Review on Energy Resilience

Computer-Aided Civil and Infrastructure Engineering 30 (2015) 583–600

Probabilistic Assessment of Civil Infrastructure Resilience to Earthquakes

Paolo Franchin* & Francesco Cavalieri

Department of Structural and Geotechnical Engineering, University of Rome, La Sapienza, Rome, Italy

Abstract: The large losses occurred in the past due to earthquakes, even in highly developed countries, as well as the ensuing prolonged inactivity of the stricken soci- eties, imparted momentum to research into regional seis- mic impact and community resilience to earthquakes. Need for comprehensive and consistent modeling is ap- parent, and this work presents a contribution in this di- rection. The extension of a recently developed civil in- frastructure simulation framework to the evaluation of resilience, as well as the introduction of a new infrastruc- ture network-based resilience metric represent the novel- ties of the article and allow one to explore the effect that sources of uncertainty and key vulnerability factors have on the probability distribution of resilience.

1 INTRODUCTION

Large earthquake-induced loss incurred by communi- ties throughout the world in the most recent events, is progressively shifting the focus of research from risk and loss assessment to their mitigation. Although the most effective loss mitigation strategy would be to act on exposure through effective land use planning (Burby et al., 2000), it seems that the tendency to settle in high seismicity regions cannot be counteracted. Thus, the main tools for loss mitigation remain vulnerability reduction and resilience improvement. Effectiveness of any improvement measure under economic constraints is maximized by attaining deeper understanding of what makes our communities and societies resilient to earthquakes.

The concept of resilience has been introduced to mea- sure not only the system’s ability to absorb externally induced changes, as implied in vulnerability, reliability, and robustness measures, but also its ability to adapt to postevent circumstances, and recover its predistur-

∗To whom correspondence should be addressed. E-mail: paolo. [email protected].

bance state. It was ecologists that initially conceptual- ized this notion of resilience. Holling (1973) defined resilience as a measure of the persistence of systems and of their ability to absorb change and disturbance and still maintain the same relationships between pop- ulations or state variables. To date, the concept of re- silience has been adapted to multiple fields. Ainud- din and Routray (2012) provided a survey of resilience definitions, whereas Paton (2005) and, later, Longstaff et al. (2010) presented comprehensive discussions fo- cusing on community resilience.

The earthquake engineering notion of resilience has started with the work by Bruneau et al. (2003), where the seismic resilience of a community is defined as its ability to:

• absorb a shock if it occurs (with an abrupt reduc- tion of performance);

• recover quickly after the shock (reestablish normal performance).

Bruneau et al. (2003) introduced Q(t), a measure of quality or functionality of a community’s infrastructure, varying with time elapsed from a disturbance from zero to one, and defined resilience as the integral of qual- ity loss between time t1 of the event and time t2 of full recovery. This definition is at the base of the MCEER “triangle” (Renschler et al., 2007), spanned by loss of Q(t) and recovery time. Quantitative measurement of resilience then requires a meaningful definition of Q(t) for the analyzed system. For systems designed to deliver physically measurable products, such as electric power and water supply networks, the definition and quantifi- cation of Q(t) is straightforward. On the other hand, for systems where the delivered service is not a sim- ple engineering unit, Q(t) is harder to define and quan- tify. An example is health-care facilities (Bruneau and Reinhorn, 2007; Lupoi et al., 2008; Cimellaro et al.,

C© 2014 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/mice.12092

584 Franchin & Cavalieri

2010), where health treatment must be analyzed from both broader societal context and engineering context.

In the literature, resilience definitions appear to have intersections and overlaps with other characteristics of the system, such as preparedness, robustness, flexibility and recovery capacity, all of which treated as indepen- dent measures. The framework by Faturechi and Miller- Hooks (2013) clarifies these intersections and pro- vides a common approach for their quantification and maximization.

Recently, a different conceptualization of resilience has been provided by Ouyang and Dueñas-Osorio (2012). Based on the concept of three-stage (i.e., disas- ter prevention, damage propagation, and recovery) re- silience, presented in detail in Ouyang et al. (2012), they pointed out that infrastructure systems are continuously evolving, together with the occurrence rates and inten- sities of some hazards. Hence, they proposed a time- dependent resilience metric, given by the ratio between the time integral of the actual performance curve and the time integral of the target performance curve. Such integrals are upper bounded to the lifetime of the sys- tem, so to allow for eventually more than one event to occur and for future evolution of both system and environment.

Besides the definition of an appropriate resilience metric, another key element of resilience evaluation is modeling the recovery process.

Reconstruction strategies were investigated, for ex- ample, by Opricovic and Tzeng (2002), who developed a multicriteria decision-making model for the analy- sis of planning strategies for reducing the future so- cial and economic costs in an area subjected to natural hazards. Mitigation of hazardous effects is proposed in the form of comprehensive reconstruction plans, includ- ing the redevelopment of urban areas and infrastruc- tures, multipurpose land use and restrictions on build- ing in hazardous areas. Something similar has been im- plemented in real-world practice in Christchurch after the 2011 earthquakes (CERA, 2013). Chang and Miles (2003) proposed an agent-based model of the recov- ery, focusing on the influence of agents’ environments, such as buildings and transportation network, on their recovery processes. The model considers the behav- iors of economic agents (households and businesses) within a community, as well as the relationships be- tween agents themselves and with their environment. A conceptual model and prototype implementation of dis- aster recovery process, according to the object-oriented paradigm, were proposed by Miles and Chang (2006). Five types of effects (i.e., dynamic, agent-attribute, in- teraction, spatial, and policy effects) were considered in the model, which explores the relationships between households, businesses, lifelines, and critical facilities.

