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A Framework for Modeling and Assessing System Resilience Using a Bayesian

Network: A Case Study of An Interdependent Electrical Infrastructure System

Article  in  International Journal of Critical Infrastructure Protection · February 2019

DOI: 10.1016/j.ijcip.2019.02.002

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international journal of critical infrastructure protection 25 (2019) 62–83

Available online at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/IJCIP

A framework for modeling and assessing system resilience using a Bayesian network: A case study of an interdependent electrical infrastructure system

Niamat Ullah Ibne Hossain a , Raed Jaradat a , ∗, Seyedmohsen Hosseini b , Mohammad Marufuzzaman a , Randy K. Buchanan c

a Department of Industrial and Systems Engineering, Mississippi State University, PO Box 9542, Mississippi 39762, USA b Industrial Engineering Technology, University of Southern Mississippi, Long Beach, Mississippi 39560, USA c Institute for Systems Engineering Research, U.S. Army Engineer Research and Development Center, Vicksburg, MS

a r t i c l e i n f o

Article history:

Received 15 September 2018

Revised 25 December 2018

Accepted 5 February 2019

Available online 13 February 2019

Keywords:

Bayesian network

Electrical infrastructure system

System resilience

Resilience capacity

a b s t r a c t

This research utilizes Bayesian network to address a range of possible risks to the electrical

power system and its interdependent networks (EIN) and offers possible options to mitigate

the consequences of a disruption. The interdependent electrical infrastructure system in

Washington, D.C. is used as a case study to quantify the resilience using the Bayesian net-

work. Quantification of resilience is further analyzed based on different types of analysis

such as forward propagation, backward propagation, sensitivity analysis, and information

theory. The general insight drawn from these analyses indicate that reliability, backup power

source, and resource restoration are the prime factors contributed towards enhancing the

resilience of an interdependent electrical infrastructure system.

© 2019 Elsevier B.V. All rights reserved.

1

U t s T t s m c

m

t d r a b p s e N

h 1

. Introduction

nited States economic sectors are highly reliant on the elec- ric power industry with generated energy being utilized to erve its people and conduct business in the global market [1] . his critical infrastructure system performs four fundamen-

al functions: the generation, transmission, distribution , and con- umption of electricity . The electric power system is linked with any other supporting infrastructures such as telecommuni-

ation, transportation, fuel distribution, and water supply [2] .

∗ Corresponding author. E-mail addresses: [email protected] (N.U.I. Hossain), [email protected]

[email protected] (M. Marufuzzaman).

e

ttps://doi.org/10.1016/j.ijcip.2019.02.002 874-5482/© 2019 Elsevier B.V. All rights reserved.

The U.S. electrical systems are susceptible to diverse hreats that can cause short-term power interruptions to long uration power outages. The Department of Energy (DOE) eports that the outages could be triggered by natural dis- sters and climate conditions such as tornados, hurricanes, lizzards, and earthquakes or man-made threats such as hysical or cyber-attacks [3] . Such disruptions may affect the ecurity, health and safety of residents and cause an annual stimated economic loss of $18-70 billion. For instance, the ortheast power blackout in 2003 caused financial losses in xcess of $6 billion [3] . Needless to say, a resilient and reliable

sstate.edu (R. Jaradat), [email protected] (S. Hosseini),

international journal of critical infrastructure protection 25 (2019) 62–83 63

electrical grid has been a central concern of national security for decades. A recent report by the National Academies of Sciences, Engineering, and Medicine (NASEM) entitled “En- hancing the Resilience of the Nations Electricity System” highlighted the potential threats including natural disaster, manmade attack, and cyber-attack of the power system and offered overarching recommendations to enhance the overall resilience of the U.S. electrical system [4] .

The conditions of any disruptive event can be broadly characterized as intense, unsettling, and severe under both pre-and post-disaster applications. The complex nature and the dynamic interactions between the system components challenge the achievement of optimal operations for system infrastructure, and this causes economic loss [5] . For instance, statistics show that the economic losses caused by natural disasters from 2000 to 2017 were around $3,312 billion across the world [6] . In 2011, when Japan was devastated by the tsunami and the massive earthquake Tohoku, economic losses soared to $440 billion [7] . In the U.S. recent hurri- canes Harvey, Irma, and Sandy caused immense damage to the economy. These three hurricanes caused an estimated $320 billion in financial losses [8] . Beyond financial losses and property damage, all these mentioned disasters have wreaked havoc on business, manufacturing and production industry, the job market, and devastation of human life. These examples would clearly highlight the importance of conducting research on systems resilience.

The term resilience comes from the Latin word “resiliere” which means “bounce back”. Resilience is an intrinsic prop- erty of a system that describes the system’s ability to absorb the shock of a disruptive event and recover to a pre-defined level of performance. The concept of resilience integrates four fundamental concepts, namely: robustness, resourcefulness, speed of recovery, and adaptability. These four concepts are addressed in risk management approaches during the differ- ent stages of the disruptive event [9] . The consequences of dis- ruptions often lead to unanticipated system behaviour and re- duced overall system resilience [10] . Several research studies have been conducted to reduce the likelihood of the occur- rence of the catastrophic event by applying security manage- ment tools, known as pre-disaster or contingency strategy . Be- yond the contingency strategy, a fast response, a high level of preparedness, and a quick recover are of paramount im- portance in minimizing the disruption caused by the event. The combined approach of response and recovery are often referred to as post-disaster strategy or mitigation strategy .

The purpose of this research paper is to quantify the re- silience of interdependent electrical systems by building an effective Bayesian network model. The model is specifically developed to deal with risks and uncertainties associated with the complex network of electrical infrastructure systems un- der disruption. The underlying factors related to the resilience of electrical infrastructure systems are identified, and the model is developed based on expert judgement and histori- cal data. Washington, D.C. is used as a case study to illustrate the quantification of the resilience of an electrical system and its interdependent network.

The following subsection discusses the literature pertain- ing to the quantification of resilience and the state-of-the- art Bayesian approach in risk and resilience engineering.

Section 2 discusses various factors related to design in the resilience of EIN. Section 3 provides background information about the Bayesian structure. Quantification of resilience factors associated with the Bayesian network for EIN is presented in Section 4 . Various kinds of analysis such as for- ward propagation, backward propagation, sensitivity analysis and information theory are described in Section 5 . Finally, Section 6 ends the paper with concluding remarks and future recommendations.

1.1. Related research

This section has two primary purposes. The first is to show some of the related methods used in quantifying system re- silience and to present the general thread running through these methods. The different methods are then mathemati- cally presented. The second is to discuss the existing litera- ture related to the use of the Bayesian network in risk and resilience engineering and to present current gaps in the liter- ature. To address these gaps, this research identifies the basic factors of resilience associated with the interdependent elec- trical infrastructure system in Washington, D.C. and then pro- poses a conceptual framework to quantify the resilience based on the Bayesian network.

1.1.1. Quantification of resilience In recent years, research pertaining to system resilience in critical infrastructure has significantly increased and quan- tification of resilience has become a central factor. Despite an increased importance on system resilience in various sectors over the past few years, substantial differences exist among the definitions and descriptions of resilience. Different re- searchers attempted to quantify resilience in different man- ners. For instance, Youn et al. [11] developed a metric for mea- suring engineering resilience in terms of passive survival rate and proactive survival rate where resilience is the summa- tion of passive survival rate and proactive survival rate . Passive survival rate refers to the reliability of the system and proac- tive survival rate represents the restoration of the system (see Eq. (1) ). Although this approach is most applicable for earth- quakes, it still can be utilized to quantify resilience for other systems.

