# Distribution Network Risk Assessment Using Multicriteria Fuzzy Influence Diagram

Authors: Aleksandar Janjic
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

Risk assessment of distribution assets is one of the most important factors in the process of network development or maintenance planning decision-making. The process of decision-making is faced with uncertainties, involving technical, financial, safety, environmental, and other operational issues that make standard risk assessment techniques insufficient. Probabilistic uncertainties require appropriate mathematical modeling and quantification when predicting future state of the nature or the value of certain parameters. The paper is proposing a new methodology for the multicriteria risk assessment of the distribution network assets, based on influence diagrams and fuzzy probabilities. Influence diagram has been used to determine all relevant factors concerning risks and their interdependencies are depicted. Fuzzy probabilities are represented by triangular fuzzy numbers with constraints on feasibility of elicited probabilities. This methodology enables the decision process in uncertain environment, with the impact evaluation of each particular distribution asset, or the asset component. The methodology is illustrated on the example of a distribution substation circuit breaker maintenance strategy selection.


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## Body

## 1. Introduction

Maintenance planning, development, and reconstruction of distribution networks are playing the crucial role in the asset management of distribution networks [1, 2]. One of the main problems of asset management in distribution companies is to find the best maintenance strategy out of following actions: do nothing and repair only after the breakdown, overhaul, or do the complete replacement of asset. Some activities, like minor or major maintenance, can be performed in a regular time interval, or depending on condition of an asset, but the problem is becoming more complex as alternatives must be evaluated on the basis of several criteria [3]. Some of them are easy to measure (costs and profit), while others can be very difficult to evaluate (public opinion, consequences of outages).Risk and uncertainties are also present in the process of decision-making, whether it is in presupposed data (consumption increase rate, prices, and preferences) or in decision factors of business environment that affect the process of decision-making. One of the latest approaches is the risk management based maintenance, which evaluates the risk of equipment failure and consequences such failure can produce on the system [4–6]. With the quantification of risk, the most efficient strategy and the optimal risk level for distribution networks assets management can be obtained [6]. In all these approaches, risk is defined as a combination of probability indices and the consequences of failure in the network.Decision about the optimal level of maintenance depends on several criteria of different nature:(i)
technical criteria(ii)
economic criteria(iii)
health and safety criteria(iv)
environmental impact(v)
public opinion and customer satisfaction(vi)
regulatory requirementsThe number and structure of these categories is changing, depending on particular conditions (legislative, regulatory requirements, etc.) but these are basic attributes out of which the others can be derived. Furthermore, the asset management problem is facing the probabilistic uncertainty and imprecision when modeling problem structural parameters, including the required goals, constraints, and external influences.Various theories of imprecise probability include the Dempster-Shafer evidence theory [7, 8], the coherent lower prevision theory [9], probability bound analysis [10], and the fuzzy probability [11]. Stochastic nature of parameters and subjective probabilities are often described with interval probabilities or fuzzy sets. Interval and fuzzy probabilities are used when it is hard to model uncertainty by point value probabilities: when little or no information to evaluate them is available, or when several information sources (sensors, individual experts in group decision-making) are combined [12]. Fuzzy modeling can be understood as an extension to interval modeling, and fuzzy probabilities can be characterized by a possibility distribution of probability, representing degree of confidence in that probability expressed by an individual [13, 14].Bayesian networks and Influence diagrams are used as a convenient tool for the large class of engineering problems, while the inherent uncertainty has been modeled by the fuzzification of random variables, and/or prior and conditional probabilities. A comprehensive review of development dealing with imprecise probabilities for the solution of various engineering problems is given in [15]. Fuzzy probabilities are treated as an extension of interval probabilities, emphasizing the correspondence between different α–levels and probability boxes. Various engineering analyses are then enabled using min–max operator and extension principle as the basis for the processing of fuzzy information.In Bayesian networks, uncertainty embodies both sources: aleatoric (random events or uncontrollable variation) and epistemic (as the absence of complete knowledge). Furthermore, fuzzy probabilities, grouped in several fuzzy sets, can be denoted with linguistic terms: “extremely low”, “very low”, “medium”, etc. [16–19]. These terms represent the information granules that are in great extent influenced by the psychological profile of the decision-maker.In the deterministic case, alternatives and consequences are directly related in terms of criteria. In the presence of uncertainties, there may exist many possible outcomes that can be described quantitatively or qualitatively (through verbal descriptions).Approaches like Bayesian networks, fault, and events trees are often used to understand and model random events and outcomes, but issues like interdependencies of different criteria in the decision-making process require further attention. New form of description, the influence diagram, that is both a formal description of the problem that can be treated by computers and a simple, easily understood representation is presented in this paper. The formal theory of Influence diagram is given in [20, 21], with the evaluation, or solving of influence diagram based on Bayesian networks.This work introduces a new methodology for the risk assessment in distribution network based on the extension of Influence diagrams with the fuzzy probabilities and different consequence evaluation. Risk assessment is performed in two steps. In the first step, influence diagram has been used to determine all relevant factors influencing risks with the depiction of their interdependencies, together with all possible alternative decisions. In the second step, the set of each particular risk values is calculated as the combination of risk factor occurrence and their consequences. Subjective probabilities are represented as information granules described by linguistic terms and modeled as triangular fuzzy numbers.In the next section of this paper, both steps of a risk assessment methodology using Bayesian networks and Multicriteria Influence diagram are presented. Building of an influence diagram and the way of solving it are presented. Using joint probability rule, the risk of particular event, for different risk categories, is calculated. In Section3, the notion of fuzzy probability has been explained and in Section 4 the methodology is illustrated on the case study of the choice of circuit breakers maintenance strategy in one transformer substation.


