ZHOU Hang(周 行), YU Le(于 樂(lè)), PENG Wei(彭 偉), PENG Wei-wen(彭衛(wèi)文), HUANG Hong-zhong(黃洪鐘)

School of Mechanic, Electronic, and Industrial Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Recursive Algorithm Based Reliability Analysis of Multiphase Satellite Systems with Propagated Failures

ZHOU Hang(周 行), YU Le(于 樂(lè)), PENG Wei(彭 偉), PENG Wei-wen(彭衛(wèi)文), HUANG Hong-zhong(黃洪鐘)*

SchoolofMechanic,Electronic,andIndustrialEngineering,UniversityofElectronicScienceandTechnologyofChina,Chengdu611731,China

Modern satellite systems are generally designed to fulfill multiphase-missions. Component/subsystem redundancies are commonly used to achieve high reliability and long life of modern satellite systems. These characteristics have leaded to a critical issue of reliability analysis of satellites that is how to deal with the reliability analysis with multiphase-missions and propagated failures of redundant components. Traditional methods based on the binary decision diagram (BDD) can hardly cope with these issues efficiently. Accordingly, a recursive algorithm method was introduced to facilitate the reliability analysis of satellites. This method was specified for the analysis of static fault tree and it was implemented by generating combination of component failures and carrying out a backward recursive algorithm. The effectiveness of the proposed method was demonstrated through the reliability analysis of a multiphase satellite system with propagated failures. The major advantage of the proposed method is that it does not need composition of BDD and its computational process is automated.

satellitereliability;recursivealgorithm;multiphasesystem(MPS);propagatedfailures;reliabilityanalysis

Introduction

In modern industry, many systems, such as satellite systems, are designed to perform a series of consecutive and non-overlapping phased-missions[1-5]. During these phases, the systems are constructed by their components in different configurations to fulfill multiple missions. The components are then subject to various environmental conditions, external stresses, and internal failures. As a result, system configuration, system reliability, and components’ failure behavior are varied from phase to phase for these systems. Meanwhile, the systems are generally unrepairable and demand for high reliability, such as the satellite systems. Component redundancies are then adopted to achieve the requirements of high reliability of these systems. However, propagated failures may occur due to failures originated from components of common cause group (CCG) for these systems. Xing and Levitin[6]proposed a method for reliability evaluation of multiphase systems(MPSs) with internal/external common cause failures (CCFs) based on a binary decision diagram (BDD) method. However, this method is not quite effective when dealing with the reliability analysis of complex systems with propagated failures. A recursive algorithm based method for this situation is more suitable. In addition, the category of CCF includes internal and external failures. Many studies have shown that internal and/or external failures can increase the system’s probability of failure (POF) and the overall system reliability is then decreased significantly[7-9]. In this paper, the internal failures of satellite systems are considered and analyzed in detail.

The operation of a satellite system can be divided into several phases, such as launch phase, cruise phase, and orbit-flight flight phase. These phases are consecutive and non-overlapping. We assume that the components among these phases have only two possible states: functioning state (denoted by 1) and failed state (labeled by 0). There are various methods for reliability analysis of MPS. These methods can be divided into two classes,i.e., analytical methods and simulation methods. Most of them focus on analytical methods, such as the combinatorial method[10], the state-space oriented method[11], and the structure modular method[12]. However, major problems of these methods are that they often suffer from the state-space explosion problem and their computational burdens are generally very high. To deal with these problems, we introduce a recursive algorithm based method for reliability analysis of satellite systems by taking consideration of static fault tree analysis and propagated failures. This paper is based on the assumption that the probability density functions of components and the related parameters of these functions are known or can be derived.

The proposed method is based on the conditional POF, combinations of failures, and a backward recursive algorithm. We further assume that failures of components will cause propagated failure in CCG. This method has no restriction on the types of components’ failure distributions as well as the specific configurations of system structures, and it can be easily realized through programming.

1 Reliability Indicator of Binary Systems with Binary State Components

In this paper, we only consider binary state systems composed by binary state components. A binary state system has only two states (functioning or failed). The system will fail if the system’s components fail during a mission phase. If the components fail after a mission phase and these components will not be used in the rest of mission phases, the system will not experience a failure caused by the failures of these components.

For a binary state system, we usento represent the number of components in this system. LetMdenote the number of mission-phases that the system is going to experience. Since every component has two states, the state of componentjin phasem, denoted byxj(m), 1≤j≤n, 1≤m≤M, can be “0” or “1”. In phasem, we can get the state vector of components denoted asXm=(x1(m),x2(m), …,xn(m)). For a non-repairable MPS,xj(m) is a non-increasing function ofm. At the end of phasem, the acceptability functionφm(Xm) can be obtained. Thus,φm(Xm)=1 represents that the system does not fail during phasem. Andφm(Xm)=0 represents that the system experiences a failure during phasem[13].

