A Novel Methodology to Assess Seismic Resilience (SR) of Interconnected Infrastructures

Abstract

Modern and smart cities are significantly vulnerable to natural hazard, and their functionality is based on resilient infrastructure systems. In particular, seismic resilience may be considered the ability to deliver services during and after hazard events. Therefore, it is fundamental to identify the most critical components within a system, especially when multiple infrastructure systems are interdependent. The paper aims to propose a novel methodology that consider interconnected infrastructures to assess seismic resilience that may be defined as a function that depends on time, and the different components are considered the functional dimensions. The proposed methodology may be applied for several typologies of infrastructures, specifically looking at the seismic resilience analyses related to transportation systems. A case study has been considered in order to apply the proposed formulation and to demonstrate the importance of considering interdependency in the assessment of the seismic resilience. Many stakeholders (infrastructure owners, public administrations, decision makers) may be interested in applying the methodology that could be used to study several applications.

1. Background

Robustness of infrastructures is a basic requirement for communities to be resilient to seismic events. Infrastructures interact with each other and a problem to one of them, may be diffused to the others that they are interdependent with, intensifying the severity and damages, as described in [1]. In this way, interdependencies consist of the mutual effects that the behavior and reliability of one network has on the other networks. In addition, infrastructure systems need to remain functional during and after earthquake events to deliver services [2]. In this regard, modern cities and communities considerably depend on many critical components, especially when the different infrastructures are interdependent. In literature, quantitative methods for assessing infrastructure seismic resilience have been proposed. For example, Ref. [3] reviewed the state of the art in the transportation network field emphasizing the importance of cross-disciplinary research. More recently, Ref. [4] proposed the analysis of perturbations caused by climate and hazard-induced loss in Vietnam. Moreover, Refs. [5,6] proposed methodologies to assess the seismic resilience of bridges and infrastructures. In particular, the importance of considering indirect losses was also considered in [7]. Other studies considered agent-based methods to model the flow of people and goods after events along transportation networks (e.g., [8,9]). More recently, Ref. [10] considered the supply chain disruptions in Tanzania from flooding hazards by identifying multiple priorities using input–output economic models resulting from agent (firm) behaviors. Moreover, Ref. [11] analyzed the perturbation effects on the delivery of and/or access to specific services using a least-cost travel model by considering economic and health facilities in Cambodia.

The previous literature allowed to assess that the interconnectivity of infrastructures may affect delivery of the services. Therefore, their failure may result in cascade effects with severe consequences. Few contributions have considered the role of interdependencies, such as [12]. Recently, Ref. [13] proposed a method that applies drop-link analysis to show the interconnectivity of service delivery and road networks. They considered three services (water, power and people) in order to define the critical cascade effects on the supply of important services (such as potable water supply, seaport materials and patient access to hospitals). Moreover, Ref. [14] considered that the functionality of infrastructures may suffer the severe consequences of cascading failures. As shown in [15], an example of interdependent networks may consist of a power grid that relies on a telecommunications network for control and at same time the telecommunications network relies on the power grid for electricity supply. In literature (i.e., [16,17]) interdependency matrices have been proposed to study the propagation of targeted attacks between the two networks.

As shown in [5], interdependencies need to be considered when for either geographical proximity or shared operations, there are interactions between the impacts on the various lifeline systems. For example, a problem or failure in the original system may compromise the product or the service delivered by the second infrastructures. In many cases, the second system can be degraded or interrupted. In particular, the functionality of a community depends on the interdependencies among its systems or components of the system. In addition, the role of interdependencies in cascading failures has been analyzed in literature. For example, Refs. [12,18] showed that most disasters are generally triggered by minor events demonstrating that the robustness of infrastructures is of fundamental importance for preventing system crashes. In [19], the importance of interdependencies was demonstrating by presenting a method for identifying service-oriented interdependencies in interconnected networks. In this regard, cascading failures have been studied intensively with the aim to investigate the changes in the structure connectivity caused by node dependence between networks [20,21,22]. In particular, Ref. [22] considered the role of interdependences between the networks in relation with recursive processes of cascading failures.

