Performance indicators for infrastructure networks

From Equations (1) and (2), the definition of the performance indicators (𝑄𝑄(𝑡𝑡)) for the infrastructure networks is required to evaluate the resilience of the infrastructure system. The network-performance indicators are suggested to be considered either the

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topology or the functionality of networks (Ghosn et al., 2016). The topology-based performance metrics study the performance from the perspectives of connectivity and efficiency; herein, the connectivity is considered as the number of the connecting paths from the supply node to the consumption nodes; the efficiency is measured as to how efficient the transmission of the utility between different nodes. However, the topology-based metrics cannot capture the functional aspect of the infrastructure networks.

The flow-based functional performance metrics combine network topology with flow patterns, which are considered as the amount of flow that a damaged network can deliver to the demand nodes comparing to what it delivers before the disruption. Such metrics consider the flow capacity and the supply and demand constraints in an optimization framework (Ghosn et al., 2016).

Interdependency categorization

With the preface to the interdependency in Section 1.1.2, interdependency illustrates the interrelations among the infrastructure systems, and it can be presented in many different aspects. Rinaldi et al. (2001) categorized interdependencies into four types:

physical, cyber, geographic, and logical interdependencies.

(i) Physical interdependency means that the state of one infrastructure system is dependent on the material output(s) of another.

(ii) Cyber interdependency implies the relationships between infrastructure systems based on information transmitted through the relevant infrastructure.

(iii) Geographic interdependency means that a local environmental event can cause state changes in all infrastructure systems.

(iv) Logical interdependency includes other state dependencies between different

infrastructure systems, which is not via the physical, cyber, or geographic connection.

It is recognized that such classification can well sort out the interdependency related issues in several practical cases.

Lee et al. (2007) identified five types of interrelationship between infrastructure systems, where these authors denoted those types of dependence as the interdependency in their studies.

(i) Input dependence indicates the infrastructure components requires the services from another infrastructure component as the input.

(ii) Mutual dependence implies that a group of infrastructure components are dependent on the activities of each other.

(iii) Shared dependence means that some infrastructure systems share the same physical components or activities.

(iv) EXCLUSIVE OR dependence illustrates the activities that some specific infrastructures are the exclusive providers.

(v) Collocated dependence specifies that the components of two or more infrastructure systems are located in a similar geographical region.

P. Zhang and Peeta (2011) also proposed a way to categorize interdependencies.

(i) Functional interdependency indicates that the functioning of one system requires inputs from or can be substituted by another system.

(ii) Physical interdependency means some infrastructure systems are coupled through shared physical attributes.

(iii) Budgetary interdependency implies that several infrastructure systems share the same resource allocation budget, especially during disaster recovery.

(iv) Market interdependency means that all of the infrastructure systems are interacting

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in the same economic system.

Ouyang (2014) reviewed the abovementioned and other types of interdependencies through studying some extreme events, such as extreme natural disaster and large-scale terrorist attack. However, the classification by Lee et al. and Zhang and Peeta does not cover some scenarios. For instance, the classification by Lee et al. cannot sort the scenario that the electric power systems and the telecommunication services are prioritized during the restoration process, and the categorization by Zhang and Peeta cannot sort the event that the debris-covered streets could block the emergency response personnel. Herein, Ouyang (2014) recognized that the classification proposed by Rinaldi et al. could well sort out the interdependency related issues in several practical cases.

Modeling interdependent infrastructure systems

In the review paper of modeling interdependent critical infrastructure systems (Ouyang, 2014), five major types of approaches have been adopted for analyzing interdependency across infrastructure systems:

(i) Empirical approaches analyze the interdependencies of the infrastructure systems through historical data and expert experience.

(ii) Agent-based approaches implement a bottom-up method that contains autonomous agents and their interactions to analyze the decision-making processes in the infrastructure systems. Herein, the reaction of the agents is based on their objectives, the pricing strategies, learning, and adaptation to the simulation environment, and the capacity expansion decisions (P. Zhang et al., 2011). However, the result of the simulation highly depends on the assumptions about the behaviors of the agent.

(iii) System-dynamics-based approaches model the dynamic behavior of the

interdependent infrastructure systems by capturing important causes, effects, and factors under the scenarios of disruption.

