# Reliability Analysis of Gasifier Lock Bucket Valve System Based on DBN Method

Authors: Ming Liu; Jiayue Ma; Yili Duo; Tie Sun
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In order to solve the problem of zero-failure data and dynamic failure in gasification system, a dynamic Bayesian network (DBN) combined with Monte Carlo simulations is proposed to analyze the reliability of the gasifier lock bucket valve system. On the basis of studying the structure of the gasifier lock bucket valve system, the reliability model of the system is built based on DBN, and the structure learning is realized. The Monte Carlo simulation is used for the timed ending test in Bayesian estimation, which effectively solves the problem of zero-failure data and realizes the parameter learning. Through the Metropolis-Hastings (M-Hs) algorithm, the prior distribution of dynamic node is randomly sampled to obtain the target distribution, which makes the reliability predictive inference for DBN of the gasifier lock bucket valve system faster and more accurate. The obtained reliability prediction is a curve varying with time. The results show that the valve frequent switch node of DBN of the gasifier lock bucket valve system is identified as the weak link by the powerful reverse inference for DBN, which needs to be paid more attention to. This method can effectively improve the maintenance level of the gasifier lock bucket valve system and can effectively reduce the possibility of accidents.


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## Body

## 1. Introduction

The development of the modern coal chemical industry makes modern coal gasification technology develop towards high temperature, pressure gasification, large scale, and diversification. The success of coal gasification technology selection is directly related to the long-term stable operation of coal chemical production unit [1]. Both coal water slurry gasification and pulverized coal gasification have been commercialized. In particular, SE gasification technology showed great advantages over traditional gasifiers in scaling up the installations, and it has the advantages of wide adaptability of coal, low consumption of raw materials, high carbon conversion rate, maturity, reliability, and so on. As shown in Figure 1, it is composed of coal preparation, high-pressure coal feed and transport, gasification and washing, and slag water treatment units. A high efficiency of carbon conversion of more than 99% was achieved. The technology has been combined with sulphur conversion, carbon dioxide capture, and integrated gasification combined cycle, which provides a total solution of environmental pollution in the use of coals and realizes the effective separation of fine ash from high ash coal in current industrial installations. Due to the high solid content in water and the great wear force on the spool, the leakage inside the scum lock valve needs to be replaced regularly, which is a technical problem in the industry. In the early stage, internal leakage happened in New Energy Phoenix Energy Co., Ltd., for failing to close the locking bucket pressure relief valve and the flushing water valve of bucket pressure relief pipeline. Gasifier lock bucket valve system is the core of the coal gasification unit, and its reliability study has guiding significance to ensure the safe and stable operation of the unit. The process of gasification system is complex and the correlation between devices is strong. Therefore, the research on the reliability of gasifier lock bucket valve system can provide an important theoretical basis for the reliability analysis of large coal gasification unit, and it is of great significance to quickly and effectively determine the weak links of gasification system. Thus, the research will not only improve the system design but also ensure the stable operation of gasification system and reduce the maintenance cost [2–4].Figure 1
Process flowchart of pulverized coal gasification.At present, the research on the reliability of systems using dynamic Bayesian network (DBN) is mainly in the case of known system failure data; observation data and hidden variables are discrete or subject to Gaussian distribution or by transforming fault tree model into DBN model. Ming Liu [5] applied Bayesian network to carry out static risk analysis on gasifier feeding system. Yuan Zhi [6] developed a probability estimation of dust explosions domino effect based on Bayesian networks. Leng David [7] presented a methodology to use deterministic steady-state process models to derive Bayesian networks based on alarm event detection. Khakzad Nima [8] used imprecise probabilities in Bayesian network. In order to solve the problem of risk and reliability evaluation of systems with redundant equipment, O’Connor Andrew [9] proposed a reliability evaluation method based on Bayesian network. Zarei Esmaeil [10] proposed a method combining Bow-tie diagram with Bayesian network to evaluate the security of dynamic systems. Scutari Marco [11] used DBN model to process incomplete and dynamic data. Zhi Jiang Mei [12] introduced a DBN model to facilitate the estimation of the dynamic emergency risk in sea lanes. Khakzad Nima [13, 14] and Xian Guo Wu [15] analyzed the reliability of the system based on the known system parameters and the method of transforming the fault tree model into a DBN model. S Montani [16] proposed the method of transforming dynamic fault tree into DBN.However, there are relatively few research studies on DBN with zero-failure data, continuous non-Gaussian observation data, and hidden variable data. Therefore, a method of DBN combined with Monte Carlo simulations is put forward to dynamically analyze the reliability of gasifier lock bucket valve system under zero-failure data. The dynamic reliability analysis of zero-failure data system is solved. The Metropolis-Hastings (M-Hs) algorithm amplifies acceptance rateα and contributes to the fast convergence of Markov chains. For analyzing DBN of continuous system, using M-Hs algorithm to discretize the prior distribution of dynamic node, inference for DBN can get faster and more accurate results.