## 2. Risk Assessment Using Influence Diagrams

### 2.1. Risk Assessment

Risk assessment, as the first step in the risk management process, attempts to identify possible failure events, evaluate their consequences, determine the probability of their future occurrence, and reduce the detrimental consequences. The usual definition of risk associated with an eventE is defined as the product of event probability p(E) and its consequence Cons(E) [22, 23]:(1)RiskE=pE·ConsEMore complex relationships between values introducing empirical scaling parametersx,y, and w are presented in the following [24]:(2)RiskE=pEy·w·ConsExCalculated value of risk became a crucial factor when deciding about the actions to be performed on distribution asset. However, decisions have to be made in a very uncertain environment. In this paper, a new graphical tool based on Bayesian networks—influence diagrams for risk assessment and decision-making under uncertainty—is proposed. The definition of Bayesian networks is given in the sequel, before proceeding to the risk assessment methodology.


### 2.2. Bayesian Networks

Bayesian network (BN) is a directed acyclic graph represented with pairsN={(V,E),P}. NodeVrepresents random variables (events) and linksE between nodes represent a causal dependency. A link from variable X to variable Y indicates that X can cause Y, or, in BN terminology, X is a parent of Y, and Y is a child of X.P is a probability distribution overV. Discrete random variables V={X1,X2,…,Xn} are assigned to the nodes variables representing a finite set of mutually exclusive states and annotated with a Conditional Probability Table (CPT) that represents the conditional probability of the variable given the values of its parents in the graph.The simple Bayes net is presented in Figure1 with two independent variables, X1 and X2, and dependent variable Y with appropriate CPT representing probabilities for each possible state of the nature of variable Y, or event, in the risk assessment terminology.Figure 1
Bayes net with two independent variables.The solving of BN is based on four rules, including conditional independence, joint probability, marginalization, and Bayesian rule, presented respectively in the following expressions [21, 22].Conditional Independence (3) P X 1 , X 2 , … , X n = ∏ i = 1 n P X i / P a r e n t s X i  Joint Probability (4) P Y = y i , X = x i = P X = x i · P Y = y i / X = x i  Marginalization Rule (5) P Y = y i = ∑ i P X = x i · P Y = y i / X = x i  Bayesian Rule (6) P X = x i / Y = y j = P X = x i · P Y = y j / X = x i P Y = y jUsing expressions (3)–(5), the probability of each possible statej out ofn possible states of variableY can be determined. After the calculation of probabilities, the following step in the methodology is to calculate the risk, using expressions (1) and (2), or more generally:(7)Ri=fCYi,PYiIn order to incorporate the risk, BN is extended by two more nodes: consequence nodeC(Y) and risk value nodeR (Figure 2).Figure 2
Extension of the Bayes net for the risk assessment.For instance, consequences grades for personal safety can be expressed by numerical grades, described in Table1.Table 1
Consequences grading scale.
Safety consequences Grade Description  1 No harmful consequences 2 Minor: failure results in minor system damage but does not cause injury to personnel or allow any kind of exposure to operational or service personnel 3 Major: failure results in a low level of exposure to Personnel, or activates facility alarm system 4 Critical: failure results in minor injury to personnel 5 Catastrophic: failure results in major injury or death of personnelRisk value node is represented by the n-dimensional array, with elements calculated from (7).