Whentheconditionalprobabilityisused,theconditionalreliabilitypj(m)andPOFqj(m)ofbinarystatecomponentjinphasemareexpressedasfollows.

pj(m)=1-qj(m).

2 Assumptions

Before introducing the proposed method, the assumptions used in this paper are given as follows.

(1) The system consists ofMconsecutive and non-overlapping phases.

(2) The system’s components ares-independent.

(3) System and components are non-repairable.

(4) Some components are used in all phases, whereas some components are used in some specific phases.

(5) System reliability depends on components’ performance in all phases.

(6) If there is no conditional propagated failure, letεj(m) denote conditional propagated failure, thenεj(m)=0.

3 Backward Recursive Algorithm Method

In CCGs,since the components are generally identical, they may have the same failure rates and phase-dependent characteristics. The phase-dependent characteristic mainly refers to the concept of equivalent age and POF with cumulative exposure model[14]. For a component, letFj(m,t) represent the failure distribution of componentjin phasem. Following the work presented by Amari and Bergman[14], the acceleration factorαj(m) is used to derive the failure distribution among different mission-phases. The failure distribution is then obtained asFj(m,t)=Fj(αj(m)t). The cumulative POF of componentjin phasemis expressed as follows:

Φj(m)=Fj(αj(1)t+αj(2)t+…+αj(m)t).

(1)

From Eq. (1), the POF of componentjin phasemcan be calculated asfj(m)=Φj(m)-Φj(m-1). Therefore the equation of conditional reliabilitypj(m) and POFqj(m) of binary state componentjin phasemcan be updated as follows.

(2)

(3)

Once a component’s failure occurs randomly and it belongs to a CCG, the propagated failure will happen. If we know the propagated POF of componentjdenoted asεj(m), independent POFvj(m) and POF of CCGwj(m) can be calculated as follows.

vj(m)=qj(m)(1-εj(m)),

(4)

wj(m)=qj(m)εj(m).

(5)

In fact, when we consider CCF together with propagated failures, there is a special phenomenon that is if some components belonging to CCG remain functioning at the end of phasem, no propagated failure will occur in this CCG in any phasekwithk

(6)

The proposed algorithm has two processes,i.e., generating combination of component failures and backward recursive algorithm.

3.1 Process 1:generating combination of component failures

LetXmrepresent the state vector in phasem, letYmbe the implementation of random binary vector includingszeros (sout ofn), andYmis defined asYm=(y1(m),y2(m), …,yn(m)). The zero components have position numberc(i)(1≤i≤s). Thus, we can obtain the set of position numbers of zero components. For example, givenYm=(1, 0, 1, 0, 1), the number of zero is 2, and the positions of these zero numbers are 2 and 4. Then, we can haves=2,c(1)=2,c(2)=4, andδ(Ym)={c(1),c(2)}={2, 4}. Sincexj(m) is a non-increasing function of phasem, and anyyj(m)=1 implies thatxj(m)=1, the expressionXm-1≠Ymimplies that the stateYmcan only be obtained by replacing “0” with “1”. It means that some component that remains functioning during phasem-1 experiences a failure at phasem. To fulfill all the possible implementations of state ofXm-1, we should enumerate all the possible combinations. The total number of combinations is 2s. In this enumeration process, we can run the integer parameter “σ” from “0” to “2s-1”. Our assumptions are as follows: if [σ/2i-1] mod 2 =1 and the component of positionc(i) fails during the phasem, then the number “0” of the corresponding position can be replaced by “1”, otherwise the component of this position keeps working and the number remains its original value. Through this way, we can get the operatorπ(Y,σ) and then determineQm(σ) fromπ(Y,σ) toYat the end of the phasem. The equation ofQm(σ) is expressed as follows.

(7)

3.2 Process 2: backward recursive algorithm

LetZm, Yrepresent the probability of the event whenXm=Y, andφl(shuí)(Xl)=1 for allh>1. Then the equation ofZm, Yis given as follows.

Zm, Y=Pr{Xm=Ym, φm-1(Xm-1)=
1, φm-2(Xm-2)=1, …, φ1(X1)=1}.

(8)

Since the conditional distribution ofXmonly depends on the most recent value ofXm-1, it follows the Markov property. Equation (8) can be generalized as follows.

(9)

(10)

The overall system reliability for theMphases is then obtained as follows.

(11)

4 An Example

In this section, an example is presented by incorporating CCGs to illustrate the proposed method and the effect of propagated failures. The satellite system consists of 3 binary elements. A parallel connection of elements means that they perform the same task, then the cumulative performance of this parallel subsystem is equal to the sum of performances of these elements. Similarly, the performance of a serial connection subsystem equals the minimum of the performances of its elements.