## 2. Theoretical Basis for DBN and M-Hs Algorithm

### 2.1. Introduction of DBN

DBN is an extension of static Bayesian network on the time dimension; in other words, the constraint of time attribute is added into the original network structure to reflect the dynamic change trend of variables [13, 14]. DBN can efficiently process the timing information of the system based on the hidden Markov model. The initial network B0 and transition network B⟶ must be defined first.The distribution probability of the current initial state can be expressed as(1)PX0=∏i=1NPX0i|PaX0i.Betweent and t+Δt moment, the state transition probability is given as the following functional form:(2)PXt+Δt|Xt=∏i=1NPXt+Δti|PaXt+Δti.Similarly, the joint distribution of any node in DBN can be derived as presented as follows:(3)PX1:T1:N=∏i=1NPB0X1i|PaX1i∏t=2T∏i=1NPB⟶Xti|PaXti,where Xit is the value at the time t of the i’th node in DBN, PaXti is the parent node of this node, and N is the number of node variables in the network.The time attribute of Bayesian network in Figure2(a) is extended to obtain the DBN in Figure 2(b); the states of nodes A, B, and C at time t+Δt are A t+Δt, B t+Δt, and C t+Δt. DBN can be decomposed into multiple single time periods, and the characteristics of the whole time period can be obtained from the characteristics of the single time period [17].Figure 2
Instance of dynamic Bayesian model. (a) Static Bayesian model. (b) Dynamic Bayesian model.
(a)(b)


### 2.2. DBN Learning

DBN learning includes network structure learning and parameter learning. The establishment of network structure mainly includes four ways: graphic element transformation modeling, using expert knowledge to model manually, learning modeling, and mixed modeling. Parameter learning mainly includes prior probability of parent node and conditional probability, and its prior distribution algorithm mainly includes maximum likelihood estimation method, Bayesian estimation method, and maximum expectation algorithm [18].For solving DBN, the key lies in the establishment of conditional probability, and the conditional probability is divided into two kinds, the first is the static conditional probability of child node in the same time slice, and the second is conditional transition probability of transition network between different time slices. For the same time slice, the relationship between parent and child node is mainly the logical relationship of“AND” and “OR,” the establishment of the conditional probability of child node is relatively simple, and thus, it will not be described in detail. The conditional transition probability of dynamic node is mainly based on hidden Markov process. When the event only has two states of normal and failure, the conditional transition probability of dynamic node is expressed as(4)PXit+Δt=0|Xit=0=1−∫tt+Δtfxtdt,PXit+Δt=1|Xit=0=∫tt+Δtfxtdt,PXit+Δt=0|Xit=1=∫tt+Δtgxtdt,PXit+Δt=1|Xit=1=1−∫tt+Δtgxtdt,where Xi=0 represents that node Xi is in the nonoccurring state, Xi=1 represents that node Xi is in the occurring state; fxt is the failure probability density function of node Xi, and gxt is the maintenance density function of node Xi [19].


### 2.3. Inference for DBN

The main purpose of inference for DBN is to reason about the probability of the maximum possible value of the hidden variable based on a large amount of observed data [20]. The observed data are the basis of inference. Observation data can be divided into discrete observation data and continuous Gaussian observation data, and hidden variables can also be divided into discrete data [21, 22] and continuous Gaussian data [12]. In this paper, the observed data and the hidden variables are not Gauss state; therefore, the M-Hs sampling in Monte Carlo simulations is used to discretize them first, and then the reliability probability of the whole system is calculated, MATLAB R2018a programming language combined with FullBNT toolbox to achieve automatic solution.


### 2.4. Introduction of M-Hs Algorithm

Markov Chain Monte Carlo (MCMC) combines the modeling capabilities of Markov methods with the computational capabilities of Monte Carlo simulations. It is an effective tool to deal with complex statistical problems and often used in the field of Bayesian analysis for complex high-dimensional integral operations. MCMC can always obtain a convergent Markov chain, and the limit distribution of the chain will be the desired target distribution.M-Hs algorithm is the core of MCMC, of which the idea is to construct a Markov chain featuring an invariant target distributionπϑ and target distribution. According to Bayesian theory, for a given set of t data sets, the general structure is as follows:(5)πϑ|t∝Lt|ϑpϑ,where Lt|ϑ is the likelihood function of the t data sets with known parameters, which depends on the function form of the selected model; pϑ is the prior distribution of parameters; L⋅|⋅ and p⋅ are assumed to be known functions; the proportionality notation indicates that the target distribution πϑ|t is proportional to its normalization factor.Markov chains randomly selectϑ0 as the initial value. The chain sequence ϑ1,ϑ2,ϑ3,...,ϑn is generated by an appropriate kinetic equation Kϑ′|ϑ, which can generate new proposal values from the previous value. In practice, symmetric Kϑ′|ϑ is often used, which makes Kϑ′|ϑ = Kϑ|ϑ′. Metropolis et al. made the same choice in their original algorithm and asymmetric dynamics are summarized by Hastings.Here,Kϑ′|ϑ is also called the proposal distribution, and it should be selected as close to the target distribution as possible.In the j’th chain iteration, ifϑj is known, it is generally assumed that the proposal value ϑ′ becomes the new ϑj+1 value in the chain with α probability, where α is given as the following functional form:(6)α=min1,πϑ′|t⋅Kϑj|ϑ′πϑj|t⋅Kϑ′|ϑj.Otherwise, the proposal valueϑ′ will be discarded, and the new proposal value ϑj+1 will be equal to the previous proposal value; that is to say, ϑj+1=ϑj[23–28].


