# Reliability Analysis of the High-Voltage Power Battery System Based on the Polymorphic Fuzzy Fault Tree

Authors: Haonan Jiang; Changbo Lin; Gaoshan Feng; Enyong Xu; Weiguang Zheng
Journal: Journal of Sensors
 (2022)

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## Abstract

Accidents caused by the failure of high-voltage power battery systems are rising with the increase of pure electric commercial vehicles. The fault tree analysis method based on traditional reliability is no longer suitable for quantitative evaluation of polymorphic systems. In this paper, the polymorphic fuzzy fault tree of the high-voltage power battery system for pure electric commercial vehicles is established and analyzed qualitatively and quantitatively based on a combined theory of the polymorphic theory, fuzzy mathematical theory, group decision theory, and fault tree analysis theory. The results showed that the multistate reliability-analysis method of the fuzzy fault tree could describe the various fault states of the high-voltage power battery system. Through quantitative evaluation of the reliability of system, the low-temperature environment and CAN high and low reverse connection were the weakest links of the system, and the problem of the occurrence probability of each state of the unknown polymorphic bottom event in the sub-fault tree of the deteriorated-state mode was solved quickly using group decision-making to deal with fuzzy probability. It provides theoretical reference for system design and detection process, which has important practical significance for the improvement of high-voltage power battery system.


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## Body

## 1. Introduction

The problems of energy and environmental protection are becoming more and more serious with the increased number of automobiles. At present, BYD, Dongfeng LiuQi, BAIC, and other major vehicle enterprises have focused on pure electric commercial vehicles with energy conservation, environmental protection, and large cargo capacity. With promoted pure electric commercial vehicles, the accidents on the reliability of the high-voltage power battery system also increase year by year. According to the statistics of the GGII (Gao Gong Industrial Research Institute), there were 34 fire accidents caused by battery problems in domestic electric vehicles from January to May 2021, involving up to 38 vehicles. Each accident is a severe challenge to the reliability of the high-voltage power battery system. The system produces hundreds of amperes of charge and discharge currents in the working process, which seriously threatens the safety of vehicles and drivers.Reliability analysis methods have been widely used in engineering fields in continuous development. Huang et al. [1] proposed the fault tree analysis (FTA) and applied it to the design of the militia missile launch control system to predict the failure probability of missile launch in the 1960s. Subsequently, Dhillon et al. [2] of Boeing Company developed a program of the fault tree analysis method, which reduces the workload of data processing in the reliability research at the early stage of aircraft design and promotes fault-tree analysis algorithms. Rahman et al. [3] of MIT applied the fault tree analysis method in the risk assessment of nuclear power plants and listed all the causes of nuclear power plant failures in 1974. Yang et al. [4] proposed a diagnosis method that combined the fault tree analysis with an expert system to develop a fault diagnosis expert system to meet the fault diagnosis function of the DC charging pile. Lanza et al. [5] analyzed the reliability of feedback fault information to establish the machine-tool fault diagnosis and prediction system. Besides, the reliability guarantee system is built in the designing, manufacturing, and assembly process of NC machine tools. The reliability analysis method is also applied to the field of fault diagnosis, and the fault diagnosis method proposed by Xin et al. [6–8] can effectively solved the problem of mechanical reliability. The fuzzy fault tree analysis method is proposed to solve fuzzy probability in practical engineering.The fuzzy fault tree analysis was first proposed by Tanaka et al. [9]. Fuzzy probability is used to replace accurate probability to solve the uncertain probability in the fault tree analysis. Singer [10] used L-R-type fuzzy numbers to represent the occurrence probability of bottom events and further developed the fuzzy fault tree analysis method. Yuhua and Datao [11] combined expert heuristics with the fuzzy set theory to solve the fuzziness of failure probability of some bottom events in oil and gas transmission pipelines. Liu et al. [12] proposed a reliability analysis method based on the Bayesian network and T-S (Takagi-Sugeno) fault tree. The nodes in the Bayesian network are represented by the fuzzy probability, and the fault degree of the system is represented by fuzzy variables. This technology is applied to the reliability analysis of the injection system of the chemotherapy robot. Yin et al. [13] proposed a safety evaluation method for natural gas storage tanks based on the fuzzy fault-tree analysis method of similarity aggregation.The theory of the multistate system (MSS) provides a new way for the reliability analysis of complex systems. Satyanarayana and Prabhakar [14] studied the reliability of networked systems to propose the idea of solving the reliability of a system by simply computing the sum of the probabilities of acyclic subgraphs in the 1970s. Iscioglu [15] proposed a new method to evaluate the MRL (mean residual lifetime) function of a one-unit, three-state system using the conditional residual function. Borges et al. [16] proposed the Monte Carlo parallel reliability analysis method, which has high efficiency in testing the actual power system model and can evaluate the reliability of a large-scale power grid. Mahadevan et al. [17] used the Bayesian network to compare with the traditional reliability analysis methods of series and parallel systems, with its effectiveness verified. Considering the impact of component degradation and different maintenance strategies, Mohammadhasani [18] used the Markov maintenance model to quantify the effectiveness of maintenance activities. These models are coupled with the fault tree method, providing a more practical and accurate fault tree analysis tool. Research on the smoothness of commercial vehicles has expanded the application of reliability in engineering [19, 20].Among the numerous causes of the functional failure of the high-voltage power battery system, the occurrence probability of power battery system failure caused by insulation failure is assumed to be between 0.02 and 0.03. Namely, the accurate occurrence probability is a fuzzy number. Among the reasons for the degradation of the high-voltage power battery system, the oxidation and reduction of battery electrolyte is a dynamic process, that is, a multistate event. Therefore, after considering the limitation of the traditional fault tree analysis method on the fuzziness and polymorphism of the high-voltage power battery system of pure electric commercial vehicles, the multistate fuzzy fault tree analysis method is used to analyze its reliability.The work took a high-voltage power battery system of pure electric commercial vehicles as an example and applied theories such as the multistate theory and fuzzy mathematics to establish a multistate fuzzy fault tree for the high-voltage power battery system. Minimum reliability was 0.4607, and maximum reliability was 0.5370 for the high-voltage power battery system. Furthermore, low-reliability events directly affecting the reliability of the system were identified.The following items have been accomplished: (1) A complete multistate fuzzy fault tree for high-voltage power battery systems of pure electric commercial vehicles was established to solve the functional-failure mode, deteriorated-state mode, and subfault tree models to analyze the reliability of high-voltage power battery systems. (2) Qualitative and quantitative analyses were performed on all bottom events of the subfault tree causing the functional failure mode. The weak links causing the system failure were identified by the fuzzy-probability importance of each bottom event. (3) For the problem of unknown probability of multistate bottom events in each state, group decision-making was used to deal with fuzzy probabilities, thus solving the subfault trees in the state deterioration mode.The remainder of the work is organized as follows. Section2 introduces the basic theory of the polymorphic fuzzy fault tree analysis method. The polymorphic fuzzy fault tree of the high-voltage power battery system of pure electric commercial vehicles is established and analyzed in Section 3. Section 4 solves the fuzzy-probability importance of each bottom event of the subfault tree in the functional failure mode. Finally, the conclusions are summarized in Section 5.


## 2. Basic Theory of the Polymorphic Fuzzy Fault Tree Analysis Method

### 2.1. Fault Tree Analysis

The fault tree analysis (FTA) [21] can analyze all the possible causes of system failure in the system design process. A “tree” logic block diagram is drawn to identify the basic events causing system failure. Qualitative analysis is used to solve the probability of system failure and the importance of the bottom event.


#### 2.1.1. Qualitative FTA

(1) Cut Set and Minimum Cut Set. The set composed of the bottom events causing the occurrence of the top event of the fault tree is called the cut set. The set consisting of the minimum number of bottom events causing the occurrence of the top event of the fault tree is called the minimum cut set.(2) Method of Solving the Minimum Cut Set. There are two methods of solving for the minimum cut set, the downward method (Fussell-Vesely) and the upward method (Semanderes) [22]. The downward method starts from the top event layer by layer down and lists “OR” gate events and top events into different lines. Besides, “AND” gate events and top events are listed in the same line. The non-minimum cut set is eliminated to obtain the minimum cut set until the basic events can no longer be decomposed. The upward method, i.e., proceeding from the bottom to the top, replaces the corresponding logic gate symbols with set symbols and expresses the fault tree structure-function in terms of event symbols and set operators. Finally, the expressions for the top event are listed to find the minimum cut set.


#### 2.1.2. Quantitative FTA

(1) Calculation of the Occurrence Probability of Top Events. In the quantitative analysis of the fault tree, the probability of the top event can be calculated according to the occurrence probability of the bottom event and structural function. The probability of the occurrence of base event Xii=1,2,⋯,n is assumed to be PXii=1,2,⋯,n. When the logic gate is “OR” and “AND,” and the occurrence of the base event is independent of each other, the calculation method of the probability of the top event is as follows:
(1)
When the fault-tree logic gates are logical “OR” gates, the probability of at least one bottom event occurring is(1)P∪=1−1−PX1×1−PX2×⋯×1−PXn=1−∏i=1n1−PXi(2)
When the fault-tree logic gates are logical “AND” gates, the probability ofn bottom events occurring simultaneously is
(2)P∩=PX1×PX2×⋯×PXn=∏i=1nPXi(2) Calculation of probability’s Importance of the Bottom Event. Importance is the magnitude of the contribution of the minimum cut set or bottom event to the occurrence of the top event, and the importance of the bottom event to the top event is positively related to its probability importance. If the probability’s importance of a bottom event is close to 1, then the system must fail once that bottom event occurs.Probability’s importance is defined as(3)Iit=∂gQt∂Qit,where Q is the unreliability of the system.(3) Calculation of Subsystem Reliability. According to the traditional reliability theory, the general expression of the function between reliability and failure probability is
(4)Rt=e−∫0tλtdt,where λt is the failure probability.Whenλt is a constant, i.e., λt=λ. Then, the reliability of the system is
(5)Rt=e−λ.According to the relationship between reliability and unreliability, the relationship between unreliability and failure probability is(6)Ft=1−Rt=1−e−λ.