Resilience (�) = Reliability (R ) + Restoration (ρ ) (1)

Bruneau et al. [9] designed a resilience triangle model for civil infrastructure by incorporating four dimensions of resilience: robustness, resourcefulness, rapid recovery, and adaptability. The authors proposed a deterministic static metric for measuring the resilience loss in terms of quality of degraded infrastructure. In this approach, resilience loss is calculated by the quality of the infrastructure before dis- ruption, which is assumed to be 100, minus the quality of the disrupted infrastructure after recovery over time period t 0 to t 1 where t 0 represents the time when the disruption occurred and t 1 refers the time when the infrastructure returns to its normal pre-disruption state. This approach is presented as a mathematical expression in Eq. (2) . Let RL is defined as the resilience loss and the average disrupted scenario is exhibited

64 international journal of critical infrastructure protection 25 (2019) 62–83

Fig. 1 – System performance and state transition to describe resilience (adapted from [13] ).

a

R

w t r r N r

D

p s I i t l f ( e e i i

�

s d O s o d t i e t F

t w a [ t S e [ a t s

1 a T fi p p m b n o a p o t t r l N r d d t t a q o i w w t

s a function of Q ( t ).

L = ∫ t 1

t 0 (100 − Q (t )) dt (2)

Rose [12] provided a definition of dynamic resilience ( DR ) hich incorporated the concept of time-dependent charac-

eristics of recovery. The author further computed dynamic esilience in terms of output of the system under hastened ecovery ( SO HR ) and without hastened recovery ( SO WR ) where is the number of time steps and t i is the i th time step during

ecovery (see Eq. (3) ).

R = N ∑

i =1 SO HR (t i ) − SO WR (t i ) (3)

Another time-dependent resilience approach was pro- osed by Henry and Ramirez-Marquez [13] . This approach imply computed the resilience as a ratio of recovery to loss. n this approach, the performance of the system at a point n time is measured through performance function φ( t ), and hree different transition states are considered: ( i ) the base- ine stable state ( S 0 ) functions under normal conditions be- ore any disruption e j occurs at time t e , ( ii ) the disruptive state S d ) at time t d due to disruptive event e

j , and ( iii ) the recov- red state ( S f ) that represents the new state after the recov- ry action started at time t s . The resilience equation ( � ( t | e j )) s the ratio of recovery to loss and is presented in Eq. (4) and llustrated in Fig. 1 .

(t | e j ) = φ(t | e j ) − φ(t d | e j )

φ(t 0 ) − φ(t d | e j ) (4)

There are several other approaches such as graph theory, imulation, and optimization techniques that have been con- ucted to quantify resilience in different ways. For instance, mer et al. [14] propose a resilience metric for infrastructure ystem resilience where resilience is computed as the ratio f the closeness centrality of the network for pre-and post- isruptive scenarios. Soni et al. [15] use graph theory to de- ermine the supply chain resilience in terms of the determin- stic modeling approach. Carvalho et al. [16] apply discrete vent simulation to compare different scenarios to enhance he resilience of a supply chain network. In another research, aturechi et al. [17] develop a mathematical model in order

o maximize the resilience of an airport’s taxiway and run- ay system. An optimization model and heuristic solution

pproach is proposed by Khaled et al. [18] and Vulgrin et al. 19] to maximize the resilience of the U.S. transportation sys- em. Interested readers can refer to the works of Chang and hinozuka [20] , Cimellaro et al. [21] , Murray-Tuite [22] , Berche t al. [23] , Heaslip et al. [24] , Dorbritz [25] , Miller-Hooks et al. 26] , and Hosenni et al. [27] to understand different techniques pplied to quantifying and assessing resilience. The different echniques used in quantifying and modeling resilience are ummarized in Table 1 .

.1.2. Existing literature related to Bayesian network in risk nd resilience engineering he Bayesian network (BN) has a wide range of usage in the eld of reliability, resilience engineering, and decision sup- ort systems. Hosseni and Barker [47] develop a resilient sup- lier selection method based on the Bayesian approach which odeled a Bayesian framework that can assess and select the

est supplier based on primary and green criteria. Constanti- ou et al. [48] develop a robust Bayesian structure to select the ptimal decision for a complex medical support system. The uthors develop a realistic BN model that can handle both ex- ert knowledge and data-driven interviews with patients in rder to provide decision support for a forensic medical sys- em. Khan et al. [49] examine the risk associated with marine ransportation in arctic waters by conducting a quantitative isk assessment via BN. The authors predict the risk of col- ision between oil tankers and ice floes in the waters of the orthern Sea Route. Perez-Minana et al. [50] conduct an envi-

onmental risk assessment using the BN to illustrate the un- erlying risk associated with biodiversity. The authors initially evelop mind-maps and the information obtained through he minds-map is fed into the BN to better manage the uncer- ainty associated with functions of biodiverse ecosystems. In nother study, Amunddson et al. [51] demonstrate a Bayesian uantitative approach to handle the risk in the development f sustainable biomass supply chain networks. The authors

dentify risk drivers related to a biomass supply chain net- ork and translated these factors into the Bayesian frame- ork to assess how risk factors influence each other and how

hey impact the overall resilience of the biomass feedstock

international journal of critical infrastructure protection 25 (2019) 62–83 65

Table 1 – Different techniques for quantifying and modelling resilience.

Approach References Application areas

Conceptual framework Vlacheas et al. [28] Telecommunication Labaka et al. [29] Nuclear plant Sterbenz et al. [30] Communication network

Semi quantitative Shirali et al. [31] Community Bruyelle et al. [32] Process industry

Probabilistic approach Barker et al. [33] Networks (Quantitative) Pant et al. [34] Transportation

Ouyang et al. [35] Urban infrastructure Deterministic approach Enjalbert et al. [36] Transportation (Quantitative) Orwin and Wardle [37] Soil system

Ouedraogo et al. [38] Human-machine system Brown et al. [39] Organization

Fuzzy Tadic et al. [40] Organization Azadeh et al. [41] Chemical industry

Simulation Jain and Bhunya [42] Water system Spiegler et al. [43] Supply chain Landegren et al. [44] IT network

Optimization Alderson et al. [45] Infrastructures Baroud et al. [46] Water system

supply network. Some other applications of BN available in the literature are traffic accidents [52] , customer service manage- ment [53] , manufacturing systems [54] , data classification [55] , software development projects [56] , and safety management [57] . All the results drawn from this research indicate that the BN model can effectively address mutual interdependency of incidents in risk analysis and provide recommendations to mitigate risk.

Although BN has been applied in different research, two significant gaps are identified and need to be addressed. These gaps are:

• The need for a Bayesian framework to design an inter- dependent electrical infrastructure system that takes into consideration the complex interactions that exist among different entities of the entire network.

• The lack of research assessing the resilience of EIN with re- spect to the concept of absorptive, adaptive and restorative capacities.

To address these gaps, this research paper proposes a new decision making approach based on Bayesian network theory that addresses the risk and uncertainty associated with EIN. A Bayesian network is an analytical tool that demonstrates all the causal relationships among the different qualitative and quantitative variables and allows practitioners to under- stand the interdependencies among the variables and how the change in one variable affects the others. The main contribu- tions of the research are summarized below:

• Proposing a new conceptual framework for designing elec- trical system and its interdependent network.