In this paper, the original concept of resilience by [23] considered the functionality of one system by proposing a single variable definition. The concept of resilience has been applied in [24] by considering several infrastructures that belong to the same typology (in this case road networks) and in the case of seismic events. Therefore, the novelty of this paper consists of proposing a multidimensional definition of resilience that overcomes the previous one by [23] for 1D systems. The proposed formulation herein aims to include the various dimensions (or variables) that may describe the interconnections between the various lifelines that affect the functionality of the global system. The calculation of resilience needs to account for the complex of infrastructures that may modify the functionality of the community. In this regard, the main novelties of the paper consist of: (1) developing a multi-dimensional definition of seismic resilience that may consider the role of several systems instead of a single infrastructure; (2) proposing a formulation that include interdependencies inside the resilience formulation; and (3) presenting a case study that applies the proposed definition and may be considered as a first attempt to implement the framework.

The structure of the paper consists of seven sections. In Section 2, interdependencies are defined to consider their significance in the assessment of resilience for infrastructures. Section 3 shows the original definition of resilience by explaining the limitations of considering a 1D definition, since it cannot consider the various interdependencies that occurs between the several systems that concur to generate the final resilience of the system (and then of the community). Section 4 proposes the novel formulation that may include the presence of interdependencies and overcome the previous ones, allowing more realistic estimations of resilience of systems. Section 5 discusses how the general formulation may be applied if linear hypotheses are assumed to simplify and deduce important considerations in implementing the proposed formulation in several realistic applications. Section 6 shows a case study where two infrastructures have been considered and the resilience has been calculated with the proposed formulation.

2. Interdependencies

This paper aims to propose a definition of resilience in a multiple dimensions scale to account for the different typologies of systems that are interdependent. In this regard, interdependencies consist of the interactions between different lifelines (i.e., lifeline types such as energy, transportation, communication and water) and are fundamental to be considered since they may produce negative consequences of direct and indirect losses. In particular, the seismic resilience may be significantly underestimated if interdependence is neglected, as shown in [14,25,26,27,28]. In this regard, Ref. [29] demonstrated the importance of considering the impact of coastal hazard countermeasures on community resilience for the Tōhoku region of Japan following the 2011 tsunami. In addition, Ref. [30] proposed a quantitative assessment framework and method to accurately assess the seismic resilience of interdependent lifeline networks.

When an event occurs, the lifelines that enable to distribute services become ways to distribute the consequences of the loss of services along the system. Following [31], several points need to be considered.

The consequences due to infrastructure interdependencies are more severe than those presented exclusively on the isolated system, because they may be responsible of cascading failures and collapses of entire regions.

• Impacts due to the disaster may be reducing by considering the propagation of failures inside the network;
• The reduction of the severe impacts of disaster may be enchanted by considering network dependencies;
• As shown in [7,24], interdependencies play a significant role in indirect losses (such as lifeline system failures and loss of services).

Interdependencies may affect the resilience of the system in several ways. First, a problem or failure in one lifeline may affect the product or service delivered by the entire system, that can be damaged, degraded or interrupted. This type of interaction may occur when the cellular phone towers are not equipped with backup systems when power is lost, and a failure on individual cell towers may affect the entire electric service.

Another example occurs when two lifelines are connected so that the first system provides a service to a second that supplies the first with a critical product, as in the case of failure of the electric distribution systems, which stops the delivery of coal to power plants affecting, in turn, the electric generation [31]. Figure 1 considers two typologies of failures specifically due to interconnections between lifelines. In Figure 1a, cascading that generates dependent relationships amongst lifelines and consists of a change in state in the first lifeline that propagates changes in a second, and thus in all the connected ones in turn, causes a change in state in a third lifeline. A common cause (Figure 1b) consists of disruptions of different systems due to a single event. For example, a tree blown over by the wind taking down electric distribution lines, television cables and telephone cables with it.

Interdependencies may be of particular importance for those stakeholders such as public- and private-sector owners deputed to the protection of lifelines and other critical infrastructures. The disruption or the failure to one of such systems may cause cascade effects upon other lifelines. The operators need to be prepared and aware of the role of interdependencies to assess the response, prioritize recovery plans and organize emergency countermeasures. For example, the role of the interactions among the different systems needs to be considered as a significant part in designing the various component, in preventing cascade effects and in planning for redundancy. In other words, interdependency may be the difference between a controlled event and an economic disaster, due to catastrophic consequences occurring with no proper plannings.

