Electrical Fire Dynamic Risk Assessment for High-Rise Buildings Based on Variable Fuzzy Set Theory and Bayesian Network

High-rise buildings fres are far more harmful than ordinary fres. In this regard, fre risk assessment is an important way to control fre risk and reduce losses. Tis study presents a comprehensive model to electrical fre dynamic risk assessment of high-rise buildings based on a Bayesian network (BN) and a variable fuzzy set theory (VFST). Firstly, electric system, safety management, and other factors were comprehensively analyzed based on three categories: hazard sources identifcation (HSI), fault tree (FT) analysis, and VFST. A high-rise building electrical fre dynamic risk assessment model was established based on a BN. Secondly, the prior probability of BN root nodes was determined by VFST, and the conditional probability table (CPT) was determined by the analytic hierarchy process (AHP) and decomposition method. On that basis, the quantitative inference and sensitivity analysis can be performed on the electrical fre risks of high-rise buildings in combination with the variable fuzzy Bayesian network (VFBN) inference. Finally, a high-rise building in Wuhan, China, was used as an example for verifcation. Te results show that the proposed method can realize dynamic risk assessment of electrical fres in high-rise buildings. Tis study provides a new method for fre risk assessment of high-rise buildings to reduce the possibility of fre.


Introduction
A growing number of high-rise buildings have been constructed in recent years, due to the increasing scarce land for urban construction.Tese buildings are characterized by multiple foors, high height, large volume, and concentrated personnel, making them exposed to higher fre risk than normal buildings [1][2][3].Meanwhile, high-rise buildings are multifunctional, highly electrifed and automated, equipped with lots of electrical facilities, and consume a large amount of power, making them vulnerable to electric leakage, short circuit, and other faults and prone to fre accidents [4,5].According to the statistics of high-rise buildings fre cases in the recent decade, the fres caused by electrical system faults accounted for about 31% [6].For example, the fres of Windsor Tower [7], the Plasco building [8], and the Grenfell Tower [9] were due to a short circuit, which resulted in severe human injuries and property damage.Terefore, it is necessary to establish an electrical fre risk assessment system for high-rise buildings and quantify the electrical fre risk level.
To reduce the hazards and economic losses caused by fres, a number of scholars have carried out research in this feld [10][11][12][13].In general, fre risk assessment methods are based on a particular application scenario.For diferent assessment scenarios, some assessment methods based on system theory have been proposed previously.For example, Liu et al. established a fre risk assessment system for largescale commercial buildings to evaluate the risk of fre protection system [14].Li et al. analyzed fre characteristics of high-rise buildings under construction and put forward a fre risk assessment method for high-rise buildings under construction based on unascertained measure theory [15].
Aiming at the difculty in carrying out efective rescue in case of fre in super high-rise buildings, Sun and Luo evaluated the risk of fre in super high-rise buildings [16].
Teir research results indicate that fre accidents in super high-rise buildings are mainly caused by electrical fre.Terefore, in order to reduce fre accidents, it is of great importance to assess the risk of electrical fre.Li and Zhu proposed a method for quantitatively detecting electrical fre hazards in high-rise buildings based on a modifed interpretive structural model (ISM) [4].Te research works of electrical fre risk assessment mainly focus on two aspects: electrical fre risk assessment and electrical fre earlywarning algorithm.For instance, Wang et al. improved the accuracy of electrical equipment detection, as well as smoke and fame detection in the warning system, by using convolutional neural network fre early-warning algorithm [17].Consequently, the studies of electrical fre risk assessment failed to focus on buildings fre risk assessment, particularly in high-rise buildings where fre accidents are frequent due to electrical factors.Te occurrence of electrical fres involves an extremely complex process, and the electrical fre risk factors have considerable uncertainty.In addition, there is a certain interaction relationship between fre risk factors.More importantly, many electrical parameters associated with fre risks in the electrical system are time varying, which poses a challenge for quantitative assessment methods dominated by expert opinion.
BN can be employed to handle multistate variables and dependencies between variables and update probabilities based on new evidence and reasoning in case of uncertainty [18][19][20].Tus, it can be considered a robust risk assessment technique [21].Given the infuence of uncertainty, Pei and Wang proposed a revised BN assessment model and applied it to high-rise buildings fre risk assessment [22].Under the circumstances of lacking detailed data and complete knowledge, it is difcult to utilize the traditional BN to perform accurate quantitative analysis, especially for highrise buildings projects that are seriously damaged.Furthermore, it is also impossible to adopt the traditional BN to obtain accurate data.Terefore, many scholars have applied fuzzy logic in combination with the BN theory to risk assessment projects on uncertain occasions.For example, Zarei et al. put forward a fuzzy Bayesian network (FBN) method for the safety assessment of process systems [23].Zhang et al. performed a risk assessment for pit collapse in subway stations based on FBN and fuzzy hierarchy analysis [24].Due to the fact that the advantages of fuzzy set theory (FST) and BN are integrated into FBN, it can clearly represent the complex relationship between each risk factor, deal with the uncertainty of risk data more efectively, and obtain more accurate quantitative assessment results [25].However, whether FBN or BN, obtaining the prior probability and CPT under the circumstances of lacking detailed data and complete knowledge is one of the key challenges that needs to be eliminated [24].
With the advancement of electrical monitoring and Internet of Tings (IoT) technologies, a lot of technical support has been provided for intelligent detection, information monitoring, and personnel positioning.Hence, a method for dynamic risk assessment of electrical fres in high-rise buildings was proposed in this study by combining engineering applications of electrical monitoring and IoT technology.Specifcally, the electrical fre risk factors were determined based on three categories of hazard sources and FT analysis, and a high-rise building FT assessment model was established by combining dynamic risk factors and static risk factors.Besides, the FT assessment model was integrated into the BN model for risk inference.In addition, the prior probability of root nodes was determined based on the VFST.Compared with the fuzzy analytic hierarchy process (FAHP) [26], this method could reduce the subjectivity of expert assessment to some extent.Meanwhile, the membership function of the variable fuzzy set has a correspondence with the actual physical quantity, which improves the credibility that it is not an a priori condition under the circumstances of lacking detailed data and complete knowledge.Moreover, the decomposition method was adopted to determine the CPT of intermediate nodes, which could reduce the subjectivity of experts in multistate assessment.Finally, a sensitivity analysis based on indicators identifed the key risk factors for electrical fres in high-rise buildings.
Te remainder of this paper is organized as follows: Section 2 introduces the theoretical background of BN and VFST.Section 3 describes the procedure of the proposed method.Section 4 combines a case study to demonstrate the risk assessment process for electrical fres in high-rise buildings.Te discussion and conclusions are presented in Sections 5 and 6, respectively.

