A Dynamic Assessment Methodology for Accident Occurrence Probabilities of Gas Distribution Station

Abstract

Gas distribution stations (GDSs), pivotal nodes in long-distance natural gas transportation networks, are susceptible to catastrophic fire and explosion accidents stemming from system failures, thereby emphasizing the urgency for robust safety measures. While previous studies have mainly focused on gas transmission pipelines, GDSs have received less attention, and existing risk assessment methodologies for GDSs may have limitations in providing accurate and reliable accident probability predictions and fault diagnoses, especially under data uncertainty. This paper introduces a novel dynamic accident probability assessment framework tailored for GDS under data uncertainty. By integrating Bayesian network (BN) modeling with fuzzy expert judgments, frequentist estimation, and Bayesian updating, the framework offers a comprehensive approach. It encompasses accident modeling, root event (RE) probability estimation, undesired event (UE) predictive analysis, probability adaptation, and accident diagnosis analysis. A case study demonstrates the framework’s reliability and effectiveness, revealing that the occurrence probability of major hazards like vapor cloud explosions and long-duration jet fires diminishes significantly with effective safety barriers. Crucially, the framework acknowledges the dynamic nature of risk by incorporating observed failure incidents or near-misses into the assessment, promptly adjusting risk indicators like UE probabilities and RE criticality. This underscores the importance for decision-makers to maintain a heightened awareness of these dynamics, enabling swift adjustments to maintenance strategies and resource allocation prioritization. By mitigating assessment uncertainty and enhancing precision in maintenance strategies, the framework represents a significant advancement in GDS safety management, ultimately striving to elevate safety and reliability standards, mitigate natural gas distribution risks, and safeguard public safety and the environment.

1. Introduction

The gas distribution station (GDS) constitutes a vital component of the long-distance gas transmission network, pivotal in ensuring the stable and secure supply of natural gas to residential, commercial, and industrial consumers situated along the transmission pipeline. Typically, numerous such stations are deployed: for instance, China’s West–East gas pipeline boasts 17 GDSs, while the Sichuan–East gas pipeline comprises 11. The multifaceted functions of these stations encompass gas impurity separation, pressure regulation, and flow metering, as well as the reception and dispatch of pipeline cleaner. An unintended release of natural gas from a GDS could have serious consequences for the surrounding populace, infrastructure, and the environment alike. Consequently, assessing the risk level of these stations and pinpointing their vulnerabilities is paramount. Such an assessment can serve as a cornerstone for devising risk mitigation strategies tailored to the unique needs of each station.

The majority of previous studies have focused on safety assessment of gas transmission pipelines [1,2,3,4,5], while few studies are concentrated on quantitative risk assessment (QRA) of gas transmission stations, including GDSs [6,7,8]. Previous studies have primarily utilized fault tree analysis (FTA), event tree analysis (ETA), and bow-tie analysis (BTA) as QRA methodologies. However, the conventional approaches encounter limitations in capturing the dynamic nature of risk and handling data scarcity or uncertainties, in addition to difficulties in representing common cause failures and conditional dependencies among failure events. To overcome these constraints, a series of integrated methodologies have emerged. Approaches based on fuzzy set theory (FST), such as fuzzy-FTA [9,10], fuzzy-ETA [11,12], and fuzzy-BTA [13,14], seamlessly integrate expert judgment with fuzzy sets, effectively addressing the uncertainties inherent in traditional analysis methods that stem from the scarcity of foundational failure data. In other integrated methodologies, such as SHIPP [15], Bayesian-FTA [16], and Bayesian-BTA [7,17], the Bayesian updating mechanism is employed to minimize uncertainty and enhance the accuracy of quantifying failure probability and accident occurrence probability by utilizing real-time abnormal event data from the plant. More recently, Bayesian network (BN)-based techniques have gained popularity for probabilistic risk assessment (PRA) in various engineering fields [18,19,20,21,22,23]. The integration of BN with fuzzy set theory (FST) offers distinct advantages: (1) fuzzy numbers systematically quantify epistemic uncertainties arising from limited data or expert judgment [24,25], while (2) BN enables dynamic probability updating through real-time data assimilation, effectively addressing both static uncertainties and evolving risk scenarios [26,27].

Despite these advancements, critical gaps persist when applying existing methodologies to complex systems in data-scarce and operationally complex contexts, such as China’s GDSs:

(1) In data-scarce scenarios, existing frameworks assume access to sufficient failure data for probabilistic modeling. However, China’s GDSs lack localized failure databases due to their recent deployment and fragmented data collection practices. Generic reliability data cannot fully capture site-specific risks, leading to biased predictions.

(2) Traditional QRA tools (e.g., FTA, BTA) are limited to unidirectional reasoning—either predicting accident probabilities or diagnosing root causes. Few studies have explored bidirectional dynamic analysis, which is essential for systems requiring both proactive risk mitigation and reactive incident learning.

(3) While BNs enable probability updating [22], their application in China’s GDSs is hindered by the absence of systematic data integration frameworks. This gap limits the ability to dynamically refine risk profiles as new failure modes emerge.

To address these challenges, this study aims to develop a novel framework that integrates BNs with fuzzy expert judgment and Bayesian updating. This framework combines fuzzy numbers to handle epistemic uncertainty and employs Bayesian updating for real-time data assimilation. This makes it possible to conduct risk assessment when data is limited. It unifies forward prediction (accident probability estimation) and backward adaptation (root-cause diagnosis) within a single BN framework. Additionally, the model is tailored to China’s GDSs by tackling localized challenges such as rapid infrastructure changes and regulatory gaps. This research direction promises to offer new perspectives and potential solutions for the safety assessment of natural gas distribution stations, while also providing valuable insights and references for research and practice in related fields.