### 2.3. Influence Diagrams

A generalization of a BN is the Influence diagram (ID), proposed by Howard and Matheson [21], as a tool to simplify modeling and analysis of decision trees, allowing not only probabilistic inference but also the graphical representation of decision-making problems. Like in BN, the input and output values of a node in an ID are based on the Bayesian theorem, allowing a user to make inferences with limited available information. Besides the chance nodes of BN, ID also contains decision nodes and utility nodes, depicting available information at the time of making a decision, and the degree of influence of each variable on other variables and decisions. Unlike a decision tree that shows more details of possible paths, ID shows dependencies among variables more clearly. IDs are particularly useful in creating computer-based models that describe a system or as descriptions of decision maker’s mental models to assess the impact of their actions.ID tries to capture system representation in a form that can be communicated to others, through several graphical symbols. A circle depicts an exogenous variable (an external influence) whose values are not affected by previous decisions. A rectangle depicts a decision, while intermediate variables depict an endogenous variable whose values are computed as functions of decision and other variables. Chance node (an ellipse) represents a random variable defined by discrete probability distribution. Arrow shows the influence between variables, and dotted arrow shows information being communicated between elements. Finally, value node (a diamond) is a quantitative criterion representing the subject of optimization. The simple example of an ID is presented in Figure3.Figure 3
Influence diagram for the risk assessment.Methods for evaluating and solving IDs are based on Bayes theorem and can be grouped in several categories. They can be (i) converted to decision trees and solved, (ii) solved by the variable elimination algorithms, or (iii) solved by efficient algorithms using graphical structures, like the junction trees [25].When nodes in the diagram are represented by appropriate fuzzy sets, ID can be solved using fuzzy reasoning [26–28]. A commonly used technique for combining fuzzy sets is fuzzy inference system, like the Mamdani type inference. The final step for building an ID model for the risk assessment is to aggregate different risk factors into one diagram. An illustration for the risk assessment ID with two criteria: safety risk (S) and economic risk (E), is presented in Figure 4.Figure 4
Complete influence diagram with different risk factors.


## 2.1. Risk Assessment

Risk assessment, as the first step in the risk management process, attempts to identify possible failure events, evaluate their consequences, determine the probability of their future occurrence, and reduce the detrimental consequences. The usual definition of risk associated with an eventE is defined as the product of event probability p(E) and its consequence Cons(E) [22, 23]:(1)RiskE=pE·ConsEMore complex relationships between values introducing empirical scaling parametersx,y, and w are presented in the following [24]:(2)RiskE=pEy·w·ConsExCalculated value of risk became a crucial factor when deciding about the actions to be performed on distribution asset. However, decisions have to be made in a very uncertain environment. In this paper, a new graphical tool based on Bayesian networks—influence diagrams for risk assessment and decision-making under uncertainty—is proposed. The definition of Bayesian networks is given in the sequel, before proceeding to the risk assessment methodology.