(iv) Economic-theory-based approaches view the operation of the infrastructure systems as the intermediate goods in the market of the economy, where the interdependencies are analyzed through economic interdependencies.

(v) Network-based approaches exploit the network structure, a common characteristic of infrastructure systems, and they are useful for analyzing physical interdependencies and the cascading disruptions (P. Zhang et al., 2011).

Herein, network-based approaches model each single infrastructure system by a respective network and describe the interdependencies between them by inter-links.

Depending on whether particle flows in the networks are de facto modeled, network-based approaches can be further categorized into two groups: topology-network-based methods and flow-based methods.

To explicitly describe the interdependencies in the infrastructure systems, this research adopts the network flow method to capture the dynamics of system evolution in terms of how restoration units and relevant resources move across systems. Accordingly, infrastructure systems are represented as the combination of networks, and the interdependencies are modeled using logical constraints in the formulation.

Restoring interdependent infrastructure networks

The relevant literature of modeling the interdependent infrastructure networks can be generally grouped according to research goals: performance evaluation, design, mitigation, and recovery models. For recovery models, most studies focus on analyzing the changing functional states of systems upon the restoration of failed components

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(Ouyang, 2014). To optimize restoration can be viewed as a network design problem to add (restore) links to the disrupted network, while the scheduling of restoration also needs to be addressed. Lee et al. (2007) modeled the restoration of services in interdependent infrastructure systems by explicitly identifying interdependencies using network flow approaches. Nurre et al. (2012) proposed an integrated network design and scheduling problem to optimize the restoration of a single infrastructure network to maximize weighted total arrived demand. Cavdaroglu et al. (2013) optimized integrating restoration and scheduling decisions with the objective function of the performance over the horizon of the restoration plan and implemented logical constraints to describe the interdependencies. González et al. (2016) optimized the restoration strategy of selecting the components to be restored through minimizing the cost of preparation, reconstruction, surplus or deficit supply, and commodity flow, and they also developed the iterative use of the interdependent network design problem to account for the order of the reconstruction. Almoghathawi et al. (2019) proposed a resilience-driven restoration model with multiple objectives, including maximizing the resilience and minimizing the restoration cost. In their study, they used ε-constraint method to generate Pareto-optimal solutions and demonstrated the tradeoff between the resilience and the restoration cost.

Karakoc et al. (2019) integrated a resilience-driven mixed integer programming model to schedule the restoration process of the disrupted interdependent infrastructure networks with the index of geographically distributed social vulnerability. Herein, this study incorporated the concept of community resilience to the restoration process.

These studies mostly model and discuss the complication of disruption patterns over interdependent infrastructure systems at a conceptual level and focusing on the perspective of system functionality. Other than functional interdependency, however, restoration interdependency which can be manifested as the accessibility/feasibility of

components in the different system required for the deployment of restoration is rarely considered in the existing literature. That is, the disruption to the roadway network which enables the restoration crews to access the disrupted components in other infrastructure networks is rarely included in the existing literature.

Restoration with incomplete information

In Section 2.2, the cyber interdependency regards the interaction between infrastructure systems through the information. That is, if the infrastructure system transmitting the information, such as the telecommunication systems, fails after the disruption, some information in other infrastructure systems can be incomplete, influencing the decision process for the restoration. However, the relevant literature is emerging but still rare, and few studies consider the factor of incomplete information involving in the restoration process.

There is some literature analyzed the incomplete information during the restoration from different perspectives. Çelik et al. (2015) addressed incomplete information about the debris amounts along the roads in the debris clearance problem using a partially observable Markov decision model. X. Zhang et al. (2018) optimized the resilience-based network design under uncertainty and developed a nonlinear function to consider the non-deterministic case about the disrupted capacity, the restoration speed, and the degree to which the component can recover of the system component. Fang and Sansavini (2019) formulated a two-stage stochastic programming model to minimize the expected system resilience loss, considering the uncertainty of the repair time and the total amount of repair resource units.