Nejat and Damnjanovic (2012) recognized the housing reestablishment as a critical factor on the overall tim- ing of recovery. The recovery process is based on the dynamic behavior of homeowners in the aftermath of disasters. In the study, a preliminary agent-based model was proposed, accounting for homeowners’ dynamic in- teractions with neighbors’ activities, such as reconstruc- tion and relocation. Recovery models could also be de- veloped on the basis of the availability of modern tools, such as regional and local earthquake rapid loss as- sessment systems (Erdik et al., 2011). In fact, modern technology allows assessment of building damage and casualties within a few minutes after an earthquake, thus improving the reduction of losses, accelerating the recovery process and increasing the community re- silience. Many researchers developed frameworks that are useful in assisting infrastructure owners and deci- sion makers in planning mitigation strategies. Among them, Zobel (2011), starting from the resilience defini- tion given by Bruneau et al. (2003), proposed a tech- nique for adjusting the resilience prediction based on the preferences and priorities of a given decision maker, to allow for a more accurate representation of the perceived value of different resilience scenarios. As a further example in this direction, that is, supporting seis- mic rehabilitation decisions, Alesch et al. (2003) de- veloped a platform combining a probabilistic model of ground shaking, engineering fragility curves, statis- tical estimates of potential damage costs, and a financial model of the costs and benefits of rehabilitation.

Based on the above, it should be clear that resilience is too broad to be described by a single metric, and that an accepted optimal set of metrics has not emerged yet. It should be also recognized that resilience is affected by factors that are uncertain, such as the size of the initial shock (a function of hazard and systemic vulnerability), or the behavior of agents in the postshock environment, as well as by policy-related factors, such as the strategic decisions made after the event about recovery actions and priorities.

Within this vast reference frame, this article focuses on one specific aspect of community resilience, which is housing reestablishment, and presents a novel network- based metric along the lines of Asprone et al. (2013). This metric should be regarded as a candidate mem- ber of a larger set apt to give a comprehensive picture of resilience. The goal of the article is to explore the influence of different factors on this resilience metric, such as the uncertainty in seismic hazard, physical vul- nerability of considered systems (which include build- ings, road network, and interacting utilities such as wa- ter and power networks), and functional consequences. To this end, the work builds on a comprehensive model of urban environments including buildings and a set of

Probabilistic assessment of civil infrastructure resilience to earthquakes 585

Fig. 1. The input–output sequence of the developed model.

interconnected infrastructural systems (Franchin and Cavalieri, 2013; Cavalieri et al., 2012), previously lim- ited to the evaluation of the initial impact of an earth- quake, and extends it to include the recovery process.

Section 2 provides a brief overview of the base model (Cavalieri et al., 2012; Franchin and Cavalieri, 2013) and presents its extension, together with details of the eval- uation of injured, fatalities, displaced population, and the new resilience metric (hereafter termed simply re- silience). Section 3 reports results of the probabilistic assessment carried out on a realistic synthetic city, to- gether with a sensitivity study on the key factors. Fi- nally, Section 4 contains conclusions and future work.

2 METHODOLOGY

2.1 Framework for earthquake impact and resilience evaluation

Figure 1 illustrates the framework for resilience eval- uation based on the model presented in Franchin and Cavalieri (2013) and Cavalieri et al. (2012). Following backward the input–output sequence, resilience, as ex- plained in detail later on, is a function of displaced pop- ulation, road damage and recovery strategy (Nejat and Damnjanovic, 2012; Asprone et al., 2013). People may be displaced from their houses, still usable based solely on direct physical damage, because the residual service level in the utilities is too low to make them habit- able for the considered season and associated weather. Time of the day when the earthquake strikes also con- curs, through building occupancy, in determining actual impact in terms of displaced population, as well as of

injured and fatalities. Road interruption decreases the capability of people to carry out their everyday activi- ties (it can even completely impair accessibility, a rel- evant problem for rescue teams in the immediate af- termath of the event), and can be due to direct (e.g., bridge collapse) or indirect (debris from adjacent col- lapsed buildings) physical damage. The recovery strat- egy determines how damage is repaired, and, second only to the initial state of damage (a function of haz- ard and vulnerability), is the most influential parameter on resilience.

To support evaluation of the quantities in Figure 1, a comprehensive model of a set of interconnected in- frastructural systems (a system of systems, or an In- frastructure, in the terminology of PCCIP, 1997) was developed. The model includes a detailed taxonomy of an extensive, though not exhaustive, subset of the systems (SYNER-G, 2012): buildings (BDG), electric power networks (EPN), water supply systems (WSS), waste-water networks (WWN), gas and oil distribution systems (GAS-OIL), road networks (RDN), harbors (HBR) and health-care systems (HCS). A list of com- ponents (or subsystems, with components, where appro- priate) for all these systems, with associated description, fragility and functional data, was produced. As a second step, interdependencies between the systems were iden- tified, as explained in Pinto et al. (2012).

To describe the identified systems and interactions, an Object-Oriented (OO) model was developed (pre- vious examples include, e.g., Chang and Chamberlin, 2004; Schläpfer et al., 2008; Miles and Chang, 2006). Ac- cording to the OO paradigm, the problem is described as a number of interacting objects, which are instances of classes. Classes and their relationships can be rep- resented graphically, using the Unified Modeling Lan- guage (UML) (Booch et al., 2007), with so-called class diagrams, like the one shown in Figure 2. There are two central classes. The first is Environment: this is com- posed of the Infrastructure, the Hazards acting upon it, and Weather conditions. The second class is Analysis, which acts upon the Environment, and is the general- ization of all possible analysis methods (i.e., it is what is called an abstract class). Probabilistic analysis is car- ried out with a method (e.g., Monte Carlo simulation, MCS, or Importance Sampling simulation, ISS) among the available ones (simulation or nonsimulation, the lat- ter not shown) in the Analysis class.

The Infrastructure class is composed of three classes: networks, buildings, and critical facilities. Buildings, and the people within them, play a central role: they are the origin and destination of all flows of goods and services. Ordinary residential or commercial/industrial (nonhaz- ardous) buildings are modeled in an aggregate fash- ion, by considering geocells characterized in terms of

586 Franchin & Cavalieri

Fig. 2. Class diagram of the object-oriented model. Abstract classes have names italicized according to the standard UML

notation.

summary data: building volume by typology, with as- sociated fragility models, and demographic attributes (population, income, education, etc.). The aforemen- tioned information, usually fragmented in databases that adopt different subdivisions of the urban area (typ- ically, a Land Use Plan, an Urban Audit, and a Build- ing Census), is projected onto a set of mutually exclu- sive and collectively exhaustive geocells using simple area ratio rules (see Section 2.2, where the adopted ap- proach is explained). Network-like systems can be an- alyzed in terms of form and flow. At the former level, they are graphs made up of nodes and edges. All com- mon properties and methods have thus been designed and implemented just once at a higher level (exploit- ing the mechanism of class inheritance), leading to the abstract Network class: all its concrete classes (one per network type) will have properties and methods inher- ited from this class, plus their own specific methods (e.g., flow equations and component damage models will differ).