• Classifying the underlying factors of EIN with respect to the concept of absorptive, adaptive, and restorative capacities.

• Developing a probabilistic graphical model, known as Bayesian network, for assessing the resilience of EIN.

• Conducting different types of analysis such as forward propagation, backward propagation, sensitivity analy- sis and information theory to provide a better insight regarding the result of the model.

2. Problem description and model formulation

This section discusses the problem description and model for- mulation using a case study and quantifying the resilience of EIN with respect to the concept of absorptive, adaptive, and restorative capacities.

2.1. Interdependent electrical infrastructure system case study

The main objective of the research is to build a Bayesian net- work for assessing and quantifying the resilience of an inter- dependent electrical infrastructure system. For this objective, the interdependent electrical infrastructure of Washington, D.C. is chosen to serve as a case study. Washington, D.C. is selected because ( i ) it is the capital of the U.S., ( ii ) the electrical infrastructure of Washington, D.C. plays a crucial role in the U.S. economy because it promotes an opportunity to trade electricity and is correlated to the U.S. gross domestic product (GDP) [58] , and ( iii ) the electrical infrastructure is connected to the adjacent states of Maryland and Virginia which makes it a complex network to study. Washington, D.C. has a population around 65 million and the annual electric power generation is 0.1 TWh which is less than 1% of total U.S power generation [59] . The electricity is mainly generated from natural gas. The electrical infrastructure of Washington, D.C. is subjected to several disruptions mainly caused by natural disaster and human error such as executing a wrong computer command to control the equipment. The most common natural hazards

66 international journal of critical infrastructure protection 25 (2019) 62–83

Fig. 2 – Energy sector profile of Washington, D.C (Adapted from [59] ).

i s q i p

2

C t t

n Washington, D.C. are thunderstorms, lightning and winter torms [60] . This study covers all possible aspects related to uantifying the resilience of the interdependent electrical

nfrastructure system of Washington, D.C. Energy sector rofile of Washington, D.C is shown in Fig. 2 .

.2. Resilience capacity (RC)

apacity is the property of a system to achieve its objec- ives. Resilience capacity enhances the capability of a system o absorb, adapt, and recover from any shock or disruption.

international journal of critical infrastructure protection 25 (2019) 62–83 67

Fig. 3 – Fundamental structure of the resilience capacity for EIN.

Resilience capacity can take the form of resources such as the aptitude and skills of people, assets including intelligence systems, and/or actions [61] . Biringer et al. [62] proposed that the resilience paradigm can be described by using a set of re- silience capacities, namely, absorptive capacity, adaptive ca- pacity, and restorative capacity based on the different stages before, during, and after a disruption. After review of the liter- ature, we have identified some underlying factors pertaining to these three capacities for the interdependent electrical in- frastructure system of Washington, D.C. The underlying fac- tors, which appear in Fig. 3 , will be included and quantified in the developed BN framework to measure the resiliency of the interdependent electrical infrastructure system ( Fig. 4 ).

2.2.1. Absorptive capacity Absorptive capacity , an endogenous feature of a system, is the ability of a system to automatically absorb the impact of a disruption in order to minimize exposure or sensitivity to the shock. Absorptive capacity is also considered to be the first line of defense to withstand and absorb the shock due to a disrup- tive event. The absorptive capacity of a system involves a set of preventive measures and a course of strategies that must be developed before a disruption occurs in order to circumvent permanent undesirable consequences. The literature identi- fies the following eight aspects of absorptive capacity that are key factors related to the absorptive capacity of the electrical power system and its interdependent network.

68 international journal of critical infrastructure protection 25 (2019) 62–83

Fig. 4 – An example of a Bayesian network with five nodes.

2 A b c c w a o s t i

• Skilled Labour Management , an efficient response team, and strong utility management are key features of absorptive capacity. Well-trained operators and efficient resources can react quickly to the disruption and maintain operation of the electrical power system [63] . A competent response team can quickly arrive at the repair yards and take less time to repair the fault.

• A Control Strategy is a substantial factor in absorptive capac- ity. Advanced automatic control and different kinds of con- trol strategies such as Distributed Energy Resources (DER) help to prevent the severity of the outage [2] . The self- adapting and self-repairing feature of the advanced au- tomatic control increases the reliability of fault-tolerant systems and maintains the power quality.

• The deployment of Visual and Physical Protection such as tall opaque fencing, grills, protective walls, and roadblocks serve as protection from physical attack, help to prevent from unauthorized visual data collection, and ultimately maintain privacy. Vegetation management and selective undergrounding also help mitigate the potential threat of local distribution outage [64] .

• Because over-reliance on a single fuel source might lead to vulnerability in an energy supply network, it is prudent to have Alternative Fuel Sources . Incorporating diverse fuels such as renewable fuels, coal, and nuclear power enhances the robustness of the fuel supply network and ensures continued power service [63] . Redundancy of supply op- tions due to multiple fuel sources also strengthens system reliability.

• Routine preventive maintenance activities including on- time repair scheduling of electrical components and ready availability of spare parts limit the probability of a major failure and the resulting massive financial loses. For the electrical system, Periodic Maintenance consists of a set of el- ementary tasks such as data collection, visual inspection, replacing old motors, lubrication, and bolt tightening. All of these activities must be completed in accordance with

periodic maintenance guides and up-to-date safety stan- dards.

• In terms of Reliability , the pieces of equipment used in electrical systems are highly sensitive and interrelated so that failure in one component may ripple through the entire system and affect the whole facility. Reliability of an electrical system is defined as the probability that electrical components continue to operate normally for a given amount of time under normal operating condi- tions. Redundancy of critical components, a standby power source, and a current risk analysis enhance the reliabil- ity of the electrical system. Lessening the probability of failure would greatly decrease the interruption of power generation and distribution.

• Having in place an effective plan for Information and Com- munication can reduce the impact of a disruption. Deploy- ment of sensors, advanced data analytics, development of the Internet of Things (IoT) enable seamless communica- tion among the generation, distribution and transforma- tion networks [63] . Detailed pre-register information and register checklist related to the emergency power must be stored in a centralized, accessible database to ensure immediate action during the power interruption.

• A strong Physical Cyber Critical Infrastructure can reduce the consequences of a disruption. For an electrical system, a cyber-attack can be classified as inadvertent or deliberate. Implementing the Supervisory Control and Data Acquisi- tion (SCADA), a Distributed Control System (DCS), smart meter, corporate network communication, strong pass- word control, and secure software updates can all reduce the likelihood of an individual or group attack [2] .

.2.2. Adaptive capacity daptive capacity adjusts the perturbations due to the shock rought on by a disruptive event. Adaptive capacity, which is onsidered to be the second line of defense , is defined as the apability of a system to adapt itself and attempt to cope ith the adverse consequences or moderate potential dam-

ge without any recovery activity. It is considered to be part f a post-disaster strategy also known as “capacity of re- ponse” [65] . Following is a list of four key factors related to he adaptive capacity of the electrical power system and its nterdependent network

• Stockpiling regular equipment and contracts for the provi- sion of emergency power equipment reduces the severity of adverse consequences during power outages [63] .

• Provision of advanced technology such as advanced meter- ing infrastructure and smart inverter can improve the robustness of the entire electrical system and limit the consequences of any disruption [66] .