## 2.1. Introduction of DBN

DBN is an extension of static Bayesian network on the time dimension; in other words, the constraint of time attribute is added into the original network structure to reflect the dynamic change trend of variables [13, 14]. DBN can efficiently process the timing information of the system based on the hidden Markov model. The initial network B0 and transition network B⟶ must be defined first.The distribution probability of the current initial state can be expressed as(1)PX0=∏i=1NPX0i|PaX0i.Betweent and t+Δt moment, the state transition probability is given as the following functional form:(2)PXt+Δt|Xt=∏i=1NPXt+Δti|PaXt+Δti.Similarly, the joint distribution of any node in DBN can be derived as presented as follows:(3)PX1:T1:N=∏i=1NPB0X1i|PaX1i∏t=2T∏i=1NPB⟶Xti|PaXti,where Xit is the value at the time t of the i’th node in DBN, PaXti is the parent node of this node, and N is the number of node variables in the network.The time attribute of Bayesian network in Figure2(a) is extended to obtain the DBN in Figure 2(b); the states of nodes A, B, and C at time t+Δt are A t+Δt, B t+Δt, and C t+Δt. DBN can be decomposed into multiple single time periods, and the characteristics of the whole time period can be obtained from the characteristics of the single time period [17].Figure 2
Instance of dynamic Bayesian model. (a) Static Bayesian model. (b) Dynamic Bayesian model.
(a)(b)


## 2.2. DBN Learning

DBN learning includes network structure learning and parameter learning. The establishment of network structure mainly includes four ways: graphic element transformation modeling, using expert knowledge to model manually, learning modeling, and mixed modeling. Parameter learning mainly includes prior probability of parent node and conditional probability, and its prior distribution algorithm mainly includes maximum likelihood estimation method, Bayesian estimation method, and maximum expectation algorithm [18].For solving DBN, the key lies in the establishment of conditional probability, and the conditional probability is divided into two kinds, the first is the static conditional probability of child node in the same time slice, and the second is conditional transition probability of transition network between different time slices. For the same time slice, the relationship between parent and child node is mainly the logical relationship of“AND” and “OR,” the establishment of the conditional probability of child node is relatively simple, and thus, it will not be described in detail. The conditional transition probability of dynamic node is mainly based on hidden Markov process. When the event only has two states of normal and failure, the conditional transition probability of dynamic node is expressed as(4)PXit+Δt=0|Xit=0=1−∫tt+Δtfxtdt,PXit+Δt=1|Xit=0=∫tt+Δtfxtdt,PXit+Δt=0|Xit=1=∫tt+Δtgxtdt,PXit+Δt=1|Xit=1=1−∫tt+Δtgxtdt,where Xi=0 represents that node Xi is in the nonoccurring state, Xi=1 represents that node Xi is in the occurring state; fxt is the failure probability density function of node Xi, and gxt is the maintenance density function of node Xi [19].


## 2.3. Inference for DBN

The main purpose of inference for DBN is to reason about the probability of the maximum possible value of the hidden variable based on a large amount of observed data [20]. The observed data are the basis of inference. Observation data can be divided into discrete observation data and continuous Gaussian observation data, and hidden variables can also be divided into discrete data [21, 22] and continuous Gaussian data [12]. In this paper, the observed data and the hidden variables are not Gauss state; therefore, the M-Hs sampling in Monte Carlo simulations is used to discretize them first, and then the reliability probability of the whole system is calculated, MATLAB R2018a programming language combined with FullBNT toolbox to achieve automatic solution.