Preliminaries
2.1.Bayesian Network.BN is a modeling method based on a probability graph model.In this method, qualitative analysis methods are combined with quantitative analysis methods to make efective inferences, and objective evidence and prior probabilities are employed to analyze uncertain problems in complex systems.Tis method enables the study of causal relationships among several factors as a research objective, and the degree of dependence among these factors, refected by the conditional probability distribution, has been widely used in many felds.Electrical fres in high-rise buildings are highly complex and uncertain.Terefore, the risk of electrical fres for a characterized building cannot be predicted with the assessment based on a frequency interpretation approach.In this study, the dynamic assessment model of electrical fre risks in high-rise buildings based on BN combined with VFST can be used to analyze the uncertainty in electrical fres and identify the relationship between each risk factor.
Te whole concept of BN is built on Bayesian theorem, which indicates the relationship between the prior probability and the posterior probability of an event through Bayesian formula as follows: where P (A) and P (B) are the prior probabilities, P (A) > 0, P (A|B) is the conditional probability, and P (B|A) is the posterior probability.

Mathematical Problems in Engineering
Suppose, there is a BN node x i and Pa [x i ] is the parent node set of x i .Te number of BN nodes is n.Te calculation formula of the BN joint probability distribution function can be expressed as follows: Te probability that system A occurs can be calculated directly by utilizing the joint probability distribution through prior probabilities and conditional probabilities in the network.