In the next section, an analytical formulation of resilience has been proposed in order to account for the functionalities of the various dimensions. In particular, the proposed formulation may consider several dimensions of the problem and therefore the interdependencies among the various systems concurring to the calculation of resilience.

3. Seismic Resilience

In this section, seismic resilience is considered with a new development that is necessary to include interdependencies. In particular, seismic resilience has been object of several contributions, such as [32,33,34,35,36]. In this this regard, Ref. [37] proposed an adaptive and compositional modelling framework to model the resilience by examining the effectiveness of the rapid-response. In addition, Ref. [38] investigated the impacts of repair sequence and repair/replacement approaches regarding different macroseismic intensity by proposing a Monte Carlo simulation-based method to select search the optimal repair strategy. In [39], a novel seismic resilience assessment method for urban systems has been proposed based on post-earthquake loss (casualties and functionality loss) and recovery time by applying the analytical hierarchy process.

Traditionally, resilience was defined at the beginning of 2000 by [32]. In addition, Ref. [40] defined resilience as the ability to maintain or restore the flows, which are important ways of approaching community resilience. The traditional formulation by [23] implemented in [24] quantifies the seismic resilience as:

$$\mathrm{SR}={{\displaystyle \int }}_{{\mathrm{t}}_{0\mathrm{E}}}^{{\mathrm{t}}_{0\mathrm{E}}+\mathrm{RT}}\frac{\mathrm{Q}\left(\mathrm{t}\right)}{\mathrm{RT}}\mathrm{dt}$$

(1)

where

t_0E is the time of occurrence of the earthquake E,

RT is the repair time (RT) that is necessary to recover the original functionality,

Q(t) is the recovery function that describes the recovery process necessary to return to the pre-event level of functionality and may be represented in 1D as in Figure 2.

In literature, different types of recovery functions were proposed and depend on many variables (such as community preparedness and available resources) with disparities among different geographic areas, in the same community or state and showing different rates and quality of recovery [41]. In particular, Ref. [23] proposed linear formulation that can be applied with no information, Ref. [42] proposed exponential, while [43] trigonometric.

However, these formulations cannot account for the interaction of the different systems that necessarily interact each other. In other words, the definition of seismic resilience needs to consider issues that may not be easily quantified, but are often the most significant [40]. Therefore, it is important to keep in mind that seismic resilience needs a multidisciplinary approach and requires a holistic point of view. In this regard, Ref. [40] considered the lessons from the Christchurch disaster by proposing six categories: time, money, resources, governance and management, social community and business community. The conclusions insisted on the fact that community resilience needs to consider two aspects: (1) the flows between the community and its surrounding environments; and (2) the community resources (commercial, capital, human, social, physical and information) [40]. In other words, the community needs to be considered as a system with many correlated interdependencies, and thus the definition of resilience must quantify at the same time several dimensions.

4. Multidimensional Seismic Resilience

In this section, the definition of the seismic resilience is proposed in terms of a multidimensional function with N variable in the R^N space. Recently, some contributions considered the multidimensional nature of the problem. For example, Ref. [44] demonstrated that the agent-based modeling approach may represent the recovery process of integrated civil infrastructure systems and the dynamic interactions among participants of the recovery process. Other contributions considered data on the vulnerability and recovery of communities [45], combined community and infrastructure systems [46]. However, an analytical formulation is herein proposed in order to derive a general procedure to assess the seismic resilience in a multidimensional space.

In particular, the previous Equation (1) is a function depending on time: R(t), that is the quantity that is used to define the recovery procedure. It is worth herein to consider that the mobilization time [47] (including building inspection, site preparation, providing engineering services) needs to be considered since generally recovery does not develop instantaneously and the recovery function does not start at the time of occurrence (t_0E), as described in [48]. Therefore, resilience depends on the functionality Q_N(t) that is a function with N variable in the R^N space and depends mainly on time (t):

$$\mathrm{Q}=\mathrm{Q}\left(\mathrm{t},{\mathrm{Q}}_1,\dots .,{ \mathrm{Q}}_{\mathrm{N}}\right)$$

(2)