## 2.4. Introduction of M-Hs Algorithm

Markov Chain Monte Carlo (MCMC) combines the modeling capabilities of Markov methods with the computational capabilities of Monte Carlo simulations. It is an effective tool to deal with complex statistical problems and often used in the field of Bayesian analysis for complex high-dimensional integral operations. MCMC can always obtain a convergent Markov chain, and the limit distribution of the chain will be the desired target distribution.M-Hs algorithm is the core of MCMC, of which the idea is to construct a Markov chain featuring an invariant target distributionπϑ and target distribution. According to Bayesian theory, for a given set of t data sets, the general structure is as follows:(5)πϑ|t∝Lt|ϑpϑ,where Lt|ϑ is the likelihood function of the t data sets with known parameters, which depends on the function form of the selected model; pϑ is the prior distribution of parameters; L⋅|⋅ and p⋅ are assumed to be known functions; the proportionality notation indicates that the target distribution πϑ|t is proportional to its normalization factor.Markov chains randomly selectϑ0 as the initial value. The chain sequence ϑ1,ϑ2,ϑ3,...,ϑn is generated by an appropriate kinetic equation Kϑ′|ϑ, which can generate new proposal values from the previous value. In practice, symmetric Kϑ′|ϑ is often used, which makes Kϑ′|ϑ = Kϑ|ϑ′. Metropolis et al. made the same choice in their original algorithm and asymmetric dynamics are summarized by Hastings.Here,Kϑ′|ϑ is also called the proposal distribution, and it should be selected as close to the target distribution as possible.In the j’th chain iteration, ifϑj is known, it is generally assumed that the proposal value ϑ′ becomes the new ϑj+1 value in the chain with α probability, where α is given as the following functional form:(6)α=min1,πϑ′|t⋅Kϑj|ϑ′πϑj|t⋅Kϑ′|ϑj.Otherwise, the proposal valueϑ′ will be discarded, and the new proposal value ϑj+1 will be equal to the previous proposal value; that is to say, ϑj+1=ϑj[23–28].


## 3. Case Studies

### 3.1. Construction of DBN for the Gasifier Lock Bucket Valve System

For familiar fields, the structure of DBN can be constructed with common sense and expert knowledge, while for less familiar fields, the algorithm of mining network structure from mass data is used. This paper is based on expert knowledge and literature [29, 30]; the logic relationship between gasifier lock bucket valve system and events is obtained. At the same time, the time attribute is added to build the DBN for the gasifier lock bucket valve system, as shown in Figure 3, and the nodes in Figure 3 are described in Table 1 based on expert knowledge from the previous studies [2, 31–37].Figure 3
The DBN structure of the gasifier lock bucket valve system.Table 1
The DBN nodes of the gasifier lock bucket valve system.
NodeName of the nodeReferenceTGasifier lock bucket valve[2]M1Sealing surface wear[31]M2Switch not in place[32, 33]X1High-pressure pulverized coal scour[2, 31, 34]X2Valve ball and ball seat wear[31]X3Ball seat stick together[2]X4Deformation and wear of interior parts[31, 34]X5Pulverized coal accumulation[2]X6The actuator output torque is insufficient[33, 34]X7Bearing internal adhesion under high pressure[2, 31, 34]X8The pulverized coal enters during the switching process[35]X9Water quality is poor[31]X10Valve frequent switch[2, 33]X11Short time to collect slag[32, 36, 37]X12Pressure reducing too fast results in three-phase current scour[32]


### 3.2. Prior Parameters of DBN of the Gasifier Lock Bucket Valve System

According to the failure form of the parent nodes, the failure mode of each node of the gasifier lock bucket valve system follows exponential distribution, and the conditional probability distribution of the child node follows the Weibull distribution. In the case of zero-failure data of the gasifier lock bucket valve system, the prior parameter of the gasifier lock bucket valve node is given by the expert knowledge and is the failure rate of the gasifier lock bucket valve asλT=0.0002. The prior parameters of other nodes are obtained by Monte Carlo simulation [23] combined with Bayesian estimation with zero-failure data [38–41]. Monte Carlo simulation is used in the timed ending test of Bayesian estimation. The following are the steps after adding Monte Carlo simulation to the timed ending test:Step 1. Set n, tk, and tii=1,2,3,…,k; here, 0<t1<t2<...<tk.Step 2. From the uniform distribution of [0,1] random sampling n times, the failure probability pn is obtained.Step 3. If pn≤pk, then the sample is reliable. The number of statistically reliable samples is called ni. It is derived Sj from(7)Sj=∑i=jknij=1,2,3,…,k,wheren is the number of samples per group; tk is the maximum duration of the timed ending test; ti is the time when the timed ending test stops; i is the number of groups of samples; Pk is the upper limit of failure probability; Sj is the data sets of reliable samples of the system.Since long-time operational performances show that the lifetime of IGSP burners is about 150∼200 days [42], and thus, the experiment maximum duration is tk=4000h at the timed ending test, and the node state is observed at every 400 h interval. In Bayesian estimation, the upper bound of failure probability Pk is the failure probability of gasifier lock bucket valve node at the time tk, and thus, Pk=0.5507. The data sets Sj obtained from simulation are put into Bayes estimation to calculate the prior parameters except node T, and the calculated results are in Tables 2 and 3 according to Table 1.Table 2
Failure rates of prior distribution of each node of the gasifier lock bucket valve system.
NodeName of the nodeFailure rate of prior distribution per hourλ/hTGasifier lock bucket valveλT=2E−4M1Sealing surface wearλM1=1.300006E−4M2Switch not in placeλM2=3.062017E−4X1High-pressure pulverized coal scourλX1=1.162172E−4X2Valve ball and ball seat wearλX2=1.295640E−4X3Ball seat stick togetherλX3=9.560840E−5X4Deformation and wear of interior partsλX4=9.465855E−5X5Pulverized coal accumulationλX5=9.561313E−5X6The actuator output torque is insufficientλX6=2.403717E−4X7Bearing internal adhesion under high pressureλX7=2.409631E−4X8The pulverized coal enters during the switching processλX8=1.487188E−4X9Water quality is poorλX9=1.369641E−4X10Valve frequent switchλX10=1.182760E−4X11Short time to collect slagλX11=2.571400E−4X12Pressure reducing too fast results in three-phase current scourλX12=1.337814E−4Table 3
Prior parameters of conditional probability distribution of child node of the gasifier bucket valve system.
NameScale parameterηShape parametermM1|X1116674.0689580.578119M1|X28198.9069861.699551M1|X338654.6686090.814707M2|X412986.5796951.060321M2|X528675.0895170.6756069M2|X6262490.8149390.368299M2|X79884.3707801.266581T|X810842.8283851.108357T|X911372.3327261.227113T|X10525041.6301310.337885T|X111436454.727950.301676T|X1212398.8192821.177966T|M190198.3269920.535338T|M263058.3549260.584732Assuming that each node includes two states of work (0) and failure (1), the prior distribution of parent node of X1 is randomly sampled by the direct sampling method. And the prior probability of parent node of X1 is obtained as(8)PX1=0=0.965554,PX1=1=0.034446.Sampling timet0 = 301.619201 h is substituted into prior distributions of other nodes. The nodes in the gasifier lock bucket valve system are independent of each other, and thus, the conditional probability in the same time slice of child node can be obtained by means of OR gate logic relation and Bayes theorem [43]:(9)PM|X1,X2,…,Xn=PM|X1.PM|X2...PM|Xn.


