An Improved Prediction Method for Failure Probability of Natural Gas Pipeline Based on Multi-Layer Bayesian Network

Abstract

The failure probability of a pipeline is a quantification of the likelihood of an accident occurring in the pipeline, which is an indispensable part of the pipeline risk assessment process. To solve the problems of strong subjectivity, low feasibility, and low accuracy in the existing pipeline failure probability calculation methods, a three-layer Bayesian network topology model of “pipeline failure–failure cause–influencing factor” is proposed, with the pipeline failure as the subnode, the type of pipeline failure as the intermediate node, and the factors affecting the pipeline failure as the parent node of the network. Based on data fitting and fuzzy theory analysis methods, the functional relationship between the impact factor and the failure frequency of various pipelines is quantified. Using the mean value theorems for definite integrals and the analytic hierarchy process, the conditional probability of the directed edge in the network is calculated. The proposed function relationship provides a method to calculate the prior probability according to the parameters of the pipeline and its surroundings and a new idea to train the network model even without sufficient data.

1. Introduction

Natural gas pipelines play a very important role in the Chinese energy supply system. It is of great significance to prevent failures and ensure efficient and stable operation of the natural gas pipeline system [1,2]. Accurate assessments of failure probability will help design and maintain pipelines, which is a crucial part of pipeline risk assessment [3,4].

In the last few decades, numerous studies have been carried out on the failure mechanism of pipeline systems. Shan et al. [5] proposed a method for evaluating the failure probability of gas pipelines based on failure data and correction factors. In their work, the basic failure probability of the pipeline is calculated through statistical analysis of pipeline failure data; an index system for pipeline correction is established, and the indicators are classified and quantified according to the nature of the indicators and the difficulty of quantification. Vianello et al. [6,7] analyzed the failure data information in the European Gas Pipeline Incident Data Group (EGIG) database, revealed the relationship between failure frequency and pipeline parameters, and proposed a novel risk assessment method that has been applied to Italy’s pipe network. Witek et al. [8] employed Det Norske Veritas (DNV) criteria and limit state functions to determine the ultimate pressure and the failure probability of individual defects and the Monte Carlo method to estimate the failure probability of pipelines. Ferdous et al. [9] analyzed the factors that may cause the failure of the pipe system, established a logical relationship between events by “and” gate and “or” gate, and developed a top-down fault tree model.

For the failure probability assessment methods proposed above, each of them has its advantages and disadvantages. The index system method is simple and easy to use, but it is highly subjective. The statistical correction method based on failure data has been widely recognized, but the determination of the correction factor is not uniform and ignores the influence of indicators without statistical data on pipeline failure frequency. The Monte Carlo method and other mathematical analysis approaches are more accurate in risk assessment, but the evaluation models are complex, professional, and difficult to understand. Based on the fault tree analysis method, the layered factors affecting the occurrence of accidents can be reflected and the basic set of pipeline failure events can be located, but the fault tree is limited by static properties and can only be analyzed by dimorphism [10,11].

To objectively and accurately calculate the failure probability of new pipelines and old pipelines with missing data, an improved method to calculate the failure probability of natural gas pipelines is proposed. Based on the Bayesian network inference [12,13,14,15,16], this paper uses the analytic hierarchy process to calculate the conditional probability of directed edge based on the integral median value theorem and provides a method for calculating the prior probability of the parent node in the case of insufficient data.

2. Bayesian Network Model of Gas Pipeline Failure Probability

Bayesian network was initially proposed and developed to solve the problem of uncertainty reasoning, which mainly uses a directed acyclic graph to represent the dependence of variables, where nodes and directed edges are employed to indicate the variable and function relationships or dependence between them, respectively [17,18,19]. Based on Bayes’ theorem, the function relationship between variables is explored, the probability of a variable is calculated in the case of a new event [20], and the probability distribution is used to predict new data. The mathematical expression of Bayes’ theorem [21] can be expressed as

$$P(y|x_i)={\displaystyle \frac{P(x_i|y)\times P(y)}{P(x_i)}}$$

(1)

where P(y|x_i) is the probability of event y occurring under the condition that a new event x_i is observed; P(x_i|y) is the probability of event x_i occurring under the condition that event y occurs; P(y) is the prior probability of event x_i, the probability of its occurrence without considering any aspect of x_i; and P(x) is the prior probability of event x_i. Equation (1) describes the probability of an event occurring when relevant information is known.