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Restoration interdependency

Sharkey et al. (2016) identified restoration interdependencies by analyzing several news reports/articles about the restoration efforts after Hurricane Sandy. This study provides a classification scheme including five distinct classes of restoration interdependency: traditional precedence, effectiveness, options precedence, time-sensitive options, and competition for resources. Herein, the most frequently observed restoration interdependency is traditional precedence. It means that the restoration task in an infrastructure system cannot be started until the restoration task in another one is complete. That is, the feasibility of restoring the specific component requires the connectivity between the depot of the restoration crews and the location of that component through the roadway network.

In the existing literature about the interdependent network design problem introduced in Section 2.4, the interdependencies are all revealed in the form of logical constraints indicating the functional association between different infrastructure components. However, the restoration of the roadway network, which the connectivity evolves through the restoration of the road links, has not considered. In this study, the restoration interdependency is reflected by limiting the restoration act to components accessible for restoration units from the roadway network.

Summary

In this chapter, the measuring approaches for resilience are first reviewed. Second, since the interdependencies among infrastructure systems can complicatedly influence the performance of the infrastructure systems, the methods to classify the interdependencies are reviewed, which can assist this thesis in inferring and modeling the

interdependencies existed in the infrastructure systems. Third, to analyze the resilience of the infrastructure restoration after severe disruption, the conventional approaches to modeling the infrastructure systems and the similar existing studies for optimizing the restoration process are reviewed. Last, in the existing literature, some aspects, including the incomplete information due to the failure of the telecommunication service and the restoration interdependency, have not studied and explored in depth. This research thus focuses on closing the abovementioned research gap.

CHAPTER 3

MODEL DEVELOPMENT

In this chapter, a problem about recovering the disrupted interdependent infrastructure networks is first proposed. Then, the mixed integer quadratic programming model to schedule the restoration of infrastructure systems is developed, seeking to maximize the combined resilience after a severe disruption.

Problem statement

This study seeks to develop an interdependent network restoration problem considering resilience optimization over multiple infrastructure systems to provide relevant Emergency Management Agencies (EMA) with a holistic perspective for disaster response. After the disaster strikes the infrastructure systems, each layer of the infrastructure systems can be partially disrupted. Hence, the manager, such as the authorities in the area, would start scheduling the restoration of the infrastructure to recover its performance. Herein, the problem proposed in this section is to optimize the restoration process considering the resilience loss.

In order to highlight the importance of factoring the interdependency across different infrastructure systems, a problem context of three-layer infrastructure upon disaster impact is established, which consists of the roadway network, electric power network, and telecommunication network. As explicitly accounting for interdependency, the infrastructure systems are modeled using network-based approach, and the characteristics of each infrastructure system as a network are detailed in this section.

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3.1.1 Objective

The objective of the problem is to minimize the weighted sum of two components, where the formulation of the objective is introduced in the objective (8) in Section 3.4.2:

(i) Resilience loss is the summation of the ratio of the performance loss over the modeling horizon as introduced in Section 2.1.1. Herein, the performance loss in this problem is defined as the expected unmet demand on each demand node at all infrastructure network layers, and the performance loss would be constrained to be positive or zero through the constraints to avoid surplus demand.

(ii) Penalty for incomplete information is defined as the ratio of the amount of demand in the roadway network which is without the telecommunication service. This part of the objective is to examine the influence of the incomplete information to the manager of the restoration schedule. Herein, if the demand information is known for a demand node, its expected unmet demand is a deterministic value.

3.1.2 Infrastructure networks

In this study, three infrastructure networks are considered, which are the roadway network, the electric power network, and the telecommunication network.

(i) Roadway network

In the roadway network, a link represents a section of road between two intersections, and a node represents an intersection. The roadway network is indispensable for emergency logistics, including the delivery of relief materials, rescue teams, and restoration units for affected infrastructure systems. If the failure or capacity reduction of a system component occurs due to disaster impact, it may cause severe delay to the logistics mentioned above for disaster response or even disrupt the network and isolate

some areas from outer supports.

In this study, the roadway network is used to transport emergency relief and the restoration crews for all the infrastructure networks. Herein, this study regards the restoration of the basic functionality of the infrastructure, and thus it only considers the recovery of the infrastructure to the level of fulfilling the basic needs of the community.