In the subsequent sections, we will elaborate on the designed methodology (Section 2) and demonstrate the practical applicability of our proposed method through a case study (Section 3). Finally, Section 4 will summarize the main conclusions of this study and offer suggestions for future research.

2. Methodology

The framework for the developed methodology aimed at dynamically assessing the accident occurrence probabilities of GDSs is illustrated in Figure 1. Initially, a Bayesian network (BN) model is constructed to represent the GDS accident scenario. Subsequently, based on the availability and quality of failure data, various methods (including fuzzy expert judgment and frequentist estimation) are employed to determine the prior probabilities of the REs within the BN model. Following this, a predictive analysis utilizing the BN is conducted to quantify the occurrence probabilities of undesired events (UEs). Thereafter, a diagnostic analysis is undertaken, leveraging importance measure indices, to pinpoint the key factors contributing to accident risks. Depending on the emergence of new failure-related data, a decision is made regarding whether to reiterate the Bayesian analysis. In the event of new failure sample data being presented, the RE probabilities are updated according to Bayesian updating theory, subsequently prompting a recalculation of the UE probabilities. Alternatively, if near-misses or incidents are observed, a probability adaptation process is applied to the BN, adjusting the probabilities of REs and intermediate events, followed by a repeat of the diagnostic analysis. Conversely, if no new failure-related events are observed, the entire analysis process concludes.

2.1. Accident Model Construction Using Bayesian Network

BN is a widely recognized graphical modeling technique that employs a directed acyclic graph to represent a set of variables and their conditional dependencies. A GDS accident model constructed using BN provides a concise visual portrayal of the intricate relationship between accident causes and consequences within a unified graphical framework. This model facilitates reasoning under uncertainty by integrating prior knowledge with observed data, enabling the inference of probabilities for unobserved variables. One of the most effective strategies for constructing an accident model utilizing BN is through the mapping of a bow-tie (BT) structure [28]. Figure 2 illustrates the streamlined procedure for transforming a BT model into a BN [22].

2.2. Probability Estimation Methods for REs

In order to perform BN quantitative analysis, it is necessary to determine the prior probabilities (i.e., initial probabilities) of REs in advance. For those REs related to mechanical and electrical equipment failure, their occurrence probabilities can be obtained from some references or reliability databanks such as OREDA [29], EXIDA [30], etc. For other REs, some probabilistic estimation methods, such as frequentist estimation, fuzzy-based methods, etc., are good choices depending upon the amount of plant-specific failure data collected.

In order to conduct quantitative analysis using BN, it is essential to predetermine the prior probabilities (i.e., initial probabilities) of REs. For REs pertaining to mechanical and electrical equipment failures, their occurrence probabilities can be sourced from various references or reliability databases, including OREDA [29], EXIDA [30], and CCPS [31], among others. For other REs, the selection of probabilistic estimation methods, such as frequentist estimation, fuzzy-based methods [2,10], and others, depends on the availability and quantity of plant-specific failure data collected.

2.2.1. Frequentist Estimation Methods

For REs with abundant failure data, the results of probabilistic estimation derived from frequentist estimation, specifically maximum likelihood estimation, are considered reliable. Sample data can be categorized into homogeneous and non-homogeneous types, where homogeneous samples refer to those failure samples sharing the same or similar operating conditions and working environments [32].

For equipment with homogeneous large sample failure data, the point estimate ($\widehat{\lambda }$) and 90% confidence intervals [λ_0.05, λ_0.95] for the running failure rate can be calculated as follows:

$$\left\{\begin{array}{l}\widehat{\lambda }=n/\tau ={\displaystyle {\sum }_i{n}_i}/{\displaystyle {\sum }_i{\tau }_i}\\ {\lambda }_{0.05}={\chi }_{0.05}^2(2n)/2\tau \\ {\lambda }_{0.95}={\chi }_{0.95}^2(2n+2)/2\tau \end{array}\right.$$

(1)

where n denotes the total number of failures; τ denotes the total time in service; the subscript i represents a single sample; and χ denotes the chi-squared distribution. And the calculation equations for the probability of failure on demand ($\widehat{p}$, [p_0.05, p_0.95]) with homogeneous large sample failure data are as follows:

$$\left\{\begin{array}{l}\widehat{p}=m/M={\displaystyle {\sum }_j{m}_j}/{\displaystyle {\sum }_j{M}_j}\\ p_{0.05}={\displaystyle \frac{mF{}_{0.05}(2m,2M-2m+2)}{(M-m+1)+mF{}_{0.05}(2m,2M-2m+2)}}\\ p_{0.95}={\displaystyle \frac{(m+1)F{}_{0.95}(2m+2,2M-2m)}{(M-m)+(m+1)F{}_{0.95}(2m+2,2M-2m)}}\end{array}\right.$$

(2)

where m denotes the observed number of failures; M denotes the observed number of demands; and the subscript j denotes a single sample.

Non-homogeneous samples, on the other hand, encompass failure samples from diverse stations, each characterized by unique operating conditions and environments [32]. To address the complexities arising from the variability in failure rates and data volumes among these samples, as evident in Figure 3, we necessitate a probabilistic statistical theory-based approach. This approach assumes that each sample’s failure rate follows a distinct probability density function, facilitating the initial estimation of failure rates. Subsequently, quantifying the variability among samples becomes a crucial step in enhancing the accuracy and reliability of the consolidated estimation. Ultimately, through the application of advanced statistical techniques, we derive a robust and reliable “average” failure rate, along with its corresponding 90% confidence interval [29]. This methodology serves as a solid foundation for subsequent probabilistic risk analyses, providing robust data support that underscores their validity and reliability. The specific calculation process is outlined below.