• Mode flexibility (substitution) is one of the key factors to maintaining continuity of an electrical system operation. Backup interdependence management such as electric ve- hicles, locomotives, and other non-standard power sources can be connected to the grid to provide limited electric ser- vice during outages. Advanced technologies such as smart grid can serve as a strong media to connect and regu- late the operation between vehicle to grid (V2G) to ensure

international journal of critical infrastructure protection 25 (2019) 62–83 69

interoperability (two-way flow of electricity) between two systems [66] .

• A backup power source is regarded as an adaptive measure. When the grid is down due to any disruption, backup gen- erators can serve as an emergency measure during crisis management [63] . Backup power sources are an affordable option and can be permanent, standby or portable with ease of use.

2.2.3. Restorative capacity Restorative capacity is the degree of ease with which a system can recover permanently from a disruption. Restorative ca- pacity is considered as the last line of defense . Restorative ca- pacity is highly dependent on the restoration of the budget and restoration of technical resources. Restorative capacity might not be fully achieved if the stakeholders fail to provide adequate financial and technical support. Within restorative capacity, three factors are identified.

• Resource restoration , which involves the repair or recovery of damaged equipment or facilities through post-disaster strategy. Resource restoration can be done through either human-based assistance such as trained engineers and a repair task force or non-human based support such as re- pair equipment and repair vehicles.

• In order to restore or repair the disrupted electrical infras- tructure, budget restoration or monetary capital is one of the primary factors of resilience-enhancing investments [67] . For an electrical system, the damaged equipment can be repaired or restored depending on the severity of disrup- tion and budget availability.

• The last factor, restoration of cyber control is expected to go faster compared to physical attacks. Cybersecurity staff and resource services often struggle to predict all vulnera- bilities and threats related to cybersecurity. For instance, if the entire system is affected by the malicious virus, reinstallation might take more time than expected [68] .

3. Background of the Bayesian network

This section provides background information on the Bayesian Network (BN), which is a powerful tool for risk assessment, reliability prediction, and decision making un- der the stochastic conditions of a complex system. The BN makes statistical inference in a rational way by updating the prior beliefs of an elementary event. Prior beliefs or probabilities are set based on subjective judgement (e.g., expert knowledge, historical data) or through a frequentist approach . BN is a Directed Acyclic Graph (DAG), developed based on the Bayes theorem [69] , that helps in addressing the cause and effect relationship (edges) among the set of interacting variables (nodes). The complete network represents a full joint probability distribution where the cause to effect and effect to cause relationships are mathematically equivalent, even though the direction of the underlying network depicts a unidirectional impact [69] .

3.1. Bayes theorem

Bayes theorem, proposed by Thomas Bayes [70] , is a mathe- matical expression that enables us to reason about belief un- der the condition of uncertainty. According to Bayes rule, the probability that A and B both would occur is the product of the probability of A and probability of B given A and this can be presented using the following equation.

P(A ∩ B ) = P(A ) × P(B | A ) (5)

where, P(A ∩ B ) = probability of A and B both would occur (joint probability); P(A ) = initial probability of A (prior probability); and P(B | A ) = probability of B given that A already occurred (posterior probability). Eq. (5) can be modified by symmetry and written as follows:

P(A | B ) = P (B | A ) P (A ) P(B )

(6)

The common terminology associated with BN is described below, followed by a detailed mathematical expression.

• Node : Node is also known as vertices , represents a random variable.

• Edge : Edge is also known as an arc , represents the condi- tional interdependencies between the variables.

• Directed graph : The underlying topology of the BN structure consisting of a set of variables and a set of arcs.

• Parent node : Parent nodes are without root nodes. • Intermediate nodes : Intermediate nodes are node with

parent and child node. • Child node : Child nodes are without leaf nodes. • Node Probability Table (NPT) : Every node possesses probabil-

ity tables, known as node probability table (NPT). NPT can be developed manually or achieved by eliciting the distribu- tion or related expression. For a node without its parent node, the NPT would be simply the probability distribution of that specific node.

3.2. Mathematical expression for Bayesian network

Suppose a BN consists of n variables Y 1 , Y 2 , Y 3 , . . . , Y n . The full joint probability distribution of BN can be written as follows:

P(Y 1 , Y 2 , Y 3 , . . . , Y n ) = P(Y 1 | Y 2 , Y 3 , . . . , Y n ) P(Y 2 | Y 3 , . . . , Y n ) . . . P (Y n −1 | P n ) P (Y n ) (7)

The above equation can be further simplified and stream- lined as follows:

P(Y 1 , Y 2 , Y 3 , . . . , Y n ) = n ∏

i =1 P(Y i | Y i +1 , Y i +2 , . . . , Y n )

= n ∏

i =1 P(Y i | Parents (Y i )) (8)

The above concepts can be represented with a simple ex- ample in Fig. 7 where a BN consists of a set of variables S = { Y 1 , Y 2 , Y 3 , Y 4 , Y 5 } and a set of edges to show the interdepen- dencies among the variables. An outgoing edge from Y i to Y j signifies a relationship where the value of Y j is conditioned on the value of Y i and Y i is the parent of Y j and Y j is the child of

70 international journal of critical infrastructure protection 25 (2019) 62–83

Y i T c

P

O d o a s v b Y

P

d w i f

P

t t c o t

4

T t e f s a a t

4

4

A a p u t t b t a t c

i o u p

i . Based on this definition, Y 1 and Y 2 are the parent nodes, Y 5 s the child node, and Y 3 and Y 4 are the intermediate nodes. hen, according to Eq. (8) the full joint probability distribution an be written as follows:

(Y 1 , Y 2 , Y 3 , Y 4 , Y 5 ) = P (Y 1 ) P (Y 2 ) P (Y 3 | Y 1 ) P(Y 4 | Y 2 , Y 3 ) P(Y 5 | Y 4 ) (9)

nce we compute the full joint probability, then the marginal istribution of each node can be calculated by the process f marginalization . Marginal distribution provides the prob- bilities of different values of the random variables in the ubset without explicitly referring to the values of the other ariables. For instance, we are interested in calculating P ( Y 3 ) y the marginalization approach. Marginalization of variable 3 can be calculated as follows:

(Y 3 ) = ∑

Y 1 ,Y 2 ,Y 4 ,Y 5

P (Y 1 ) P (Y 2 ) P (Y 3 | Y 1 ) P(Y 4 | Y 2 , Y 3 ) P(Y 5 | Y 4 ) (10)

Marginalization in belief function theory corresponds to a istributive operation over combinations which specifies that e can marginalize the global joint probability by marginal-

zing local NPTs [69] . From Fig. 7 , P ( Y 3 ) can be calculated as ollows:

(Y 3 ) = ( ∑

Y 1

P (Y 1 ) P (Y 3 | Y 1 ) ( ∑

Y 4

( ∑ Y 2

P(Y 4 | Y 2 , Y 3 ) P(Y 2 )

( ∑ Y 5

P(Y 5 | Y 4 ) ))))

(11)

It is important to note that Eqs. (9)–(11) are considered o be true when all the variables in the BN structure have wo possible binary outcomes: true or false . However, in many ases, such as the Washington, D.C. case study, different types f variables including continuous and fixed variables must be aken into consideration during the computation.

. Quantifying resilience capacity

he following subsections demonstrate the quantification of he resilience of the system as a function of the various el- ments of the BN through the interdependent electrical in- rastructure of our Washington, D.C. case study. AgenaRisk oftware [69] is used to show the different states of the vari- bles to quantify the resilience. Various kinds of nodes such s discrete, continuous, rank node, label node can be designed hrough AgenaRisk.