Consequently:

$$\mathrm{SR}={{\displaystyle \int }}_{{\mathrm{t}}_{0\mathrm{E}}}^{{\mathrm{t}}_{0\mathrm{E}}+\mathrm{RT}}\frac{\mathrm{Q}\left(\mathrm{t},{\mathrm{Q}}_1,\dots .,{ \mathrm{Q}}_{\mathrm{N}}\right)}{\mathrm{RT}}\mathrm{dt}$$

(3)

That expresses the dependency of SR on time and the various (N) functionalities. In other words, this definition of SR in the R^N space, may be displayed as a multidimensional function with the main variable t. The seismic resilience is graphically represented by the normalized volume underneath the recovery function Q_N(t) represented by the solid (Figure 3). In addition, any plane at a constant t:

$$\mathrm{Q}\left(\mathrm{t}=\mathrm{k},{\mathrm{Q}}_1,\dots .,{ \mathrm{Q}}_{\mathrm{N}}\right)$$

(4)

Represents a plain surface connecting the points on the recovery function corresponding at t = k for all the variable Qi. In other words, the plain figure represents the cross section of the solid represented in Figure 2 and the perimeter represent the sum of the various functionalities Qi reached by the system.

It is worth noting that Figure 3 represents a case where the functionality at the time of occurrence t_0E is considered 0 for all the quantities. This is not a necessary assumption for the formulation that was presented herein, but assumed only to make the figure clearer. However, the repair time could also be different among the different quantities and thus the figure may complicate more, but still the general formulation is still valid and be applicable to represent these cases.

5. Linear Hypothesis

The hypothesis introduced by [23], which considers that in the case of no information, the recovery function may be represented with a linear function, are herein discussed. In particular, such assumptions may be particularly important for the definition of interdependencies. Figure 4 shows the case of linear hypotheses of Q_N, considering the functionality at the time of occurrence t_0E 0 for all the quantities (as above specified).

Such formulation may be described as:

$${\mathrm{Q}}_{\mathrm{N}};\mathrm{t}=\sqrt{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{N}}{\left(\frac{{\mathrm{Q}}_{\mathrm{i}}}{{\mathrm{c}}_{\mathrm{i}}}\right)}^2}+{\mathrm{t}}_{0\mathrm{E}}$$

(5)

That represents a conical solid with different angles with the t-axis, depending on the coefficient c_i. In case of:

$${\mathrm{Q}}_{\mathrm{i}}\ne 0{ \mathrm{and} \mathrm{Q}}_{\mathrm{j}=\mathrm{i}-1,\mathrm{N}}=0$$

(6)

Then, Equation (5) simplifies as:

$${\mathrm{Q}}_{\mathrm{i}}={ \mathrm{c}}_{\mathrm{i}}\left(\mathrm{t}-{\mathrm{t}}_{0\mathrm{E}}\right)$$

(7)

That is the formulation represented in Figure 2 for the linear case.

Now, it is worth noting that the coefficients ${\mathrm{c}}_{\mathrm{i}}$ represent the trends of reparation for the various dimensions since the rapidity of the recovery may differs among the various dimensions. Therefore, the angles of the surface of the cone with the t-axis (vertical axis) are described by ${\mathrm{c}}_{\mathrm{i}}$. In addition, such parameter may be calculated as:

$${\mathrm{c}}_{\mathrm{i}}=\frac{{\mathrm{Q}}_{\mathrm{F},\mathrm{i}}}{{\mathrm{RT}}_{\mathrm{i}}}$$

(8)

where

${\mathrm{Q}}_{\mathrm{F},\mathrm{i}}$ is the final value of the functionality that system i reaches at the end of the recovery process. This final functionality may vary from 1% to 100% respect with the originally functionality (that it in paper is considered 0). Sometimes recovery procedures allow an improvement relative to the original functionality, and this value can be higher than 100%, as described in [7]. Therefore, Equation (5) may be expanded in:

$${\mathrm{Q}}_{\mathrm{N}};\mathrm{t}=\sqrt{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{RT}}_{\mathrm{i}}\frac{{\mathrm{Q}}_{\mathrm{i}}}{{\mathrm{Q}}_{\mathrm{F},\mathrm{i}}}\right)}^2}+{\mathrm{t}}_{0\mathrm{E}}$$

(9)