(ii) Electric power network

A typical electric power network is composed of facilities at three levels: power generation, power transmission, and power distribution. The analysis of the electric power network in this study focuses on the restoration of power distribution from substations to each household in the disaster-affected areas. Here, the substation plays the role as an interface to transfer power from the transmission system to the distribution system of an area. The disruption of power distribution can significantly impact people’s lives, as it can cause the malfunction of any electricity-dependent systems. On the other hand, restoring electric power can help households accelerate the recovery of the standard of living and capture the latest information, which is virtual but another critical form of relief.

(iii) Telecommunication network

This study considers both mobile and data services for the telecommunication network. The telecommunication network transmits data or communication needs, where the internet service provider is at supply nodes, and base stations act as demand nodes to provide service to surrounding area wirelessly. However, base stations require electric power to transmit a signal through antennas. Although they are generally equipped with emergency power generators, when the fuel in the generator is exhausted, even if the facility is intact, it cannot provide telecommunication service.

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(iv) Summary

In summary, the characteristics of each infrastructure network are listed in Table 3.1.

Table 3.1 Summary of the considered infrastructure networks

Infrastructure

network Transmitted utility Supply node Demand node

Roadway Emergency logistics Dispatch center Townships Electric power Electric power Power generator and

major substations Substations Telecommunication Telecommunication service Internet service

provider Base stations

3.1.3 Interdependency

In this study, the interdependencies among the infrastructure networks are considered following the classification by Rinaldi et al. introduced in Section 2.2.

(i) Physical interdependency

The physical interdependency between electric power and telecommunication network is accounted, as the functionality of the telecommunication network (particularly the mobile network) is electricity-dependent. Although the facilities in the telecommunication system, such as the base stations, may be equipped with the backup electric power sources (i.e., the emergency generators), when the backup electric power is exhausted, even if the base station is functional and connected to the supply nodes through the telecommunication network can it not provide the telecommunication service.

(ii) Geographical interdependency

Natural disasters can generally cause geographically-related disruption areas (such as flooded areas) and thereby impact the associated infrastructure networks. This type of

interdependency is manifested through the outcome of the natural disaster, and it will be presented in the numerical experiment, Chapter 4.

(iii) Cyber interdependency

As addressed in Section 2.5, the cyber interdependency describes the transmission of the demand information in the roadway network through the telecommunication network. If the telecommunication service of a demand node in the roadway network is failed, the demand information of that node is uncertain to the manager of the restoration process. Hence, in this situation, the manager can only optimize the restoration schedule based on the prior probability of the demand information about the telecommunication-service-blocked nodes in the roadway network rather than the deterministic demand information.

In this study, the probability distribution of the emergency demand in the roadway network is assumed to be known to the manager of the restoration schedule; besides, the study assumed a sectioned uniform probability distribution to accommodate the low, medium, and high estimation to the possible amount of demand with the probability of 𝑃𝑃_i,2^r , 𝑃𝑃_i,3^r , and 𝑃𝑃_i,4^r respectively. The assumed distribution of the demand is presented in Figure 3.1.

Figure 3.1 Probability distribution for the demand in the roadway network 𝑑𝑑_𝑖𝑖,0^𝑟𝑟 𝛿𝛿

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(iv) Logical interdependency

Additionally, this study seeks to address the research gap by factoring the effect of the restoration interdependency between the roadway network and other networks in the system recovery phase. When system restoration is implemented after the disruption due to disaster impact, restoration interdependency, which can be viewed as the logical interdependency defined by Rinaldi et al., becomes a critical issue affecting how restoration tasks should be scheduled within or across the interdependent network.

Assumptions

The assumptions for the developed model are listed as follows.

 The disruption to the infrastructure networks is given at the beginning of the planning horizon for the restoration.

 The failure of the component in the networks primarily occurs on the links.

 The performance of each infrastructure network is time-dependent and evaluated on a staged basis.

 The functional states of the links in the network are assumed to have two state: fully functional and fully disrupted.

 The restoration of each component in the networks takes a single time stage and single restoration crew.

 The links in each network are bidirectional, but the variable for the functional state of a link is unidirectional. Hence, the restoration of a link recovers the functionality of links for both directions.

 The incompleteness of the demand information is only considered in the roadway

 The incompleteness of the demand information is only considered in the roadway