### 2.2. Fuzzy Fault Tree Analysis

#### 2.2.1.L-RFuzzy Number

The occurrence probability of the bottom event of the high-voltage power battery system of pure electric commercial vehicles is fuzzy. A fuzzy number is used to analyze the fault tree.Generally, fuzzy numbers are described by reference functions. LetL and R be the reference functions of fuzzy numbers, if
(7)μA~x=Lm−xα,x≤m,α>0,Rx−mβ,x≥m,β>0.Then, the fuzzy numberA~ in Equation (7) is defined as L-R fuzzy number. It is recorded as A~=m,α,βLR, where m is the mean value of fuzzy number A~, α and β are the upper and lower confidence limits of fuzzy number A~, and μA~x is the membership function of A~. When α and β are equal to 0, A~ is no longer a fuzzy number but a regular clear number. Moreover, larger α and β indicate more ambiguous A~ [22]. αandβ are generally taken as 10 to 20% of fuzzy mean m [23]. The commonly used membership functions of L-R fuzzy numbers are triangular and normal.(1) Triangular. The triangular membership function can be expressed as
(8)μA~x=0,x<m−α,1−m−xα,m−α≤x≤m,1−x−mβ,m<x≤m+β,0,x>m+β.Figure1 shows the triangular membership function.Figure 1
Triangular membership functionTheλ-cut set interval of triangular fuzzy number A~=m−α,m,m+β is
(9)A~λ=m−α+α∙λ,m+β−β∙λ.If there are two triangular fuzzy numbersA~ and B~, there are four algorithms for expansion according to the classical extension principle [24] for any λ in interval [0, 1].
(1)
Fuzzy-number addition:(10)A~λ+B~λ=LAλ,RAλ+LBλ,RBλ=LAλ+LBλ,RAλ+RBλ.(2)
Fuzzy-number subtraction:(11)A~λ−B~λ=LAλ,RAλ−LBλ,RBλ=LAλ−LBλ,RAλ−RBλ.(3)
Fuzzy-number multiplication:(12)A~λ×B~λ=LAλ,RAλ×LBλ,RBλ=LAλ×LBλ,RAλ×RBλ.(4)
Fuzzy-number division:(13)A~λ/B~λ=LAλ,RAλ/LBλ,RBλ=LAλLBλ,RAλRBλ.(2) Normal. The normal membership function can be expressed as
(14)μB~x=m−xα2,x∈m−α−lnλ,m,Rx=e−x−m/β2,x∈m,m+β−lnλ,1,x=m,0,others.The interval of theλ-cut set of normal type membership function B~ can be obtained as
(15)B~λ=m−α−lnλ,m+β−lnλ.Figure2 shows the normal membership function.Figure 2
Normal membership function.


#### 2.2.2. Fuzzy Operator of the Fault Tree Analysis

The logic gate operator is used to calculate the probability of the basic bottom event, thus obtaining the probability of the top event of the traditional fault tree [25]. The system’s occurrence probability of the top event, namely, the failure probability of the top event, can be accurately obtained by the bottom-event occurrence probability and structure function. Besides, system reliability can be obtained. Fuzzy number F~i is introduced to represent the occurrence probability of the basic bottom event in the process of analyzing the fuzzy fault tree, and the fuzzy logic gate operator is used to replace the traditional one, which can obtain the fuzzy number of the occurrence of the top event [26].ProbabilityF~i of the bottom event represented by a triangular fuzzy number that λ-cut set is
(16)F~1λ=m1−α1+λα1,m1+β1−λβ1,F~2λ=m2−α2+λα2,m2+β2−λβ2,⋮,F~iλ=mi−αi+λαi,mi+βi−λβi,⋮,F~nλ=mn−αn+λαn,mn+βn−λβn.According to the characteristics of fault-tree logic “AND” and “OR” gates, the fuzzy operators of a triangular fuzzy number “AND” and “OR” gates are as follows.(1)
Logic “AND” gate:(17)F~S∩=∏i=1nF~iλ=F~iλ∙F~2λ∙F~3λ∙⋯∙F~n−1λ∙F~nλ=∏i=1nmi−αi+λαi,∏i=1nmi+βi+λβi.(2)
Logic “OR” gate:(18)F~S∪=1−∏i=1n1−F~iλ=1−∏i=1n1−mi−αi−λαi,1−∏i=1n1−mi+βi+λβi.ProbabilityF~i of the bottom event represented by a normal fuzzy number that λ-cut sets is
(19)F~1λ=m1−α1−lnλ,m1+β1−lnλ,F~2λ=m2−α2−lnλ,m2+β2−lnλ,⋮,F~iλ=mi−αi−lnλ,mi+βi−lnλ,⋮,F~nλ=mn−αn−lnλ,mn+βn−lnλ.Similarly, the fuzzy operators of “AND” and “OR” gates of normal fuzzy numbers are as follows.(1)
Logic “AND” gate:(20)F~S∩=∏i=1nF~iλ=F~iλ∙F~2λ∙F~3λ∙⋯∙F~n−1λ∙F~nλ=∏i=1nmi−αi−lnλ,∏i=1nmi+βi−lnλ.(2)
Logic “OR” gate:(21)F~S∪=1−∏i=1n1−F~iλ=1−∏i=1n1−mi−αi−lnλ,1−∏i=1n1−mi+βi−lnλ.


### 2.3. Polymorphic Fuzzy Fault-Tree Analysis

#### 2.3.1. Multistate Fault Gate

When the input event of the system top event contains one or more multistate events, this logic gate is called a polymorphic fault gate. Supposing there aren polymorphic input events Xii=1,2,⋯,n in the fault gate, i.e., the state of the input event meets SXi∈0,0.5,1, SXi∈normal,degradation,failure. Then the state of top event U satisfies the following equation:
(22)SU=0,∑i=1nSXi=0,0.5,SXi≤0.5and∑i=1nSXi≠0,1,others.Equation (22) shows that when the states of input polymorphic events are all 0, that is, all input polymorphic events are normal, and the state of output events is also normally 0. When the state of the input polymorphic event is 0.5 or a combination of 0 and 0.5, namely, the input polymorphic event is in a degraded state or a combination of the degraded state and normal state, and the state of the output event is also degraded. When at least one of the input polymorphic events is in state 1, that is, the state of at least one input event is in a failure state, and the output of the whole system is in a failure state.


#### 2.3.2. Analysis of the Deteriorated-State Mode

Aiming at the problem to determine the exact probability of polymorphic bottom events in different states in the deteriorated-state mode, the work used group decision-making, expert experience, and fuzzy numbers. The probability of polymorphic bottom events in each state was obtained by averaging, defuzzifying, and normalizing the obtained fuzzy probability.(1) Fuzzy Probability Obtained according to the Language Variables. Seven linguistic variables are introduced to combine fuzzy probability with linguistic variables [27]. Table 1 shows the relationship between language variables and the fuzzy probability.Table 1
Correspondence between language variables and the fuzzy probability.
Serial numberFuzzy languageSemantic symbolsTriangular fuzzy number1Very highVH(0.9, 1.0, 1.0)2HighH(0.7, 0.9, 1.0)3Generally highGH(0.5, 0.7, 0.9)4MediumM(0.3, 0.5, 0.7)5Generally lowGL(0.1, 0.3, 0.5)6LowL(0, 0.1, 0.3)7Very lowVL(0, 0, 0.1)The last column of Table1 represents the triangular fuzzy number corresponding to the language variable, and this method can convert the expert experience into a fuzzy probability expressed in triangular fuzzy numbers (see Table 1). Namely, it is a constant under the membership function, and its value corresponds to the fuzzy probability and solved the problem of probability unknown.Ifq experts are participating in the decision-making that the language variable of the bottom event Xi given by the k-th decision-maker in the jj=1,2,and3, state is transformed into the fuzzy probability as P~ijk=αijk,mijk,βijkk=1,2,⋯,q.(2) Fuzzy-Probability Averaging. The obtained fuzzy probability is mathematically averaged. Then, the fuzzy probability of bottom event Xi in state j is
(23)P~ij’=P~ij1+P~ij2+⋯+P~ijqq.(3) Defuzzification. After the fuzzy probability is processed by the mean area method, the accurate probability of each bottom event Xi in state j is obtained as follows:
(24)Pij′=P~ij’1+2P~ij’2+P~ij’34.(4) Normalization. The accurate probability of the bottom event in each state is normalized to ensure that the sum of the probabilities of the bottom event in each state is 1. After processing, the accurate probability of each bottom event Xi in state j is obtained as
(25)Pij=Pij’∑j=13Pij’.


## 2.1. Fault Tree Analysis

The fault tree analysis (FTA) [21] can analyze all the possible causes of system failure in the system design process. A “tree” logic block diagram is drawn to identify the basic events causing system failure. Qualitative analysis is used to solve the probability of system failure and the importance of the bottom event.


### 2.1.1. Qualitative FTA

(1) Cut Set and Minimum Cut Set. The set composed of the bottom events causing the occurrence of the top event of the fault tree is called the cut set. The set consisting of the minimum number of bottom events causing the occurrence of the top event of the fault tree is called the minimum cut set.(2) Method of Solving the Minimum Cut Set. There are two methods of solving for the minimum cut set, the downward method (Fussell-Vesely) and the upward method (Semanderes) [22]. The downward method starts from the top event layer by layer down and lists “OR” gate events and top events into different lines. Besides, “AND” gate events and top events are listed in the same line. The non-minimum cut set is eliminated to obtain the minimum cut set until the basic events can no longer be decomposed. The upward method, i.e., proceeding from the bottom to the top, replaces the corresponding logic gate symbols with set symbols and expresses the fault tree structure-function in terms of event symbols and set operators. Finally, the expressions for the top event are listed to find the minimum cut set.


