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Dynamic fault tree is often used to analyze system reliability. The Markov model is a commonly used method, which can accurately reflect the relationship between the state transition process and the dynamic logic gate transfer in the dynamic fault tree. When the complexity or scale of system is increasing, the Markov model encountered a problem of state space explosion leading to increase troubles. To solve the above problems, a modular approach is needed. Based on the modular approach, a hybrid fault module was researched in this paper. Firstly, the stackable fault subtree containing complex static/dynamic logic gate is transformed into four common combinational logic gates through preprocessing of the dynamic gate in the module. Then, the complexity of the model was reduced by incorporating four common combinational logic gates and using the binary decision graph to solve variable ordering in the calculation of failure probability of static subtree. Moreover, the calculating process of complex mixed logic gate fault tree can be simplified. An example of the ammonium nitrate/fuel explosive production system for BCZH-15 explosive vehicle was used to verify the feasibility of the presented method.

Fault tree analysis (FTA) is a common method for the reliability modeling and evaluating large safety-critical systems as discussed in [

However, after the modularization, the applicability of the model should be considered for relatively complex systems. Zhang et al. [

In recent years, the focus on field mixed explosive vehicle research has been performed to improve or design new electrical control systems to enhance the safety of equipment; to study the advantages of equipment in special geographic locations; to improve the blasting efficiency by changing the composition of the mixed explosive. There are only a few literatures on the reliability of the equipment, and only the reliability of the equipment can meet the operating requirements and can the safety, superiority, and efficiency of the equipment be demonstrated. With the dynamic characteristics becoming more and more obvious, it is urgent to evaluate equipment reliability considering the dynamic characteristics.

The remainder of the paper is organized as follows: Section

Rauzy et al. [

Calculate the traverse node for the fault tree and list the results

Sort the root node, leaf nodes, and the intermediate nodes and perform depth-first leftmost traversal for them

Collect the information of the internal event V for the third traversal node and collect the first and the last period of each subtime to provide strong evidence for modularizing the event.

Module division for the fault tree is shown in Figure

The fault tree.

First traversal result.

S | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

V | T | M1 | M2 | M4 | A1 | A2 | M4 | M5 |

S | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

V | A3 | A4 | M5 | M2 | M3 | M5 | M6 | A5 |

S | 17 | 18 | 19 | 20 | 21 | 22 | ||

V | A6 | M6 | M3 | M1 | A7 | T |

Final traversal result.

Module name | T | M1 | M2 | M3 | M4 | M5 | M6 |
---|---|---|---|---|---|---|---|

Min | 2 | 3 | 4 | 8 | 5 | 9 | 16 |

Max | 21 | 19 | 14 | 18 | 6 | 10 | 17 |

Y/N | Y | Y | N | N | Y | Y | Y |

In Table

Table

From the operations given in Table

After modularizing the fault tree, it is necessary to further determine the module state (dynamic or static). The BDD method is used to calculate the subtree for the static submodule. The Markov model is applied to mold for the dynamic submodule. Since the system is divided into small modules, state explosion problem will not occur during the model processing. Due to logic gate nesting, the static submodule may contain dynamic submodule after the module division in a complex fault tree. In this case, the corresponding algorithm cannot be applied to the fault tree submodule. Preprocessing of dynamic logic gate is required for the basic fault tree, and the complex hybrid dynamic logic gate is transformed into an easy solution model, which can greatly simplify the subsequent calculation.

Pretreatment of the logic gate includes the following: copretreatment of AND gate by OR gate, pretreatment of AND gate or PAND gate, pretreatment of two kinds of PAND gate stacking, and pretreatment of FDEP gate (Figures

AND gate and OR gate preprocessing.

PAND gate and OR gate preprocessing.

Priority and gate preprocessing.

FDEP gate preprocessing.

The above is the basic preprocessing process of logic gates.

The BDD method is generally applied to solve the problem of the static fault tree. The BDD method plays a great role in promoting the analysis of the static subtree. The BDD method was proposed by the American scholar Akers in 1978, as discussed in [

In the BDD conversion process, the main operation is based on the if-then-else (ite) structure of the Shannon decomposition, shown in the following equations:

Simple coding and index ordering are performed using ite for a simple fault tree. The influence of different index sorting on BDD generation is analyzed.