.1. Types of variables used

• Boolean variables (BV): A Boolean variable is expressed in forms of exactly two states, true and false , to present positive and negative outcomes, respectively. For instance, in Fig. 5 , the node for periodic maintenance (bottom of the figure) shows True = 0.71453 and False = 0.28547, meaning that the periodic maintenance of the electrical system is suc- cessful 71.453% and fails 28.547% of the time, respectively. In other words, the chance of being a successful periodic maintenance ( true state) is 71.453% while the probability of being a failed state is 28.547%. Similarly, the prior distri- bution of the management variable with two states of True

= 0.82 and False = 0.18 means that there is a 82% chance that a strong management policy, administered by author- ities, can effectively thwart the adverse impacts of disrup- tive events according to expert opinion; on the other hand, there is a 18% chance that it may fail. In another example, while 92% of the time a strong cyber critical infrastructure may positively contribute towards adaptive capacity, there is an 8% chance that it might fail.

• Continuous variables (CV): Continuous variables can take continuous realizations via a probability distribution of random variables. An example of a continuous variable is the availability of spare parts (see Fig. 5 ). The node of the continuous variable, “availability of spare parts” is mod- eled using a truncated normal distribution (TNORM) with a mean ( μ) of 87%, variance ( σ 2 ) of 2%, and a lower bound ( LB ) and upper bound ( UB ) set as 70% and 100%, respectively. This is represented in Eq. (12) .

Availability of spare parts ∼ T NORM (μ = 0 . 87 , σ 2 = 0 . 02 , LB = 0 . 70 , UB = 1 . 0) (12)

The above equation represents that in the worst possible scenario, the availability of spare parts is not lower than 70% and in the best possible scenario all the spare parts (100%) are available to conduct the periodic maintenance work. Since truncated normal distribution is a simple mod- ification of a normal distribution that confines the mean values between lower and upper bounds, it is one of the best possible ways to represent the continuous variables related to the electrical system and its interdependent net- work. All the parameters for continuous normal distribu- tion are generated through collecting and analyzing the historical data.

• Qualitative variables : Qualitative variables, which are also known as categorical variables , capture ordinal categories used for the weight of different factors pertaining to ab- sorptive, adaptive, and restorative capacity.

• Labelled variables : These variables possess a number of dis- crete states. Weighted value node is an example of Labelled variables.

.2. Quantifying absorptive capacity

s discussed earlier, eight important factors were identified s contributing to the absorptive capacity of EIN ( Fig. 3 ). The rior probability distribution for six of the variables i.e., skill tility management, control strategy, visual and physical pro- ection, alternative fuel source, information and communica- ion, and strong cyber-physical infrastructure are represented y two states through Boolean expression. In other words, hese six variables follow the same rules of Boolean variables s discussed in Section 4.1 . The posterior probability distribu- ion for the reliability and periodic maintenance variables are omputed based on Boolean logic.

To calculate the reliability of EIN, mean time to failure (MTTF) s computed in terms of operating hours. MTTF can be simply btained from historical data and is an example of a contin- ous variable. If the MTTF is greater than or equal to the ex- ected MTTF of EIN, then the electrical system and its related

international journal of critical infrastructure protection 25 (2019) 62–83 71

Fig. 5 – Base model of the Bayesian network for measuring the resilience of EIN.

Table 2 – Boolean expression used to define the posterior probability distribution of reliability and periodic.

Variable name Prior distribution Boolean expression Significance of boolean expression

Reliability MMTF ∼ TNORM (μ = 8 , 064 , σ 2 = 75 , LB = 0 , UB = 8 , 400)

IF (MTTF ≥ 8,016, “On”, “Fail”) If the MTTF is greater than or equal to 8,016 hr of EIN, then EIN is reliable (On state); otherwise, it fails (Fail state)

Maintenance On time repair ∼ TNORM ( μ= 0.85, σ 2 = 0 . 01 , LB = 0 . 50 , UB = 0 . 95) Availability of spare parts ∼ TNORM ( μ= 0.87, σ 2 = 0 . 02 , LB = 0 . 70 , UB = 1 . 0)

IF (Ontime repair ≥ 0.85 || Availability of spare parts ≥ 0.85, “True”, “False”)

If the probability of on time repair scheduling is greater than or equal to 85% AND the the probability of availability of spare equipment is greater than 85%, then the maintenance mission will succeed (True state); otherwise not (False state)

networks are reliable ( on state); otherwise, they are failures ( fail state). The same logic is applied to determine the posterior probability value for the continuous variable on time repair scheduling and spare parts availability. The Boolean expres- sions for these “reliability” and “periodic maintenance” nodes are summarized in Table 2 .

The posterior probability distribution of absorptive capac- ity of EIN is computed by the weighted sum of probabilities of its parent nodes. In order to calculate the absorptive capacity of EIN, a labelled node named “weighted value” is created to show the weight of each variable contributing to the absorp- tive capacity of EIN. Such weights can be obtained through dif- ferent decision-making processes such as Analytical Hierar- chy Process (AHP) [71] , swing weights, or from survey data. The general equation associated with a weighted mean (WMEAN) is presented in equation (13) , where i is the number of variables connected (eight in this case) to the weighted average node of absorptive capacity (see Fig. 5 ) and W i is the weight associated with the i th variable. This same weighted average approach has been applied to the weighted value of adaptive capacity

and restorative capacity as well.

WMEAN = ∑

W i X i = 1 , 2 , . . . , n, ∀ i = 1 ; 0 < W i < 1 ; ∑

i

W i = 1

(13)

4.3. Quantifying adaptive and restorative capacity

To set the prior probability of four nodes under adaptive ca- pacity and three nodes under restorative capacity, the same Boolean logic will be applied. Adaptive capacity and restora- tive capacity contribute to post-disaster strategy; however, their significance might be different. Adaptive capacity is the sec- ond line of defense in recovering the lost capacity whereas restorative capacity is the last line of defense. If adaptive ca- pacity is not able to withstand the shock due to the disrup- tion, restorative capacity restored the damage, but it takes a longer time compared to adaptive capacity. The fundamen- tal structure of Bayesian networks depicts the causal relation- ship between the nodes. The causal relationships between

72 international journal of critical infrastructure protection 25 (2019) 62–83

t t e t o e h i s v a s

a

N

a c w a b

P

l w e o m c b [ t p

N

4

A s m a h i a a r o t a b e

Table 3 – NPT for lost production capacity.

Absorptive capacity False True

Expression PDO × APC 0

Table 4 – NPT for recovered production capacity.

Post disaster strategy False True

Expression 0 LPC × 0.95

P

P

P

4 c

B D T d n d t o o o c L l e d d i

4

R a a p f p

he nodes are shown by drawing a connection between them hrough arcs. The posterior probability of post-disaster strat- gy can be calculated through Boolean logic. However, other han adaptive and restorative capacity, there might be some ther hidden factors contributing toward post-disaster strat- gy. This can be better described by NoisyOR function. These idden or missing parameters are known as “leak parameters”

n NoisyOR function. For instance, if there are n causal factors uch as Y 1 , Y 2 , . . . , Y n are conditioned on Z , with a probability alue for Z being true when one and only one Y 1 is true , and ll causes other than Y 1 are false . The NoisyOR function is pre- ented in Eq. (14) where for each i , S i = P(Z = true | Y i = true, Y j =

f alse ; ∀ j � = i ) is the probability of the conditional being true if nd only if that causal factor is true [69] .

oisyOR (Y 1 , S 1 , Y 2 , S 2 , . . . , Y n , S n , l ) (14)

Leak factor l can be defined as the extent to which there re missing factors from the model that can contribute to the onsequence being true . It is the probability that Z will be true hen all of its causal factors are false . The conditional prob-

bility of Z obtained with the NoisyOR function is presented elow in Eq. (15) .