### 2.1.2. Quantitative FTA

(1) Calculation of the Occurrence Probability of Top Events. In the quantitative analysis of the fault tree, the probability of the top event can be calculated according to the occurrence probability of the bottom event and structural function. The probability of the occurrence of base event Xii=1,2,⋯,n is assumed to be PXii=1,2,⋯,n. When the logic gate is “OR” and “AND,” and the occurrence of the base event is independent of each other, the calculation method of the probability of the top event is as follows:
(1)
When the fault-tree logic gates are logical “OR” gates, the probability of at least one bottom event occurring is(1)P∪=1−1−PX1×1−PX2×⋯×1−PXn=1−∏i=1n1−PXi(2)
When the fault-tree logic gates are logical “AND” gates, the probability ofn bottom events occurring simultaneously is
(2)P∩=PX1×PX2×⋯×PXn=∏i=1nPXi(2) Calculation of probability’s Importance of the Bottom Event. Importance is the magnitude of the contribution of the minimum cut set or bottom event to the occurrence of the top event, and the importance of the bottom event to the top event is positively related to its probability importance. If the probability’s importance of a bottom event is close to 1, then the system must fail once that bottom event occurs.Probability’s importance is defined as(3)Iit=∂gQt∂Qit,where Q is the unreliability of the system.(3) Calculation of Subsystem Reliability. According to the traditional reliability theory, the general expression of the function between reliability and failure probability is
(4)Rt=e−∫0tλtdt,where λt is the failure probability.Whenλt is a constant, i.e., λt=λ. Then, the reliability of the system is
(5)Rt=e−λ.According to the relationship between reliability and unreliability, the relationship between unreliability and failure probability is(6)Ft=1−Rt=1−e−λ.


## 2.1.1. Qualitative FTA

(1) Cut Set and Minimum Cut Set. The set composed of the bottom events causing the occurrence of the top event of the fault tree is called the cut set. The set consisting of the minimum number of bottom events causing the occurrence of the top event of the fault tree is called the minimum cut set.(2) Method of Solving the Minimum Cut Set. There are two methods of solving for the minimum cut set, the downward method (Fussell-Vesely) and the upward method (Semanderes) [22]. The downward method starts from the top event layer by layer down and lists “OR” gate events and top events into different lines. Besides, “AND” gate events and top events are listed in the same line. The non-minimum cut set is eliminated to obtain the minimum cut set until the basic events can no longer be decomposed. The upward method, i.e., proceeding from the bottom to the top, replaces the corresponding logic gate symbols with set symbols and expresses the fault tree structure-function in terms of event symbols and set operators. Finally, the expressions for the top event are listed to find the minimum cut set.


## 2.1.2. Quantitative FTA

(1) Calculation of the Occurrence Probability of Top Events. In the quantitative analysis of the fault tree, the probability of the top event can be calculated according to the occurrence probability of the bottom event and structural function. The probability of the occurrence of base event Xii=1,2,⋯,n is assumed to be PXii=1,2,⋯,n. When the logic gate is “OR” and “AND,” and the occurrence of the base event is independent of each other, the calculation method of the probability of the top event is as follows:
(1)
When the fault-tree logic gates are logical “OR” gates, the probability of at least one bottom event occurring is(1)P∪=1−1−PX1×1−PX2×⋯×1−PXn=1−∏i=1n1−PXi(2)
When the fault-tree logic gates are logical “AND” gates, the probability ofn bottom events occurring simultaneously is
(2)P∩=PX1×PX2×⋯×PXn=∏i=1nPXi(2) Calculation of probability’s Importance of the Bottom Event. Importance is the magnitude of the contribution of the minimum cut set or bottom event to the occurrence of the top event, and the importance of the bottom event to the top event is positively related to its probability importance. If the probability’s importance of a bottom event is close to 1, then the system must fail once that bottom event occurs.Probability’s importance is defined as(3)Iit=∂gQt∂Qit,where Q is the unreliability of the system.(3) Calculation of Subsystem Reliability. According to the traditional reliability theory, the general expression of the function between reliability and failure probability is
(4)Rt=e−∫0tλtdt,where λt is the failure probability.Whenλt is a constant, i.e., λt=λ. Then, the reliability of the system is
(5)Rt=e−λ.According to the relationship between reliability and unreliability, the relationship between unreliability and failure probability is(6)Ft=1−Rt=1−e−λ.


## 2.2. Fuzzy Fault Tree Analysis

### 2.2.1.L-RFuzzy Number

The occurrence probability of the bottom event of the high-voltage power battery system of pure electric commercial vehicles is fuzzy. A fuzzy number is used to analyze the fault tree.Generally, fuzzy numbers are described by reference functions. LetL and R be the reference functions of fuzzy numbers, if
(7)μA~x=Lm−xα,x≤m,α>0,Rx−mβ,x≥m,β>0.Then, the fuzzy numberA~ in Equation (7) is defined as L-R fuzzy number. It is recorded as A~=m,α,βLR, where m is the mean value of fuzzy number A~, α and β are the upper and lower confidence limits of fuzzy number A~, and μA~x is the membership function of A~. When α and β are equal to 0, A~ is no longer a fuzzy number but a regular clear number. Moreover, larger α and β indicate more ambiguous A~ [22]. αandβ are generally taken as 10 to 20% of fuzzy mean m [23]. The commonly used membership functions of L-R fuzzy numbers are triangular and normal.(1) Triangular. The triangular membership function can be expressed as
(8)μA~x=0,x<m−α,1−m−xα,m−α≤x≤m,1−x−mβ,m<x≤m+β,0,x>m+β.Figure1 shows the triangular membership function.Figure 1
Triangular membership functionTheλ-cut set interval of triangular fuzzy number A~=m−α,m,m+β is
(9)A~λ=m−α+α∙λ,m+β−β∙λ.If there are two triangular fuzzy numbersA~ and B~, there are four algorithms for expansion according to the classical extension principle [24] for any λ in interval [0, 1].
(1)
Fuzzy-number addition:(10)A~λ+B~λ=LAλ,RAλ+LBλ,RBλ=LAλ+LBλ,RAλ+RBλ.(2)
Fuzzy-number subtraction:(11)A~λ−B~λ=LAλ,RAλ−LBλ,RBλ=LAλ−LBλ,RAλ−RBλ.(3)
Fuzzy-number multiplication:(12)A~λ×B~λ=LAλ,RAλ×LBλ,RBλ=LAλ×LBλ,RAλ×RBλ.(4)
Fuzzy-number division:(13)A~λ/B~λ=LAλ,RAλ/LBλ,RBλ=LAλLBλ,RAλRBλ.(2) Normal. The normal membership function can be expressed as
(14)μB~x=m−xα2,x∈m−α−lnλ,m,Rx=e−x−m/β2,x∈m,m+β−lnλ,1,x=m,0,others.The interval of theλ-cut set of normal type membership function B~ can be obtained as
(15)B~λ=m−α−lnλ,m+β−lnλ.Figure2 shows the normal membership function.Figure 2
Normal membership function.


### 2.2.2. Fuzzy Operator of the Fault Tree Analysis

The logic gate operator is used to calculate the probability of the basic bottom event, thus obtaining the probability of the top event of the traditional fault tree [25]. The system’s occurrence probability of the top event, namely, the failure probability of the top event, can be accurately obtained by the bottom-event occurrence probability and structure function. Besides, system reliability can be obtained. Fuzzy number F~i is introduced to represent the occurrence probability of the basic bottom event in the process of analyzing the fuzzy fault tree, and the fuzzy logic gate operator is used to replace the traditional one, which can obtain the fuzzy number of the occurrence of the top event [26].ProbabilityF~i of the bottom event represented by a triangular fuzzy number that λ-cut set is
(16)F~1λ=m1−α1+λα1,m1+β1−λβ1,F~2λ=m2−α2+λα2,m2+β2−λβ2,⋮,F~iλ=mi−αi+λαi,mi+βi−λβi,⋮,F~nλ=mn−αn+λαn,mn+βn−λβn.According to the characteristics of fault-tree logic “AND” and “OR” gates, the fuzzy operators of a triangular fuzzy number “AND” and “OR” gates are as follows.(1)
Logic “AND” gate:(17)F~S∩=∏i=1nF~iλ=F~iλ∙F~2λ∙F~3λ∙⋯∙F~n−1λ∙F~nλ=∏i=1nmi−αi+λαi,∏i=1nmi+βi+λβi.(2)
Logic “OR” gate:(18)F~S∪=1−∏i=1n1−F~iλ=1−∏i=1n1−mi−αi−λαi,1−∏i=1n1−mi+βi+λβi.ProbabilityF~i of the bottom event represented by a normal fuzzy number that λ-cut sets is
(19)F~1λ=m1−α1−lnλ,m1+β1−lnλ,F~2λ=m2−α2−lnλ,m2+β2−lnλ,⋮,F~iλ=mi−αi−lnλ,mi+βi−lnλ,⋮,F~nλ=mn−αn−lnλ,mn+βn−lnλ.Similarly, the fuzzy operators of “AND” and “OR” gates of normal fuzzy numbers are as follows.(1)
Logic “AND” gate:(20)F~S∩=∏i=1nF~iλ=F~iλ∙F~2λ∙F~3λ∙⋯∙F~n−1λ∙F~nλ=∏i=1nmi−αi−lnλ,∏i=1nmi+βi−lnλ.(2)
Logic “OR” gate:(21)F~S∪=1−∏i=1n1−F~iλ=1−∏i=1n1−mi−αi−lnλ,1−∏i=1n1−mi+βi−lnλ.


## 2.2.1.L-RFuzzy Number

The occurrence probability of the bottom event of the high-voltage power battery system of pure electric commercial vehicles is fuzzy. A fuzzy number is used to analyze the fault tree.Generally, fuzzy numbers are described by reference functions. LetL and R be the reference functions of fuzzy numbers, if
(7)μA~x=Lm−xα,x≤m,α>0,Rx−mβ,x≥m,β>0.Then, the fuzzy numberA~ in Equation (7) is defined as L-R fuzzy number. It is recorded as A~=m,α,βLR, where m is the mean value of fuzzy number A~, α and β are the upper and lower confidence limits of fuzzy number A~, and μA~x is the membership function of A~. When α and β are equal to 0, A~ is no longer a fuzzy number but a regular clear number. Moreover, larger α and β indicate more ambiguous A~ [22]. αandβ are generally taken as 10 to 20% of fuzzy mean m [23]. The commonly used membership functions of L-R fuzzy numbers are triangular and normal.(1) Triangular. The triangular membership function can be expressed as
(8)μA~x=0,x<m−α,1−m−xα,m−α≤x≤m,1−x−mβ,m<x≤m+β,0,x>m+β.Figure1 shows the triangular membership function.Figure 1
Triangular membership functionTheλ-cut set interval of triangular fuzzy number A~=m−α,m,m+β is
(9)A~λ=m−α+α∙λ,m+β−β∙λ.If there are two triangular fuzzy numbersA~ and B~, there are four algorithms for expansion according to the classical extension principle [24] for any λ in interval [0, 1].
(1)
Fuzzy-number addition:(10)A~λ+B~λ=LAλ,RAλ+LBλ,RBλ=LAλ+LBλ,RAλ+RBλ.(2)
Fuzzy-number subtraction:(11)A~λ−B~λ=LAλ,RAλ−LBλ,RBλ=LAλ−LBλ,RAλ−RBλ.(3)
Fuzzy-number multiplication:(12)A~λ×B~λ=LAλ,RAλ×LBλ,RBλ=LAλ×LBλ,RAλ×RBλ.(4)
Fuzzy-number division:(13)A~λ/B~λ=LAλ,RAλ/LBλ,RBλ=LAλLBλ,RAλRBλ.(2) Normal. The normal membership function can be expressed as
(14)μB~x=m−xα2,x∈m−α−lnλ,m,Rx=e−x−m/β2,x∈m,m+β−lnλ,1,x=m,0,others.The interval of theλ-cut set of normal type membership function B~ can be obtained as
(15)B~λ=m−α−lnλ,m+β−lnλ.Figure2 shows the normal membership function.Figure 2
Normal membership function.