Step 1: Assuming that the failure rate of each sample obeys the distribution of probability density function π(λ), the average failure rate θ and the variance ${\sigma }^2$ are as follows:

$$\theta ={\displaystyle {\int }_0^{\infty }\lambda ·\pi (\lambda )d\lambda }$$

(3)

$${\sigma }^2={\displaystyle {\int }_0^{\infty }{(\lambda -\theta )}^2·\pi (\lambda )d\lambda }$$

(4)

Step 2: Calculate an initial value ${\widehat{\theta }}_1$ of the failure rate for non-homogeneous samples:

$${\widehat{\theta }}_1={\displaystyle \frac{totalnumberoffailures}{Totaltimeinservice}}={\displaystyle \sum _{i=1}^k{n}_i}/{\displaystyle \sum _{i=1}^k{\tau }_i}$$

(5)

Step 3: Calculate the immediate parameters S₁, S₂, and ν:

$$S_1={\displaystyle \sum _{i=1}^k{\tau }_i}$$

(6)

$$S_2={\displaystyle \sum _{i=1}^k{\tau }_i^2}$$

(7)

$$\nu ={\displaystyle \sum _{i=1}^k\frac{{(n_i-{\widehat{\theta }}_1{\tau }_i)}^2}{{\tau }_i}}={\displaystyle \sum _{i=1}^k\frac{n_i^2}{{\tau }_i}-}{\widehat{\theta }}_1^2{S}_1$$

(8)

Step 4: Estimate the variation ${\widehat{\sigma }}^2$ between samples:

$${\widehat{\sigma }}^2={\displaystyle \frac{\nu -(k-1){\widehat{\theta }}_1}{S_1^2-S_2}}·S_1$$

(9)

If the calculated value of the above formula is less than 0, then,

$${\widehat{\sigma }}^2={\displaystyle \sum _{i=1}^k\frac{{\left(n_i/{\tau }_i-{\widehat{\theta }}_1\right)}^2}{k-1}}$$

(10)

Step 5: Calculate the mean failure rate ${\theta }^{*}$:

$${\theta }^{*}={\displaystyle \frac{1}{{\displaystyle \sum _{i=1}^k\frac{1}{{\widehat{\theta }}_1/{\tau }_i+{\widehat{\sigma }}^2}}}}·{\displaystyle \sum _{i=1}^k\left(\frac{1}{{\widehat{\theta }}_1/{\tau }_i+{\widehat{\sigma }}^2}·\frac{n_i}{{\tau }_i}\right)}$$

(11)

Step 6: As the distribution of the running failure rate for a non-homogeneous sample remains unknown prior to analysis, it is conventionally assumed to follow the probability density function of a gamma distribution, parameterized by α and β. Subsequently, the specific values of α and β can be calculated as detailed below:

$$\left\{\begin{array}{l}\widehat{\beta }={\theta }^{*}/{\widehat{\sigma }}^2\\ \widehat{\alpha }=\widehat{\beta }·{\theta }^{*}\end{array}\right.$$

(12)

Step 7: The 90% confidence intervals for the running failure rate are calculated as follows:

$$\left\{\begin{array}{l}{\theta }_{0.05}={\chi }_{0.05}^2(2\widehat{\alpha })/2\widehat{\beta }\\ {\theta }_{0.95}={\chi }_{0.95}^2(2\widehat{\alpha })/2\widehat{\beta }\end{array}\right.$$

(13)

2.2.2. Fuzzy Set Theory-Based Expert Judgment Method

Since there exist REs for which failure sample data are scarce or non-existent, a fuzzy-based method that incorporates expert judgments is proposed to address the data uncertainties and quantify their occurrence probabilities. This methodology, which has been previously introduced in our work [10], is distinguished by its utilization of two crucial indicators: the relative agreement degree (RAD) and the importance degree (ID). These indicators facilitate the aggregation of expert opinions, leading to a more rational and comprehensive assessment. The fundamental steps of this approach are outlined below:

• Evaluate the likelihood of occurrence for each RE using natural language terms.
• Convert the experts’ linguistic assessments into fuzzy numbers to quantify their subjective evaluations.
• Aggregate the individual experts’ assessments into a unified, comprehensive fuzzy possibility score.
• Transform the fuzzy possibility score into a probability value for subsequent analysis or application.

2.3. Dynamic Bayesian Network Analysis

When performing a BN analysis, we frequently encounter the challenge of limited or incomplete failure data, given the rarity of hazardous events. Under such conditions of uncertainty, the occurrence probabilities of UEs within the GDS can only be initially estimated in an approximate manner. Upon the observation of new specific failure data, a dynamic BN analysis can be performed to derive more reliable outcomes. Unlike conventional static methods (e.g., FTA/ETA), which rely solely on historical failure data and lack real-time adaptability [28], BN-based probability adaptation integrates fuzzy expert judgment with incremental data assimilation, effectively addressing data scarcity in station systems like GDSs. This hybrid approach reduces epistemic uncertainty compared to purely data-driven methods [8], making it particularly suitable for systems with limited operational histories.

2.3.1. Bayesian Updating Estimation of RE Probabilities

In scenarios where failure data are inadequate, the aforementioned methods can be utilized to derive the initial, or prior, probability of REs. Nevertheless, it is widely acknowledged that generic failure data do not accurately reflect the operational conditions and environments specific to GDS facilities. Furthermore, the relatively short operational history of GDSs in China contributes to the scarcity of plant-specific data, which, in turn, influences experts’ linguistic assessments of RE occurrence probabilities. Consequently, the direct use of initial probabilities inevitably introduces uncertainties into the PRA results for the GDS. However, upon the acquisition of new specific failure data pertaining to the GDS, the prior probabilities of REs can be updated through Bayesian estimation techniques.