Interactions are modeled as associations between the classes corresponding to the interacting objects. These

relationships between objects can be visualized as de- pendency edges in a supergraph. One example is the link between each geocell and its reference nodes (one for each network, to evaluate utility loss, see Section 2.2 and Franchin and Cavalieri, 2013) and each network node to a number of tributary cells (to evaluate the de- mand of goods based on population). Another example is the link between a pumping station and an electrical substation (WSS and EPN node types, respectively, not shown in Figure 2) that causes loss of function in the former due to physical damage in the latter.

Hazards are currently limited to seismic hazard. The Seismic class is composed of three subclasses, to repre- sent sources, events, and local intensity at each site of interest. The latter is represented as a vector s of in- tensity measures (IMs). These are the IMs in terms of which fragility models for the components at the site are formulated and are used in the direct damage eval- uation stage: they include ground shaking intensity on rock outcrop (abstract class implementing ground mo- tion prediction equations, GMPEs, which in Figure 2 is the generalization of the concrete class Akkar and Bommer, 2010, used in the example to follow), at the surface of soil deposits (Amplification models) and per- manent ground deformation (PGD) due to relevant geotechnical hazards. The collection of vectors s at all sites in the analysis region is a realization of a multi- variate random field (see Section 2.4 and Franchin and Cavalieri, 2013).

2.2 Injured, fatalities, displaced population and data granularity

All the considered measures of social impact (displaced population, injured, and fatalities) employed in the evaluation of resilience, require knowledge of pi t , the population in each cell by building typology. Evaluation of the latter quantity is carried out differently based on the granularity of data in each of the data sources. Pop- ulation data, coming from an urban audit, can be given at the city, district, or building level, by increasing reso- lution. Building information from the building census, in terms of volume, typology and usage, can be simi- larly given at the city, district, and building levels. Dis- tricts in the building census and in the urban audit, as it has already been mentioned, will not coincide. Table 1 shows that some combinations of inputs are not realis- tic, that is, it is unlikely that one has detailed building- level population data, without knowing any physical characteristics of the buildings. On the other hand, it may be possible that more refined data about buildings are known, while only generic population density infor- mation is available at the city, or at the (urban audit) district level.

Probabilistic assessment of civil infrastructure resilience to earthquakes 587

Table 1 Population and building data granularity

Building census

Building District Vb, Ibt , Ibi , Ibu

Vkt = ∑

b

Vb Ibk Ibt City

Urban audit Building pb Case 1 N/A N/A District Pj =∑

b pb Ib j Case 2 Case 4 N/A

City P = ∑ j Pj Case 3 Case 5 Case 6 Note: Symbols are defined in the text.

Fig. 3. Distribution of population by hour, for working days (left) and holidays (right) (from Zuccaro and Cacace, 2011).

Starting with Case 1 where refined data are known for both population and physical properties/usage of build- ings, the population in cell i in buildings of type t is given by

Pi t = ∑

u Pi t u = Pi t 1 + ∑

u>1 Pi t u = = ∑b pb Ibi Ibt f1 (h, d ) + ∑u>1 Pu Vi t uVu = = ∑b pb Ibi Ibt f1 (h, d )+

+ ∑u>1 [

fu (h, d ) ∑

b pb ] ∑

b Vb Ibi Ibt Ibu∑ b Vb Ibu

(1)

where pb (from the urban audit) and Vb (from the build- ing census) are building b population (zero unless use is residential, i.e., Ib1 = 1) and volume; Ibi, Ibt, and Ibu are indicator functions equal to one if building b belongs to cell i, typology t, and usage u, respectively, and zero otherwise. Furthermore, at any given time of the day the functions fu(h, d), satisfying the property∑

u fu (h, d ) = 1 (2)

describe the distribution of city population among dif- ferent activities. Four activities are considered, corre- sponding one-to-one to the land use codes: (1) home– residential, (2) shop–commercial, (3) work–industrial, and (4) outdoor–green. Figure 3 shows an example of functions of this type, used in the application (Section 3) and obtained by fitting cubic splines to data in Zuccaro

and Cacace (2011). Such functions vary depending on the type of weekday (working day or holiday).

Equation (1) states that, in a way of simplification, the population in the ith cell in buildings of type t at any given hour h of day d is the sum of the proportion f1 of the resident population that remains at home, plus a share of the total city population not staying at home, proportional to the volume of buildings of type t and use u (u > 1, i.e., other than residential) in cell i over the total volume of buildings for use u in the city. This is done to avoid a rigorous determination of the popu- lation in the cell involved in other activities, that would require a distribution similar to that employed in the se- quential model for transportation (some of the residents will shop/work, etc., in the same cell, others will move to another and similarly people from other cells will be moving to the current cell, with the overall constraint that

∑ i ∑

t Pi t = P ). The second case is when refined data at building level

are known from the building census whereas population data are known in an aggregated way at the urban au- dit district level. Correspondingly, population in cell i in buildings of type t is given by

Pi t = Pi t 1 + ∑

u>1 Pi t u = = Pi Vi t 1Vi 1 f1 (h, d ) + P

∑ u>1 fu (h, d )

Vi t u Vu

= = ∑ j Pj A j iA j

∑ b Vb Ibi Ibt Ib1∑

b Vb Ibi Ib1 f1 (h, d ) +

+ ∑ j Pj ∑u>1 fu (h, d ) ∑

b Vb Ibi Ibt Ibu∑ b Vb Ibu

(3)

where Aj and Aji are the district area and its intersection with cell i area, and are used to project as explained the district population to the cell. Equation (3) has the same meaning as Equation (1), with only a slight modification in the Pit1 evaluation.

