24.159800 [4] DUGAN J B, SULLIVAN K J, COPPIT D. Developing a low-cost high-quality software tool for dynamic fault-tree analysis[J]. IEEE Transactions on Reliability, 2000, 49 (1) : 49 –59. DOI:10.1109/24.855536 [5] SMOTHERMAN M K, ZEMIUDEH K. A non-homogenedous Makrov model for phased-mission reliability analysis[J]. IEEE Transaciton on Reliability, 1989, 38 (5) : 585 –590. DOI:10.1109/24.46486 [6] DUGAN J B.Galileo:A tool for dynamic fault tree analysis[C]//Proceedings of the 11th International Conference on Computer Performance Evaluation:Modelling Techniques and Tools.Berlin:Springer-Verlag,2000:328-331. [7] GULATI R,DUGAN J B.A modular approach for analyzing static and dynamic fault trees[C]//Reliability and Maintainability Symposium.Piscataway,NJ:IEEE Press,1997:57-63. [8] AMARI S,DILL G,HOWALD E.A new approach to solve dynamic fault trees[C]//Annual Reliability and Maintainability Symposium.Piscataway,NJ:IEEE Press,2003:374-379. [9] BOUDALI H,DUGAN J B.A new Bayesian network approach to solve dynamic fault trees[C]//Reliability and Maintainability Symposium.Piscataway,NJ:IEEE Press,2005:451-456. [10] RAO K D, GOPIKA V, RAO V V S S, et al. Dynamic fault tree analysis using Monte Carlo simulation in probabilistic safety assessment[J]. Reliability Engineering and System Safety, 2009, 94 (4) : 872 –883. DOI:10.1016/j.ress.2008.09.007 [11] XING L D, SHRESTHA A, DAI Y. Exact combinatorial reliability analysis of dynamic systems with sequence-dependent failures[J]. Reliability Engineering and System Safety, 2011, 96 (10) : 1375 –1385. DOI:10.1016/j.ress.2011.05.007 [12] XING L D, TANNOUS O, BECHTA DUGAN J. Reliability analysis of nonrepairable cold-standby systems using sequential binary decision diagrams[J]. IEEE Transactions on Systems Man and Cybernetics, 2012, 42 (3) : 715 –726. DOI:10.1109/TSMCA.2011.2170415 [13] TANNOUS O, XING L,DUGAN J B.Reliability analysis of warm standby systems using sequential BDD[C]//Reliability and Maintainability Symposium.Piscataway,NJ:IEEE Press,2011:1-7. [14] DUGAN J B. Fault trees and imperfect coverage[J]. IEEE Transactions on Reliability, 1989, 38 (2) : 177 –185. DOI:10.1109/24.31102 [15] BRYANT R E. Graph-based algorithms for boolean function manipulation[J]. IEEE Transactions on Computers, 1986, 35 (8) : 677 –691. [16] YUGE T, YANAGI S. Quantitative analysis of a fault tree with priority AND gates[J]. Reliability Engineering & System Safety, 2008, 93 (11) : 1577 –1583.

#### 文章信息

LI Peichang, YUAN Hongjie, LAN Jie, CHENG Ming

Dynamic fault tree analysis using sequential binary decision diagrams

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(1): 167-175
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0036