Variable Fuzzy Set Teory.
In view of the uncertainty and fuzziness of electrical fre in high-rise buildings, this study adopts VFST method for risk assessment [27,28].At present, VFST has been applied in many felds and has broad application prospects [29][30][31].Te details are as follows: (1) Each index of the assessment system was divided into assessment intervals according to m levels, with 1 level as the worst level and m level as the best level.
Ranking the interval values at each level in turn results in a matrix of indicator evaluation intervals [32].
where [a nm , b nm ] is the standard value interval of index n under the m level; a nm and b nm represent the upper and lower limits of the interval, respectively.
where [c nm , d nm ] is the variable range value interval of index n under the m level.c nm and d nm represent the upper and lower limits of the interval, respectively.
(2) According to matrix I ab , the point matrix of index i corresponding to [a, b] ik , whose relative membership degree is equal to 1 is expressed as follows: where, It is assumed that the actual grade of electrical fre assessment index of high-rise buildings is X � [x 1 , x 2 , . .., x n ], and compare the actual score x i with the corresponding point-value matrix M ik (k � 1, 2, 3, 4, 5) [33].
If x i ≤ M ik , the membership formula is as follows: If x i > M ik , the membership formula is as follows: Using ( 8) and ( 9), the relative membership matrices of assessment index in diferent levels were obtained as follows [34]: According to the characteristics of electrical systems and high-rise buildings, three categories of hazard sources are proposed in the literature [1]. Figure 2 shows three categories of hazard sources in high-rise electrical systems.Based on the HIS technology of electrical fre in high-rise buildings, combined with the buildings fre protection standards and fre risk assessment guidelines of the United States, Britain, and China [35][36][37][38], referenced from the previous research results [39] and real fre case investigation reports [7][8][9]40], the main risk factors and the FT model can be determined.

Establishment of BN Model.
Establishing BN structure directly from existing fre case data is difcult.Fortunately, transformation methods based on FT or event tree to BN structure have been applied in engineering [41].After the key risk factor of electric fres were identifed and the FT relationships were established in the previous step, a BN risk assessment model was established based on the combination of dynamic and static assessment indexes by FT.FT was directly converted to the equivalent BN model according to the transformation relationship between FT and BN, as listed in Table 1 [19].In the numerical transformation principle, the probability for the occurrence of a major event serves as the prior probability at the root node.CPT was obtained by Boolean gate of FT [24].

Variable Fuzzy Prior Probabilities and CPT.
After a BN assessment model is established, the model parameters, including the prior probability of root nodes and the CPT of intermediate nodes, need to be imported.If enough available data are collected, the prior probability and CPT of BN can be obtained based on shrinkage methods [42] or hill climbing algorithms [43].However, due to the severe loss caused by high-rise buildings fres, it is difcult to provide enough available data for the construction and parameterization of BN models in engineering practice.Terefore, during BN modeling, the establishment of probabilistic parameters of BN still relies on the inspiration of experts.However, the involvement of human judgment inevitably brings subjectivity and ambiguity.To date, many scholars have introduced fuzzy BN for risk assessment analysis.However, there are some defciencies in the BN based on the fuzzy set theory, such as membership function fxation and poor adjustability.In this study, a VFST was adopted to deal with the uncertainty and ambiguity of the criteria and judgment process.decomposition methods, and improved weight calculations, the prior probability and CPT of BN can be established as follows.

Generation of Root Nodes' Prior Probabilities.
Assume that the root node N has n states, S 1 , S 2 , ..., S n , the probability of the occurrence of state S i is P (S i ).In the traditional risk assessment methods, P (S i ) is usually obtained from FAHP or AHP by expert assessment, which is limited by the efect of the number of states.Under the circumstances of a large number of states and a complex BN model structure, the accuracy of the assessment results cannot be ensured.Given the subjective assessment of experts, there would also be strong subjectivity for the risk factor parameters in an assessment model.In this study, root node correspondence bottom factor are assessed by VFST to determine the probabilities of each state of root nodes.
Each basic event corresponds to each root node, and the factors contributing to the occurrence of that event can be subdivided, namely bottom risk factor.Besides, objectivity shall be considered for risk factors, which preferably correspond to the actual physical quantities.It is also an advantage of VFST compared with FST.Assume that the root node n corresponds to m assessment factors D 1 , D 2 , ..., D m , the weight coefcients of each assessment factor can be expressed as w 1 , w 2 , . . ., w m w ( 1 + w 2 , . . .+ w m � 1).Te weight calculation by AHP is a common method used in quantitative risk assessment [44], but this method yields a weight value that is constant and subjective.Terefore, the variable weight comprehensive theory with equilibrium function is adopted to optimize the weight coefcient [45], and the formula for calculating variable weights can be expressed as follows: where α refects the equilibrium of variable weight theory and it is generally determined according to practical engineering experience.A large amount of engineering experience indicates that α = 0.2 is suitable for general engineering [46].w 0 i is the constant weight of the i-th index; x i is the standardized score corresponding to the i-th index, and standardized index scores used extreme value method [47]; m is the number of assessment indexes.
After the variable weight coefcient is calculated, the risk score criteria for each factor can be divided into k (k � n) levels, which is the same as the number of states at the root node.Specifc division boundaries refer to industry specifcation standards with the incorporation of expert experience.Ten, the factors are scored by some experts according to the established risk score criteria.After the means are counted, the eigenvalues of each factor can be obtained.Combined with the variable fuzzy set principle introduced in Section 2.2, the eigenvalues of each index are calculated sequentially corresponding to I ab , I cd , m, and μ A .Te afliation vector R of each index can be obtained by normalizing μ A .According to the univariate assessment model of fuzzy integrated risk assessment, R can be substituted into (12) to obtain a comprehensive risk vector as follows: where " ∘ " is weighted average fuzzy composition operator [48]  According to the method proposed in a previous research [49], for a node N with n states (S 1 , S 2 , ..., S n ), it has a parent node count of m (m ≥ 2), that is, (T 1 , T 2 , ..., T j , ..., T m ).Among them, the parent T j has k states, namely, S Tj (1) , S Tj (2) , . .., S Tj