The third considered case is that in which population data are known very roughly only at the city level. In this case, the distribution proportionally to cell volume in a given typology and use applies to all uses, and thus the formula simplifies to

Pi t = ∑

u Pi t u = P ∑

u fu (h, d ) Vi t u Vu

= = P ∑u fu (h, d )

∑ b Vb Ibi Ibt Ibu∑

b Vb Ibu

(4)

Case 4 is the most likely in practical situations, with consistently rough data on buildings and population available only at the district level. In this case, popu- lation in cell i in buildings of type t is given by

Pi t = Pi t 1 + ∑

u>1 Pi t u = = Pi Vi t 1Vi 1 f1 (h, d ) + P

∑ u>1 fu (h, d )

Vi t u Vu

= = ∑ j Pj A j iA j

∑ k Vkt

Aki 1 Ak∑

t

∑ k Vkt

Aki 1 Ak

f1 (h, d ) +

+ ∑ j Pj ∑u>1 fu (h, d ) ∑

k Vkt Aki u Ak∑

t

∑ k Vkt

Aku Ak

(5)

588 Franchin & Cavalieri

where Ak, Aku, and Akiu are the areas of the kth build- ing census district and of its intersections with the uth land use district and with the uth district and ith cell, respectively (the latter, as explained, are determined at preprocessing with polygon intersection tools).

Finally, Cases 5 and 6 have simple expressions for Pit

Pi t = Pi t 1 + ∑

u>1 Pi t u = = Pi Vi t 1Vi 1 f1 (h, d ) + P

∑ u>1 fu (h, d )

Vi t u Vu

= = ∑ j Pj A j iA j

∑ k Vkt

Aki 1 Ak∑

t

∑ k Vkt

Aki 1 Ak

f1 (h, d ) +

+ ∑ j Pj ∑u>1 fu (h, d ) ∑

k Vkt Aki u Ak∑

t

∑ k Vkt

Aku Ak

(6)

Pi t = P Vt V

Ai A

(7)

but are unlikely to be employed because the resolution of input data is probably too coarse for the results to be useful.

Once Pit is known, social impact metrics can be eval- uated as follows. Casualties are estimated based on a model, proposed by Zuccaro and Cacace (2011) in ac- cordance with Coburn and Spence (1992), and modified herein to account for the fact that all data are referred to the cells. In particular, the number of injured people PI and the number of dead people (fatalities) PF are given by

PI (h, d ) = ∑

i

∑ t

Pi t (h, d ) ∑

s Ii t s Q I t s (8)

PF (h, d ) = ∑

i

∑ t

Pi t (h, d ) ∑

s Ii t s Q F t s (9)

where Iits = 1 if buildings of type t in cell i are in damage state s, and zero otherwise, while factors QIts and QFts are experimentally derived casualty ratios. The damage state of buildings is determined with an approach sim- ilar to ATC-58 (ATC, 2007), with probability masses of each state obtained as differences of fragility values of the associated limit states given the current seismic intensity at the building site. More details are given in Cavalieri et al. (2012), where the issue of associating the same damage state to all buildings of a given type in a given cell, as well as that of using the centroid seis- mic intensity rather than an average value over the cell, are also discussed. The Q-ratios account for the propor- tion of the population living in type t buildings experi- encing damage s that is either injured or killed in the event.

Evaluation of displaced population proceeds as fol- lows. As already mentioned, according to the model, population can be displaced from their homes either be- cause of direct physical damage (building usability) or because of lack of basic services/utilities (building hab- itability) resulting from damage to other interconnected

systems. The proportions of the cell population being in fully usable, partially usable, and nonusable buildings are obtained as

Pi,FU = ∑

t Pi t (h = 4am, d = any)

× ∑

s Ii t s UFUt s (10)

Pi,PU = ∑

t Pi t (h = 4am, d = any)

× ∑

s Ii t s UPUt s (11)

Pi,NU = ∑

t Pi t (h = 4am, d = any)

× ∑

s Ii t s UNUt s (12)

where the hour of the day is obviously set to 4 AM, when f1 = 1 and all residents are at home, and the func- tions U are experimentally derived usability ratios giv- ing the proportion of fully, partially, and nonusable type t buildings experiencing damage state s.

Habitability will be expressed as a function of Utility Loss UL, that is, the weighted average

UL = ∑Nutilities

n wn (1 − SSIn) (13)

where SSIn is the nth utility system serviceability index and wn is the weight associated with the importance of loss in utility system n’.

Although nonusable buildings are obviously nonhab- itable, the proportion of nonhabitable buildings among fully or partially usable ones will be given by

H i,FU = 1 − H (ULi − ULFU) (14)

H i,PU = 1 − H (ULi − ULPU) (15) where H is the Heaviside step function and ULFU and ULPU are tolerable threshold values, depending on weather conditions (good/bad weather) at the time of the event.

The total displaced population is then evaluated as

Pd = ∑

i

( Pi,FU H i,FU + Pi,PU H i,PU + Pi,NU

) − PF (16)

2.3 Resilience

Asprone et al. (2013), aiming at reconciling the en- gineering (Bruneau et al., 2003) and the ecosystem (Holling, 1973) visions of resilience, introduced a novel metric defined in a broad sense with reference to the whole urban system as its ability to recover the full func- tional level, in terms of housing reestablishment, exist- ing prior to the event even if in a new, different state. Their definition of resilience is based on the notion of efficiency of a hybrid physical–social network, that is,

Probabilistic assessment of civil infrastructure resilience to earthquakes 589

the spatially distributed physical network of infrastruc- tures and facilities, enriched with extra nodes represent- ing citizens. The model presented in Sections 2.1 and 2.2 represents one such hybrid network.

According to Latora and Marchiori (2001), the notion of efficiency of communication E between two nodes in a network can be defined as inversely proportional to their shortest distance dij. The shortest distance is the minimum sum of distances (arc lengths) over all possi- ble paths between nodes i and j. The average efficiency over the network (simply “efficiency” in the following) can be defined in the form:

E = 1 N (N − 1)

∑ i �= j

1 di j

(17)

where N is the number of nodes of the network and the product N(N − 1) is the maximum number of direct connections between the N nodes in a directed graph (where the arc between i and j does not imply an arc between j and i). Efficiency is maximum when all direct connections exist and dij = lij, the geodesic distance be- tween nodes i and j.