## 2.2.2. Fuzzy Operator of the Fault Tree Analysis

The logic gate operator is used to calculate the probability of the basic bottom event, thus obtaining the probability of the top event of the traditional fault tree [25]. The system’s occurrence probability of the top event, namely, the failure probability of the top event, can be accurately obtained by the bottom-event occurrence probability and structure function. Besides, system reliability can be obtained. Fuzzy number F~i is introduced to represent the occurrence probability of the basic bottom event in the process of analyzing the fuzzy fault tree, and the fuzzy logic gate operator is used to replace the traditional one, which can obtain the fuzzy number of the occurrence of the top event [26].ProbabilityF~i of the bottom event represented by a triangular fuzzy number that λ-cut set is
(16)F~1λ=m1−α1+λα1,m1+β1−λβ1,F~2λ=m2−α2+λα2,m2+β2−λβ2,⋮,F~iλ=mi−αi+λαi,mi+βi−λβi,⋮,F~nλ=mn−αn+λαn,mn+βn−λβn.According to the characteristics of fault-tree logic “AND” and “OR” gates, the fuzzy operators of a triangular fuzzy number “AND” and “OR” gates are as follows.(1)
Logic “AND” gate:(17)F~S∩=∏i=1nF~iλ=F~iλ∙F~2λ∙F~3λ∙⋯∙F~n−1λ∙F~nλ=∏i=1nmi−αi+λαi,∏i=1nmi+βi+λβi.(2)
Logic “OR” gate:(18)F~S∪=1−∏i=1n1−F~iλ=1−∏i=1n1−mi−αi−λαi,1−∏i=1n1−mi+βi+λβi.ProbabilityF~i of the bottom event represented by a normal fuzzy number that λ-cut sets is
(19)F~1λ=m1−α1−lnλ,m1+β1−lnλ,F~2λ=m2−α2−lnλ,m2+β2−lnλ,⋮,F~iλ=mi−αi−lnλ,mi+βi−lnλ,⋮,F~nλ=mn−αn−lnλ,mn+βn−lnλ.Similarly, the fuzzy operators of “AND” and “OR” gates of normal fuzzy numbers are as follows.(1)
Logic “AND” gate:(20)F~S∩=∏i=1nF~iλ=F~iλ∙F~2λ∙F~3λ∙⋯∙F~n−1λ∙F~nλ=∏i=1nmi−αi−lnλ,∏i=1nmi+βi−lnλ.(2)
Logic “OR” gate:(21)F~S∪=1−∏i=1n1−F~iλ=1−∏i=1n1−mi−αi−lnλ,1−∏i=1n1−mi+βi−lnλ.


## 2.3. Polymorphic Fuzzy Fault-Tree Analysis

### 2.3.1. Multistate Fault Gate

When the input event of the system top event contains one or more multistate events, this logic gate is called a polymorphic fault gate. Supposing there aren polymorphic input events Xii=1,2,⋯,n in the fault gate, i.e., the state of the input event meets SXi∈0,0.5,1, SXi∈normal,degradation,failure. Then the state of top event U satisfies the following equation:
(22)SU=0,∑i=1nSXi=0,0.5,SXi≤0.5and∑i=1nSXi≠0,1,others.Equation (22) shows that when the states of input polymorphic events are all 0, that is, all input polymorphic events are normal, and the state of output events is also normally 0. When the state of the input polymorphic event is 0.5 or a combination of 0 and 0.5, namely, the input polymorphic event is in a degraded state or a combination of the degraded state and normal state, and the state of the output event is also degraded. When at least one of the input polymorphic events is in state 1, that is, the state of at least one input event is in a failure state, and the output of the whole system is in a failure state.


### 2.3.2. Analysis of the Deteriorated-State Mode

Aiming at the problem to determine the exact probability of polymorphic bottom events in different states in the deteriorated-state mode, the work used group decision-making, expert experience, and fuzzy numbers. The probability of polymorphic bottom events in each state was obtained by averaging, defuzzifying, and normalizing the obtained fuzzy probability.(1) Fuzzy Probability Obtained according to the Language Variables. Seven linguistic variables are introduced to combine fuzzy probability with linguistic variables [27]. Table 1 shows the relationship between language variables and the fuzzy probability.Table 1
Correspondence between language variables and the fuzzy probability.
Serial numberFuzzy languageSemantic symbolsTriangular fuzzy number1Very highVH(0.9, 1.0, 1.0)2HighH(0.7, 0.9, 1.0)3Generally highGH(0.5, 0.7, 0.9)4MediumM(0.3, 0.5, 0.7)5Generally lowGL(0.1, 0.3, 0.5)6LowL(0, 0.1, 0.3)7Very lowVL(0, 0, 0.1)The last column of Table1 represents the triangular fuzzy number corresponding to the language variable, and this method can convert the expert experience into a fuzzy probability expressed in triangular fuzzy numbers (see Table 1). Namely, it is a constant under the membership function, and its value corresponds to the fuzzy probability and solved the problem of probability unknown.Ifq experts are participating in the decision-making that the language variable of the bottom event Xi given by the k-th decision-maker in the jj=1,2,and3, state is transformed into the fuzzy probability as P~ijk=αijk,mijk,βijkk=1,2,⋯,q.(2) Fuzzy-Probability Averaging. The obtained fuzzy probability is mathematically averaged. Then, the fuzzy probability of bottom event Xi in state j is
(23)P~ij’=P~ij1+P~ij2+⋯+P~ijqq.(3) Defuzzification. After the fuzzy probability is processed by the mean area method, the accurate probability of each bottom event Xi in state j is obtained as follows:
(24)Pij′=P~ij’1+2P~ij’2+P~ij’34.(4) Normalization. The accurate probability of the bottom event in each state is normalized to ensure that the sum of the probabilities of the bottom event in each state is 1. After processing, the accurate probability of each bottom event Xi in state j is obtained as
(25)Pij=Pij’∑j=13Pij’.


## 2.3.1. Multistate Fault Gate

When the input event of the system top event contains one or more multistate events, this logic gate is called a polymorphic fault gate. Supposing there aren polymorphic input events Xii=1,2,⋯,n in the fault gate, i.e., the state of the input event meets SXi∈0,0.5,1, SXi∈normal,degradation,failure. Then the state of top event U satisfies the following equation:
(22)SU=0,∑i=1nSXi=0,0.5,SXi≤0.5and∑i=1nSXi≠0,1,others.Equation (22) shows that when the states of input polymorphic events are all 0, that is, all input polymorphic events are normal, and the state of output events is also normally 0. When the state of the input polymorphic event is 0.5 or a combination of 0 and 0.5, namely, the input polymorphic event is in a degraded state or a combination of the degraded state and normal state, and the state of the output event is also degraded. When at least one of the input polymorphic events is in state 1, that is, the state of at least one input event is in a failure state, and the output of the whole system is in a failure state.


## 2.3.2. Analysis of the Deteriorated-State Mode

Aiming at the problem to determine the exact probability of polymorphic bottom events in different states in the deteriorated-state mode, the work used group decision-making, expert experience, and fuzzy numbers. The probability of polymorphic bottom events in each state was obtained by averaging, defuzzifying, and normalizing the obtained fuzzy probability.(1) Fuzzy Probability Obtained according to the Language Variables. Seven linguistic variables are introduced to combine fuzzy probability with linguistic variables [27]. Table 1 shows the relationship between language variables and the fuzzy probability.Table 1
Correspondence between language variables and the fuzzy probability.
Serial numberFuzzy languageSemantic symbolsTriangular fuzzy number1Very highVH(0.9, 1.0, 1.0)2HighH(0.7, 0.9, 1.0)3Generally highGH(0.5, 0.7, 0.9)4MediumM(0.3, 0.5, 0.7)5Generally lowGL(0.1, 0.3, 0.5)6LowL(0, 0.1, 0.3)7Very lowVL(0, 0, 0.1)The last column of Table1 represents the triangular fuzzy number corresponding to the language variable, and this method can convert the expert experience into a fuzzy probability expressed in triangular fuzzy numbers (see Table 1). Namely, it is a constant under the membership function, and its value corresponds to the fuzzy probability and solved the problem of probability unknown.Ifq experts are participating in the decision-making that the language variable of the bottom event Xi given by the k-th decision-maker in the jj=1,2,and3, state is transformed into the fuzzy probability as P~ijk=αijk,mijk,βijkk=1,2,⋯,q.(2) Fuzzy-Probability Averaging. The obtained fuzzy probability is mathematically averaged. Then, the fuzzy probability of bottom event Xi in state j is
(23)P~ij’=P~ij1+P~ij2+⋯+P~ijqq.(3) Defuzzification. After the fuzzy probability is processed by the mean area method, the accurate probability of each bottom event Xi in state j is obtained as follows:
(24)Pij′=P~ij’1+2P~ij’2+P~ij’34.(4) Normalization. The accurate probability of the bottom event in each state is normalized to ensure that the sum of the probabilities of the bottom event in each state is 1. After processing, the accurate probability of each bottom event Xi in state j is obtained as
(25)Pij=Pij’∑j=13Pij’.