For continuous variables, the equation of Bayesian theorem is as follows [32]:

$$f\left(\lambda |E\right)={\displaystyle \frac{f\left(\lambda \right)·L\left(E|\lambda \right)}{{\displaystyle {\int }_0^{\infty }f\left(\lambda \right)·L\left(E|\lambda \right)d\lambda }}}\propto f\left(\lambda \right)·L\left(E|\lambda \right)$$

(14)

where f(λ|E) is posterior probability density function (PDF) of the failure rate (λ); f (λ) is the prior PDF of λ; and L(E|λ) is likelihood function of λ based on plant-specific data (E).

For discrete variables, the Bayesian equation is as follows [32]:

$$f\left({\lambda }_i|E\right)={\displaystyle \frac{f\left({\lambda }_i\right)·L\left(E|{\lambda }_i\right)}{{\displaystyle {\sum }_{j=1}^{m}f\left({\lambda }_i\right)·L\left(E|{\lambda }_i\right)}}}\propto f\left({\lambda }_i\right)·L\left(E|{\lambda }_i\right)$$

(15)

where m denotes the number of variables. λ_i denotes the ith variable.

The schematic illustration of the Bayesian updating estimation for the occurrence probability of REs is presented in Figure 4. The detailed methodology of this updating approach can be found in our previous research [7]. As the volume of specific failure data collected increases, the uncertainty associated with the estimated results will decrease correspondingly.

2.3.2. Predictive Analysis

The primary aim of BN predictive analysis is to determine the occurrence probabilities of UEs, specifically critical events (CEs) and outcome events (OEs). This is accomplished through deductive reasoning, also known as forward prediction, which relies on the available probabilities of REs, conditional probability tables (CPTs), and the topology of the network structure. The probability distribution for a given UE, denoted as P(UE = 1), is calculated using Equation (16) [33]. Upon the acquisition of new observed failure evidence pertaining to REs, a dynamic predictive analysis can be performed to obtain updated probabilities of UEs.

$$P(UE=1)={\displaystyle \sum _{i=1}^{2^n}\left[P(UE=1|X_1=x_1,X_2=x_2,\cdots ,X_n=x_n)\times P(X_1=x_1,X_1=x_2,\cdots ,X_n=x_n)\right]}$$

(16)

where $x_i=\{x_i^1,x_i^2,\cdots ,x_i^{Q_i}\},i=1,2,\cdots ,n$; Q_i is the number of the states for a single RE X_i; n is the number of the REs; and x_i is the state of RE X_i. Since each RE has two states “Yes/No”, n REs contribute to 2ⁿ combinations; P(UE = 1|X₁ = x₁, X₂ = x₂, …, X_n = x_n) denotes the CPT of UE; and P(X₁ = x₁, X₂ = x₂, …, X_n = x_n) is the joint probability distribution of X_i.

2.3.3. Probability Adaptation

Probability adaptation allows the BN model to dynamically adjust the probabilities of REs and IEs when new safety-related observations, such as near-misses or incidents, are recorded. Near-misses, which are defined as events that came close to causing accidents but revealed critical vulnerabilities in the system, are especially valuable for refining risk assessments. When a near-miss occurs, it offers indirect evidence of potential system weaknesses. For instance, a near-miss resulting from a faulty valve in a GDS may not lead to an accident due to timely intervention, but it indicates a higher probability of valve failure.

To incorporate near-misses into the framework, we utilize abductive reasoning (as shown in Equation (17)) to update the BN. The observed near-miss is regarded as soft evidence (e.g., via likelihood weighting or virtual evidence) to adjust the probabilities of associated REs and intermediate events. For example, if a near-miss is associated with a specific RE (e.g., manual shutdown valve failure), the conditional probability tables (CPTs) of the corresponding nodes are modified to reflect the increased risk. This adjustment is proportional to the severity and frequency of the near-miss, as determined by expert judgment or historical data. After this adjustment, the diagnostic analysis (Section 2.3.4) is repeated to re-evaluate the criticality of REs and prioritize risk-mitigation strategies.

$$P(X_i=x_i|UE=1)={\displaystyle \frac{P(X_i=x_i)\times P(UE=1|X_i=x_i)}{P(UE=1)}},i=1,2,\cdots ,n$$

(17)

where P(X_i = x_i|UE = 1) denotes the self-adapted probability of root event X_i; X_i is more likely to become the direct cause of a UE when P(X_i = x_i|UE = 1) is close to 1.

2.3.4. Diagnostic Analysis

Diagnostic analysis is instrumental in uncovering the most critical REs that contribute to the occurrence of the UEs [8,22]. The identified critical REs constitute the weakest links in the GDS, thereby playing a pivotal role in informing risk-management decisions.

To determine the most critical REs, two key importance measures are introduced: the Birnbaum measure (BM) and the risk reduction worth (RRW). Specifically, the indices for BM and RRW of each RE, denoted as I^BM(RE_i) and I^RRV(RE_i), respectively, can be computed by integrating Equations (18) and (19) with BN predictive analysis.

$$I^{BM}(R{E}_i)=P(UE=1|R{E}_i=1)-P(UE=1|R{E}_i=0),i=1,2,\cdots ,n$$

(18)

$$I^{RRW}(R{E}_i)=P(UE=1)-P(UE=1|R{E}_i=0),i=1,2,\cdots ,n$$

(19)

where P(UE = 1|RE_i = 1) and P(UE = 1|RE_i = 0) show the conditional failure probability of the GDS given the occurrence and non-occurrence of the RE_i. The total weighting ($R_j^{*}$) of the jth RE is calculated as follows, and the critical REs can be obtained by ranking them.

$$R_j^{*}=\left({\displaystyle \frac{I_j^{BM}}{{\displaystyle \sum _{i=1}^n{I}_j^{BM}}}}+{\displaystyle \frac{I_j^{RRW}}{{\displaystyle \sum _{i=1}^n{I}_j^{RRW}}}}\right)\times 100\%$$