In general, the plain definition of efficiency can be adapted to any physical network simply by replacing the geodesic distance with a system-specific distance, for ex- ample, electrical distance (Arianos et al., 2009). In this study, the efficiency of the city road network is appro- priately weighted to account for population density, so that it becomes the efficiency of communications be- tween citizens. Within the presented model, the expres- sion of efficiency in terms of cell population takes the form

E = 1 P ( P − 1)

∑ i

Pi

[ ( Pi − 1) +

∑ l �=i

Pl d ei l di l

] (18)

where de and d are the Euclidean distance and the length of the shortest path along the network between the ith and jth cells (the latter set to � when the nodes are disconnected). The ratio de/d � 1 is a normaliza- tion adopted for road networks. Efficiency is an upper bounded quantity. For any set of nodes, the maximum of efficiency is attained when the graph is complete, that is, any node is directly connected to all other nodes (the adjacency matrix is a matrix of ones). In the case of straight arcs, de/d = 1 for any pair of nodes and maxi- mum efficiency becomes

max (E ) = ∑

i Pi [( Pi −1)+ ∑

l �=i Pl ] P ( P−1) =

= ∑

i Pi ( ∑

i Pi −1) P ( P−1) = 1

(19)

In all other cases max(E) < 1. Efficiency can be computed in preearthquake condi-

tions, and termed E0 < 1, and then, after an event, em- ploying the damaged network and considering that a

Fig. 4. Evolution of efficiency for two damage cases and associated resilience values.

fraction Pd of the population has been displaced. Al- though displaced population is reallocated according to the chosen recovery strategy (Figure 1), which governs the repair of buildings and roads (and utilities), effi- ciency can be computed over time defining a recovery curve E(t). Following Asprone et al. (2013), use of the reallocated population Pr as a measure of the progress of the recovery process (in place of elapsed time from the event) avoids economic considerations. The pro- posed resilience metric is then defined as

R = 1 Pd E0

∫ Pd 0

E ( Pr) d Pr (20)

where the normalization by Pd and E0 is required to obtain a measure bounded between zero and one. The upper integration bound equal to Pd corresponds to the case where the entire displaced population is re- allocated in the region affected by the event (in the preshock buildings or eventually in different ones). The above definition, on the other hand, covers also those cases where part of the population relocates to other re- gions, that is, max(Pr) < Pd, resulting in a permanent efficiency shift.

Figure 4 illustrates the evolution of efficiency with Pr/Pd for two cases of identical damage to buildings and different damage to the road network. In the first case, where roads are not damaged nor blocked by debris, ef- ficiency increases linearly (see later). In the second case, there are step jumps corresponding to rehabilitation of edges, which are more pronounced when the repaired edges play the role of “bridges” (in the “graph theo- retical” sense) for the road network, and thus increase

590 Franchin & Cavalieri

Fig. 5. Representation of uncertainty as a network of random variables.

sharply the de/d ratios for most cells. The figure also shows the associated resilience values.

2.4 Uncertainty

Uncertainty is described in the model in terms of ran- dom variables that describe single random quantities or discretized random fields.

The full set of random quantities describing uncer- tainty in the model is exemplified in Figure 5, with reference to the simple example of two interconnected systems (system 2 depends on system 1) with two com- ponents each. In the figure, each circle represents a random variable and arrows represent statistical de- pendence between the variables. Variables and arrows form a network, which is a graphical (partial) repre- sentation of their joint probability distribution. Such a representation is commonly termed a Bayesian Net- work (BN), but the attribute Bayesian is meaningful only as long as Bayesian operations are carried out on it by reevaluating the distribution of a subset of variables given observations on another subset of vari- ables. In this case, the network is just the represen- tation of a single run in a forward simulation: so far,

plain Monte Carlo and Importance Sampling enhanced with K-means clustering (Jayaram and Baker, 2010) are implemented.

The hazard portion of the network is similar to the BN model of seismic hazard presented by Straub et al. (2008). In each simulation run (one event or scenario), magnitude M is sampled first, its marginal density given by the weighted sum over the active seismic sources of the magnitude densities given an event, with weights equal to the mean annual rates of events on each source, λ0. Given magnitude, the source S is sampled from the subset of those that can generate such magnitude (dis- crete distribution, with probability mass proportional to λ0). The location L is then sampled within source S (with probability depending on the source model, uni- form for area sources). Given M and L, the inter- and intraevent (the latter spatially correlated) errors η and ε in a ground-motion prediction equation are sampled to arrive at a chosen local intensity, for example, peak ground acceleration (PGA), on a rectangular grid on rock Sgr. Interpolation to the components’ sites gives bedrock or rock outcrop values of the chosen inten- sity (in the figure each intensity at a component site, Sr, is a function of those at the closest four grid points, e.g., Sr1 is a function of Sgr1, Sgr2, Sgr4, and Sgr5), condi- tional on which all further needed IMs can be sampled. Surface intensity values Ss are then obtained with ran- dom amplification functions A. Secondary, geotechni- cal hazards Sg are finally obtained with relevant models (e.g., the FEMA/HAZUS, 2003, model where landslid- ing probability is a function of surface PGA). Details can be found in Franchin and Cavalieri (2013). Once values for the appropriate IMs at the surface are known at each site, the physical damage state of each compo- nent is sampled. This is done differently for point-like and line-like components (Der Kiureghian, 2009), with fragility curves used for the former, and Poisson rates as a function of the intensity for the latter. The Poisson rate λ is usually employed to describe the rate of faults per unit length, and is experimentally derived from re- pair rates obtained in postearthquake damage surveys. For pipes, based on the rate one can sample a number of leaks NL and determine whether the pipe is broken B (Figure 5). Figure 5 also shows a number of variables (gray filled) denoted by θ that are used to model epis- temic uncertainty. These express uncertainty on the dis- tribution parameters of other variables and are usually sampled first (hierarchical model). Not all the θ vari- ables in the figure have currently been included in sim- ulations carried out with the model, but, for instance, log-mean and log-standard deviation of the yield and collapse fragilities of buildings (for each of the 16 build- ing categories in the identified taxonomy) have been modeled as joint variables (Crowley et al., 2014; Lago- marsino and Cattari, 2014).

Probabilistic assessment of civil infrastructure resilience to earthquakes 591

Fig. 6. The synthetic city. Black dots and empty circles denote demand nodes and sources, respectively. The empty square in the EPN layer is the generator chosen as balance

node.