The calculation of the failure probability of natural gas pipelines based on the Bayesian network includes two parts: constructing a Bayesian network framework and calculating the Bayesian network parameters [22,23]. The constructing Bayesian network framework includes computing natural gas pipeline failure factors and drawing the gas pipeline failure factor tree map. The calculation of the Bayesian network parameters includes computing the conditional probability of the directed edge and the prior probability of the parent node.

3. Construction of the Bayesian Network Structure

The United States Pipeline and Hazardous Materials Safety Administration database, the European natural gas pipeline accident data group database [24], and the domestic pipeline failure database were employed as the databases of this work. The single and combined causes of pipeline system failures are investigated [25,26,27,28,29,30]. Figure 1 shows the constructed natural gas pipeline failure factor tree map. As seen in the figure, failure causes included corrosion, external interference, construction defects/material failure, natural force damage, misoperation, and others. All the influencing factors that may lead to pipe failure were taken into account, resulting in a total of 51 influencing factors from the right to the left of the figure. Moreover, the pipeline failure was taken as the target node, and the influence factors were set as parent nodes.

According to the correspondence between the failure factor tree map and the Bayesian network, the Bayesian network model was constructed, and the detailed steps are as follows:

(1)

In the failure factor tree map, the pipeline failure is mapped to the subnode, the failure causes are mapped to the intermediate node, and the impact factors are mapped to the parent nodes, and each node is labeled;

(2)

The function relationship in the failure factor dendrogram is mapped to the directed edge in the Bayesian network;

(3)

The state of the influencing factor is the prior probability of the node in the Bayesian network, and the function relationship between the nodes is the conditional probability of the edges in the Bayesian network.

Based on the above steps, a three-layer Bayesian network topology model of “pipeline failure–failure causes–influencing factors” was established, as shown in Figure 2.

4. Determination of Bayesian Network Parameters

A flowchart for calculating the failure probability of a natural gas pipeline based on a Bayesian network is shown in Figure 3. The Bayesian network parameters that were determined were the conditional probability of the directed edge and the prior probability of the parent node [4]. The detailed steps for calculating the parameters are as follows:

4.1. Solution Procedure

(1)

Categorical impact factors: According to the nature of the impact factors and the difficulty of quantification, 51 impact factors were classified. Pipe diameter, wall thickness, buried depth, pipe age, anticorrosion layer type, and pipeline steel grade were the quantitative factors, which can directly quantify the function relationship between this factor and pipeline failure frequency, and the rest were nonquantitative factors.

(2)

The functional relationship between the impact factors and the pipeline failure frequency was quantified. For quantitative factors, the factors were fit as a function of the pipeline failure frequency; for nonquantitative factors, we referred to the relevant standard manuals [26,27,28,29,30] and the literature to divide the evaluation criteria of the impact factor and establish the membership function correspondence between the grade evaluation criteria and the fuzzy number [31]. We then converted the fuzzy number into fuzzy probability through Equations (2) and (3) and fit the function relationship between the fuzzy number and the fuzzy probability.

(3)

The conditional probabilities were calculated. The optimal fitting function was filtered, the median value of the fitting function was calculated, the median value judgment matrix was constructed, the maximum eigenvalue and the corresponding eigenvector of the matrix were calculated, and the conditional probability of the directed edge from the parent node to the subnode was calculated.

(4)

The pipeline prior probability was calculated. According to the pipeline parameters and the surrounding environment of the pipeline, the failure probability of the corresponding influencing factor in the parent node was calculated according to the optimal fitting function in step (2), which is the prior probability.