### 3.3. Sampling by M-Hs Algorithm

In this paper, the dynamic nodes of the gasifier lock bucket valve system areX1,X2,...,X12. To highlight the changing trend of system reliability and the importance of the dynamic node, the maintenance factor is not considered in this part, so the maintenance density function is 0. According to (4) and Table 2, t moment to t+Δt conditional transition probability of dynamic node can be obtained as follows:(10)PXit+Δt=0|Xit=0=e−λΔt,PXit+Δt=1|Xit=0=1−e−λΔt,PXit+Δt=0|Xit=1=0,PXit+Δt=1|Xit=1=1.According to (10) and Bayes formula [39], the probability density function (PDF) of the target distribution of the dynamic nodes of the gasifier lock bucket valve system can be found as(11)πλ∝λe−λt.The normal distribution is used as proposal distribution forϑ to generate samples from the given target distribution. The following steps are required to generate posterior samples of the target distribution using M-Hs algorithm:Step 1. Select an initial guess of ϑ and as ϑ0.Step 2. Set t=0,1,2,3...T.Step 3. Generate ϑ′ using the proposal Nϑt−1,σ2 distribution.Step 4. Compute α=min1,πϑ′|λ⋅Kϑt−1|ϑ′/πϑt−1|λ⋅Kϑ′|ϑt−1.Step 5. Then, generate a sample u from the uniform distribution of [0, 1].Step 6. If u≤α, accept the proposal and set ϑt=ϑ′, else set ϑt=ϑt−1.Step 7. Repeat steps (3∼6) T times and collect an adequate number of samples.Taking X7 as an example, it can be seen from Table2 that the failure rate of node X7 is λX7=2.409631E−4 and the number of iterations is set as 1E5. In Figure 4, the black dashed line is the posterior distribution after sampling of target distribution of X7, and the red solid line is the target distribution before sampling of X7. The target distribution after sampling is highly consistent with the target distribution before sampling in the early stage, and the accuracy of exact inference for DBN can be further improved through DBN updating. Figure 5 shows the convergence function curve of the Markov chain changing with the number of iterations and converges after 5E4∼6E4 iteration. Figure 6 shows the target distribution PDF after sampling, which can accurately describe the curve of the probability density function of X7.Figure 4
Target distribution of X7 is compared before and after sampling.Figure 5
X7’s Markov chain converges with the number of iterations.Figure 6
The target distribution PDF of X7 sampled data.