3 APPLICATION

The evaluation of resilience, according to the proposed metric and based on the presented model, is exempli- fied in this section with reference to the “synthetic” city shown in Figure 6. The city has a 15 by 15 km square footprint and includes utilities (electric power and wa- ter supply networks) and a transportation (road) net- work. Size, properties, and topology of the component systems mimic those of real systems. Seismicity in the area is described by means of a number of distributed seismic zones.

Sections 3.1, 3.2, and 3.3 provide a detailed descrip- tion of the input, of the results and of sensitivity analy- ses and the associated insights, respectively.

3.1 Case study description

Figure 6 shows several layers, from top to bottom: BDG, WSS, EPN, RDN and two seismic sources, sub- divided into three (1, 2, and 3) and two (4 and 5) simple rectangular areas, respectively.

The square footprint of the city is subdivided into nine geometrically coincident Subcity Districts (SCDs) and Building Census Areas (BCAs), each of size 5 by 5 km.

Figure 7 shows the 10 areas identified by the fic- titious land use plan, with labels indicating their use type: residential (R), commercial (C), industrial (I), and

Fig. 7. City discretization according to the land use plan, with indication of use type.

green (G). The scheme mimics a typical layout with an old historical city, that is, the central (mainly) res- idential area, surrounded by a park, possibly outside the remnants of former city walls and modern construc- tions in new suburbs, with residential areas on one side, and commercial/industrial areas on the other. The to- tal population amounts to 1,800,000, corresponding to an average density equal to 8,000 inhab./km2, which is in the high range for Italian cities. The total population has been concentrated in the four SCDs overlapping with residential areas (see Figure 7), each of them thus characterized by a population of 450,000. Hence, the actual density results to be 18,000 inhab./km2 (Rome, for instance, has an average density of about 2,000 inhab./km2, with peaks of 17,000 inhab./km2).

Two typologies are used for buildings, deemed rep- resentative of “older” (e.g., pre-WWII in the European context) and “modern” constructions: unreinforced ma- sonry (URM, Type I) and reinforced concrete (RC, Type II) buildings. The central BCA includes only Type I buildings, to reproduce the city historical center, whereas the remaining BCAs contain Type II buildings only.

Buildings in each typology generally exhibit variabil- ity in height and other properties, but for the purpose of this application they are assigned central values (see Table 2).

With these data, the masonry total volume and number of buildings in the central BCA amount to 135,000,000 m3 and 45,000, respectively, whereas for the three remaining residential BCAs they amount to

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Table 2 Properties of the two building typologies

Property Type I Type II

No. floors 3 6 Interstory (m) 4 3 Average height (m) 12 18 Average footprint (m2) 250 500 Volume per person (m3) 300 150

Table 3 Parameters of fragility curves for buildings

Yield Collapse

μln σ ln μln σ ln

Type I Mean −2.029 0.572 −1.320 0.552 CoV (%) 19 16 28 24

Type II Mean −1.832 0.474 −1.091 0.485 CoV (%) 33 21 48 24

67,500,000 m3 and 7,500, respectively. The latter val- ues have been assigned also to the nonresidential BCAs containing Type II buildings.

The corresponding seismic fragility models are as- signed on the basis of a comprehensive literature sur- vey conducted within the SYNER-G (2012) project and focusing on reinforced concrete and masonry buildings. About 50 studies have been reviewed, each one usu- ally investigating more than one building typology and identifying a number of fragility function sets. To com- pare all the different fragility functions, harmonization has been carried out with reference to: (1) IM type (all the curves have been expressed in terms of PGA), (2) limit states (yield and collapse limit states have been adopted), and (3) building typology (based on the taxonomy for European buildings proposed within the project). As a result of this work, for each typology a set of two lognormal fragility curves is provided, whose parameters are themselves characterized as joint log- normal variables (epistemic uncertainty). Table 3 re- ports the mean and CoV of fragility parameters for Types I and II buildings (low-rise, bearing wall ma- sonry buildings and mid-rise, seismically designed, non- ductile bare frames, respectively, according to Crow- ley et al. 2014 and Lagomarsino and Cattari, 2014). Table 4 reports the corresponding correlation matrices for each typology, necessary to ensure that yield and collapse fragility curves do not cross. The lowest cor- relation values, between log-mean and log-standard de- viations of different fragilities, have negligible statistical significance owing to the size of the sample used to eval-

Table 4 Correlation among parameters of fragility curves

Yield Collapse

μln σ ln μln σ ln

Type I Yield μln 1 −0.220 0.647 −0.051 σ ln −0.220 1 0.119 0.709

Collapse μln 0.647 0.119 1 0.199 σ ln −0.051 0.709 0.199 1

Type II Yield μln 1 0.158 0.783 0.033 σ ln 0.158 1 0.118 0.614

Collapse μln 0.783 0.118 1 −0.453 σ ln 0.033 0.614 −0.453 1

uate them and have therefore been set to zero for the analysis.

The values employed for casualty ratios in Equa- tions (8) and (9) come from Zuccaro and Cacace (2011) and are calibrated for the 2009 L’Aquila event. Assum- ing that only the collapse damage state causes injured people and fatalities, the values are: QI = 0.84 and QF = 0.19 for Type I buildings, QI = 0.62 and QF = 0.38 for Type II buildings.

As far as the three networks are concerned, they are all characterized by grid or mesh-like topological struc- ture, typical in urban areas of the main links connecting suburbs or districts, and can be considered as transmis- sion/distribution (TD) systems. Dueñas-Osorio (2005) provided a network model to represent real TD sys- tems, based on the ideal class of the d-lattice-graph, an unweighted, undirected, regular graph of dimension d with vertices joined to their lattice neighbors according to specified rules.

For the case at hand, a modified two-lattice with ver- tex degree (number of edges incident at the vertex) d(v) = 8 for all vertices is used as a substrate graph, that is, a template from which TD models (realizations) for the network typologies of interest can be sampled. To allow the sampling of TD models, it is first needed to remove the periodicity of the boundary vertices of the lattice, so that such vertices will have a smaller vertex degree than the internal ones. Because TD models exist on adjacency matrices of square topologies, the num- ber of vertices in the two directions must be the same. Hence, an aperiodic TD substrate of order n is gener- ated first, and then m edges of the substrate are retained with a probability of existence equal to pm. The ran- dom creation of the graph is repeated until a connected graph is obtained. The probability pm can be estimated empirically for each network typology as the ratio be- tween the edge densities of real networks and aperi- odic TD substrates, with the edge density being the ratio

Probabilistic assessment of civil infrastructure resilience to earthquakes 593

between the number of edges of a graph with n ver- tices, and the number of edges in a complete graph of order n. The ratio between densities of real graphs and TD substrates reduces to the ratio between their num- ber of edges, m. Empirical relations have been derived for expressing the number of edges m as a function of n for real graphs, thus allowing for obtaining pm as a function of n (Dueñas-Osorio, 2005). The following de- scribes in more detail the three lifelines generated for the case study.