Based on the natural gas pipeline failure data sample in the EGIG database, the function relationship between the quantitative factor and the pipeline failure frequency was fitted. Figure 4 illustrates the fitting diagram of the relationship between the pipe age and the corrosion failure frequency, and Figure 5 demonstrates the fitting diagram of the relationship between the pipe diameter and the failure frequency of misoperation. Since the nonquantitative impact factor rating could not be directly fitted, the detailed steps of quantitative analysis based on a five-level natural language expression scale and fuzzy theory [32] analysis were as follows:

(1)

Based on a five-level natural language expression scale, namely “excellent”, “good”, “medium”, “poor” and “inferior”, the corresponding evaluation grades were “I”, “II”, “III”, “IV”, and “V”.

(2)

The quantization level was an ambiguous number. A link was established between rank and triangular fuzzy numbers [31,32], with grade I corresponding to 0.1, grade II corresponding to 0.3, grade III corresponding to 0.5, grade IV corresponding to 0.7, and grade V corresponding to 0.9. The fuzzy number was converted into fuzzy probability using Equations (2) and (3), and

Table 1. Nonquantitative impact factor fuzzy number and fuzzy probability.

Impact Factor Rating	Fuzzy Number	Fuzzy Possibility (km−1·year−1)
I	0.1	5.62 × 10−11
II	0.3	2.88 × 10−7
III	0.5	1.17 × 10−5
IV	0.7	1.91 × 10−4
V	0.9	4.27 × 10−3

shows the results after the transformation.

(3)

Fitting function relationships involved taking the fuzzy number as the independent variable and the fuzzy probability as the dependent variable; then, the function relationship between the fuzzy number and the fuzzy probability was fitted. Figure 6 shows the fuzzy probability corresponding to the fuzzy number of the interval [0, 1].

(4)

Based on the fitting results of the function relationship, the optimal fitting function was screened after a comprehensive analysis of the function characteristics, the rationality of the fitting function, and the goodness of fit (

Table 2. The optimal fitting function of the factor and the failure frequency.

Reason for Invalidation	Impact Factor	Judging Criteria	Optimal Fitting Function
Corrosion	Wall thickness	Fitting function	y = 9.23x−2.07
	Pipe age	Fitting function	y = 5.66 × 10−8x3.56
	Type of antiseptic layer	Discrete relationships	/
External interference	Pipe diameter	Fitting function	y = 2.66 × 10−3 + 0.99e−x/147.05
	Wall thickness	Fitting function	y = 8.68 × 10−4 + 5.82e−x/1.84
	Buried deep	Fitting function	y = 5.71 × 10−2 + 4.67e−x/22.67
Destruction by natural forces	Pipe diameter	Fitting function	y = 5.43 × 10−7x2.87
Construction defects/material failures	Construction—Pipe age	Fitting function	y = 1.20 × 10−7x3.07
	Material—Pipe age	Fitting function	y = −7.38 × 10−3 + 0.066e−x/596.56
	Pipe steel grade	Discrete relationships	/
Misoperation	Pipe diameter	Fitting function	y = 127.90x−1.65
Fuzzy number and fuzzy probability			y = −4.41 × 10−4 + 5.62 × 10−4ex/0.17

). Due to the discrete relationship between the steel grade and the type of anticorrosion layer and the failure frequency, it cannot be expressed by the fitting function, and the corresponding relationship between the influencing factor and the failure probability is shown in

Table 3. The value of the pipeline steel grade and the corresponding impact factor.

Steel Grade	Failure Frequency (1000−1 km−1·year−1)
Grade A	0.04221
Grade B	0.01587
X42	0.01356
X46	0.02583
X52	0.02873
X56	0.01392
X60	0.00616
X65	0.00409
X70 and above	0.04975

and

Table 4. The value of the anticorrosive layer type and the corresponding impact factor.