## 3. Fault-Tree Analysis of the High-Voltage Power Battery System

### 3.1. Working Principle of the High-Voltage Power Battery System

The high-voltage power battery pack provides electricity for pure electric commercial vehicles. Its high-voltage DC output flows the high-voltage power equipment such as the steering power motor, brake air pump motor, and DC/DC motor inverter. The high-voltage current flowing through the motor inverter is converted into a three-phase alternating current and flows into the high-voltage drive motor and electric air conditioner motor through the high-voltage wiring harness. Figure3 shows the working principle of the high-voltage power battery system.Figure 3
Working principle of the high-voltage power battery system.


### 3.2. Establishment of the Fault Tree for the High-Voltage Power Battery System

The fault types of the high-voltage power battery system of pure electric commercial vehicles mainly include the battery-module fault and battery-management-system fault. Battery module faults are mainly single-cell and battery-pack ones. Single-cell faults are manifested by electrolyte decomposition, active substance shedding, and the internal short circuit of the battery. Battery pack faults are manifested by single-cell inconsistency, and charging faults and battery-management-system faults are manifested by no communication, electromagnetic interference, and low drive voltage.For the high-voltage power battery system, the power battery system fault was selected as the top event, and two intermediate events were established—the functional failure mode and state deterioration mode. The functional-failure mode was divided into battery module failure and battery management system failure to analyze the bottom event, and these two events were regarded as intermediate ones of a lower level. Then, the battery module fault and battery management system fault were analyzed until they were decomposed into events that could not be decomposed, i.e., the bottom event of the subfault tree of the functional failure mode. Meanwhile, the subfault tree of the deteriorated-state mode was decomposed and analyzed. Besides, the events of the established polymorphic fuzzy fault tree of the high-voltage power battery system were coded. The fault of the power battery system was recorded asT, and Table 2 shows the event definition and code.Table 2
Event definitions and codes.
Event nameEvent codeEvent nameEvent codeFunctional-failure modeU1Deteriorated-state modeU2Battery module failureV1Management system failureV2Battery unit failureW1Battery pack failureW2No CAN communicationW3Inaccurate signalW4No executionW5OvertemperatureS1Short circuitS2Low voltageS3Monomer inconsistencyS4Charging faultS5Fan failureS6Main relay faultS7Cannot be chargedK1High charge voltageK2Electrolyte redoxE1Positive material phase changeE2Negative electrode material agingE3Aging of diaphragms and increased porosityE4Insulation failureE5Fan not ONE6Loose connectionE7Increased internal resistance of the batteryE8Plate damageE9Short circuitE10Low SOCE11Capacity attenuationE12Inconsistent voltageE13Inconsistent temperatureE14Charger faultE15Shedding of active substancesE16Battery deformationE17Low-temperature environmentE18Late lifeE19High SOCE20No external internal resistanceE21CAN high and low reverse connectionE22Software bugE23Impact faultE24Bias faultE25Electromagnetic interferenceE26Sampling circuit faultE27Connection line failureE28Line damageE29Damaged cooling fanE30No working power supplyE31Low driving voltageE32According to Table2, the polymorphic fault tree of the high-voltage power battery system of pure electric commercial vehicles is simplified as shown in Figure 4.Figure 4
Multistate fault tree of the high-voltage power battery system.


### 3.3. Qualitative Analysis of the Fault Tree

When the minimum cut set of the polymorphic fault tree is calculated, only the qualitative analysis of the subfault tree of functional failure mode is carried out due to polymorphic events and polymorphic fault gates in the subfault tree of the deteriorated-state mode. The fault tree established in the work uses the downward method according to the principle of multiplication of the “AND” gate and the addition of the “OR” gate.(26)U1=E5+V1+V2,V1=W1+W2,V2=W3+W4+W5,W1=S1+S2+S3,W2=S4+S5,W3=E21+E22+E23,W4=E24+E25+E26+E27+E28,W5=S6+S7,S1=E6+E7+E8,S2=E9+E10,S3=E11+E12,S4=E13+E14,S5=E15+K1+K2,S6=E29+E30,S7=E31+E32,K1=E16+E17,K2=E18+E19+E20.Then,(27)U1=E5+E6+E7+E8+⋯+E30+E31+E32.Therefore, the minimum cut set of the functional failure mode’s subfault tree of the power battery system isE5,E6,E7,E8,⋯,E31,E32, where the minimum cut set contains 28 basic base events.


### 3.4. Quantitative Analysis of Fault Tree

Two intermediate events of the functional failure mode and deteriorated-state mode are selected for analysis to more accurately reflect the failure mode of the high-voltage power battery system. The subfault tree of functional failure mode is solved by the fuzzy fault tree analysis, and the subfault tree of deteriorated-state mode is solved by group decision-making and fuzzy probability.


#### 3.4.1. Quantitative Analysis of the Subfault Tree of the Functional Failure Mode

According to the data and relevant experience and the probability of subfault tree bottom event of the functional failure mode (see Table3).Table 3
Probability of the bottom event of the functional failure mode.
Event codemαβE50.0260.00520.0052E60.0460.00920.0092E70.0160.00320.0032E80.0160.00320.0032E90.0260.00520.0052E100.0180.00360.0036E110.10.020.02E120.050.010.01E130.0160.00320.0032E140.0160.00320.0032E150.010.0020.002E160.010.0020.002E170.060.0120.012E180.120.0240.024E190.0160.00320.0032E200.050.010.01E210.0280.00560.0056E220.140.0280.028E230.020.0040.004E240.0140.00280.0028E250.0250.0050.005E260.0160.00320.0032E270.0260.00520.0052E280.020.0040.004E290.0180.00360.0036E300.0160.00320.0032E310.0140.00280.0028E320.0120.00240.0024In Table3, m represents the mean of the fuzzy number, which is a dimensionless number, indicating the size of the fuzzy probability of the occurrence of the bottom event. The larger the mean of m, the greater the fuzzy probability of the occurrence of the bottom event, and the smaller of m, the smaller the fuzzy probability of the bottom event occurring. According to the reference of [23], the upper and lower limits of αandβ take 10 to 20% of the fuzzy mean m, considered the harshness of the actual working environment of the high-voltage power battery system and the ambiguity of the probability of causing system failure, and according to Figure 1, the larger the value of α and β, the larger the fuzzy probability interval, and the reverse is the opposite, when α and β take 10% of the mean of m, and the probability interval is contained in the 20% probability interval of α and β taking m, so in order to make the probability interval better represent the fuzzy probability of the event occur and then selected α and β as 20% of m in combination with the data.The interval of each bottom event can be obtained using Table3, Equation (16), and the triangular membership function.Since top eventU1=E5+E6+E7+E8+⋯+E30+E31+E32, Equation (18) is used to select the triangular membership function. The probability section of top event U1 varies with λ (see Figure 5 for the scattering curve).Figure 5
Probability interval distribution of top events changing withλ under the triangular membership function.The interval of each bottom event can be obtained from Table3, Equation (19), and the normal membership function.Similarly, when the normal membership function is selected, the probability of top eventU1 varies with λ (see Figure 6 for the scattering curve).Figure 6
Probability interval distribution of top events changing withλ under the normal membership function.By comparing Figures5 and 6, it can be seen that whether triangular membership function or normal membership function is selected, the probability interval of the top event becomes smaller and smaller with increased λ. When λ is equal to 1, the interval is an accurate number, consistent with the theory. However, for the normal membership function, when λ tends to 0, the fuzzy operator of the top event tends to infinity. The closer it is to 0, the greater the fuzzy probability interval of the top event. When λϵ0,1, the whole probability interval about the change of λ is nonlinear, and the absolute value of the slope of the upper and lower bounds of the top-event probability curve decreases as λ increases. That is, the closer it is to 0, the greater the changing trend of the curve; the closer it is to 1, the more gentle the change of the curve.Whenλ takes the special value of 0.01, Equation (21) is used to obtain that the probability interval of top-event occurrence is [0.4230, 0.7644] and the corresponding reliability interval of the top event is [0.4656, 0.6551]. Compared with the triangular fuzzy operator, the interval is too fuzzy and has no practical significance. Therefore, considering the fuzziness of the occurrence probability of the bottom event and the rationality of operation, the triangular fuzzy operator is selected to calculate the fuzzy probability interval of the top event.Figure5 shows that the upper and lower limits of the probability of top-event occurrence are approximately linear. Hence, the least square linear fitting and the intercept interval for the occurrence of the top event of the functional failure mode are used to obtain
(28)F~U1=0.0838λ+0.5415,−0.0694λ+0.6947.Therefore, please see as follows:(1)
Whenλ=1, the probability of the bottom event of the functional failure mode is a certain value, and that of the functional failure U1 of the high-voltage power battery system of the top event is
(29)F~U1,λ=1=0.6253Whenλ=1, the reliability of functional failure mode U1 of the high-voltage power battery system of the top event can be obtained from Equation (5).
(30)RU1,λ=1=0.5351.(2)
Whenλ=0, the probability of the bottom event of the functional-failure mode is a fuzzy number and F~U1 is an interval range, i.e., the probability interval of function-failure U1 of the high-voltage power battery system of the top event is
(31)F~U1,λ=0=0.5415,0.6947That is, the minimum probability of top eventU1 is 0.5415, and the maximum is 0.6947.Whenλ=0, the reliability interval of functional failure mode U1 of the high-voltage power battery system of the top event can be obtained from Equation (5).
(32)RU1,λ=0=0.4992,0.5819.