(k)
. Te prior probability that each state of n is conditional on the diferent state combinations of its parent nodes can be expressed as follows: For a node A, which has two parent nodes B and C, its conditional probability on B and C can be approximated according to (14) as follows: Combined with (13), it can be simplifed as follows: where α is a normalized constant to ensure that 3.5.Risk Analysis Based on VFBN 3.5.1.Inference Analysis.Te assessment results based on BN can be mainly analyzed by the forward inference and backward inference methods.In terms of the forward inference method, the probability that a leaf node corresponds to the occurrence of a risk event can be calculated by the joint probability distribution, as (2) described in Section 2.1, which can be used to perform a quantitative assessment for the overall risk level of high-rise buildings.Te backward inference calculation for the BN model is fault diagnosis, which is the calculation of the posterior probability of each risk factor by (1) described in Section 2.1.When an outcome event is bound to occur, the key factors that lead to the occurrence of a disaster can be derived.Based on that, critical risk factors for the occurrence of an electrical fre can be identifed scientifcally.

Sensitivity Analysis.
Sensitivity analysis, which is performed to determine the contribution rate of each risk factor to a risk event, plays an important role in probabilistic risk assessment.In actual fre risk assessment, it is often required to identify the most important factors, which conduces to further fre risk control.Te sensitivity analysis of the BN model could identify the most important factors.

Case Study
4.1.Background.An ofce building in Wuhan, China, with a height of 32 m and a total of 9 foors, is selected as a case sample.Te main function of the building is to conduct power testing and to provide power advisory services.On the frst foor, there is a high-voltage power distribution room with an area of 60 m 2 , and a low-voltage power distribution room with an area of 115 m 2 , 3 administrative power distribution rooms with an area of 19 m 2 , and two laboratories with high-power electric equipment.Te rest of rooms are regular ofces, and the whole foor can be regarded as a fre prevention zone, as shown in Figure 3.All foors of the ofce building are equipped with automatic sprinkler systems, fre alarm devices, and fre hydrants.Case buildings part actual situation is shown in Figure 4; Figure 4(a) shows the overall structure of the building.Te high-voltage supply cable and the low-voltage distribution cabinet are, respectively, shown in Figures 4(b) and 4(c).A high voltage laboratory in the case is shown in Figure 4(d).Tis ofce building is located in a transportation hub, with convenient transportation, humid, and hot climate.Within 2 km, there are two fre brigades with perfect frefghting equipment.