## 2.2. Bayesian Networks

Bayesian network (BN) is a directed acyclic graph represented with pairsN={(V,E),P}. NodeVrepresents random variables (events) and linksE between nodes represent a causal dependency. A link from variable X to variable Y indicates that X can cause Y, or, in BN terminology, X is a parent of Y, and Y is a child of X.P is a probability distribution overV. Discrete random variables V={X1,X2,…,Xn} are assigned to the nodes variables representing a finite set of mutually exclusive states and annotated with a Conditional Probability Table (CPT) that represents the conditional probability of the variable given the values of its parents in the graph.The simple Bayes net is presented in Figure1 with two independent variables, X1 and X2, and dependent variable Y with appropriate CPT representing probabilities for each possible state of the nature of variable Y, or event, in the risk assessment terminology.Figure 1
Bayes net with two independent variables.The solving of BN is based on four rules, including conditional independence, joint probability, marginalization, and Bayesian rule, presented respectively in the following expressions [21, 22].Conditional Independence (3) P X 1 , X 2 , … , X n = ∏ i = 1 n P X i / P a r e n t s X i  Joint Probability (4) P Y = y i , X = x i = P X = x i · P Y = y i / X = x i  Marginalization Rule (5) P Y = y i = ∑ i P X = x i · P Y = y i / X = x i  Bayesian Rule (6) P X = x i / Y = y j = P X = x i · P Y = y j / X = x i P Y = y jUsing expressions (3)–(5), the probability of each possible statej out ofn possible states of variableY can be determined. After the calculation of probabilities, the following step in the methodology is to calculate the risk, using expressions (1) and (2), or more generally:(7)Ri=fCYi,PYiIn order to incorporate the risk, BN is extended by two more nodes: consequence nodeC(Y) and risk value nodeR (Figure 2).Figure 2
Extension of the Bayes net for the risk assessment.For instance, consequences grades for personal safety can be expressed by numerical grades, described in Table1.Table 1
Consequences grading scale.
Safety consequences Grade Description  1 No harmful consequences 2 Minor: failure results in minor system damage but does not cause injury to personnel or allow any kind of exposure to operational or service personnel 3 Major: failure results in a low level of exposure to Personnel, or activates facility alarm system 4 Critical: failure results in minor injury to personnel 5 Catastrophic: failure results in major injury or death of personnelRisk value node is represented by the n-dimensional array, with elements calculated from (7).


## 2.3. Influence Diagrams

A generalization of a BN is the Influence diagram (ID), proposed by Howard and Matheson [21], as a tool to simplify modeling and analysis of decision trees, allowing not only probabilistic inference but also the graphical representation of decision-making problems. Like in BN, the input and output values of a node in an ID are based on the Bayesian theorem, allowing a user to make inferences with limited available information. Besides the chance nodes of BN, ID also contains decision nodes and utility nodes, depicting available information at the time of making a decision, and the degree of influence of each variable on other variables and decisions. Unlike a decision tree that shows more details of possible paths, ID shows dependencies among variables more clearly. IDs are particularly useful in creating computer-based models that describe a system or as descriptions of decision maker’s mental models to assess the impact of their actions.ID tries to capture system representation in a form that can be communicated to others, through several graphical symbols. A circle depicts an exogenous variable (an external influence) whose values are not affected by previous decisions. A rectangle depicts a decision, while intermediate variables depict an endogenous variable whose values are computed as functions of decision and other variables. Chance node (an ellipse) represents a random variable defined by discrete probability distribution. Arrow shows the influence between variables, and dotted arrow shows information being communicated between elements. Finally, value node (a diamond) is a quantitative criterion representing the subject of optimization. The simple example of an ID is presented in Figure3.Figure 3
Influence diagram for the risk assessment.Methods for evaluating and solving IDs are based on Bayes theorem and can be grouped in several categories. They can be (i) converted to decision trees and solved, (ii) solved by the variable elimination algorithms, or (iii) solved by efficient algorithms using graphical structures, like the junction trees [25].When nodes in the diagram are represented by appropriate fuzzy sets, ID can be solved using fuzzy reasoning [26–28]. A commonly used technique for combining fuzzy sets is fuzzy inference system, like the Mamdani type inference. The final step for building an ID model for the risk assessment is to aggregate different risk factors into one diagram. An illustration for the risk assessment ID with two criteria: safety risk (S) and economic risk (E), is presented in Figure 4.Figure 4
Complete influence diagram with different risk factors.