(20)

when near-misses or incidents are observed, the framework further employs the ratio of variation (RoV) metric (Equation (21)) to dynamically reassess critical REs [8]. The RoV quantifies the relative change in the probability of an undesired event (UE) before and after incorporating near-miss data. For example, if a near-miss indicates a malfunction in a pressure sensor, the RoV evaluates how significantly this observation alters the UE probability (e.g., gas leakage). REs with higher RoV values are deemed more critical, as their contributions to system risk are amplified by the near-miss. Engineers can then prioritize these REs for maintenance or redesign, thereby proactively reducing the likelihood of future accidents.

$$RoV(X_i)={\displaystyle \frac{\pi (X_i)-\theta (X_i)}{\theta (X_i)}}$$

(21)

where π(X_i) is the adapted probability of RE_i; θ(X_i) is the non-adapted probability of RE_i.

BM/RRW are primarily used for baseline risk prioritization based on static or historical data. RoV complements BM/RRW by capturing the dynamic impact of near-misses, enabling real-time risk re-evaluation. Engineers can thus prioritize REs with both high BM/RRW values (inherent criticality) and elevated RoV (near-miss-driven sensitivity) for targeted interventions.

3. Application of the Methodology

3.1. Constructing a Bayesian Network Model for GDS Accident Scenarios

The bow-tie (BT) model previously established for GDS accident scenarios in our earlier work [7] clearly outlines the logical interdependencies among basic events, critical events, and outcome events. This structured representation serves as a robust foundation for constructing the corresponding BN model in the present study. By strategically adapting the original BT model and adhering to the established mapping algorithm of BT to BN [28], we have developed a BN structure tailored specifically for analyzing GDS accident scenarios, as illustrated in Figure 5.

This BN model begins with various REs that culminate in the release of natural gas (designated as the CE). It then progresses through potential accident scenarios (OEs), contingent upon the occurrence or non-occurrence of safety-barrier failure events (SBFEs) and the spatial confinement event (SCE) within the station. As depicted in Figure 5, the BN model encompasses 78 REs (filled in green), 1 CE (filled in yellow), 2 SBFEs (filled in blue), 1 SCE (also filled in blue), 51 intermediate events (IEs) (filled in white), and 6 OEs (filled in pink). The model employs distinct visual indicators to represent the conditional dependencies among events. Specifically, red arrows indicate situations where the simultaneous occurrence of all lower-level events triggers an upper-level event. In contrast, black arrows represent conditional relationships that do not require the concurrent occurrence of events. Moreover, the red dotted arrows associated with the SBFEs or SCE indicate that the occurrence of these events is negated or prevented.

3.2. Calculating Prior Probabilities of REs, CE, and OEs

The occurrence probabilities of REs within the accident BN model are ascertained utilizing generic reliability databases [29,31], FST-based expert judgment, frequentist estimation techniques, and the relevant literature [10,15]. The comprehensive probability outcomes of all REs are presented in

Table 1. The prior probabilities of occurrence of root events.

RE	Description	Prior Probability	RE	Description	Prior Probability
X1	Improper inspection and maintenance	2.47 × 10−3	X41	Vent valve of pig trap failure	4.67 × 10−2
X2	Solid impurities in natural gas	1.90 × 10−4	X42	Pig indicator failed	8.00 × 10−4
X3	Separator outlet valve fails to shutdown	2.70 × 10−2	X43	Safety valve fails to active	1.70 × 10−2
X4	Non-compliance with operating procedure	4.50 × 10−2	X44	Gas erosion abrasion	2.37 × 10−3
X5	Non-professional operation	4.30 × 10−4	X45	Seal failure of blind plate	1.70 × 10−4
X6	Separator drain valve fails to open	9.57 × 10−2	X46	Flange gasket breakage	4.32 × 10−3
X7	Separator inlet valve fails to close	3.12 × 10−2	X47	Improper selection of flange gasket	2.30 × 10−4
X8	Separator inlet pressure gauge failure	8.64 × 10−2	X48	Flange gasket aging	2.69 × 10−3
X9	Separator pressure gauge failure	8.64 × 10−2	X49	Flange connection bolt failure	1.70 × 10−4
X10	Separator manual vent valve failure	4.67 × 10−2	X50	Poor installation quality	1.90 × 10−4
X11	Separator high pressure alarm failure	4.22 × 10−2	X51	Valve body defect	1.30 × 10−3
X12	No human response to high voltage alarm	8.00 × 10−4	X52	Seal packing failure	1.04 × 10−3
X13	Seal ring of manhole cover failure	4.32 × 10−3	X53	Poor seal between valve body and cover	9.50 × 10−4
X14	Fastening bolt of manhole cover failure	1.70 × 10−4	X54	Artificial blockage leakage point failure	2.50 × 10−2
X15	Equipment material defect	9.80 × 10−4	X55	Automatic ball valve fails to close	1.18 × 10−2
X16	Equipment manufacturing defect	1.00 × 10−2	X56	Manual shutdown valve failure	1.31 × 10−3
X17	Insufficient strength design	2.30 × 10−4	X57	No response to manual valve	4.00 × 10−2
X18	Calculation method error	2.50 × 10−4	X58	Automatic emergency shut−off valve failure	4.67 × 10−2
X19	Outdated design specifications	1.80 × 10−4	X59	Automatic emergency cut−off sensor failure	2.89 × 10−2
X20	Poor coating repair	1.84 × 10−3	X60	Automatic emergency cut−off controller failure	2.50 × 10−1
X21	Material damaged in handling	4.90 × 10−4	X61	Manual emergency cut−off valve failure	3.12 × 10−2
X22	Material damage during installation	1.80 × 10−4	X62	No response to manual emergency valve	4.00 × 10−2
X23	Weld defect	9.30 × 10−4	X63	Lightning spark	1.80 × 10−4
X24	Earthquake	1.30 × 10−4	X64	Accidental external impact	1.81 × 10−4
X25	Flood	1.20 × 10−4	X65	Sparks caused by short circuit	1.80 × 10−4
X26	Subsidence	1.20 × 10−4	X66	Non−explosion−proof monitoring equipment	1.80 × 10−4
X27	Corrosion inhibitor failure	5.10 × 10−4	X67	Other non−explosion−proof electrical appliances	2.20 × 10−4
X28	Inner coating thinning	9.60 × 10−4	X68	Clothing friction electrification	1.95 × 10−3
X29	Internal coating peeling	9.80 × 10−4	X69	Other electrostatic	9.80 × 10−4
X30	With water	1.22 × 10−3	X70	Station domestic fire	1.90 × 10−4
X31	Acid medium	2.30 × 10−4	X71	Smoking	1.10 × 10−4
X32	Harsh environment	1.90 × 10−4	X72	Vehicles not fitted with spark arrestor	4.20 × 10−4
X33	Outer coating thinning	1.32 × 10−3	X73	Friction at the leak	4.80 × 10−4
X34	Outer coating peeling	1.66 × 10−3	X74	High speed collisions of solid impurities	4.60 × 10−4
X35	Residual stress	1.80 × 10−4	X75	Tool impact sparks	1.30 × 10−4
X36	Stress concentration	1.90 × 10−4	X76	Hot work without permission	3.30 × 10−2
X37	Lack of periodic corrosion detection	5.00 × 10−2	X77	Not following hot work permit	4.50 × 10−2
X38	Corrosion detection instrument failure	2.40 × 10−4	X78	Insufficient supervision	3.40 × 10−2
X39	Electric ball valve of pig trap outlet fails to open	1.18 × 10−2	SCE	Spatial confinement event	1.21 × 10−4
X40	Drain valve of pig traps fails to close	3.86 × 10−2