Fire Risk Factor Analysis.
Tere are many risk factors for electrical fres in high-rise buildings, and there is also a certain interaction relationship between various fre risk factors that afect fre risks.Li et al. analyzed the coupling relationships among various factors of electrical fres in high-rise buildings by ISM [4], including the source of fre, the fre environment, the victims, and the fre drivers.To realize the dynamic risk assessment in this paper, based on Li's study results and fre accident investigations [7][8][9], the electrical fre risk factors in high-rise buildings are identifed from the aspect of the failure of electrical lines, the failure of electrical equipment, and the fre protection ability of building by using FT analysis.Electrical fre risk factors are mainly classifed into two categories: (1) Dynamic risk factors: Trough the analysis of data from cases of electrical fre accidents, abnormal sign information such as increased current, voltage fuctuation, and temperature rise will occur before the occurrence of electrical fre.Terefore, the dynamic risk factors select electrical line failure (B 1 ) as the risk factor, which specifcally includes four types of electrical failure that mainly lead to the formation of an ignition source: line short circuit (C 1 ), overload (C 2 ), ground failure (C 3 ), and failure arc (C 4 ).Electrical failure (C 1 , C 2 , . .., C 4 ) occurrence is the result of the combination of factors, and fre can only be stated if there is a certain degree of abnormality in multiple parameters.To achieve a dynamic risk assessment in conjunction with the electrical fre monitoring system, the occurrence of an electrical failure was associated with a bottom risk factor (D    8 Mathematical Problems in Engineering can be found by the feedback of the electrical fre system.To enable the generalizability of assessment methods, the eigenvalues of dynamic risk factors are set as the ratio of real values to baseline values (rated values).Since static risk factors nodes would not be changed in a short period, static nodes are mainly dominated by expert assessment.First, the risk states division of the VFST of static bottom risk factor can be established based on the analysis of electrical fre cases in recent years, relevant specifcations in the electrical industry and construction industry, and the survey of previous studies (Table S2).Ten, the bottom risk factors corresponding to each static node can be scored by domain experts on site.In this study, the prior probabilities and CPT for VFBN are derived by three domain experts based on the collected information and their experience.
Taking dynamic root node C 1 as an example, based on the measured data of the electrical fre monitoring system, it can be obtained that the eigenvalue of the dynamic risk factor C 1 corresponding to the bottom factors (D 1 , D 2 , and D 3 ) are X 1 � [0.88, 0.9, 1.4].Te index scores are standardized, and the standard rank intervals in Table 3 S1, the risk assessment standard interval matrix I ab can be expressed as follows: (3) Te variable interval matrix I cd can be expressed as follows: (5) Te relative membership matrix of the grade of the secondary risk factors can be calculated based on (8) and ( 9) and then normalized to obtain the membership matrix of the secondary risk factors of the assessment system as follows: Finally, after the variable weight coefcient is obtained by Step (1) and equation ( 12), the coupling factor weight with the membership matrix is selected as a fuzzy operator, and the prior probability under the feedback value corresponding to the dynamic node C 1 can be obtained as follows: Based on that, the prior probabilities of other root nodes can be provided specifcally in Table 3.
As described in Section 3.4.3, the method combining AHP with the decomposition method can be used to calculate the CPT of each node.Te comparison matrix can be derived by opinion assessment from 3 experts.Te frst expert is a senior engineer at Hubei Academy of Electric Power Sciences, the second one is a professor at China Petroleum University, and the third one is an associate professor at China Petroleum University.Examples would not be provided for calculations due to space limitations.Te specifc calculation process of determining CPT based on the combination of the decomposition method and AHP can be consulted in the previous literature [49].

Case Result Analysis
4.5.1.Inference Analysis Result.After the prior probability and CPT of nodes are calculated, Bayesian inference is performed with Netica.As shown in Figure 6, the probability of an electrical fre in a high-rise building is P (A high) � 16.5%, is higher than 10%.Terefore, it can be argued that there are some risks for the occurrence of electrical fres in this high-rise building and decision makers should take measures to conduct risk containment.With rather limited posterior knowledge, one can use forward inference to make a preliminary judgment on the overall risk of an electrical fre in a high-rise building and then determine the optimal plan to eliminate fre hazards.
In the event of an electrical fre accident in a high-rise building, it is common to cut the whole power supply system frstly, and then, the concerned department immediately organizes experts to perform an on-site analysis to determine the causes inducing the fre.
Due to a larger variety of features in a high-rise building, too long assessment time also causes certain economic losses.With the help of the inference of the posterior probability, the reverse inference analysis is conducted for electrical fre accidents to realize the dynamic detection of the accident cause.Meanwhile, the posterior probability can also be used to determine the important infuence degree of each fre risk factor on the occurrence of electrical fres.Te posterior probability of each risk factor can be calculated with (1) under the electrical fre occurrence condition of P (A � High) � 1.As shown in Figure 7, the top six fre risk factors can be ranked as on their posterior probabilities.C 4 (arc fault), C 6 (operation risk of electrical equipment), and C 3 (grounding fault) are the causes with the largest possibility inducing electrical fres.Terefore, safety inspection and risk management should be performed on arc faults, line-to-ground faults, and electrical equipment operation faults.Trough inspection, it is found that there are some problems of excessive contact resistance at the connection points of electrical lines and some nonstandard grounding.In the static risk assessment methods based on traditional expert experience, it is difcult to identify these dynamic risk factors with physical quantity changes.In addition, buildings structure risk (C 8 ) and plane layout risk (C 9 ) are factors with a large posterior probability among nonelectrical factors, which is the same as the building's fndings in a recent fre inspection report.Terefore, the risk assessment method proposed in this study can be used to scientifcally and reasonably assess the real-time changing electrical system.