The 49 nodes/buses of the EPN are subdivided into five gate stations, deriving power from the high-voltage network at national scale, and 44 load/demand nodes (or buses). The gate stations, considered nonvulner- able, are the source nodes for the city distribution network. The buses are enclosed into distribution sub- stations, modeled in detail with all their internal compo- nents to capture short-circuit propagation and interme- diate operational states (Franchin and Cavalieri, 2013; Cavalieri et al., 2013). Nodes are connected by a total of 70 overhead lines, some of which perform voltage transformation. The generation nodes collectively sup- ply 4,800 MW of power at high voltage (380 kV). The balance node or slack bus, which is one of the source nodes, supplies the power imbalance between genera- tion and demands. Smaller voltage (220 and 60 kV) lines convey energy to distribution substations, where the sin- gle power demands are aggregated.

The 49 nodes of the WSS are subdivided into 44 sink/demand nodes and 5 source nodes, two of which are constant-head water sources, whereas the remain- ing three are modeled as variable-head water tanks served by pumping stations (linked to the closest EPN substation). Thus, pumping stations can be out of ser- vice, making the water source unavailable, as a result of failure of the reference substation feeding the pump with power. The pump–tank systems are dimensioned to cover the necessary water requirements under nor- mal operational conditions (5.2 m3/s are delivered, cor- responding to a daily water equipment of 250 liters per inhabitant per day). A total of 71 pipes connect the nodes.

The RDN is composed of 64 nodes and 114 bidirec- tional edges. All nodes are centroids of traffic anal- ysis zones (TAZs), that is, points where citizens ac- cess and leave the road network. Edges are subdivided for the purpose of fragility evaluation into embank- ments, trenches, unstable slopes, bridges, and plain road segments.

The implemented models for EPN, WSS, and RDN allow capacitive (flow) analysis in steady-state condi- tions. However, the actual flows are computed in the simulation only for the EPN (in alternating current, AC) and the WSS, given the lack, at the current stage

of modeling, of a postearthquake travel demand model for the RDN, which is thus studied in terms of pure connectivity.

Vulnerable components in the EPN are all the sub- station components, as described in Vanzi (1996), with related fragility curves being lognormal (function of PGA), specified in terms of their two parameters (Cav- alieri et al., 2014). Notably, seismically induced damage to the components of a substation can have nonlocal consequences, leading to a short-circuit that may prop- agate within the substation and eventually further away from that substation to adjacent ones, generating in ex- treme cases large-scale blackouts (Franchin and Cava- lieri, 2013; Cavalieri et al., 2013).

In the WSS, only pipelines are vulnerable and the fragility model is given in terms of two Poisson repair- rates per kilometer, functions of peak ground velocity (PGV, in cm/s) and PGD (in m), respectively (ALA, 2001):

λrepair (PGV) = K1 · 0.0024 · PGV λrepair (PGD) = K2 · 11.224 · PGD0.319

(21)

where K1 and K2 are functions of the pipe material, soil, joint type and diameter and λrepair is returned in km−1. The number of leaks/breaks (variable NL in Figure 5) for the generic pipe is randomly generated us- ing the highest repair-rate as the mean of the Poisson distribution (variable λ in Figure 5). If NL > 0, a number NL of standard uniform numbers is sampled and com- pared with the rupture probability (a function of λ). If at least one is lower than the rupture probability, the pipe is broken and removed from the network, the asso- ciated outflow being attributed to the end nodes of the removed pipe as additional demands. If no rupture oc- curs, the total leakage area is determined as the number of leaks times the area of one leak and the correspond- ing outflow distributed to the end nodes.

For the RDN, the edges are the vulnerable com- ponents. Lognormal fragility curves in terms of PGA characterize all the edge typologies but road segments, whose fragility model is formulated in terms of PGD (Tsionis and Fardis, 2014; Argyroudis and Kaynia, 2014). For each typology, different limit states are taken into account in the original models, but only the collapse fragility curve has been considered for this example, as the network connectivity alone is concerned. Road blockage caused by building collapse is also accounted for, so that RDN edges can be deleted from the graph not only for a seismically induced damage, but also for being blocked by debris. The implemented road block- age model, described in Gehl et al. (2011), considers for the generic road segment three different functionality levels: open, open for emergency, and closed. Because

594 Franchin & Cavalieri

buildings are modeled through geocells, the projection of roads onto cells is taken into account. The probabil- ity of exceeding a given functionality level is obtained as a function of a number of parameters, such as percent- age, average height and damage state of each building typology in the cell, length and width of the projected road segment in the cell, presence of buildings on one or both sides of the road, etc.

Concerning the seismic environment, two sources are considered to affect the city. They are discretized into three and two rectangular seismogenetic areas, respec- tively, for a total of five areas, displayed and numbered in Figure 6. Their activity is characterized in terms of the parameters for the truncated Gutenberg–Richter recur- rence law. In particular, λ0 (i.e., the mean annual rate of the events in the source with M greater than the lower limit ML) is set to 0.014, 0.016, 0.020, 0.018, and 0.022 for the five areas (such values have been obtained by de- composing one λ0 value through area ratios within each of the two sources), while the magnitude slope β and lower and upper magnitude limits ML and MU are set to 2.72, 4.5 and 6.5 for the first three areas, and 1.70, 5.0 and 6.5 for the remaining two areas. The λ0 values have been calibrated to produce an average seismicity in the area grossly equivalent to Italian first hazard zone (500- year return period PGA equal to 0.35g).