(Z = True | Y 1 , Y 2 , . . . , Y n ) = 1 − n ∏

i =1 [ (1 − P(Z = True | Y i = True )

(1 − P(l )) ] (15)

As discussed in the proposed BN model, in order to calcu- ate the posterior probability of the “post-disaster strategy”, e have used NoisyOR function, represented in Eq. (14) . This

quation means that the chance of successful achievement f a post-disaster strategy is 70% if only adaptive capacity is et, while this value increases to 95% when only restorative

apacity is met and the leak parameters are set as 0.02 (shown elow in Eq. (16) ). This approach is supported by Vugrin et al.

72] where the authors stated that during a disruption, restora- ive capacity is needed to attain a higher level of recovery com- ared to adaptive capacity.

oisyOR ( Adaptive capacity , 0 . 7 , Restorative capacity , 0 . 95 , 0 . 02)

(16)

.4. Disruption modeling

n electrical system and its interdependent network (EIN) are ubject to different types of disasters. The three most com- on types of disasters are natural disasters, human threats,

nd cyber-attacks. The most common natural disasters are urricanes, tornados, and snow storms. Human threats can be

ntentional (sabotage) or electromagnetic while cyber-attacks re often in the form of denial of service, cross-site scripting, nd arbitrary code generation. In Fig. 5 , the likelihood of occur- ence of these threats is represented through True state based n historical data. We have used NoisyOR function to compute he posterior probability of natural disaster, human threat, nd cyber-attack (see equations (17)–(19) ). Finally, the proba- ility of disruption is calculated based on the weight value of ach individual disaster.

( Natural disaster ) = NoisyOR ( Hurricane , 0 . 15 , Snow storm , 0 . 10 , Tornados , 0 . 05 , 0 . 1) (17)

( Human threat ) = NoisyOR ( Electromagnetic , 0 . 12 , Sabotage , 0 . 09 , 0 . 08) (18)

( Cyber-attack ) = NoisyOR ( Arbitrary code , 0 . 15 , Denial of services , 0 . 1 , Cross site scripting , 0 . 05 , 0 . 1)

(19)

.5. Actual production capacity and lost production apacity

ased on the available data source for the city of Washington, .C., yearly production capacity is considered as 0.1TWh [59] . he production capacity of the electrical facility may hamper ue to either man-made attacks or natural disasters, such as atural disaster, human attack, or cyber-attack. The lost pro- uction capacity is highly dependent on whether the absorp- ive capacity is capable of absorbing shocks (the probability f being a True state) or not (the probability of being False). In ur model, the lost production capacity variable is conditioned n three variables including the probability of disruption oc- urrence, the absorptive capacity, and the actual production. ost production capacity is computed as the product of the ikelihood of disaster occurrence and actual production. The lectrical facility does not lose its production if the shock of isruption can be absorbed (True-state). Thereby, the lost pro- uction capacity is set to zero. NPT for lost production capacity

s shown in Table 3 .

.6. Recovered lost production capacity

ecovered lost production capacity is a function of two vari- bles: post-disaster strategy and lost production capacity (LPC). We ssume that an electrical facility will recover 95% of its lost roduction capacity, if the post-disaster strategy is success- ul ( True -state); zero otherwise ( False -state). NPT for recovered roduction capacity is represented in Table 4 .

international journal of critical infrastructure protection 25 (2019) 62–83 73

Fig. 6 – Different approaches to model interdependencies [74] .

Fig. 7 – Triangular Fuzzy Numbers representation of ASP [75] .

Fig. 8 – Forward propagation analysis of Bayesian network for measuring resilience of EIN.

4.7. Resilience

Resilience is the ratio of recovery (recovered production capac- ity) to loss (lost production capacity). Based on this calculation, the expected resilience is 0.87 as depicted in Fig. 5 .

4.8. Other modeling and quantification techniques

Holistic and reductionist approaches, mixed holistic- reductionist paradigm, and multiple formalism are some techniques that can be used to model infrastructure in-

74 international journal of critical infrastructure protection 25 (2019) 62–83

Fig. 9 – Backward propagation analysis of Bayesian network for measuring the resilience of EIN.

Table 5 – Comparative scenarios among different capacities.

Scenario periodic maintenance

Backup interdependence management

Restoration resource

Absorptive capacity (%)

Adaptive capacity (%)

Restorative capacity (%)

Expected resilience (%)

Base Case – – – 82.00 84.44 87.00 86.70 1 false – – 68.55( ↓ ) 84.44 87.00 85.70( ↓ ) 2 false false – 68.55 71.25( ↓ ) 87.00 84.80( ↓ ) 3 false false false 68.55 71.25 55.50( ↓ ) 71.64( ↓ )

↓ indicates that the value of the corresponding variable reduces compare to the value of the base case.

t i o a s t t c ( d t e a

t l t t t b f μ

f u s a

μ

a c p c p m T 0 l u

erdependencies [73,74] . Holistic approaches treat critical nfrastructures as a whole unit by simplifying the appearance f infrastructure interdependencies(see Fig. 6 a). Reductionist pproaches on the other hand identify different fundamental ystems components of the complex infrastructures, and hen describe the evolutionary development of the entire sys- em based on the aggregate behavior of the identified system’s omponents(see Fig. 6 b). In the mixed holistic-reductionist MHR) approach, modeling of infrastructure interdepen- encies is performed through reductionist approaches and he logical and functional dependencies between the het- rogeneous infrastructures are performed through holistic pproaches. Fig. 6 (c) shows the MHR modelling approach.

With regards to the quantification of different variables, he extension of this work can be performed using triangu- ar fuzzy approach. For example, the conditional probability able of variables using expert knowledge can be expressed in erms of triangular fuzzy numbers. Fuzzy approach is one of he easiest ways to handle data uncertainty and a more flexi- le technique to define a prior probability [75] . Let us define a uzzy number M on Z to be a triangular fuzzy number and let

M ( y ): Z → [0, 1] be its elemental functions as represented in

p

ollowing Eq. (20) where, l ≤ m ≤ u, l and u denote lower and pper value of of the support for M, respectively, and m repre- ents the mid value of the fuzzy triangular number. This tri- ngular fuzzy number can be donated by ( l,m,u ) [76] .

M (y ) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

y m − l −

l m − l , y ∈ [ l, m ] ,

y m − u −

u m − u , y ∈ [ m, u ] ,

0 , otherwise

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(20)

For our base model as depicted in Fig. 5 , the NPT of avail- bility of spare parts (ASP) for the periodic maintenance of EIN an be defined in terms of triangular fuzzy number. For exam- le, the expert opinions suggest that in order to successfully onduct the periodic maintenance, the probability of spare arts availability should range from a minimum of 70% to a aximum of up to 100% while most likely may fall into 85%.

his can be represented by a triangular fuzzy number where .70 is the minimum value (lower bound, l), 0.85 is the most ikely value (m), and 1.0 is the maximum value (upper bound, ). The mathematical representation of this statement is de- icted in Fig. 7 .

international journal of critical infrastructure protection 25 (2019) 62–83 75

Fig. 10 – Sensitivity analysis of absorptive capacity.