Critical Analysis.
During quantitative risk assessment, the analytical identifcation of signifcance for the basic event C i is an important step.Obtaining critical parameters of risk factors through sensitivity analysis of the BN model is an important step during risk assessment.Terefore, sensitivity analysis is required in identifying the most critical variables or factors.With the sensitivity analysis technique of BN, the risk factors afecting electrical fres in high-rise buildings can be adjusted and optimized in real time, which contributes to fre risk control.Te RAW, RRW, and AVG for each basic event of an electrical fre can be obtained according to equations ( 11)- (13).Te calculation results are shown in Figure 8.As can be seen from the rank results of AVG values in Figure 8, faults arc (C 4 ) and electrical equipment operation faults (C 6 ) are the top two risk factors.Based on this rank, fre digital simulation analysis can be conducted on  Mathematical Problems in Engineering four common electrical fre risk factors: short circuit, overloaded circuit, arc fault, and ground fault.After the of thermal release smoke density transformation, CO concentration change, and CO 2 concentration change are considered comprehensively, the simulation results can be obtained, as shown in Figure 9.As shown in Table 4, it can be concluded that the four common electrical faults can be ranked based on their hazards: arc fault (C 4 ) > ground fault (C 3 ) > short circuit (C 1 ) > overload (C 2 ), which is diferent from those in Figure 8 . Te reason is that the overloaded circuit in Figure 8 is associated with the electrical device state, which would cause their corresponding AVG bigger.Terefore, the risk assessment model proposed in this study is verifed to be objective and reasonable.

Discussion
Electrical fres have been the leading cause of fre outbreaks in high-rise buildings.However, it is difcult to perform an accurate quantitative risk assessment for electrical fres due to the complexity and uncertainty of causes.FBN, which combines the uncertainty of BN and the ambiguity and uncertainty of FST, has been widely used in the safety assessment of engineering.In this paper, a combination of VFST and BN, in which membership function can be adjusted and correspond to the physical meaning of risk factors, is applied to the dynamic risk assessment of electrical fres in high-rise buildings.Based on the variable fuzzy probabilities and electrical quantity monitoring values from expert assessment, this method can be used for electrical fre risk assessment.Besides, it can also be employed to quantify electrical fre risk probabilities and identify the most sensitive cause for the occurrence of electrical fres.Some questions would be discussed as follows.

Dynamic Uncertainty Assessment.
For an electrical fre, it is common that an electrical fault occurs in conjunction with a mutation in the electrical quantity in the circuit.Tat is to say, the electrical fre risk is variable in real time.Meanwhile, it is difcult to collect the risk data from electrical fres in high-rise buildings, which causes a blockage in logistic analysis calculations.In this study, a BN risk assessment model is established based on the combination of dynamic and static assessment risk factor.For dynamic risk factors, the electrical fre risks in high-rise buildings can dynamically assessed through the monitoring, state identifcation, and inference of medium electrical information by an electrical fre monitoring system.For static risk factors, the eigenvalues of bottom risk factors can be judged based on the knowledge and experience of domain experts, and a few indicators are related to building physical properties.Based on that, the same reasoning and inference algorithm as conventional BN is adopted for predictive analysis and probability update in this paper.Trough Bayesian inference, the probability of an electrical fre and the probability distribution of fre risk factors can be calculated in real time based on prior knowledge and evidence updates.Furthermore, the static node will not change in a short period after the prior probability is obtained through expert assessment, and VSFT assessment does not need to change in a short period.Hence, dynamic risk factor can be   Mathematical Problems in Engineering identifed through the real-time monitoring of the electrical fre monitoring system.Moreover, this method can realize real-time evolutionary inference, dynamic risk assessment, and management.