, serving as foundational prior failure data for the subsequent BN predictive analysis. Here, X₆ is taken as an example to illustrate the calculation process of occurrence probability by the frequentist estimation (see Section 2.2.1), as shown in

Table 2. The prior probabilities of X₆ by frequentist estimation method.

Parameter	Values	Parameter	Values
n1	1 failures	τ1	43,800 h
n2	0 failures	τ2	26,280 h
n3	1 failures	τ3	61,320 h
n4	0 failures	τ4	43,800 h
${\widehat{\theta }}_1$	1.14 × 10−5	${\theta }^{*}$	1.09 × 10−5
S1	1.75 × 105	$\widehat{\beta }$	7.90 × 104
S2	8.29 × 109	$\widehat{\alpha }$	8.63 × 10−1
ν	1.63 × 10−5	θ0.05	6.52 × 10−7
${\widehat{\sigma }}^2$	1.38 × 10−10	θ0.95	3.79 × 10−5
$({\theta }_{0.05},{\theta }^{*},{\theta }_{0.95})$ = (6.52 × 10−7, 1.09 × 10−5, 3.79 × 10−5) hr−1 = (5.63 × 10−3, 9.57 × 10−2, 3.28 × 10−1) yr−1

. Notably, the application of FST-based expert judgment for RE probability calculation has been reported in our previous research [7]. Concerning UEs, namely CE and OEs, their occurrence probabilities are derived through rigorous BN deductive reasoning. The corresponding results are compiled in

Table 3. The prior probabilities of occurrence of CE and OEs.

Undesired Event	Description	Prior Probability
CE	Natural gas release from GDS	5.23 × 10−2
OE1	Minor material losses	4.54 × 10−2
OE2	Flash-fire	6.98 × 10−7
OE3	Short-duration jet fire	5.77 × 10−3
OE4	Major material losses	9.93 × 10−4
OE5	Vapor cloud explosion	1.53 × 10−8
OE6	Long-duration jet fire	1.26 × 10−4

. Specifically, the probability of leakage stemming from GDS failure (CE) is determined to be 5.23 × 10⁻². Among the OEs, minor material losses (OE₁) and short-duration jet fires (OE₃) exhibit relatively higher occurrence probabilities, at 4.54 × 10⁻² and 5.77 × 10⁻³, respectively. Conversely, the probabilities for major hazard scenarios, such as vapor cloud explosions (OE₅) and long-duration jet fires (OE₆), are significantly lower, at 1.53 × 10⁻⁸ and 1.26 × 10⁻⁴, respectively. This pronounced disparity primarily stems from the effective mitigation measures implemented by the multi-layered safety barriers installed in the station.

3.3. Determining Contributions of REs to GDS Accident

Identifying the most critical REs based on their contributions to the occurrence of the GDS accidents is of paramount importance for the formulation of effective prevention measures. To achieve this, the total weighting index method outlined in Section 2.3.4 is employed to determine the criticality of REs within the BN accident model. The outcomes of this analysis, illustrating the contributions of REs to GDS leakage, are presented in Figure 6 (specifically, the blue bars represent the prior contribution weight values of the REs). As evident from the figure, X₄ (non-compliance with operating procedure) emerges as the most significant RE contributing to the occurrence of leakage at the station, closely followed by X₁₆ (equipment manufacturing defect). Furthermore, Figure 7 details the measurement of RE contributions to a specific accident scenario, taking OE₆ (long-duration jet fire) as an illustrative example, under the condition of GDS leakage (again, the blue bars indicate the contributions). According to the criticality ranking presented in Figure 7, X₆₀ (automatic emergency cut-off controller failure) holds the highest contribution to the occurrence of the major jet-fire accident, followed by X₆₂ (no response to manual emergency valve).