Type of Antiseptic Layer	Failure Frequency (1000−1 km−1·year−1)
Coal tar	0.04286
Bitumen	0.07
Polyethylene	0.00819
Epoxy resin	0.01959
Other	0.0855

.

$$FFR=\left\{\begin{array}{cc}1/{10}^k& FPS\ne 0\\ 0& FPS=0\end{array}\right\}$$

(2)

$$k={\left({\displaystyle \frac{1-FPS}{FPS}}\right)}^{{\displaystyle \frac{1}{3}}}\times \lg \left({\displaystyle \frac{1}{E_{rM}}}\right)$$

(3)

where k is the corresponding failure probability when the membership value is the largest (equal to 1); E_rM is the most likely failure rate of the pipeline, and after investigating the failure database of natural gas pipelines at home and abroad, lg(1/E_rM) was taken as 4.93.

4.2. Calculation of Conditional Probabilities

The probability that a subnode has different attribute values under the conditions of different values and combinations of parent nodes. The occurrence of the parent node will affect the subnode to produce a certain result, which is not inevitable but is a probabilistic description. The essence of calculating the conditional probability of a subnode with n parent nodes is to solve the maximum eigenvalue of the judgment matrix and its corresponding eigenvectors. The detailed steps of calculating the conditional probability based on the analytic hierarchy process [33] are as follows:

(1)

The median value of the fitting function is calculated according to the integral median value theorem (Equation (4)).

(2)

A judgment matrix of the median value is constructed. If the node has n parent nodes, an n × n square matrix is established first, which is the judgment matrix. The values in row i, and column j in the matrix indicate the importance of the ith evaluation index relative to the jth evaluation index, as shown in

Table 5. The judgment matrix.

	a11	a12	a13	…	aij
a11	a11/a11	a11/a12	a11/a13	…	a11/aij
a21	a21/a11	a21/a12	a21/a13	…	a21/aij
a31	a31/a11	a31/a12	a31/a13	…	a31/aij
…	…	…	…	…	…
aij	aij/a11	aij/a12	aij/a13	…	aij/aij

.

(3)

The maximum eigenvalue λ_max and the corresponding eigenvector ω of each judgment matrix are solved. The maximum eigenvalue λ_max of the corrosion judgment matrix is 11, and the eigenvector ω is shown in Equation (5).

(4)

A consistency test is performed, and the judgment matrix is adjusted. According to the consistency ratio of the judgment matrix, the CR test judges whether the contradiction degree of the matrix is reasonable, and if the CR < 0.1, the consistency of the matrix is considered acceptable; otherwise, the matrix needs to be adjusted. Equations (6) and (7) are the formulas for testing the consistency of the matrix [33].

(5)

Assuming that the features are independent of each other, and the conditional probabilities between the features are also independent of each other in the case of a given category, Equation (8) is the conditional probability formula for calculating multiple parent nodes:

$$f(\xi )={\displaystyle \frac{{\displaystyle {\int }_a^{b}f(x)}}{b-a}}$$

(4)

where f(ξ) is the mean of all function values of the function f(x) in the interval range; ${\int }_a^{b}f\left(x\right)dx$ is the integral of the function f(x) in the range of the interval [a, b]; a is the lower limit; and b is the upper limit.

$$\omega ={\left(2.61\times {10}^{-2},1.08\times {10}^{-1},1.05\times {10}^{-1},3.49\times {10}^{-1},3.49\times {10}^{-1},3.49\times {10}^{-1},3.49\times {10}^{-1},3.49\times {10}^{-1},3.49\times {10}^{-1},3.49\times {10}^{-1},3.49\times {10}^{-1}\right)}^T$$

(5)

$$Cl={\displaystyle \frac{{\lambda }_{max}-n}{n-1}}$$

(6)

where Cl is the consistency index of the judgment matrix; λ_max is the maximum eigenvalue of the judgment matrix; and n is the order of the judgment matrix.

$$CR={\displaystyle \frac{Cl}{RI}}$$

(7)

where CR is the consistency ratio, and if CR < 0.1, the matrix consistency is considered acceptable; the RI is an average random consistency metric, and the value of RI is shown in

Table 6. Mean stochastic consistency indicator RI.

n	1	2	3	4	5	6	7	8	9	10	11	12
RI	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46	1.49	1.52	1.54

when n is constant.

$$P(y|x)=P(y|x_1)\times P(y|x_2)\times \cdots \times P(y|x_n)$$

(8)

where P(y|x) is the multi-parent node x₁, x₂... the probability of event y occurring under x_n conditions; P(y|x_i) is the probability of event y occurring under the condition that event x_i occurs.