## 3. Fuzzy Probability

Uncertain and subjective probabilities can be granulated in different terms like: improbable or doubtful, but we will draw our attention to special form of probability granulation, focusing on point value with inherent uncertainty: around 30%, around 75%, etc. These probabilities are defined starting from previous works on linguistic probability [29–32] defining similar probability measure for fuzzy probabilities.Klir [31, 32] introduced a notion of a fuzzy interval defined on [0,1] as a probability granule, that is, a normal fuzzy set on [0,1] with α cuts for all as closed subintervals on [0,1]. These fuzzy probabilities will be denoted as P1,P2,…,Pn and they may be expressed by the canonical form:(8)Pix=fix,x∈0,b1,x∈b,cgix,x∈c,d0,otherwisewhere a,b,c,d∈[0,1], a≤b≤c≤d, andfi andgi are strictly increasing right continuous and strictly decreasing left continuous real valued function, respectively. The α–cuts ofPi are expressed for all a∈[0,1]:(9)Piα=fi-1α,gi-1α,α∈0,1b,c,α=1In this paper, we will investigate the elicitation of triangular fuzzy set support—left and right bounds of triangular fuzzy numbers. The support of fuzzy setA is the set of all pointsx inX such that μAx>0.Consider a discrete random variableX with values in the set X=xi,i∈Nn. We will assume that the probabilities of this random variablesP(xi) are assessed approximately, by a triangular fuzzy number:(10)μPix=0,x<aix-aibi-ci,ai<x<bici-xci-bi,bi<x<ci0,x>ciWe can interpret these fuzzy numbers as fuzzy probabilities as follows.Definition 1.
Fuzzy numbersPi=ai,bi,ci,i=1,…,n are called fuzzy probabilities ofX if there are x1∈a1,c1,…,xi∈ai,ci,…,xn∈an,cn such that:(11)∑i=1nxi=1,∑i=1nbi=1The set of fuzzy numbers P satisfies (11) if and only if the following conditions hold [33]:(12)ci+a1+…+ai-1+ai+1+…+an≥1,∀iai+c1+…+ci-1+ci+1+…+cn≥1,∀iIf there are only two fuzzy probabilities a1,b1,c1 and a2,b2,c2, then a1+c2=1, a2+c1=1 and b1+b2=1.Let us consider a set of fuzzy numbers FP=FPi=ai,bi,ci,i=1,…,n. The interval of probability values for every α–cut will be denoted as [aα,i,cα,i]. We can interpret these fuzzy numbers as fuzzy probabilities as follows.Definition 2.
Fuzzy numbersFPi=ai,bi,ci are called fuzzy probabilities ofX if for ∀α∈[0,1] and ∀xi∈[aα,i,cα,i] there are x1∈aα,1,cα,1,…,xi-1∈aα,i-1,cα,i,xi+1∈aα,i+1,cα,i+1,…,xn∈aα,n,cα,n such that:(13)∑i=1nxi=1Lemma 3.
The set of fuzzy numbersFP satisfies (13) if and only if the following conditions hold:(14)cα,i+aα,1+…+aα,i-1+aα,i+1+…+aα,n≤1,∀α,∀iaα,i+cα,1+…+cα,i-1+cα,i+1+…+cα,n≥1,∀α,∀i.Proof.