3.4. Updating Probabilities and Impact Contributions of REs

According to the proposed methodology, the initial occurrence probabilities of REs as well as UEs, including CEs and OEs, need to be updated upon the acquisition of new plant-specific failure data pertaining to specific REs. The updated probabilities, as calculated and tabulated in

Table 4. The updated probabilities of REs and UEs in the BN accident model.

RE	Cumulative Plant-Specific Failure Data	Number of Samples	Updated Probability	UE	Updated Probability
X3	2 times within 56,160 h	8	3.34 × 10−2	CE	6.18 × 10−2
X6	1 times within 17,520 h	2	1.69 × 10−1	OE1	5.35 × 10−2
X7	0 times within 105,500 h	8	2.42 × 10−2	OE2	8.22 × 10−7
X10	1 times within 73,270 h	8	8.94 × 10−2	OE3	6.80 × 10−3
X11	1 times within 73,270 h	8	7.17 × 10−2	OE4	1.34 × 10−3
X39	1 times within 87,640 h	10	2.12 × 10−2	OE5	2.06 × 10−8
X40	1 times within 98,480 h	10	6.38 × 10−2	OE6	1.70 × 10−4
X41	1 times within 121,040 h	10	6.43 × 10−2
X43	1 times within 103,680 h	12	2.45 × 10−2
X55	1 times within 141,080 h	12	1.99 × 10−2
X56	1 times within 109,820 h	10	4.11 × 10−3
X58	1 times within 165,600 h	14	4.93 × 10−2
X59	0 times within 146,300 h	14	2.22 × 10−2
X61	1 times within 107,900 h	10	4.24 × 10−2

, reveal a notable variation from their initial estimates, albeit remaining within the same order of magnitude. This discrepancy in probabilities underscores the dynamic nature of PRA, reflecting real-time shifts in risk profiles. In instances where the probability of occurrence of a CE or OE exceeds the acceptable threshold set forth by regulatory bodies or governing authorities, the implementation of integrity maintenance measures becomes imperative and must be undertaken without delay.

The contribution of each RE to the occurrence of UEs exhibits slight variations, as clearly demonstrated in Figure 6 and Figure 7 (refer to the red bars, signifying the updated contribution weight values assigned to the REs). To ensure that the updated results remain timely and can effectively guide the allocation of maintenance resources for the subsequent year, it is advisable to restrict the maximum time interval for probability updating to one year. The optimal application of this “dynamic” approach lies in its integration with the station’s integrity data management platform, thereby facilitating the dynamic updating of probabilities based on real-time failure sample data. The reduction in uncertainty in prior probabilities, which is a notable advantage of the Bayesian update estimation method, is expected to intensify as the amount of plant-specific data from GDSs continues to accumulate. This trend, as thoroughly explained in previous studies [7], highlights the significance of continuous updating. Consequently, the dynamic updating of probabilities using real-time data enables a more precise pinpointing of critical monitoring points within the station.

3.5. Adapting Probabilities for Accident Cause Diagnosis

Utilizing the dynamic BN model of the GDS, the prior probabilities of REs can be adapted given the occurrence of near-misses or incidents. Through continuous updating and adapting the RE probabilities, the reliability and accuracy of the assessment results generated by the dynamic prediction model can be enhanced [28].

Table 5. The adapted probabilities of the REs.

RE	Adapted Probability	RE	Adapted Probability	RE	Adapted Probability
X1	2.47 × 10−3	X19	2.91 × 10−3	X37	5.17 × 10−2
X2	1.92 × 10−4	X20	2.98 × 10−2	X38	2.48 × 10−4
X3	1.75 × 10−1	X21	7.93 × 10−3	X39	3.99 × 10−2
X4	2.75 × 10−1	X22	2.91 × 10−3	X40	1.20 × 10−1
X5	2.62 × 10−3	X23	1.51 × 10−2	X41	1.68 × 10−1
X6	1.83 × 10−1	X24	2.10 × 10−3	X42	2.09 × 10−3
X7	4.11 × 10−2	X25	1.94 × 10−3	X43	3.57 × 10−2
X8	1.60 × 10−1	X26	1.94 × 10−3	X44	4.17 × 10−3
X9	1.60 × 10−1	X27	5.10 × 10−4	X45	2.75 × 10−2
X10	1.65 × 10−1	X28	9.60 × 10−4	X46	6.99 × 10−2
X11	1.33 × 10−1	X29	9.80 × 10−4	X47	3.72 × 10−3
X12	1.48 × 10−3	X30	1.22 × 10−3	X48	4.35 × 10−2
X13	6.99 × 10−2	X31	2.30 × 10−4	X49	2.75 × 10−3
X14	2.75 × 10−3	X32	1.90 × 10−4	X50	3.08 × 10−3
X15	1.59 × 10−2	X33	1.32 × 10−3	X51	2.10 × 10−2
X16	1.62 × 10−1	X34	1.66 × 10−3	X52	1.68 × 10−2
X17	3.72 × 10−3	X35	1.80 × 10−4	X53	1.54 × 10−2
X18	4.05 × 10−3	X36	1.90 × 10−4	X64	2.93 × 10−2

shows the adapted probabilities of the REs subsequent to a leakage event at the GDS. A comparison of the RE probabilities before and after the update reveals the most critical REs that significantly contribute to the occurrence of the GDS leakage accident. As depicted in Figure 8, the REs X₁₃~X₂₅, X₄₅~X₅₃, and X₆₄ are identified as the most critical, indicating that they have the greatest impact on the likelihood of the GDS leakage accident. Consequently, these REs should be prioritized in the risk-management plan to mitigate the failure probability of the GDS.