Risk
Inference.BN risk analysis features that it yields a posterior probability based on the actual observation of the node.Diferent from the prior probability, the posterior probability could exert an important impact on diferentiating the extent to which individual essential events contribute to this top event in electrical fre accidents.Te current investigative diagnosis of electrical fre accidents highly relies on the experience and knowledge of experts.Once an electrical fre occurs, the power supply system of the entire high-rise building would be typically cut of, and then, experts would be organized for site survey analysis.Tis presents a certain hazard to the personal safety of experts.In addition, prolonged power outages time would afect the daily use of high-rise buildings and cause more economic losses.With the assistance of the reverse inference technique of BN, fre risk factors can be reversely analyzed and the posterior probability of each risk factor can be calculated.Furthermore, fre risk control can be performed by prioritizing diagnoses for those infuencing factors with large posterior probabilities and quickly determining the source of fres.Tis improves the accuracy of identifying the fre source, accelerates the recovery in the normal operation of high-rise buildings, and reduces the loss caused by fres.

Key Factor Identifcation.
In electrical fre risk assessment, decision makers usually pay more attention to which factors play a crucial role in the occurrence of electrical fres in high-rise buildings.However, key factor identifcation is often dominated by the subjective judgment of experts.Te assessment method proposed in this paper can quickly identify the most dangerous fre factors and facilitate the adjustment and optimization of fre protection measures in real time by calculating the extent to which each fre risk factor has an impact on fre accidents.Tus, the method can identify key sensitive risk factors, which would facilitate real-time adjustment and optimization by measuring the impact of each risk factor C i on the top risk event A.

Conclusions
To reduce the hazards of electrical fres in high-rise buildings, a quantitative analytical model combining VFST and BN is proposed in this study to assess the risk probability of electrical fres in high-rise buildings.A BN risk assessment model based on dynamic risk factors and static risk factors is also established.Te prior probability of root nodes can be calculated with the membership function, a more fexible VFST method instead of the traditional FBN.Besides, the decomposition and hierarchical analysis methods are employed to determine the CPT.Te electrical fre cases in high-rise buildings are also analyzed, and the results verify the feasibility of the proposed method in engineering practice.Te prediction results of electrical risk probability in high-rise buildings by Bayesian forward inference demonstrate that electrical fres have a certain probability during their occurrence, and the overall risk level of electrical fres in high-rise buildings can be quantitatively assessed.By backward inference, the diagnosis can be made for the factors with larger efects that cause electrical fres, which can quickly strengthen the preparedness and control for these risk factors and lower the probability of an electrical fre accident.In addition, sensitivity analysis is conducted to identify key infuencing factors, as an attempt to control the risk of electrical fres in advance, and to ensure the safety of people in tall buildings.
Nevertheless, there are still several limitations in the methodology of this study.For example, this method relies heavily on domain experts in establishing prior probabilities of VFST and CPT.Although the subjective bias and uncertainty have been reduced to some extent by VFST-based risk states division, the efects of subjectivity cannot be eliminated.Moreover, the dynamic risk factors of electrical fre in high-rise buildings among the risk factors considered in this study are limited by the infuence of electrical monitoring technology.Electrical equipment-related dynamic risk factors were not considered, and more electrical fre cases are required for improvement.

Figure 1 :
Figure 1: Te overall procedure for risk assessment approach based on VFBN.

4. 4 .
Variable Fuzzy Prior Probabilities and CPT.In terms of dynamic root nodes (C 1 , C 2 , . .., C 4 ), the electrical system design and electrical equipment information for evaluating high-rise buildings are collected frstly to determine the risk states division of bottom risk factors.Te number of level intervals is the same as the number of states of the root node.Standard interval divisions for the assessment of the dynamic bottom risk factors (D 1 , D 2 . .., D 7 ) are shown in

( 4 )
According to(7), the point-value matrix M ik of the membership degree of 1 of each assessment interval of the secondary risk factors is as follows:

Figure 6 :
Figure 6: BN of electrical fre risk for high-rise building under prior conditions.

Figure 9 :
Figure 9: Results of electrical fre simulation: (a) heat release rate change curve; (b) smoke density change curve; (c) CO concentration change curve; (d) CO 2 concentration change curve.