Sufficient Conditions. If the first part of Lemma 3 holds, then:(15)∀α,∀ixi+aα,1+…+aα,i-1+aα,i+1+…+aα,n≤cα,i+aα,1+…+aα,i-1+aα,i+1+…+aα,n≤1∀α,∀ixi+cα,1+…+cα,i-1+cα,i+1+…+cα,n≥aα,i+cα,1+…+cα,i-1+cα,i+1+…+cα,n≥1.Then, the following expression holds:(16)xi+aα,1+…+aα,i-1+aα,i+1+…+aα,n≤1≤xi+cα,1+…+cα,i-1+cα,i+1+…+cα,n. The expression shows that there exist aα,j≤xj≤cα,j,j∈1,…n,j≠i that satisfies (13).
Necessary Conditions. If the first part of Lemma 3 does not hold, then:(17)∀α,∃icα,i+aα,1+…+aα,i-1+aα,i+1+…+aα,n>1∀α,∃iaα,i+cα,1+…+cα,i-1+cα,i+1+…+cα,n<1.Then, takingxi as aα,i or cα,i (13) cannot hold.
An alternative definition of fuzzy probabilities can be formulated from two extreme cases ofα = 0 and α = 1.Definition 4.
Fuzzy numbersFPi=ai,bi,ci are called fuzzy probabilities ofX if for and ∀xi∈[ai,ci] there are x1∈a1,c1,…,xi-1∈ai-1,ci,xi+1∈ai+1,ci+1,…,xn∈an,cn such that:(18)∑i=1nxi=1,∑i=1nbi=1Bayesian networks with fuzzy numbers replacing point value probabilities are proposed [30] defining “Bayesian fuzzy probability” as convex, normal fuzzy set of [0,1]. Complementation law has been relaxed in order to extend a partially defined linguistic probability measure, and this method has been successfully used in forensic statistics [29] and risk analysis [18]. More possible scenarios for fuzzifying the Bayesian approach are presented in [15] using nonfuzzy algorithmically efficient reformulation of the Bayesian formula. Although time-consuming, we will implement the corresponding fuzzy version of Bayesian formulas.
The fuzzy counterparts to the standard arithmetic operators are defined using the extension principle. It is possible to derive these operators by examining the effects of interval based calculations at eachα–cut. The extended operators are defined by (19), using a circled arithmetic operator symbol for the extension of a real arithmetic operator.Definition 5.
For alla,b∈RFa, the extended operators are defined by(19)μA⊕Bz=supminμAxx+y=z,μByμA⊗Bz=supminμAxxy=z,μByμA-Bz=supminμAxx-y=z,μByμA∅Bz=supminμAxx/y=z,μByFrom previous definition, two fuzzy Bayes rules analogue to classical crisp number relations are formulated. Operator “≅” stands for “=” operator.Fuzzy Joint Probability (20) P Y = y j , X = x i ≅ P X = x i ⊗ P Y = y j ∖ X = x i  Fuzzy Bayes Rule (21) P X = x i ∖ Y = y j ≅ P X = x i ⊗ P Y = y j ∖ X = x i P Y = y jBased on the law of total probability another rule for the fuzzy marginalization can be added, represented by (22).Fuzzy Marginalization Rule (22) P Y = y j ≅ ∑ i P X = x i ⊗ P Y = y j ∖ X = x iFinally, risk can be calculated from(23)R=∑inPCi⊗CiIt is possible to use any other form of additive, multiplicative, or tabular risk aggregation function. The influence diagram with fuzzy probabilities will be illustrated on a simple case study of maintenance strategy selection.