The fault tree is shown in Figure

The fault tree transformation diagram.

The BDD conversion graph of the fault tree is drawn by the above relationship.

At present, the optimal ordering of BDD is still under study. The mainstream BDD optimization algorithms are precise sorting algorithm and dynamic heuristic algorithm. The optimal index ranking is shown in Figure

Reordering BDD.

Since the different ordering, the original six sides change to twelve sides. The calculation process is increased from two sets of equations to four groups, and variable ordering is especially critical when dealing with more complex models. It is shown that a good variable ordering is related to the complexity of the BDD.

After sorting the variables, the path pointing to 0 or 1 can be obtained, where the point 1 represents the top event occurring and the point 0 represents not occurring.

Determining all paths point to 1, the path is recorded as

Disjoint expressions of the fault tree can be represented as follows:

The probability of occurrence of the top event can be calculated using probability formula of the mutual exclusion event as below:

The probability of failure of the bottom event can be expressed as

The above content is verified by an example. A, B, C, D, and E are bottom events (Figure

Modular fault tree.

Bottom event failure probability.

Bottom event code | A | B | C | D | E |
---|---|---|---|---|---|

Failure probability | 0.2 | 0.1 | 0.2 | 0.3 | 0.15 |

After determining the probability of the leaf node, the fault tree can be modularly divided. The fault tree is divided into three parts: M1, M2, M3, where the submodules are M1 and M2. The submodule M1 is converted into BDD, as shown in Figure

Model M1 and model M2.

The probability of occurrence of module M2 is calculated based on the following BDD (Figure

After obtaining the probability of occurrence of the modules M1 and M2, probability of occurrence for M3 can be further calculated (Figure

Model M3.

Obtain the probability of M1, M2, and A, and the probability of M3 can be calculated as 0.657.

Continuous-time Markov decision processes provide a very powerful mathematical framework to solve widely used decision problems, as discussed by Bartocci [

The sequence is shown as follows:

In order to make the fault tree model better deal with the sequential logical relationship between the parts in the dynamic system, Dugan et al. [

The process of converting the four common dynamic logic gates into Markov chain will be described. The four common dynamic logic gates are PAND gate, SEQ, SPARE, and FDEP, and the relevant transition diagrams are shown in Figures

PAND logical gates transform to MC.

SEQ logical gates transform to MC.

CSP logical gates transform to MC.

FDEP logical gates transform to MC.

The quantitative analysis method of the Markov model generally consists of five steps: first, define the system state; second, the transition probability matrix construction; third, solve the spherical transition probability matrix; fourth, solve the differential equations; fifth, solve the fault probability state at any time.

The system state set, the fault state set, and the working state set are defined as follows:

When defining the random process, the corresponding time point

Defining

The derivative column of the state probability and the column vector of the probability derivative are, respectively, expressed as

In the process of the Markov chain transfer, the complexity of state transition usually increases with the increase of chain length. The transition probability from

The transfer probability matrix can be written as follows:

The state transition process of chain length 2 can be deduced by the formula of chain length 1. When the event from 0 to

The following equation represents the state transition of chain length is

The main work of the pharmaceutical system of explosive vehicle is to manufacture explosives. The working environment of the explosive vehicle is harsh, and the failure rate is high. Through existing data, the failure probability of bottom event was obtained for the explosive vehicle.

Figure

The model of dynamic fault tree.

The FDEP gate exists under the OR gate of module N1. The FDEP gate transformation process is as follows: the subordinate modules of N2 are all static gates. The module M1 under N2 can be combined due to a static module. Fault tree after preprocessing is shown in Figure

Dynamic fault tree module division.

The fault tree can be divided into four basic modules S1, S2, S3, and S4. In module S1, there is a submodule C consisting of static logic gates, which can be divided into static module. Modules S2 and S3 are static modules. Module S4 is divided into dynamic submodule because its subtree root node is the spare parts gate.

According to BDD, its nodes are firstly divided. Sorting the bottom events uses a heuristic sorting method and defined the order

The binary decision diagrams.