Table 1 :
Mapping relationship from a FT to a BN.
[49][b 1 , b 2 ,. ..b k ] is the probability of each state of root node N.3.4.2.CPT for aNode with One Parent.Similarly, for a node N (S N1 , S N2 , ..., S Nn ) with n states, it can be assumed that it has only one parent node T (S T1 , S T2 ,. .., S Tm ) with m states.Under this circumstance, the main idea is based on T state of each node to estimate the probability of each state of node N, namely, P(S Ni |S tm ) (i � 1, 2, ..., n, j � 1, 2, ..., m).Based on the idea of AHP, experts are asked to determine which state S In this study, a method proposed in a previous literature[49]is adopted.AHP and decomposition can be combined to calculate the CPT for cases with multiple parents, and the basic idea of this method is to decompose the multiple parent nodes into multiple CPTs for single parent nodes, as described in Section 3.4.2.
[49]nd S Nk (k � 1, 2, ..., n) corresponding to N node is more likely to occur in state S Mj .Meanwhile, the 1-9 scale of AHP was used for comparison.After the comparison matrix is obtained, P(S Ni |S Tn ) � w ij can be further calculated.Te calculation process and the commonly used AHP to calculate index weights are the similar as reference[49].Since node M has m states, M matrices should be constructed to obtain all wij(i � 1, 2, ..., n; j � 1, 2, ..., m), and the CPTof the single parent node can be obtained.3.4.3.CPT for a Node with Multiple Parents.Under the condition that one node has multiple parents, the results of estimating multiple states by experts are extremely inaccurate.
Mathematical Problems in Engineeringfactor C i has a large impact on fre risks, it corresponds to a larger RAW (C i ).Similarly, a larger RRW (C i ) indicates that smaller changes in the indicator C i can signifcantly change a fre event.Birnbaum measure (BM) is the mean of RAW and RRW.Tese three indicators can be obtained by the following formulas: [50]t al.proposed the calculation method of sensitivity indicators based on the risk achievement worth (RAW) and the risk reduction worth (RRW)[50].In this study, this method was adopted to analyze sensitivity factors.If a risk 6 1 , D 2 , . .., D 7 ) as shown in Table 2. Te bottom risk factors are correspond to electrical parameters, and the specifc descriptions and risk states divisions are given in Table S1 ( Supplementary Materials).Dynamic bottom risk factor eigenvalues are obtained through the electrical fre monitoring system and served as an evaluation indicator for the probability of risk factors (C 1 , C 2 , . .., C 4 ) quantifed by VFST.
2 , . .., S 27 ) are shown in Table 2 and S2.4.3.Establishment of BN Model.After the establishment of the risk factor FT, there are a total of 14 root risk factors (C 1 , C 2 , . .., C 14 ) listed in the FT assessment model of electrical fre risks in high-rise buildings, which correspond to the root Mathematical Problems in Engineering node of the BN model.Te 5 factors categories correspond to intermediate nodes (B 1 , B 2, ..., B 5 ), and the top event A corresponds to the leaf node.To present the advantages of fexible and variable membership functions of VSFT, the risk status of each root node can be divided into four levels: H (high), M (medium), L (low), and S (safe); each intermediate node can be divided into two levels: H (high) and L(low).Te dynamic risk assessment BN model of electrical fre risks in high-rise buildings established in this study is shown in Figure 5.

Table S1
. It can be seen that the dynamic bottom risk factors are all physical quantities related to electrical faults, which can be applied to extreme value normalization.Eventually, x 1 � [0.824, 0.45, 0.72] can be obtained.

Table 2 :
Basic events for electrical fre risk assessment of high-rise buildings.Line short circuit risk level C 1 D 1 , D 2 , and D 3 Line overload risk level C 2 D 1 , D 4 , and D 5 Risk level of line grounding fault C 3 D 2 and D 6 Hidden danger of line fault arc C 4 D 2 and D 7 Electrical equipment failure B 2 Operation risk of distribution equipment C 5 S 1 , S 2 , and S 3 Operation risk of electrical equipment C 6 S 4 , S 5 , and S 6 Risk of electrical equipment placement C 7 S 7 , S 8 , and S 9 Building fre prevention capacity B 3 Building's structure risk C 8 S 10 , S 11 , and S 12 Plane layout risk C 9 S 13 , S 14 , S 15 , and S 16 Fire protection facilities capability B 4 Fire station risk C 10 S 17 and S 18 Reliability risk of fre protection system C 11 S 19 , S 20 , and S 21 Ability of evacuation safety management B 5 Safety management system C 12 S 22 and S 23 Evacuation routes and evacuation facilities C 13 S 24 , S 25 , and S 26 Fire emergency plan and exercise C 14 S 23 and S 27 According to the risk states division of dynamic risk factors of high-rise buildings in Table

Table 3 :
Prior probabilities of each root node.

Table 4 :
Rank of importance of infuence.