### 3.4. System Reliability Analysis

The dynamic prediction of the reliability change of the gasifier lock bucket valve system within 0∼300 h is carried out. Take the number of time slices as 10, and then the time interval of each slice is 30 h. Based on the prior probability of parent nodes, the conditional probability of child nodes, and the conditional transition probability of dynamic nodes, the reliability prediction of the gasifier lock bucket valve system is obtained by the predictive inference for DBN. As shown in Figure7, the reliability of the gasifier lock bucket valve system and its subsystems gradually decreases with the increase of time, and the decrease rate is in turn T > M2 > M1. The reliability prediction of the gasifier lock bucket valve system when maintenance factors are taken into account is shown in Figure 8. It can be seen that maintaining the system equipment can reduce the possibility of accidents occurring in the operation process, improve the reliability of the system, and extend the operation cycle of the equipment. The importance of maintenance measures for safe operation of equipment can be seen.Figure 7
Reliability dynamic changes.Figure 8
Reliability dynamic changes with and without maintenance.According to the powerful reverse inference for DBN, the failure probability of the nodeT of the gasifier lock bucket valve system is set as 1, the posterior probability of each node of the system is obtained, and the weak links are analyzed. After 1000 h of operation, prior probability, posterior probability, and their differences of parent nodes are compared, as shown in Figure 9. In order to improve the reliability of the gasifier lock bucket valve system, the order of attention of each node is X10 > X11 > X6 > X8 > X9 > X7 > X4 > X1 > X3 > X2 > X5 > X12. Nodes X6, X8, X10, and X11 have a great impact on system reliability, and X10 (valve frequent switch) is the weak link of the system.Figure 9
Comparison of prior probability, posterior probability, and their differences of parent nodes.In order to verify the feasibility and accuracy of the method proposed in this paper, GeNIe 2.1 software [26] is used to compare and analyze gasifier lock bucket valve system.The conditional transition probability of dynamic nodes is obtained by (10). Taking dynamic node X1 as an example and making Δt=1h, the conditional transition probability of dynamic node X1 from moment t to t+Δt is shown in Table 4.Table 4
Conditional transition probability table of dynamic node X1.
No. 1X1(t) = 0X1(t) = 1X1(t+1) = 00.9998840X1(t+1) = 10.0001161The prior probability of parent nodes, the conditional probability of child nodes, and the conditional transition probability of dynamic nodes in Table4 are input into the DBN model of the gasifier lock bucket valve system established by GeNIe 2.1. The result is shown in Figure 10.Figure 10
DBN model of the gasifier lock bucket valve system created by GeNIe 2.1.The reliability solution results of the two methods at multiple time points are compared and analyzed as shown in Table5. As can be seen from Table 5, as time goes by, the difference between the two methods fluctuates between 0.005 and 0.0106, with the maximum difference only being 0.010501, which verifies the feasibility of DBN combined with the Monte Carlo method proposed in this paper and has higher calculation accuracy. The calculation process of this paper can be shown in Figure 11.Table 5
Comparison of solution result.
30 h120 h210h300 hDBN MC0.9582550.8366450.7262810.623723GeNIe 2.10.9533730.8261440.7158960.620365Difference value0.0048810.0105010.0103850.003358Figure 11
Technical route.


## 3.1. Construction of DBN for the Gasifier Lock Bucket Valve System

For familiar fields, the structure of DBN can be constructed with common sense and expert knowledge, while for less familiar fields, the algorithm of mining network structure from mass data is used. This paper is based on expert knowledge and literature [29, 30]; the logic relationship between gasifier lock bucket valve system and events is obtained. At the same time, the time attribute is added to build the DBN for the gasifier lock bucket valve system, as shown in Figure 3, and the nodes in Figure 3 are described in Table 1 based on expert knowledge from the previous studies [2, 31–37].Figure 3
The DBN structure of the gasifier lock bucket valve system.Table 1
The DBN nodes of the gasifier lock bucket valve system.
NodeName of the nodeReferenceTGasifier lock bucket valve[2]M1Sealing surface wear[31]M2Switch not in place[32, 33]X1High-pressure pulverized coal scour[2, 31, 34]X2Valve ball and ball seat wear[31]X3Ball seat stick together[2]X4Deformation and wear of interior parts[31, 34]X5Pulverized coal accumulation[2]X6The actuator output torque is insufficient[33, 34]X7Bearing internal adhesion under high pressure[2, 31, 34]X8The pulverized coal enters during the switching process[35]X9Water quality is poor[31]X10Valve frequent switch[2, 33]X11Short time to collect slag[32, 36, 37]X12Pressure reducing too fast results in three-phase current scour[32]