5. Examples of Application

The pipe diameter was 1016 mm, the wall thickness was 30.8 mm, and the pipe age was 10 years. X70 steel was selected, and the anticorrosion coating was epoxy resin. Firstly, the GeNIe Bayesian simulation software v.4.1 was used to construct the Bayesian network. Then, the prior probability of the influencing factor was calculated according to the pipeline parameters and the surrounding environment of the pipeline. Finally, based on the Bayesian network inference analysis, the probability of pipeline failure was deduced forward, and the importance of each factor to pipeline failure was inversely inferred.

6. Prior Probability

According to the pipe diameter, buried depth, and other pipeline parameters and fitting functions, the prior probability of the quantitative factor was calculated. The fuzzy number of the pipeline was determined according to the detection, monitoring, operation, and maintenance information of the pipeline and the surrounding environment, and the prior probability of the nonquantitative factors was calculated based on the functional relationship between the fuzzy number and the fuzzy probability. The calculation results are shown in

Table 7. A priori probability table of corrosion, external interference, construction defects material failure.

Corrosion		External Interference		Construction Defects/Material Failures
Impact Factor	Prior Probability (1000−1 km−1·Year−1)	Impact Factor	Prior Probability (1000−1 km−1·Year−1)	Impact Factor	Prior Probability (1000−1 km−1·Year−1)
Wall thickness	9.47 × 10−4	Pipe diameter	2.91 × 10−3	Pipe age	2.72 × 10−4
Pipe age	2.07 × 10−4	Wall thickness	8.68 × 10−4	Pipe grade	4.98 × 10−2
Type of antiseptic layer	1.96 × 10−2	Buried deep	5.71 × 10−2	Non-destructive testing of welds	1.13 × 10−1
Soil corrosiveness	1.13 × 10−1	Patrol frequency	1.13 × 10−1	Defective welds	1.13 × 10−1
Cathodic protection	1.13 × 10−1	Signs along the pipeline	1.13 × 10−1	Pipe deformation and mitering	1.13 × 10−1
Stray current interference	1.13 × 10−1	Pipeline positioning and excavation response	1.13 × 10−1	Horizontal clear distance	1.13 × 10−1
Anticorrosion layer detection	1.13 × 10−1	Punch and steal gas	1.13 × 10−1	Pipeline safety factor	1.13 × 10−1
Detection	1.13 × 10−1	Ground activities	1.13 × 10−1	System safety factor	1.13 × 10−1
External detection	1.13 × 10−1	Public Education	1.13 × 10−1	Design factor	1.13 × 10−1
The medium is corrosive	1.13 × 10−1	Illegal occupation situation	1.13 × 10−1	Special construction pipe sections	1.13 × 10−1
Internal corrosion protective measures	1.13 × 10−1	Above-ground facilities	1.13 × 10−1	Opening and completion reports	1.13 × 10−1

and

Table 8. A priori probability table of natural force destruction and misoperation.

Destruction by Natural Forces		Misoperation
Impact Factor	Prior Probability (1000−1 km−1·year−1)	Impact Factor	Prior Probability (1000−1 km−1·year−1)
Pipe diameter	3.51 × 10−4	Pipe diameter	1.05 × 10−3
Topography	1.13 × 10−1	SCADA system	1.13 × 10−1
Land type	1.13 × 10−1	Valve chamber failure	1.13 × 10−1
Sensitive points of geological hazards	1.13 × 10−1	Employee training and assessment	1.13 × 10−1
Pipeline laying method	1.13 × 10−1	Maintenance plan execution	1.13 × 10−1
Extreme weather	1.13 × 10−1	Protection against mechanical errors	1.13 × 10−1
Monitoring and early warning systems	1.13 × 10−1	Data and Data Management	1.13 × 10−1
Emergency precautions	1.13 × 10−1	Procedures and work instructions	1.13 × 10−1

.