Figure 1 is a fault tree (FT) model for a satellite system. This FT model shows that the satellite system has three main mission phases. These phases are consecutive and non-overlapping. At phase 1 the system structure takes the form presented in Fig.2. At phases 2 and 3 the system structures take the forms presented in Fig. 3.

Fig.1 FT of a satellite system with multiple phases

Fig.2 Subsystem structure of phase 1

(a) (b)

During these phases, there are three components,i.e., component A (labeled asx1), component B (labeled asx2), and component C (labeled asx3). From Fig.1, we find that if component A finishes its mission at launch phase (labeled as 1), there is no other mission in cruise phase (labeled as 2) and orbit-flight phase (labeled as 3). But components B and C have to finish different missions during different phases. We assume that these components share some identical elements which may lead to CCF or propagated failures.

We use symbolφm(Xm) to represent acceptability in every phase. Therefore, the reliability evaluation is implemented based the acceptability of three phases. According to Fig.1, we can obtain the expression ofφm(Xm) as follows,φ1(X)=x1(x2+x3),φ2(X)=x2x3, andφ3(X)=x2. For the acceptability of the satellite system, the foregoing equations should be equal to number “1” with binary value (0 or 1). Then, the vector of work state of phase can be obtained.

Phase 1: (111), (110), (101);

Phase 2: (011), (111);

Phase 3: (111), (110), (011), (010).

According to Eq. (11),Requals the sum of 23=8 terms. Removing the termsφ3(X)=0, we can obtain expression asR=Z3,(111)+Z3,(110)+Z3,(011)+Z3,(010).

Then, we can calculate four kinds ofZm, Yseparately.

For the calculation ofZ3,(111)withs=0(111) =0, according to Eq. (9), the process is given as follows.

According to Eq. (10),Z1,(111)=p1(1)p2(1)p3(1). Therefore,Z2,(111)=p1(2)p2(2)p3(2)p2(1)p3(1) and

Z3, (111)= p1(3)p2(3)p3(3)p1(2)p2(2)p3(2)p1(1)
p2(1)p3(1).

In the same way, we can get otherZm, Yby the backward method presented in section 3.

Z3, (110)=p1(1)p2(1)p3(1)p1(2)p2(2)p3(2)p1(3) p2(3)p3(3)(1-ε3(3)),

Z3, (011)=p2(3)p3(3)p2(2)p3(2)p2(1)p3(1) p1(1)(q1(2)(1-ε1(2))+p1(2)(q1(3))(1-ε1(3))),

Z3, (010)=p1(1)p2(1)p3(1)p2(2)p3(2) p2(3)[q1(2)(1-ε1(2))q3(3)+p1(2)(1+q1(3)(1-ε1(3))q3(3)(1-ε3(3))-(1-q1(3)ε1(3))(1-q3(3)ε3(3)))].

Becausepi(m)+qi(m)=1, the reliability evaluation expression is simplified as follows.

R=p1(1)p2(1)p3(1)p2(2)p3(2)p2(3)[1-q1(2)ε1(2)].

The baseline failure time distribution of each element follows Weibull distribution with the cumulative distribution function F(t; η, β)=1-exp(-(t/η)β). Table 1 presents the values of parameterβfor different types of elements. This table also contains the phase durations, phase-dependent parametersαj(m)/ηjand the conditional failure probabilities in each phaseqj(m).

Table 1 Parameters of system elements

ElementsβPhaseno.123Duration206080133103·αj(m)/ηj0.03.02.0qj(m)0.00000.00580.03292,31.5103·αj(m)/ηj0.51.50.8qj(m)0.00100.03020.034221103·αj(m)/ηj0.00.30.5qj(m)0.00040.01780.0392

Figure 4 presents the mission reliability of the satellite with conditional failure propagation probabilitiesεj(m). It can be seen that the mission reliability is affected greatly byε2(m).

Fig.4 Multiphase-mission reliability with conditional failure propagation probabilities for CCG1={1, 3} and CCG2={2, 3}

5 Conclusions

A recursive algorithm based approach for reliability analysis of a multiphase satellite system is proposed in the paper. The problem of computational burden suffered by the BBD based method is solved with the proposed method. The effect of propagated failures is also considered in the proposed method. In detail, the recursive algorithm and reliability evaluation equation are developed for the multi-phase system with non-repairable components considering propagated failures in CCGs. Compared with traditional analytical and simulation methods, this method does not need conversion of FT based on BDD. The efficiency of the proposed method can then be guaranteed through this recursive strategy. An illustrative example is presented to demonstrate the implementation of the proposed method. The results of this illustrative example also show that the propagated failures of non-repairable multi-phase system should be considered when carrying out a reliability analysis. Our future work will focus on the developing of a backward recursive method for dynamic fault tree models.

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TB114.3 Document code: A

1672-5220(2015)01-0136-04

Received date: 2014- 08- 08

*Correspondence should be addressed to HUANG Hong-zhong, E-mail: hzhuang@uestc.edu.cn