76 international journal of critical infrastructure protection 25 (2019) 62–83

Fig. 11 – Sensitivity analysis of adaptive capacity.

5

T b t t i o n p r t o i

p m g a c m c

d s a a w a a

. Results and analysis

his section analyzed results based on forward propagation, ackward propagation, sensitivity analysis, and information heory. During the analysis of probabilistic inference of mul- iconnected BN, the posterior probability of a set of variables s computed such that Y 1 ⊂ S at given evidence e . The feature f the BN to disseminate the effect of evidence through the etwork is defined as “propagation analysis”, and the related robability is represented by P(Y i | e ) ; ∀ Y i ∈ Y l [69] . BN offers a obust framework to compute posterior propagation probabili- ies from the experimental data. There is no strict direction f information flow; thus, queries can be made at any node

n the underlying structure. Forward propagation refers to the

ropagation of an individual or set of observed variables and easures their impact on the target node. Forward propa-

ation is a type of reasoning that refers the cause to effect nalysis. In the forward propagation analysis, predictive cal- ulations are computed by successively passing the resulting

arginal distributions from one node to one of its connected hild nodes.

In order to conduct the forward propagation analysis, three ifferent types of scenarios are designed by setting the false tate to three different variables types. Three decision vari- bles are chosen that contribute significantly toward the over- ll resilience of the electrical system and its independent net- ork. The three variables are: ( i ) maintenance , which belongs to

bsorptive capacity, ( ii ) backup interdependence management as part of adaptive capacity, and ( iii ) restoration resource which

international journal of critical infrastructure protection 25 (2019) 62–83 77

Fig. 12 – Sensitivity analysis of restorative capacity.

falls into restorative capacity. Scenario 1 accounts for the failure of periodic maintenance which if not successful ( false state) eventually increases the lost production capacity. Sce- nario 2 refers to the case when observation is made for failure of two events: periodic maintenance and backup interdependence , which ultimately drops the resiliency from 86.70% to 84.80%. Scenario 3 simulates the impact of failures of all three vari- ables: periodic maintenance, backup interdependence management , and restoration resource . Results indicate that failure of all three variables generates a larger adverse impact on the resilience which drops the resilience of EIN to 71.64%. The observations generated by these three scenarios are reported in Table 5 . Forward propagation analysis for scenarios 1, 2, and 3 is illus- trated in Fig. 8 .

On the other hand, backward propagation is the opposite ap- proach to the forward propagation analysis. Backward propa- gation enables us to conduct what-if analysis; an observation is set for a specific (descendant/target) variable and then the

BN calculates the marginal probabilities of ancestor variables by propagating the impact of the successor variable in a back- ward tactic through the entire network. In the case study, if the resilience value is set to 92%, as shown in Fig. 9 , then the absorptive, adaptive, and restorative capacities should be en- hanced from 82.00% to 87.04%, 84.44% to 86.58%, and 87.00% to 91.46%, respectively. Several analyses could also be performed for different desired outcomes as well.

Remark 1. We realized that in the real-world BN models, where different number of states exist for each variable, it becomes a daunting task to perform all the calculations man- ually. Therefore, there is a need to develop computationally- efficient algorithms which are capable of assessing fast and efficient propagation for a large class of BN models. Among the existing ones, Junction Tree (JT) is commonly used to feature factorization of the distribution for efficient inference with faster calculation [77] . This inference algorithm runs based on

78 international journal of critical infrastructure protection 25 (2019) 62–83

Fig. 13 – Mutual information between resilience and three individual capacities.

t c t t

o v p i o a u n r T c p

p i T p F c t c f p w fi p p t f w p f c i v e a t a o a t l

l [ t e

he distributive property of marginalization and conduct local omputation on the different part of the tree and propagate his calculation to other parts of the tree. Future extension of his work can delve more into this research direction.

Sensitivity analysis is a useful means to check the validity f the expert-built simulation model. Sensitivity analysis pro- ides a visual representation to understand the greatest im- act of a set of variable nodes on a selected node (target node)

n the BN. Sensitivity analysis is highly applicable in the field f analysis, quantification, and propagation of uncertainty in complex system. In order to gain more insight and a better nderstanding of the simulation model, we have used Age- aRisk software to examine the extent to which the input pa- ameters affect the output (target) of the underlying model. o examine the impact of the causal factors of the absorptive apacity, absorptive capacity is set as a target node and the im- act of its causal factors is measured in terms of conditional

Fig. 14 – Different states of mutual information b

robability. The sensitivity analysis of the absorptive capac- ty is illustrated in Fig. 10 , in the form of a tornado graph. he length of the bar in the tornado chart represents the im- act of that corresponding variable on absorptive capacity. ig. 10 (a) illustrates the impact of a set of selected nodes in- luding reliability, maintenance, visual and physical protec- ion, control strategy, alternative fuel source, information and ommunication, management and strong cyber-physical in- rastructure on the absorptive capacity when absorptive ca- acity is false . Fig. 10 (b) shows the impacts of those variables hen the absorptive capacity is true . It is evident from both gures that reliability has the highest impact and visual and hysical protection has the lowest impact on absorptive ca- acity. Fig. 10 (b) further shows that the probability of absorp- ive capacity changes from 0.613 (when reliability is false = ail ) to 0.856 (when reliability is true = on ). Compared to the idely impacted range of reliability, the impact of visual and hysical protection is limited to a narrow range which varies rom 0.777 to 0.828. This implies that improvement in electri- al system reliability will have the highest impact on improv- ng the absorptive capacity of EIN, whereas improvement in isual and physical protection will have a negligible impact on nhancing the absorptive capacity of the EIN. The sensitivity nalysis of adaptive and restorative capacities are shown in he Figs. 11 and 12 , respectively. It is evident from Fig. 11 that backup power source has the highest impact and provision f advanced technology has the lowest impact on improving daptive capacity. Further, Fig. 12 shows that resource restora- ion has the highest impact and budget restoration has the owest impact on improving restorative capacity.

In order to improve the quality of communication, we uti- ize information theory as proposed by Shannon and Weaver 78] . In information theory, entropy is one of the critical fac- ors in calculating the mutual information between the par- nt node and its child nodes. The entropy is measured by the

etween resilience and absorptive capacity.

international journal of critical infrastructure protection 25 (2019) 62–83 79

Fig. 15 – Different states of mutual information between resilience and adaptive capacity.

“mess” inherent in the variable X . Let P ( X ) and H ( X ) be the probability and entropy of a random variable X . In terms of risk analysis, entropy is a measure of uncertainty which can be computed using Eq. (21) as shown below:

H(X ) = − ∑ x ∈ X

P X (x ) log 2 P X (x ) (21)

Suppose, the entropy of the target node X is conditional on its dependent variables Y , then Eq. (22) can be used to repre- sent such relationships:

H(Y| X ) = ∑

i

P(Y i ) H(Y i | X i ) (22)

where i refers the number of states. The mutual information between the target node and its conditional node can be rep- resented by Eq. (23) as shown below:

I(X, Y ) = H(X ) − H(Y| X ) (23)

where I ( X, Y ) refers to the mutual information between the tar- get node and its dependent node; H ( X ) signifies the marginal entropy of the target node; and H ( Y | X ) refers to the conditional entropy of target node on its dependent node.