#### 3.4.2. Quantitative Analysis of the Subfault Tree in the Deteriorated State

According to the definition of a polymorphic fault gate, whenU1 does not occur and U2 is in a degraded state, the power battery system is in a degraded state and is not recognized as failure. At this time, the high-voltage power battery system of pure electric commercial vehicles does not need immediate maintenance, consistent with the actual operation condition. When U1 occurs or U2 is in a failure state, the high-voltage power battery system is in a failure state and needs to be overhauled.The decision information of the decision-maker is collected by group decision-making because the probability of each bottom event in each state of the subfault tree of the deteriorated-state mode cannot be accurately obtained. According to the conclusion of the relationship between group size and decision-making, when the number of group decision-makers is 2-5, consensus can be obtained; moreover, 5-11 people are the most effective to draw more correct conclusions.Consequently, the work collected the probability of four polymorphic bottom events in each state from five experts and combined the language variables of expert opinions with the corresponding triangular fuzzy numbers (see Table4).Table 4
Expert opinion and fuzzy probability of the bottom event in each state.
Bottom eventE1E2E3E4State00.5100.5100.5100.51Expert 1VHGLVLHLVLVHGLVLHVLVLExpert 2VHGLVLVHGLVLVHLVLVHLVLExpert 3HVLVLVHGLVLHVLVLVHGLVLExpert 4HVLVLHLVLHLVLVHMVLExpert 5VHMVLVHMVLVHMVLVHGLVL(1) Averaging of Fuzzy Probability. According to the corresponding relationship between language variables and fuzzy probability in Table 1 and Equation (23), the mean value of the fuzzy probability of each polymorphic bottom event in each state is determined as follows:
(33)P~1−0′=0.82,0.96,1.0,P~1−0.5′=0.1,0.22,0.38,P~1−1′=0,0,0.1,P~2−0′=0.82,0.96,1.0,P~2−0.5′=0.1,0.26,0.46,P~2−1′=0,0,0.1,P~3−0′=0.82,0.96,1.0,P~3−0.5′=0.08,0.2,0.38,P~3−1′=0,0,0.1,P~4−0′=0.86,0.98,1.0,P~4−0.5′=0.1,0.24,0.42,P~4−1′=0,0,0.1.(2) Defuzzification of the Fuzzy Probability. The fuzzy probability is defuzzified according to Equation (24) to convert the obtained fuzzy probability into accurate probability, that is,
(34)P1−0′=0.935,P1−0.5′=0.230,P1−1′=0.025,P2−0′=0.935,P2−0.5′=0.270,P2−1′=0.025,P3−0′=0.935,P3−0.5′=0.215,P3−1′=0.025,P4−0′=0.955,P4−0.5′=0.250,P4−1′=0.025.(3) Normalization of exact probability. According to Equation (25), the accurate probability of polymorphic bottom events occurring in each state is normalized to obtain
(35)P1−0=0.7857,P1−0.5=0.1933,P1−1=0.0210,P2−0=0.7602,P2−0.5=0.2195,P2−1=0.0203,P3−0=0.7957,P3−0.5=0.1830,P3−1=0.0213,P4−0=0.7764,P4−0.5=0.2033,P4−1=0.0203.According to the structural characteristics of a multistate fault gate, when at least one bottom event state is 1, the system’s top-event output is in the failure state. According to the calculated occurrence probability of each state of the polymorphic bottom event, the probability of each polymorphic bottom event in state 1 is substituted into Equation (1). The occurrence probability of the sub-fault tree of the top event in the deteriorated-state mode is
(36)PU2=1−1−P1−1×1−P2−1×1−P3−1×1−P4−1=0.0804.According to Equation (5), the subfault tree reliability in the deteriorated mode can be obtained.
(37)RU2=0.9228.(1)
RU1,λ=1=0.5351 when λ=1. The reliability of the high-voltage power battery system of pure electric commercial vehicles is
(38)RT=0.4938(2)
RU1,λ=0=0.4992,0.5819 when λ=0. The reliability of the high-voltage power battery system of pure electric commercial vehicles is
(39)RT=0.4607,0.5370


## 3.1. Working Principle of the High-Voltage Power Battery System

The high-voltage power battery pack provides electricity for pure electric commercial vehicles. Its high-voltage DC output flows the high-voltage power equipment such as the steering power motor, brake air pump motor, and DC/DC motor inverter. The high-voltage current flowing through the motor inverter is converted into a three-phase alternating current and flows into the high-voltage drive motor and electric air conditioner motor through the high-voltage wiring harness. Figure3 shows the working principle of the high-voltage power battery system.Figure 3
Working principle of the high-voltage power battery system.


## 3.2. Establishment of the Fault Tree for the High-Voltage Power Battery System

The fault types of the high-voltage power battery system of pure electric commercial vehicles mainly include the battery-module fault and battery-management-system fault. Battery module faults are mainly single-cell and battery-pack ones. Single-cell faults are manifested by electrolyte decomposition, active substance shedding, and the internal short circuit of the battery. Battery pack faults are manifested by single-cell inconsistency, and charging faults and battery-management-system faults are manifested by no communication, electromagnetic interference, and low drive voltage.For the high-voltage power battery system, the power battery system fault was selected as the top event, and two intermediate events were established—the functional failure mode and state deterioration mode. The functional-failure mode was divided into battery module failure and battery management system failure to analyze the bottom event, and these two events were regarded as intermediate ones of a lower level. Then, the battery module fault and battery management system fault were analyzed until they were decomposed into events that could not be decomposed, i.e., the bottom event of the subfault tree of the functional failure mode. Meanwhile, the subfault tree of the deteriorated-state mode was decomposed and analyzed. Besides, the events of the established polymorphic fuzzy fault tree of the high-voltage power battery system were coded. The fault of the power battery system was recorded asT, and Table 2 shows the event definition and code.Table 2
Event definitions and codes.
Event nameEvent codeEvent nameEvent codeFunctional-failure modeU1Deteriorated-state modeU2Battery module failureV1Management system failureV2Battery unit failureW1Battery pack failureW2No CAN communicationW3Inaccurate signalW4No executionW5OvertemperatureS1Short circuitS2Low voltageS3Monomer inconsistencyS4Charging faultS5Fan failureS6Main relay faultS7Cannot be chargedK1High charge voltageK2Electrolyte redoxE1Positive material phase changeE2Negative electrode material agingE3Aging of diaphragms and increased porosityE4Insulation failureE5Fan not ONE6Loose connectionE7Increased internal resistance of the batteryE8Plate damageE9Short circuitE10Low SOCE11Capacity attenuationE12Inconsistent voltageE13Inconsistent temperatureE14Charger faultE15Shedding of active substancesE16Battery deformationE17Low-temperature environmentE18Late lifeE19High SOCE20No external internal resistanceE21CAN high and low reverse connectionE22Software bugE23Impact faultE24Bias faultE25Electromagnetic interferenceE26Sampling circuit faultE27Connection line failureE28Line damageE29Damaged cooling fanE30No working power supplyE31Low driving voltageE32According to Table2, the polymorphic fault tree of the high-voltage power battery system of pure electric commercial vehicles is simplified as shown in Figure 4.Figure 4
Multistate fault tree of the high-voltage power battery system.


## 3.3. Qualitative Analysis of the Fault Tree

When the minimum cut set of the polymorphic fault tree is calculated, only the qualitative analysis of the subfault tree of functional failure mode is carried out due to polymorphic events and polymorphic fault gates in the subfault tree of the deteriorated-state mode. The fault tree established in the work uses the downward method according to the principle of multiplication of the “AND” gate and the addition of the “OR” gate.(26)U1=E5+V1+V2,V1=W1+W2,V2=W3+W4+W5,W1=S1+S2+S3,W2=S4+S5,W3=E21+E22+E23,W4=E24+E25+E26+E27+E28,W5=S6+S7,S1=E6+E7+E8,S2=E9+E10,S3=E11+E12,S4=E13+E14,S5=E15+K1+K2,S6=E29+E30,S7=E31+E32,K1=E16+E17,K2=E18+E19+E20.Then,(27)U1=E5+E6+E7+E8+⋯+E30+E31+E32.Therefore, the minimum cut set of the functional failure mode’s subfault tree of the power battery system isE5,E6,E7,E8,⋯,E31,E32, where the minimum cut set contains 28 basic base events.


## 3.4. Quantitative Analysis of Fault Tree

Two intermediate events of the functional failure mode and deteriorated-state mode are selected for analysis to more accurately reflect the failure mode of the high-voltage power battery system. The subfault tree of functional failure mode is solved by the fuzzy fault tree analysis, and the subfault tree of deteriorated-state mode is solved by group decision-making and fuzzy probability.


### 3.4.1. Quantitative Analysis of the Subfault Tree of the Functional Failure Mode

According to the data and relevant experience and the probability of subfault tree bottom event of the functional failure mode (see Table3).Table 3
Probability of the bottom event of the functional failure mode.
Event codemαβE50.0260.00520.0052E60.0460.00920.0092E70.0160.00320.0032E80.0160.00320.0032E90.0260.00520.0052E100.0180.00360.0036E110.10.020.02E120.050.010.01E130.0160.00320.0032E140.0160.00320.0032E150.010.0020.002E160.010.0020.002E170.060.0120.012E180.120.0240.024E190.0160.00320.0032E200.050.010.01E210.0280.00560.0056E220.140.0280.028E230.020.0040.004E240.0140.00280.0028E250.0250.0050.005E260.0160.00320.0032E270.0260.00520.0052E280.020.0040.004E290.0180.00360.0036E300.0160.00320.0032E310.0140.00280.0028E320.0120.00240.0024In Table3, m represents the mean of the fuzzy number, which is a dimensionless number, indicating the size of the fuzzy probability of the occurrence of the bottom event. The larger the mean of m, the greater the fuzzy probability of the occurrence of the bottom event, and the smaller of m, the smaller the fuzzy probability of the bottom event occurring. According to the reference of [23], the upper and lower limits of αandβ take 10 to 20% of the fuzzy mean m, considered the harshness of the actual working environment of the high-voltage power battery system and the ambiguity of the probability of causing system failure, and according to Figure 1, the larger the value of α and β, the larger the fuzzy probability interval, and the reverse is the opposite, when α and β take 10% of the mean of m, and the probability interval is contained in the 20% probability interval of α and β taking m, so in order to make the probability interval better represent the fuzzy probability of the event occur and then selected α and β as 20% of m in combination with the data.The interval of each bottom event can be obtained using Table3, Equation (16), and the triangular membership function.Since top eventU1=E5+E6+E7+E8+⋯+E30+E31+E32, Equation (18) is used to select the triangular membership function. The probability section of top event U1 varies with λ (see Figure 5 for the scattering curve).Figure 5
Probability interval distribution of top events changing withλ under the triangular membership function.The interval of each bottom event can be obtained from Table3, Equation (19), and the normal membership function.Similarly, when the normal membership function is selected, the probability of top eventU1 varies with λ (see Figure 6 for the scattering curve).Figure 6
Probability interval distribution of top events changing withλ under the normal membership function.By comparing Figures5 and 6, it can be seen that whether triangular membership function or normal membership function is selected, the probability interval of the top event becomes smaller and smaller with increased λ. When λ is equal to 1, the interval is an accurate number, consistent with the theory. However, for the normal membership function, when λ tends to 0, the fuzzy operator of the top event tends to infinity. The closer it is to 0, the greater the fuzzy probability interval of the top event. When λϵ0,1, the whole probability interval about the change of λ is nonlinear, and the absolute value of the slope of the upper and lower bounds of the top-event probability curve decreases as λ increases. That is, the closer it is to 0, the greater the changing trend of the curve; the closer it is to 1, the more gentle the change of the curve.Whenλ takes the special value of 0.01, Equation (21) is used to obtain that the probability interval of top-event occurrence is [0.4230, 0.7644] and the corresponding reliability interval of the top event is [0.4656, 0.6551]. Compared with the triangular fuzzy operator, the interval is too fuzzy and has no practical significance. Therefore, considering the fuzziness of the occurrence probability of the bottom event and the rationality of operation, the triangular fuzzy operator is selected to calculate the fuzzy probability interval of the top event.Figure5 shows that the upper and lower limits of the probability of top-event occurrence are approximately linear. Hence, the least square linear fitting and the intercept interval for the occurrence of the top event of the functional failure mode are used to obtain
(28)F~U1=0.0838λ+0.5415,−0.0694λ+0.6947.Therefore, please see as follows:(1)
Whenλ=1, the probability of the bottom event of the functional failure mode is a certain value, and that of the functional failure U1 of the high-voltage power battery system of the top event is
(29)F~U1,λ=1=0.6253Whenλ=1, the reliability of functional failure mode U1 of the high-voltage power battery system of the top event can be obtained from Equation (5).
(30)RU1,λ=1=0.5351.(2)
Whenλ=0, the probability of the bottom event of the functional-failure mode is a fuzzy number and F~U1 is an interval range, i.e., the probability interval of function-failure U1 of the high-voltage power battery system of the top event is
(31)F~U1,λ=0=0.5415,0.6947That is, the minimum probability of top eventU1 is 0.5415, and the maximum is 0.6947.Whenλ=0, the reliability interval of functional failure mode U1 of the high-voltage power battery system of the top event can be obtained from Equation (5).
(32)RU1,λ=0=0.4992,0.5819.