## 3.2. Prior Parameters of DBN of the Gasifier Lock Bucket Valve System

According to the failure form of the parent nodes, the failure mode of each node of the gasifier lock bucket valve system follows exponential distribution, and the conditional probability distribution of the child node follows the Weibull distribution. In the case of zero-failure data of the gasifier lock bucket valve system, the prior parameter of the gasifier lock bucket valve node is given by the expert knowledge and is the failure rate of the gasifier lock bucket valve asλT=0.0002. The prior parameters of other nodes are obtained by Monte Carlo simulation [23] combined with Bayesian estimation with zero-failure data [38–41]. Monte Carlo simulation is used in the timed ending test of Bayesian estimation. The following are the steps after adding Monte Carlo simulation to the timed ending test:Step 1. Set n, tk, and tii=1,2,3,…,k; here, 0<t1<t2<...<tk.Step 2. From the uniform distribution of [0,1] random sampling n times, the failure probability pn is obtained.Step 3. If pn≤pk, then the sample is reliable. The number of statistically reliable samples is called ni. It is derived Sj from(7)Sj=∑i=jknij=1,2,3,…,k,wheren is the number of samples per group; tk is the maximum duration of the timed ending test; ti is the time when the timed ending test stops; i is the number of groups of samples; Pk is the upper limit of failure probability; Sj is the data sets of reliable samples of the system.Since long-time operational performances show that the lifetime of IGSP burners is about 150∼200 days [42], and thus, the experiment maximum duration is tk=4000h at the timed ending test, and the node state is observed at every 400 h interval. In Bayesian estimation, the upper bound of failure probability Pk is the failure probability of gasifier lock bucket valve node at the time tk, and thus, Pk=0.5507. The data sets Sj obtained from simulation are put into Bayes estimation to calculate the prior parameters except node T, and the calculated results are in Tables 2 and 3 according to Table 1.Table 2
Failure rates of prior distribution of each node of the gasifier lock bucket valve system.
NodeName of the nodeFailure rate of prior distribution per hourλ/hTGasifier lock bucket valveλT=2E−4M1Sealing surface wearλM1=1.300006E−4M2Switch not in placeλM2=3.062017E−4X1High-pressure pulverized coal scourλX1=1.162172E−4X2Valve ball and ball seat wearλX2=1.295640E−4X3Ball seat stick togetherλX3=9.560840E−5X4Deformation and wear of interior partsλX4=9.465855E−5X5Pulverized coal accumulationλX5=9.561313E−5X6The actuator output torque is insufficientλX6=2.403717E−4X7Bearing internal adhesion under high pressureλX7=2.409631E−4X8The pulverized coal enters during the switching processλX8=1.487188E−4X9Water quality is poorλX9=1.369641E−4X10Valve frequent switchλX10=1.182760E−4X11Short time to collect slagλX11=2.571400E−4X12Pressure reducing too fast results in three-phase current scourλX12=1.337814E−4Table 3
Prior parameters of conditional probability distribution of child node of the gasifier bucket valve system.
NameScale parameterηShape parametermM1|X1116674.0689580.578119M1|X28198.9069861.699551M1|X338654.6686090.814707M2|X412986.5796951.060321M2|X528675.0895170.6756069M2|X6262490.8149390.368299M2|X79884.3707801.266581T|X810842.8283851.108357T|X911372.3327261.227113T|X10525041.6301310.337885T|X111436454.727950.301676T|X1212398.8192821.177966T|M190198.3269920.535338T|M263058.3549260.584732Assuming that each node includes two states of work (0) and failure (1), the prior distribution of parent node of X1 is randomly sampled by the direct sampling method. And the prior probability of parent node of X1 is obtained as(8)PX1=0=0.965554,PX1=1=0.034446.Sampling timet0 = 301.619201 h is substituted into prior distributions of other nodes. The nodes in the gasifier lock bucket valve system are independent of each other, and thus, the conditional probability in the same time slice of child node can be obtained by means of OR gate logic relation and Bayes theorem [43]:(9)PM|X1,X2,…,Xn=PM|X1.PM|X2...PM|Xn.


## 3.3. Sampling by M-Hs Algorithm

In this paper, the dynamic nodes of the gasifier lock bucket valve system areX1,X2,...,X12. To highlight the changing trend of system reliability and the importance of the dynamic node, the maintenance factor is not considered in this part, so the maintenance density function is 0. According to (4) and Table 2, t moment to t+Δt conditional transition probability of dynamic node can be obtained as follows:(10)PXit+Δt=0|Xit=0=e−λΔt,PXit+Δt=1|Xit=0=1−e−λΔt,PXit+Δt=0|Xit=1=0,PXit+Δt=1|Xit=1=1.According to (10) and Bayes formula [39], the probability density function (PDF) of the target distribution of the dynamic nodes of the gasifier lock bucket valve system can be found as(11)πλ∝λe−λt.The normal distribution is used as proposal distribution forϑ to generate samples from the given target distribution. The following steps are required to generate posterior samples of the target distribution using M-Hs algorithm:Step 1. Select an initial guess of ϑ and as ϑ0.Step 2. Set t=0,1,2,3...T.Step 3. Generate ϑ′ using the proposal Nϑt−1,σ2 distribution.Step 4. Compute α=min1,πϑ′|λ⋅Kϑt−1|ϑ′/πϑt−1|λ⋅Kϑ′|ϑt−1.Step 5. Then, generate a sample u from the uniform distribution of [0, 1].Step 6. If u≤α, accept the proposal and set ϑt=ϑ′, else set ϑt=ϑt−1.Step 7. Repeat steps (3∼6) T times and collect an adequate number of samples.Taking X7 as an example, it can be seen from Table2 that the failure rate of node X7 is λX7=2.409631E−4 and the number of iterations is set as 1E5. In Figure 4, the black dashed line is the posterior distribution after sampling of target distribution of X7, and the red solid line is the target distribution before sampling of X7. The target distribution after sampling is highly consistent with the target distribution before sampling in the early stage, and the accuracy of exact inference for DBN can be further improved through DBN updating. Figure 5 shows the convergence function curve of the Markov chain changing with the number of iterations and converges after 5E4∼6E4 iteration. Figure 6 shows the target distribution PDF after sampling, which can accurately describe the curve of the probability density function of X7.Figure 4
Target distribution of X7 is compared before and after sampling.Figure 5
X7’s Markov chain converges with the number of iterations.Figure 6
The target distribution PDF of X7 sampled data.