For the transformation of the static module S2, the above method should be used. First, the heuristic algorithm is applied to determine the subunit sequencing:

The BDD of module S2 is shown in Figure

The binary decision diagrams.

Similarly, module S3 was converted into BDD, where the sorting is defined as

The BDD of transformation is shown in Figure

The binary decision diagrams.

The module S4 contains the submodules C4 and C5, where C5 is the spare part of C4. BDD was established to solve the probability of C4 (Figure

The BDD and Markov chain.

The failure probability of parts in the bottom event is shown in Table

Bottom event failure probability.

Encoding | Event | Probability ( |
---|---|---|

Line interface failure | 0.5 | |

Fuel flow-meter failure | 0.4 | |

Sensitizer flow-meter failure | 0.2 | |

Solenoid valve failure | 0.3 | |

Connection line failure | 0.2 | |

System display failure | 0.1 | |

The leakage of fuel tank | 0.1 | |

The leakage of oil pipe joints | 0.5 | |

Oil pipe failure | 0.1 | |

Oil filter element failure | 0.8 | |

Oil-pressure meter failure | 0.2 | |

One-way valve failure | 0.2 | |

Stator failure | 0.45 | |

Bearing failure | 0.15 | |

Emulsion matrix pump failure | 0.3 | |

Flexible pipe failure | 0.7 | |

Flexible pipe joint failure | 0.2 | |

Butterfly valve failure | 0.4 | |

Sensitizer pump failure | 0.1 | |

Water tube failure | 0.5 | |

Sensitizer filter element failure | 0.2 | |

Sensitizer tank failure | 0.2 |

The probability of failure of each module at 1000 hours is obtained by the bottom event probability, as shown in Table

Failure probability of each module.

Encoding | Model | Probability |
---|---|---|

C | Fuel flow module | 0.1008 |

S1 | Flow display module | 0.1537 |

S2 | Tank module | 0.2261 |

S3 | Pump body module | 0.1513 |

S4 | Sensitizing agent system module | 0.3333 |

After obtaining the data failure probability of each module, the data are further integrated to obtain failure probability of the top event.

According to the integration, the reliability of the system after one thousand hours of running time can be obtianed. The integration diagram is shown in Figure

Integration diagram.

Finally, combining the failure probability of each module to evaluate the system reliability and failure rate can be obtained as 0.6292 and 0.3708, respectively.

The Markov model is usually used to solve the problem of dynamic fault tree. Complex systems are difficult to build using traditional Markov models. Due to the dynamic fault tree containing static subtrees, the BDD can be used to solve the problem of static subtree. The logic gate combination is simplified using the pretreatment method. For the simplified dynamic fault tree [

The information obtained from the current research results is still very limited. Because of its influence by many factors, the accuracy of the failure probability still needs further research. Therefore, the subsequent researches should go along diversification. Since the amount of information that can be obtained using only one data source is small, in the future, a multisource information fusion method will be used to more accurately evaluate the system reliability.

This article introduces how to use the modular approach to solve the probability density of basic events in the dynamic fault tree when the system has certain complexity and multiple subsystems. The module pretreatment method is adopted to simplify the submodule of the fault tree due to the excessive mixture and overlap of logic gates. This modular approach has two advantages: first the variables ordering for the module becomes easy to reduce the complexity of BDD and solve difficulty by pretreatment, and second the failure probability of the subsystem can be obtained for evaluating the reliability of the subsystem. The feasibility of the method was verified by analyzing dynamic fault tree of explosive production system for the BCZH-15 explosive vehicle. At present, due to the difficulty of individual data collection in the data collection process and the lack of a large amount of experimental data support, the failure rate of the collected products is not accurate enough. It may have an impact on the accuracy of the result evaluation. In the future, when the data are insufficient, fuzzy theory will be used to define the failure rate of the product, and the rationality of the final output result will be determined through the comparison of expert experience.

The data used to support the findings of this study are included within the article.

There are no conflicts of interest.

This work was partially supported by the National Natural Science Foundation of China under the contract no. 71761030 and the Graduate Teaching Program of Inner Mongolia University of Technology under the contract no. YJG2017013.