### 3.4.2. Quantitative Analysis of the Subfault Tree in the Deteriorated State

According to the definition of a polymorphic fault gate, whenU1 does not occur and U2 is in a degraded state, the power battery system is in a degraded state and is not recognized as failure. At this time, the high-voltage power battery system of pure electric commercial vehicles does not need immediate maintenance, consistent with the actual operation condition. When U1 occurs or U2 is in a failure state, the high-voltage power battery system is in a failure state and needs to be overhauled.The decision information of the decision-maker is collected by group decision-making because the probability of each bottom event in each state of the subfault tree of the deteriorated-state mode cannot be accurately obtained. According to the conclusion of the relationship between group size and decision-making, when the number of group decision-makers is 2-5, consensus can be obtained; moreover, 5-11 people are the most effective to draw more correct conclusions.Consequently, the work collected the probability of four polymorphic bottom events in each state from five experts and combined the language variables of expert opinions with the corresponding triangular fuzzy numbers (see Table4).Table 4
Expert opinion and fuzzy probability of the bottom event in each state.
Bottom eventE1E2E3E4State00.5100.5100.5100.51Expert 1VHGLVLHLVLVHGLVLHVLVLExpert 2VHGLVLVHGLVLVHLVLVHLVLExpert 3HVLVLVHGLVLHVLVLVHGLVLExpert 4HVLVLHLVLHLVLVHMVLExpert 5VHMVLVHMVLVHMVLVHGLVL(1) Averaging of Fuzzy Probability. According to the corresponding relationship between language variables and fuzzy probability in Table 1 and Equation (23), the mean value of the fuzzy probability of each polymorphic bottom event in each state is determined as follows:
(33)P~1−0′=0.82,0.96,1.0,P~1−0.5′=0.1,0.22,0.38,P~1−1′=0,0,0.1,P~2−0′=0.82,0.96,1.0,P~2−0.5′=0.1,0.26,0.46,P~2−1′=0,0,0.1,P~3−0′=0.82,0.96,1.0,P~3−0.5′=0.08,0.2,0.38,P~3−1′=0,0,0.1,P~4−0′=0.86,0.98,1.0,P~4−0.5′=0.1,0.24,0.42,P~4−1′=0,0,0.1.(2) Defuzzification of the Fuzzy Probability. The fuzzy probability is defuzzified according to Equation (24) to convert the obtained fuzzy probability into accurate probability, that is,
(34)P1−0′=0.935,P1−0.5′=0.230,P1−1′=0.025,P2−0′=0.935,P2−0.5′=0.270,P2−1′=0.025,P3−0′=0.935,P3−0.5′=0.215,P3−1′=0.025,P4−0′=0.955,P4−0.5′=0.250,P4−1′=0.025.(3) Normalization of exact probability. According to Equation (25), the accurate probability of polymorphic bottom events occurring in each state is normalized to obtain
(35)P1−0=0.7857,P1−0.5=0.1933,P1−1=0.0210,P2−0=0.7602,P2−0.5=0.2195,P2−1=0.0203,P3−0=0.7957,P3−0.5=0.1830,P3−1=0.0213,P4−0=0.7764,P4−0.5=0.2033,P4−1=0.0203.According to the structural characteristics of a multistate fault gate, when at least one bottom event state is 1, the system’s top-event output is in the failure state. According to the calculated occurrence probability of each state of the polymorphic bottom event, the probability of each polymorphic bottom event in state 1 is substituted into Equation (1). The occurrence probability of the sub-fault tree of the top event in the deteriorated-state mode is
(36)PU2=1−1−P1−1×1−P2−1×1−P3−1×1−P4−1=0.0804.According to Equation (5), the subfault tree reliability in the deteriorated mode can be obtained.
(37)RU2=0.9228.(1)
RU1,λ=1=0.5351 when λ=1. The reliability of the high-voltage power battery system of pure electric commercial vehicles is
(38)RT=0.4938(2)
RU1,λ=0=0.4992,0.5819 when λ=0. The reliability of the high-voltage power battery system of pure electric commercial vehicles is
(39)RT=0.4607,0.5370


## 3.4.1. Quantitative Analysis of the Subfault Tree of the Functional Failure Mode

According to the data and relevant experience and the probability of subfault tree bottom event of the functional failure mode (see Table3).Table 3
Probability of the bottom event of the functional failure mode.
Event codemαβE50.0260.00520.0052E60.0460.00920.0092E70.0160.00320.0032E80.0160.00320.0032E90.0260.00520.0052E100.0180.00360.0036E110.10.020.02E120.050.010.01E130.0160.00320.0032E140.0160.00320.0032E150.010.0020.002E160.010.0020.002E170.060.0120.012E180.120.0240.024E190.0160.00320.0032E200.050.010.01E210.0280.00560.0056E220.140.0280.028E230.020.0040.004E240.0140.00280.0028E250.0250.0050.005E260.0160.00320.0032E270.0260.00520.0052E280.020.0040.004E290.0180.00360.0036E300.0160.00320.0032E310.0140.00280.0028E320.0120.00240.0024In Table3, m represents the mean of the fuzzy number, which is a dimensionless number, indicating the size of the fuzzy probability of the occurrence of the bottom event. The larger the mean of m, the greater the fuzzy probability of the occurrence of the bottom event, and the smaller of m, the smaller the fuzzy probability of the bottom event occurring. According to the reference of [23], the upper and lower limits of αandβ take 10 to 20% of the fuzzy mean m, considered the harshness of the actual working environment of the high-voltage power battery system and the ambiguity of the probability of causing system failure, and according to Figure 1, the larger the value of α and β, the larger the fuzzy probability interval, and the reverse is the opposite, when α and β take 10% of the mean of m, and the probability interval is contained in the 20% probability interval of α and β taking m, so in order to make the probability interval better represent the fuzzy probability of the event occur and then selected α and β as 20% of m in combination with the data.The interval of each bottom event can be obtained using Table3, Equation (16), and the triangular membership function.Since top eventU1=E5+E6+E7+E8+⋯+E30+E31+E32, Equation (18) is used to select the triangular membership function. The probability section of top event U1 varies with λ (see Figure 5 for the scattering curve).Figure 5
Probability interval distribution of top events changing withλ under the triangular membership function.The interval of each bottom event can be obtained from Table3, Equation (19), and the normal membership function.Similarly, when the normal membership function is selected, the probability of top eventU1 varies with λ (see Figure 6 for the scattering curve).Figure 6
Probability interval distribution of top events changing withλ under the normal membership function.By comparing Figures5 and 6, it can be seen that whether triangular membership function or normal membership function is selected, the probability interval of the top event becomes smaller and smaller with increased λ. When λ is equal to 1, the interval is an accurate number, consistent with the theory. However, for the normal membership function, when λ tends to 0, the fuzzy operator of the top event tends to infinity. The closer it is to 0, the greater the fuzzy probability interval of the top event. When λϵ0,1, the whole probability interval about the change of λ is nonlinear, and the absolute value of the slope of the upper and lower bounds of the top-event probability curve decreases as λ increases. That is, the closer it is to 0, the greater the changing trend of the curve; the closer it is to 1, the more gentle the change of the curve.Whenλ takes the special value of 0.01, Equation (21) is used to obtain that the probability interval of top-event occurrence is [0.4230, 0.7644] and the corresponding reliability interval of the top event is [0.4656, 0.6551]. Compared with the triangular fuzzy operator, the interval is too fuzzy and has no practical significance. Therefore, considering the fuzziness of the occurrence probability of the bottom event and the rationality of operation, the triangular fuzzy operator is selected to calculate the fuzzy probability interval of the top event.Figure5 shows that the upper and lower limits of the probability of top-event occurrence are approximately linear. Hence, the least square linear fitting and the intercept interval for the occurrence of the top event of the functional failure mode are used to obtain
(28)F~U1=0.0838λ+0.5415,−0.0694λ+0.6947.Therefore, please see as follows:(1)
Whenλ=1, the probability of the bottom event of the functional failure mode is a certain value, and that of the functional failure U1 of the high-voltage power battery system of the top event is
(29)F~U1,λ=1=0.6253Whenλ=1, the reliability of functional failure mode U1 of the high-voltage power battery system of the top event can be obtained from Equation (5).
(30)RU1,λ=1=0.5351.(2)
Whenλ=0, the probability of the bottom event of the functional-failure mode is a fuzzy number and F~U1 is an interval range, i.e., the probability interval of function-failure U1 of the high-voltage power battery system of the top event is
(31)F~U1,λ=0=0.5415,0.6947That is, the minimum probability of top eventU1 is 0.5415, and the maximum is 0.6947.Whenλ=0, the reliability interval of functional failure mode U1 of the high-voltage power battery system of the top event can be obtained from Equation (5).
(32)RU1,λ=0=0.4992,0.5819.