Such formulation accounts for the fact that the various systems may reach different levels of functionalities ${\mathrm{Q}}_{\mathrm{F},\mathrm{i}}$ and that such level is reached at different repair times ${\mathrm{RT}}_{\mathrm{i}}$. In particular, applying Equation (9) instead of Equation (5) is fundamental in making decisions and defining recovery plans because the seismic resilience of the system depends on the RT of the various dimensions. Therefore, when seismic resilience is quantified, the value of RT needs to be calculated as:

$$\mathrm{RT}=\max \left\{{\mathrm{RT}}_{\mathrm{i}}\right\}$$

(10)

Consequently, (3) becomes:

$$\mathrm{SR}={{\displaystyle \int }}_{\mathrm{t}0\mathrm{E}}^{\mathrm{t}0\mathrm{E}+\mathrm{RT}}\frac{\sqrt{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{N}}{\left({\mathrm{RT}}_{\mathrm{i}}\frac{{\mathrm{Q}}_{\mathrm{i}}}{{\mathrm{Q}}_{\mathrm{F},\mathrm{i}}}\right)}^2}+{\mathrm{t}}_{0\mathrm{E}}}{\mathrm{RT}}\mathrm{dt}$$

(11)

It is worth considering that the choice to calculate RT as the maximum among the RT calculated for the various systems is a realistic assumption that we did. Another important assumption herein considered is that the functionality at the time of the event is 0 for all the cases, meaning that the solid has a vertex in V (t_0E, 0, … 0). This may be considered a limit of the present formulation that needs further development.

Overall, it is worth noting that Equation (11) may consider the interrelations between different systems, their components and subcomponents that are significant when seismic resilience is assessed. In particular, Equation (11) may consider several sources of interdependencies, such as: (1) the effects of structural vulnerability of the built environment and its components on the surrounding environment and community; (2) social component, community characteristics, social capitals; (3) the effect of city configuration and of the street network topology; and (4) the role of urban open space and its distribution. In particular, these aspects may contribute to reduce the seismic resilience of the single infrastructure, as demonstrated in the following case study.

6. Case Study

In this section, a case study is performed to apply the proposed formulation. Two infrastructures have been selected with several assumptions, such as time of occurrence of the earthquake E equal to zero for both the systems, final values of the functionality equal to 1, to simulate 100% recovery and linear recovery functions, to apply Equation (11). The coefficients c₁ and c₂ are calculated by following Equation (8). Therefore, a parametric study has been performed in order to assess the importance of considering the presence of a second infrastructures during the assessment of SR. In particular, the value of the repair time RT was considered 100 crew working days (CWD) for the first infrastructure, while for the second system, four values have been considered: 400, 200, 150 and 100 CWD. In other words, the second infrastructure may take four times, two times, one and a half and the same time to be repaired as the first one. The values of the seismic resilience (see column 5, SR) in

Table 1. SR calculation by applying Equation (11), (RT1: repair time for infrastructure 1, RT2: repair time for infrastructure 2, C1: coefficient for infrastructure 1, C2: coefficient for infrastructure 2, SR: seismic resilience).

RT1 (CWD)	RT2 (CWD)	C1	C2	SR
100	400	0.01	0.0025	0.763
100	200	0.01	0.005	0.786
100	150	0.01	0.0066	0.858
100	100	0.01	0.011	0.948

, may significantly represent the effects of a second infrastructure on the SR. In particular, it is worth noting that considering that the second infrastructure needs a longer RT, means that SR decrease significantly: −19.5%, −17.0% and −9.5% for 400, 200 and 150 CWD. These results may be extended to many other (and even more realistic) case studies, that will be object of future research studies.

7. Conclusions

The paper proposes a novel formulation of the seismic resilience that may consider the presence of interdependencies, defined as the relations that may occur between different systems, their components and subcomponents. Such dimensions are fundamental to be considered for especially for modern societies that are based on multiple systems and may be severely affected by earthquakes. In this regard, the paper discusses the main sources of interdependencies and the importance of accounting inside the resilience definition, together with the limits of the original definition of seismic resilience. The formulation herein proposed for N dimensions was also defined for linear recovery functions, and thus the simplified formulations of resilience has been introduced and discussed. This paper may be considered an attempt to generalize the original resilience formulation by proposing the analytical formulation and its application to a case study that considers the effects of a second interdependent infrastructure on the original one. Future studies are necessary to extend to new implementations and developments.