In the proposed model, resilience is conditional on absorp- tive, adaptive, and restorative capacities. These capacities are connected through the dotted line with the resilience node as illustrated in Fig. 13 . We have used dotted lines since in our model these capacities are not directly connected to resilience. We are interested in calculating the mutual information between resilience and all of these individual capacities. A different state of mutual information between resilience and absorptive capacity I ( X, Y ) is shown in Fig. 14 . The detailed calculation for mutual information between resilience and absorptive capacity I ( X, Y ) is shown below. Note that H (Resilience), H (Resilience|Absorptive capacity), and I (Resilience, Absorptive capacity) are calculated using Eqs. (21), (22) , and (23) , respectively.

From Fig. 14 , we find out that the prior probability of nodes (Resilience = Yes) = 0.867 and (Resilience = No) = 0.132. H (Resilience) and H (Resilience|Absorptive capacity) can be cmputed as follows:

H( Resilience ) = ∑ x ∈ X

P x ( Resilience ) log 2 P x ( Resilience )

= 0 . 867 log 2 (0 . 867) + 0 . 132 log 2 (0 . 132) = 0 . 5656

H( Resilience | Absorptive capacity )

= 2 ∑

i =1 P( Resilience ) × H(( Resilience | Absorptive capacity )

= P( Resilience = Yes ) H( Resilience = Yes | Absorptive capacity = Yes ) + P( Resilience = No ) H( Resilience No | = Absorptive capacity = No ) (24)

In order to calculate H( Resilience = Yes | Absorptive capacity = Yes ) and H( Resilience = No | Absorptive capacity = No ) in equation (24) , we set the absorptive capacity at True and False state, respectively, and simulate the model. The resulting outputs are reported below:

H( Resilience = Yes | Absorptive capacity = Yes ) = −{ 0 . 985 log 2 (0 . 985) + 0 . 015 log 2 (0 . 015) } = 0 . 1123

H( Resilience = No | Absorptive capacity = No ) = −{ 0 . 33 log 2 (0 . 33) + 0 . 67 log 2 (0 . 67) } = 0 . 9149

Plugging the above values into Eq. (24) yields the following:

80 international journal of critical infrastructure protection 25 (2019) 62–83

Fig. 16 – Different states of mutual information between resilience and restorative capacity.

Table 6 – Summary of information theory results.

H (Resilience|Types of capacity) Mutual information Significance of mutual information

H (Resilience|Absorptive capacity) = 21.81% I (Resilience|Absorptive capacity) = 34.75% If we have proper knowledge about absorptive capacity, we can reduce the uncertainty about resilience by 34.75%

H (Resilience|Adaptive capacity) = 31.27% I (Resilience|Adaptive capacity) = 25.29% If we have proper knowledge about adaptive capacity, we can reduce the uncertainty about resilience by 25.29%

H (Resilience|Restorative capacity) = 31.70% I (Resilience|Restorative capacity) = 24.78% If we have proper knowledge about restorative capacity, we can reduce the uncertainty about resilience by 24.78%

Concluding Remarks: I (Resilience, Absorptive capacity) > I (Resilience, Adaptive capacity) > I (Resilience, Restorative capacity). This implies that absorptive capacity , the first line of defense, has more influence in terms of uncertainty for the resilience of EIN.

H

a r

H

H

I

a

r b

( I

H

H

I

r s t T r

( Resilience | Absorptive capacity ) = (0 . 867 × 0 . 11236) + (0 . 132 × 0 . 91493)

= 21 . 87% I( Resilience | Absorptive capacity ) = H(X ) − H(Y| X )

= 0 . 5656 − 0 . 2181 = 34 . 75%

This implies that if we have proper knowledge about bsorptive capacity, we can reduce the uncertainty about esilience by 34.75%. Similarly, for adaptive capacity

( Resilience ) = ∑ x ∈ X

P x ( Resilience ) log 2 P x ( Resilience ) = 0 . 5656

( Resilience | adaptive capacity ) = 0 . 3127 = 31 . 27% ( Resilience , adaptive capacity ) = H(X ) − H(Y| X )

= 0 . 5656 − 0 . 3127 = 25 . 29%

Likewise, it implies that if we have proper knowledge bout adaptive capacity, we can reduce the uncertainty about

esilience by 25.29%. Different states of mutual information etween resilience and adaptive capacity is shown in Fig. 15 .

Finally, for restorative capacity we can compute H Resilience), H (Resilience|Restorative capacity), and (Resilience, Restorative capacity) as follows:

( Resilience ) = ∑ x ∈ X

P x ( Resilience ) log 2 P x ( Resilience ) = 0 . 5656

( Resilience | Restorative capacity ) = 0 . 3178 = 31 . 78% ( Resilience , Restorative capacity ) = H(X ) − H(Y| X )

= 0 . 5656 − 0 . 3178 = 24 . 78%

This implies that if we have proper knowledge about estorative capacity, we can reduce the uncertainty about re- ilience by 24.78%. Different states of mutual information be- ween resilience and restorative capacity is shown in Fig. 16 . he summary of results obtained from information theory are eported in Table 6 .

international journal of critical infrastructure protection 25 (2019) 62–83 81

6. Conclusion

A general framework for the resilience of electrical systems and their interdependents (EIN) is proposed in this research paper. The prime objective is to quantify the resilience of electrical systems during disruptive events. We developed a BN model for assessing resilience with respect to the con- cept of absorptive capacity, adaptive capacity, and restora- tive capacity. BN is a rigorous tool that provides a better in- sight into the uncertainty pertaining to complex models and allows the creation of future scenarios where assumptions and alterations in conditions or states can be tested and verified.

The proposed framework has been demonstrated through a case study of the interdependent electrical infrastructure system of Washington, D.C. The BN framework facilitates the identification of the different underlying factors that could potentially impact the resilience of the electrical system and its interdependent network. The information obtained from the historical data and the subjective judgment of experts is translated into BNs to provide a better understanding of the complex interaction among the different variables. The BN model is then validated through sensitivity analysis. We found that the key elements of the EIN are reliability, a backup power source, and resource restoration. The belief prepara- tion further reveals how the failing of any variable impacts the other variables. The information theory analysis was also conducted to better understand the mutual information be- tween resilience and different capacities. The contribution of this paper to the existing body of knowledge in interdepen- dent electrical infrastructure system can be summarized as follows:

• A model for designing an electrical system and its interde- pendent network was developed.

• The underlying factors pertaining to interdependent elec- trical infrastructure system were identified and classified with respect to the concept of absorptive, adaptive and restorative capacities using Bayesian structure.

• A real-world case study of the model is presented and different kinds of analysis are performed to validate the effectiveness of the proposed model. Although this framework is specifically developed for the electrical infrastructure network, it can be modified based on the structure and nature of complex systems and utilized to quantify the resilience for any other system as well. This framework can also be used as a decision support tool in assessing risk and uncertainties in a complex environment and providing better insight when designing and develop- ing strategies to offset the severity of a disruptive event.

This work can be extended in several directions. For in- stance, decision-theoretic troubleshooting for interdependent electrical infrastructure systems and corresponding improve- ment activities can be designed and executed in order to achieve higher resiliency.

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.ijcip.2019.02.002 .

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