## 3.4.2. Quantitative Analysis of the Subfault Tree in the Deteriorated State

According to the definition of a polymorphic fault gate, whenU1 does not occur and U2 is in a degraded state, the power battery system is in a degraded state and is not recognized as failure. At this time, the high-voltage power battery system of pure electric commercial vehicles does not need immediate maintenance, consistent with the actual operation condition. When U1 occurs or U2 is in a failure state, the high-voltage power battery system is in a failure state and needs to be overhauled.The decision information of the decision-maker is collected by group decision-making because the probability of each bottom event in each state of the subfault tree of the deteriorated-state mode cannot be accurately obtained. According to the conclusion of the relationship between group size and decision-making, when the number of group decision-makers is 2-5, consensus can be obtained; moreover, 5-11 people are the most effective to draw more correct conclusions.Consequently, the work collected the probability of four polymorphic bottom events in each state from five experts and combined the language variables of expert opinions with the corresponding triangular fuzzy numbers (see Table4).Table 4
Expert opinion and fuzzy probability of the bottom event in each state.
Bottom eventE1E2E3E4State00.5100.5100.5100.51Expert 1VHGLVLHLVLVHGLVLHVLVLExpert 2VHGLVLVHGLVLVHLVLVHLVLExpert 3HVLVLVHGLVLHVLVLVHGLVLExpert 4HVLVLHLVLHLVLVHMVLExpert 5VHMVLVHMVLVHMVLVHGLVL(1) Averaging of Fuzzy Probability. According to the corresponding relationship between language variables and fuzzy probability in Table 1 and Equation (23), the mean value of the fuzzy probability of each polymorphic bottom event in each state is determined as follows:
(33)P~1−0′=0.82,0.96,1.0,P~1−0.5′=0.1,0.22,0.38,P~1−1′=0,0,0.1,P~2−0′=0.82,0.96,1.0,P~2−0.5′=0.1,0.26,0.46,P~2−1′=0,0,0.1,P~3−0′=0.82,0.96,1.0,P~3−0.5′=0.08,0.2,0.38,P~3−1′=0,0,0.1,P~4−0′=0.86,0.98,1.0,P~4−0.5′=0.1,0.24,0.42,P~4−1′=0,0,0.1.(2) Defuzzification of the Fuzzy Probability. The fuzzy probability is defuzzified according to Equation (24) to convert the obtained fuzzy probability into accurate probability, that is,
(34)P1−0′=0.935,P1−0.5′=0.230,P1−1′=0.025,P2−0′=0.935,P2−0.5′=0.270,P2−1′=0.025,P3−0′=0.935,P3−0.5′=0.215,P3−1′=0.025,P4−0′=0.955,P4−0.5′=0.250,P4−1′=0.025.(3) Normalization of exact probability. According to Equation (25), the accurate probability of polymorphic bottom events occurring in each state is normalized to obtain
(35)P1−0=0.7857,P1−0.5=0.1933,P1−1=0.0210,P2−0=0.7602,P2−0.5=0.2195,P2−1=0.0203,P3−0=0.7957,P3−0.5=0.1830,P3−1=0.0213,P4−0=0.7764,P4−0.5=0.2033,P4−1=0.0203.According to the structural characteristics of a multistate fault gate, when at least one bottom event state is 1, the system’s top-event output is in the failure state. According to the calculated occurrence probability of each state of the polymorphic bottom event, the probability of each polymorphic bottom event in state 1 is substituted into Equation (1). The occurrence probability of the sub-fault tree of the top event in the deteriorated-state mode is
(36)PU2=1−1−P1−1×1−P2−1×1−P3−1×1−P4−1=0.0804.According to Equation (5), the subfault tree reliability in the deteriorated mode can be obtained.
(37)RU2=0.9228.(1)
RU1,λ=1=0.5351 when λ=1. The reliability of the high-voltage power battery system of pure electric commercial vehicles is
(38)RT=0.4938(2)
RU1,λ=0=0.4992,0.5819 when λ=0. The reliability of the high-voltage power battery system of pure electric commercial vehicles is
(39)RT=0.4607,0.5370


## 4. Fuzzy Probability Importance of the Bottom Event

Similar to the probability’s importance of the bottom event of the traditional fault tree, the fuzzy probability’s importance of the bottom event also refers to the degree to which the probability of the top event is affected when the probability of the bottom event changes. According to the definition of Equation (3), the fuzzy probability’s importance is
(40)Ihj=∂hp∂pj,j=1,2,3,⋯,n,where hp=hp1,p2,p3,⋯,pn is the unreliability function of the top event; pj the fuzzy probability of the occurrence of the jth bottom event. Triangular fuzzy numbers were selected to analyze the probability of bottom events of the subfault tree of the functional failure mode in the work. Besides, the logic gates of the subtree of the established functional failure mode are “OR” gates, so we can obtain
(41)hp=1−∏i=1n1−F~iλ1−∏i=1n1−mi−αi−λαi,1−∏i=1n1−mi+βi+λβi,(42)pj=mi−αi+λαi,mi+βi−λβi.Equations (40), (41), and (42) are used to obtain
(43)Ihj=∂hp∂pj=∏i=1,i≠jn1−mi−αi−λαi,1−mi+βi+λβi.It can be seen from the previous chapters that in the high-voltage power battery system of pure electric commercial vehicles, although the probability of each fault of the bottom event is low, the reliability of the whole system is only 0.4938. Therefore, the probability’s importance of each bottom event is vital for the prevention and improvement of top events. The bottom event of the subfault tree of the functional failure mode is taken as the research object. According to Equation (43), λ=0 is introduced to obtain the interval of probability’s importance of each bottom event of the sub-fault tree in the functional failure mode (see Table 5). Figure 7 shows the broken line of the interval of probability’s importance.Table 5
Interval of probability’s importance of the bottom event.
CodeProbability intervalCodeProbability intervalE5[0.3163, 0.4695]E6[0.3243, 0.4773]E7[0.3124, 0.4657]E8[0.3124, 0.4657]E9[0.3163, 0.4695]E10[0.3132, 0.4665]E11[0.3482, 0.4998]E12[0.3260, 0.4789]E13[0.3124, 0.4657]E14[0.3124, 0.4657]E15[0.3102, 0.4635]E16[0.3102, 0.4635]E17[0.3302, 0.4830]E18[0.3580, 0.5086]E19[0.3124, 0.4657]E20[0.3260, 0.4789]E21[0.3171, 0.4703]E22[0.3683, 0.5178]E23[0.3140, 0.4673]E24[0.3117, 0.4650]E25[0.3159, 0.4692]E26[0.3124, 0.4657]E27[0.3163, 0.4695]E28[0.3140, 0.4673]E29[0.3132, 0.4665]E30[0.3124, 0.4657]E31[0.3116, 0.4650]E32[0.3109, 0.4642]Figure 7
Broken lines of importance interval of bottom event’s probability of the high-voltage battery system.According to Table5, the fuzzy importance ranking of the bottom event can be obtained as follows.
(44)Ih22>Ih18>Ih11>Ih17>⋯>Ih16=Ih15.In terms of the event definition and code in Table2, the maximum importance interval of CAN high and low reverse connection probability is [0.3683, 0.5178], and the minimum is [0.3102, 0.4635]. The relationship between the importance of each probability is as follows: CANhighandlowreverseconnection>low‐temperatureenvironment>lowSOC>batterydeformation>⋯>sheddingofactivesubstances=chargerfault.In the light of the previous section, the higher the probability’s importance of the bottom event, the greater the impact on the high-voltage power battery system of pure electric commercial vehicles. Figure7 shows that the probability’s importance of the bottom event coded as E11,E17,E18,andE22 is higher than that of other bottom events. The corresponding bottom events are low SOC, battery deformation, low-temperature environment, CAN high, and low reverse connection. Compared with the four bottom events, the fluctuation of the probability’s importance interval of the other bottom events is relatively small, and the changing trend of the broken line chart of the corresponding probability’s importance interval is relatively flat. Therefore, these four bottom events with probability’s high importance are the weak links of the high-voltage power battery system. We should focus on the optimization and timing detection of these four bottom events.During designing, testing, and optimizing the high-voltage power battery system of pure electric commercial vehicles, the bottom event with a high probability of importance is selected according to the probability’s importance interval curve of the high-voltage battery system, which can improve the tests of the whole system to a certain extent and reduce the occurrence of serious failures.


## 5. Conclusions

Taking a pure electric commercial vehicle as an example, the work analyzed the reliability of the high-voltage power battery system. The basic theory of polymorphic fuzzy fault-tree analysis, the working principle of the high-voltage power battery system and fault type, and the polymorphic fuzzy fault tree of the high-voltage power battery system were introduced to divide the fault tree into two sub-fault trees (functional failure and deteriorated-state modes) for qualitative and quantitative analysis.The triangular membership function was selected as the fuzzy operator for the bottom event of the subfault tree in the functional failure mode by comparing the influences between the triangular membership function and the normal membership function on the probability interval of the top event. It could solve the subfault tree in a functional failure mode with better effect and obtain its reliability. For the subfault tree in the deteriorated-state mode, the expert experience and triangular fuzzy numbers were combined by the group decision theory, with the fuzzy probability averaged, defuzzified, and normalized. Accurate probability was obtained to solve the subfault tree in the deteriorated-state mode and the reliability of the high-voltage power battery system.Finally, the fuzzy probability’s importance analysis of bottom events of the functional failure mode showed that the key weak links of the system were low SOC, Battery deformation, low-temperature environment, CAN high, and low reverse connection. In system design and fault detection, checking according to the importance of probability can improve system’s fault detection. The calculation results showed that although the state deterioration mode did not play a decisive role in system failure, it is still a potential safety hazard to system failure from the reliability of the whole high-voltage power battery system as well as an indispensable detection link in the process of fault detection.



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*Source: 2455345-2022-07-13.xml*