## 3.4. System Reliability Analysis

The dynamic prediction of the reliability change of the gasifier lock bucket valve system within 0∼300 h is carried out. Take the number of time slices as 10, and then the time interval of each slice is 30 h. Based on the prior probability of parent nodes, the conditional probability of child nodes, and the conditional transition probability of dynamic nodes, the reliability prediction of the gasifier lock bucket valve system is obtained by the predictive inference for DBN. As shown in Figure7, the reliability of the gasifier lock bucket valve system and its subsystems gradually decreases with the increase of time, and the decrease rate is in turn T > M2 > M1. The reliability prediction of the gasifier lock bucket valve system when maintenance factors are taken into account is shown in Figure 8. It can be seen that maintaining the system equipment can reduce the possibility of accidents occurring in the operation process, improve the reliability of the system, and extend the operation cycle of the equipment. The importance of maintenance measures for safe operation of equipment can be seen.Figure 7
Reliability dynamic changes.Figure 8
Reliability dynamic changes with and without maintenance.According to the powerful reverse inference for DBN, the failure probability of the nodeT of the gasifier lock bucket valve system is set as 1, the posterior probability of each node of the system is obtained, and the weak links are analyzed. After 1000 h of operation, prior probability, posterior probability, and their differences of parent nodes are compared, as shown in Figure 9. In order to improve the reliability of the gasifier lock bucket valve system, the order of attention of each node is X10 > X11 > X6 > X8 > X9 > X7 > X4 > X1 > X3 > X2 > X5 > X12. Nodes X6, X8, X10, and X11 have a great impact on system reliability, and X10 (valve frequent switch) is the weak link of the system.Figure 9
Comparison of prior probability, posterior probability, and their differences of parent nodes.In order to verify the feasibility and accuracy of the method proposed in this paper, GeNIe 2.1 software [26] is used to compare and analyze gasifier lock bucket valve system.The conditional transition probability of dynamic nodes is obtained by (10). Taking dynamic node X1 as an example and making Δt=1h, the conditional transition probability of dynamic node X1 from moment t to t+Δt is shown in Table 4.Table 4
Conditional transition probability table of dynamic node X1.
No. 1X1(t) = 0X1(t) = 1X1(t+1) = 00.9998840X1(t+1) = 10.0001161The prior probability of parent nodes, the conditional probability of child nodes, and the conditional transition probability of dynamic nodes in Table4 are input into the DBN model of the gasifier lock bucket valve system established by GeNIe 2.1. The result is shown in Figure 10.Figure 10
DBN model of the gasifier lock bucket valve system created by GeNIe 2.1.The reliability solution results of the two methods at multiple time points are compared and analyzed as shown in Table5. As can be seen from Table 5, as time goes by, the difference between the two methods fluctuates between 0.005 and 0.0106, with the maximum difference only being 0.010501, which verifies the feasibility of DBN combined with the Monte Carlo method proposed in this paper and has higher calculation accuracy. The calculation process of this paper can be shown in Figure 11.Table 5
Comparison of solution result.
30 h120 h210h300 hDBN MC0.9582550.8366450.7262810.623723GeNIe 2.10.9533730.8261440.7158960.620365Difference value0.0048810.0105010.0103850.003358Figure 11
Technical route.


## 4. Conclusion

(1)
The Monte Carlo simulation sampled the timed ending test in Bayesian estimation that the prior data of the gasifier lock bucket valve system can be obtained. M-Hs algorithm samples the prior distribution of dynamic nodes and gets the target distribution, which is highly consistent with the actual target distribution and provides an effective method to solve the change of conditional transition probability with time.(2)
The reliability analysis of gasifier lock bucket valve in two directions is carried out by the exact inference for DBN. The reliability variation trend of this system and the factors causing system failure can be obtained by predictive inference. Reverse inference for DBN of the system can be used to obtain the important attention sequence and the weak links of the nodes that cause the failure of the gasifier lock bucket valve system and the weak link is the valve frequent switch.(3)
Based on DBN and Monte Carlo simulation, reliability analysis of gasifier lock bucket valve system is carried out in this paper. The practical problems such as the difficulty of reliability analysis of dynamic system, zero-failure probability, and the inconformity of observed data and hidden variables with Gaussian distribution can be effectively solved by this method. The maximum difference between this method and GeNIe 2.1 is only 0.010501.(4)
This paper focuses on the monitoring and analysis of location and attribute information of workers and mechanical equipment, while in the aspect of microbehavior monitoring of workers and machinery, only a preliminary discussion is made, and further research is needed in the future.



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*Source: 8840545-2021-03-30.xml*