## 4. Case Study

Risk assessment methodology is illustrated on substation with low oil circuit breakers. The decision has to be made about three possible alternatives: do nothing, perform minor interventions, or do the overhaul and major repair of circuit breakers. The alternatives will be assessed by the risk assessment of two criteria: safety and environment. Both criteria will be evaluated by their risk and then aggregated in the one influence diagram value node.Two failure modes and normal operating condition of a circuit breaker are taken into account: breaker is in operating conditions (OK), failure to close (Close), when breaker does not close the circuit to conduct current in one or more poles, and failure to open (FO), when breaker does not open the circuit to interrupt current. In the case of the bad weather conditions in the following year, the network condition will worsen due to the increased number of failure, the network loading will increase, and the breaker will be exposed to more severe operation conditions. Due to the uncertainty about the weather forecast, and consequently network technical condition, network maximal demand power (loading), and possible failure modes, probabilities elicited by experts are also uncertain. According to the definition of the fuzzy probabilities, possible probability grades are represented in Table2.Table 2
Fuzzy probabilities.
Triangular fuzzy probability number Description Notation [ 5 10 15] Extremely low probability EL [ 1 5 20 25] Low probability L [ 2 5 30 35] Low to medium LM [ 3 5 40 45] Medium to low ML [ 4 5 50 55] Medium probability M [ 5 5 60 65] Medium to high MH [ 6 5 70 75] High to medium HM [ 7 5 80 85] High probability H [ 8 5 90 95] Extremely high probability EHPrior fuzzy probabilities of ambient conditions and global weather forecast for the next year are given in Table3.Table 3
Prior probability of weather states.
States Description Probability Bad Severe weather conditions MM Medium No extreme temperatures below – 20 degree LM Good Good weather conditions, no extreme temperatures below -10 degree LConditional probability tables for network condition, circuit breaker failure modes, loading levels, and consequences are represented in Tables4, 5, 6, and 7, respectively.Table 4
Conditional probabilities of network conditions.
Weather States Bad conditions on MV side—no tree trimming, no maintenance, increased number of failures Good conditions on MV network, no increase in failure rate  B MH ML M MM MM G ML MHTable 5
Conditional probabilities of failure modes.
Decision NC Ok Close FO Minor  HM L EL Minor Good H EL EL Major Bad H EL EL Major Good EH EL EL Do nothing Bad MH L L Do nothing Good MH L LTable 6
Conditional probabilities of network loading levels.
Weather Low Loading Medium Loading High Loading Bad EL LM MH Medium LM MM L Good MH LM ELTable 7
Conditional probabilities of consequences.
Loading Failure mode Safety risk Environmental risk Low loading OK EH EL Impossible H EL EL Failure to close H EL EL H EL EL Failure to open HM L EL HM L EL Medium loading OK H EL EL H EL EL Failure to close HM L EL MM LM L Failure to open MH LM EL MH LM EL High loading OK HM L EL HM L EL Failure to close MH L L MH L L Failure to open MM LM L MM LM LSafety and environment criteria evaluations are expressed in numerical grades (from 1 to 5) and represented in Table8.Table 8
Safety and environment criteria grades.
Grade Safety risk Environmental risk 1 No harmful consequences No harmful consequences, Failure does not allow any release of chemicals into the environment 2 Minor: failure results in minor system damage but does not cause injury to personnel Personnel exposure to harmful chemicals or radiation or fire 3 Critical: failure results in minor injury to personnel A release of chemical to the environmentForα = 1, the influence diagram becomes the deterministic influence diagram with crisp probability values. The solved diagram with calculated values for three different scenarios and two criteria is represented in Figure 5.Figure 5
Influence diagram with crisp probabilities.Risk is calculated using the following consequences grades: 1 for no consequences, 2 for minor, and 3 for major consequences, and aggregation of two risks is presented in Table9.Table 9
Aggregated risks.
Safety 1 2 3 1 2 3 1 2 3 Environment 1 1 1 2 2 2 3 3 3 Aggregated risk 1 2 3 4 5 8 7 8 10It is visible that risk calculated values do not show great variance, and that decision about the future maintenance strategy cannot be easily determined. Therefore, the problem is solved again using fuzzy probabilities and expressions (19)–(22). Results are presented on Figures 6, 7, and 8.Figure 6
Fuzzy probabilities for no consequences state for (a) safety risk and (b) environmental risk.Figure 7
Fuzzy probabilities for minor consequences state for (a) safety risk and (b) environmental risk.Figure 8
Fuzzy probabilities for critical consequences state for (a) safety risk and (b) environmental risk.Calculated values for risks for both alternatives with crisp values (Figure5) and fuzzy probability values defuzzified using the centroid method are given in Table 10.Table 10
Aggregated risk comparison.
Crisp probability Fuzzy probability Minor 2,38 2,40 Major 2,29 2,65 Do nothing 2,48 4,12Unlike the crisp calculated values for the risk node, the fuzzy probabilities of possible consequences are highlighting the whole range of future scenarios consequences, facilitating the decision-making process. From Figure6 it is visible that the “doing nothing” strategy almost doubles the risk for minor and major consequences, while the minor and major maintenance practically do not make any difference.


## 5. Conclusions

Decision-making in maintenance planning is always confronted with several aspects: technical, financial, safety, environmental, and operational ones. Each of these factors can be modeled by the appropriate risk of possible harmful consequences. This paper introduces a new methodology for the optimization of maintenance activities in distribution network, based on the calculation of the risk of the particular component, or the overall risk of the distribution object. For this purpose, the Fuzzy Influence diagram has been used, in order to determine all relevant factors influencing risks, and depicting their interdependencies. Probabilistic uncertainties require appropriate mathematical modeling and quantification when predicting future state of the nature or the value of certain parameters and the proposed model enables evaluation of the impact of each particular component of maintenance decision process. Bayesian networks and Influence diagrams are used as a convenient tool for the large class of engineering problems, while the inherent uncertainty has been modeled by the fuzzification of random variables, and/or prior and conditional probabilities. The methodology is illustrated on the example of the choice of circuit breakers maintenance in one distribution transformer substation. This methodology is especially suited for systems with great number of unobservable components, in the presence of uncertainty and lack of operational data, like power distribution system is. The main challenges for the future research are the introduction of fuzzy multicriteria decision analysis for the choice of activity minimizing the overall risk and more flexible representation of fuzzy subjective probabilities, as information granules described by linguistic terms and modeled as triangular fuzzy numbers.



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