7. Bayesian Network Analysis

Inputting the prior probability of the parent nodes and the conditional probability of the subnodes into the corresponding nodes of the software GeNIe v.4.1, the node status “Y” indicates the occurrence of the node event, and the status “N” indicates that the node event did not occur. The Bayesian forward reasoning results are shown in Figure 7. The pipeline failure probability was 0.18 times/(thousand kilometers·years), of which the probability of corrosion accounted for 19.51%, the proportion of external interference was 18.29%, the proportion of construction defects/material failure was 18.29%, the proportion of natural force damage was 20.73%, the proportion of misoperation was 20.73%, and the failure probability of other factors was negligible.

To analyze the importance of the impact factors on pipeline failure, reversing the Bayesian network reasoning and updating evidence, the state of pipeline failure “Y” was set to 100%, assuming pipeline failure. Figure 8 shows the importance of each influencing factor, among which the importance of natural force failure was the greatest, followed by external interference, and the order from the most important to the least important was natural force failure > external interference > corrosion > misoperation > construction defect/material failure > other. In this case, the probability of natural force failure was the greatest, and the influence of this factor was the most important, so it is necessary to focus on monitoring the pipeline’s surroundings and performing extreme weather prediction, emergency protection, and treatment effectively.

The reverse diagnostic inference was used to obtain the main controlling factors affecting pipeline failure. Then, the proportional change in failure probability (R_ov value) was used to balance the degree of dependency of the parent nodes to subnodes (the pipe failure). The R_ov value was larger, and the degree of dependency of the parent nodes to subnodes was greater. The calculation is shown in Equation (3). The calculated results are shown in Figure 9. The buried depth of the pipeline was the most dependent factor, followed by the subnodes corresponding to the destruction by natural forces.

$$R_{ov}(X_i)=\left|{\displaystyle \frac{\phi \left({\mathrm{X}}_i\right)-\theta \left({\mathrm{X}}_i\right)}{\theta \left({\mathrm{X}}_i\right)}}\right|$$

(9)

where $\phi \left({\mathrm{X}}_i\right)$ is the prior probability of the parent nodes, and $\theta \left({\mathrm{X}}_i\right)$ is the posterior probability of the parent nodes.

8. Conclusions

In this paper, a three-layer Bayesian network model of the failure probability of natural gas pipelines under the influence of multiple factors was constructed, and a method for calculating conditional probability and prior probability was proposed, which provides a new idea for calculating the failure probability of new pipelines or pipelines without historical data compared with other failure probability calculation methods. Taking a gas pipeline as an example, forward inference (calculating the failure probability) and reverse inference (analyzing the importance of the impact factor) were carried out. In this research, the specific conclusions obtained are as follows:

(1)

According to the correspondence between the failure factor tree map and the Bayesian network, a three-layer Bayesian network topology model from the static failure factor tree map to the dynamic “pipeline failure–failure cause–influencing factor” was constructed with the pipeline as the subnode of the network, the type of pipeline failure as the intermediate node, and the factor affecting the pipeline failure as the parent node.

(2)

The optimal fitting function between the influencing factor and the pipeline failure frequency was screened, the median value of the fitting function integral was proposed as the data basis, and the conditional probability of the directed edge of the network was calculated by constructing the judgment matrix of the median value and solving the maximum eigenvalue and eigenvector of the matrix.

(3)

In the Bayesian network model, the prior probability of the pipeline was calculated by inputting the fuzzy number of pipeline parameters and pipeline operation and maintenance information into the optimal fitting function. Forward reasoning involves calculating the likelihood of pipeline failure, and reverse reasoning highlights the significance of the impact factor of pipeline failure and offers reference data for key protections and precise positioning of pipeline-related departments.

However, this manuscript currently has two limitations that need to be addressed in future research:

(1)

The applicability and uncertainty of failure data samples from PHMSA and EGIG about other pipelines;

(2)

The challenge of integrating failure data for a pipeline that has missing data for a certain period, specifically, how to incorporate these data into the existing method.

These issues should be resolved moving forward.