3.6. Practical and Managerial Implications

The proposed methodology offers actionable insights for maintaining GDSs. By dynamically updating probabilities of REs and UEs, operators can achieve the following:

• Prioritize Critical Components: The importance measures (e.g., BM, RRW) identify high-risk REs (e.g., X₄: non-compliance with operating procedures, X₆₀: automatic emergency cut-off controller failure), enabling targeted inspections and resource allocation.
• Real-Time Risk Mitigation: Integration with GDS integrity management platforms allows automatic updates using new failure data. For instance, if X₆₀’s failure probability increases, maintenance teams can immediately inspect controller units or schedule replacements.
• Adaptive Maintenance Plans: The probability adaptation feature (Section 2.3.3) refines maintenance schedules based on near-misses. For example, a leakage incident triggers adjustments to RE probabilities, highlighting vulnerabilities in corrosion detection (X₃₇) or valve integrity (X₅₅), prompting preemptive repairs.
• Regulatory Compliance: Updated UE probabilities (e.g., CE: 6.18 × 10⁻² post-update) provide quantifiable metrics to ensure compliance with safety thresholds, avoiding penalties and enhancing public trust.

These capabilities transform traditional reactive maintenance into a proactive, data-driven strategy, reducing downtime and operational risks.

4. Conclusions

4.1. Theoretical Contributions

This study makes some improvements in the methodological framework for the safety assessment of complex systems, and the theoretical contributions are as follows:

• By integrating Bayesian network technology, fuzzy expert judgment, and Bayesian updating methods, the framework addresses data uncertainty challenges in safety assessments. Unlike existing approaches limited to unidirectional updates (e.g., backward diagnosis), this framework uniquely supports both forward dynamic prediction (updating UE probabilities with new RE data) and backward dynamic adaptation (adjusting RE probabilities based on observed incidents), enabling comprehensive risk-evolution analysis.
• A novel scenario-deduction model explicitly maps the causal chain from root events (e.g., X₄: procedural non-compliance) to critical events (CE: gas leakage) and outcome events (e.g., OE₆: long-duration jet fires). This model fills a gap in prior studies by systematically quantifying dynamic interactions among risk factors.
• The introduction of the RoV (ratio of variation) and hybrid importance measures (BM, RRW) establishes a quantifiable linkage between component-level failures and system-level risks, enhancing the precision of causal inference under uncertainty.

4.2. Practical Contributions

The proposed methodology delivers actionable solutions for GDS safety management:

• Real-time updates of RE probabilities (e.g., X₆’s probability increasing from 9.57 × 10⁻² to 1.69 × 10⁻¹ in Table 4) enable operators to prioritize inspections on high-risk components.
• Updated UE probabilities (e.g., CE: 6.18 × 10⁻²) provide quantifiable metrics to align with safety thresholds, avoiding regulatory penalties and enhancing public trust in GDS operations.
• Integration with GDS integrity management platforms automates risk alerts (e.g., triggering controller replacements when X₆₀ exceeds 3.0 × 10⁻¹), transforming reactive protocols into adaptive strategies.
• Diagnostic analysis (Section 3.3, Section 3.4 and Section 3.5) identifies non-critical REs (e.g., X₂₉: internal coating peeling), allowing operators to reallocate maintenance budgets from low-impact areas to high-priority interventions, such as procedural training (X₄) and controller redundancy (X₆₀).

4.3. Cross-Regional Application Expansion

The proposed framework, while initially tailored for China’s natural gas distribution stations (GDSs), holds potential for broader applicability in other regions or countries. To ensure effective implementation, contextual adaptations are essential. First, regional regulatory frameworks and environmental conditions must be integrated. For instance, safety thresholds (e.g., acceptable gas leakage probabilities) should align with local standards, while natural hazards prevalent in specific areas (e.g., earthquakes in Japan or hurricanes in coastal regions) necessitate adjustments to root event probabilities (e.g., X₂₄: earthquake likelihood). Similarly, infrastructure maturity and operational practices—such as variations in equipment maintenance protocols or workforce training levels—would require the recalibration of key parameters (e.g., X₄: procedural non-compliance rates) through localized expert judgments or historical data.

Additionally, data availability and technological disparities play a critical role. In regions with sparse failure records, reliance on fuzzy expert judgment and Bayesian prior optimization would intensify to compensate for data gaps. Conversely, in areas with advanced monitoring systems (e.g., IoT-enabled pipelines in Europe), real-time data streams could enhance dynamic probability updates. Furthermore, localized safety barriers—such as explosion-proof designs in high-risk zones or region-specific emergency response protocols—must be embedded into the model’s structure (e.g., modifying SBFEs or SCE nodes). By addressing these contextual nuances, the framework can be systematically adapted to diverse operational landscapes, balancing methodological rigor with regional practicality.

4.4. Future Work

Though this study has advanced both theoretical and practical aspects of dynamic risk assessment, challenges persist in correlating multi-state failures and refining causal relationships using big data. Future research will primarily focus on three key directions. First, we intend to extend the Bayesian network (BN) framework by incorporating multi-state component degradation models. This will enable a more precise depiction of how components degrade over time, which is vital for accurate risk evaluation in intricate systems. Second, we plan to leverage machine learning to automate probability updates from heterogeneous data streams like IoT sensors and maintenance logs. This approach will allow us to efficiently extract valuable information from real-time data and dynamically adjust risk probabilities.

Another crucial aspect of our future work is to validate the framework in diverse industrial contexts, such as offshore pipelines and chemical plants. By doing so, we aim to generalize its applicability and ensure that it can be effectively utilized across different industrial settings. Overall, this research acts as a bridge between theoretical risk modeling and practical safety management, offering a scalable toolkit to enhance the resilience of complex industrial systems.