To predict the local seismic intensity, PGA is taken as the primary IM. PGV is used as a secondary IM as it is needed as an input to pipes’ fragility model. The GMPE by Akkar and Bommer (2010), recently de- veloped for the European context, is used to perform both simulation of PGA and conditional simulation of PGV. The correlation parameter (range) estimated by Esposito and Iervolino (2011), based on European records, is used in the spatial correlation model. For within-site correlation coefficient between PGA and PGV, ρPGA,PGV, the value 0.754 is used. No site amplifi- cation is considered for this example. The geotechnical hazard module of the OO framework has been used to compute PGD at the locations of RDN edges’ and WSS pipes’ centroids. The rectangular grid for the prediction of the primary IM consists of 69 × 27 = 1,863 points, corresponding to a grid size of 3 km. Such size, com- pared to the employed range value of 13.5 km, allows a proper consideration of spatial correlation of intraevent residuals.

3.2 Results

One MCS with 1,000 runs has been carried out, which yields stable estimates of all considered performance metrics. As an example, the moving average of dis- placed population (normalized to total population) sta- bilizes around 22% after about 500 runs. Similar results

Fig. 8. CDFs of fatalities (left) and displaced population (right), both normalized to the total population.

have been obtained for the remaining computed met- rics. As already pointed out above, the displaced popu- lation is a function of the weather conditions at the time of the event: the results refer to good weather condi- tions. All the analyses carried out for this work, whose results are presented in this and the following sec- tion, take into account the interdependencies between the modeled infrastructural systems, as explained in Section 2.1.

In each simulation run, the number of fatalities and the displaced population are computed using Equa- tions (9) and (16), respectively. Figure 8 shows the Cu- mulative Distribution Function (CDF) of fatalities (left) and displaced population (right), both normalized to the total population (i.e., 1,800,000). The extreme values (right tail) in the distributions correspond to almost to- tal disruption (recall that fatalities are subtracted from displaced population).

At the end of the generic simulation run, resilience is computed. To this aim, a recovery strategy for the displaced population and the road network must be adopted. The chosen one, called status quo down–up by Asprone et al. (2013), consists of subdividing the pro- cess in n steps, so that in each step a fraction 1/n of the displaced population is reallocated, reconstructing the building cells based on their number of displaced peo- ple, from the smallest to the largest. When a cell is reha- bilitated, all blocked or damaged roads incident to the reference TAZ of that cell are recovered; in the last step of the process, all the remaining damaged roads are re- opened, returning to the undamaged network adjacency matrix. Such a process is carried out under the assump- tion of unlimited resources available, to avoid tempo- ral and economic considerations. During the recovery process, in which the reallocated population Pr ranges between 0 and Pd, efficiency increases from the value attained at the time of the event to the preshock value E0 (in this case 0.697).

A subset of the recovery curves E(Pr/Pd) obtained within the simulation is shown in Figure 9.

Probabilistic assessment of civil infrastructure resilience to earthquakes 595

Fig. 9. Subset of retrieved efficiency recovery curves. The two dashed lines refer to two particular cases of intact and

damaged road network.

Fig. 10. CDF of resilience.

In any run the initial decrease in efficiency, from the preevent value E0 to the immediate aftermath one E(0), can be due to the displaced population component only of the efficiency expression in Equation (18), when no damage or blockage of roads occur, or due to both Pd and de/d. Two such cases are highlighted in Figure 9 with dashed lines: in the former case the recovery is almost linear, whereas the latter presents the steeper increases already shown in Figure 4.

Starting from the recovery curves, resilience can be obtained in each run as explained in Section 2.3. The CDF of resilience is shown in Figure 10 (log-scale is used here, because it is used later in Figure 13 to en- hance visibility of differences in the left tail). The low- est possible value, corresponding as shown later to the

Fig. 11. Scatter plot of resilience and CoV versus normalized displaced population.

extreme events in the right tail of the CDFs of Figure 8, is R = 0.45.

In Figure 11, the obtained 1,000 resilience values are plotted against the normalized displaced popula- tion. The reduction of maximum resilience with in- creasing displaced people is clearly visible. In particu- lar, this reduction appears to follow a straight line. For a generic value of Pd/P, the highest value of R corre- sponds to an earthquake that caused only the disloca- tion of people (linear recovery curve), whereas lower values refer to cases in which also RDN damage is present (i.e., the recovery curve presents some steps). The figure also shows (right axis) the coefficient of variation of R as a function of Pd/P. In general, the variability is stable and very low. It has only a small increase in the low Pd/P range, where repairs to the RDN have larger influence: this can be explained by recalling the adopted reallocation strategy, which re- covers first the roads incident to the reference TAZ of the reallocated cells; as a consequence, the rela- tive position of disrupted roads and cells to be reallo- cated becomes a relevant aspect for resilience compu- tation. For events causing intermediate values of the displaced population, the number of seismically dam- aged or blocked roads will increase leading to a smaller range of possible damage scenarios, and hence less dis- persed resilience values. When magnitude increases fur- ther, the variability of (larger) damage scenarios re- duces again.

As far as the road network performance is concerned, besides the indirect indication obtained through the re- silience estimate, Figure 12 shows the Mean Annual Frequency (MAF) of exceedance of the simple and weighted connectivity loss in the RDN (singled out among the three considered networks).

596 Franchin & Cavalieri

Fig. 12. Mean annual frequency of exceedance curves of SCL and WCL.

These global metrics (SCL and WCL) represent the average level of disconnection induced by the event in the network. The former, which accounts only for the existence of a path between two TAZs, obviously ex- hibits a stepwise behavior (many cases are judged equiv- alent based on the plain count of nonfunctional edges), whereas the latter takes into account also the increase in travel time due to the seismically induced damage suf- fered by the RDN. Complete disconnection occurs with a 1,000 years return period (for MAFs of 10−3) and cor- responds to the lowest resilience values in the left tail of Figure 10.

3.3 Sensitivity study

To explore the influence of different factors on the proposed resilience metric and on its relationship with the displaced population, a sensitivity analysis has been performed. In particular, six different cases have been analyzed:

• Case 1: complete, that is, the case discussed in the previous section;

• Case 2: exclusion of spatial correlation of in- traevent model error, ε;

• Case 3: exclusion of epistemic uncertainty on the parameters of fragility curves for buildings;

• Case 4: exclusion of road blockage (but road seg- ments are still vulnerable to ground shaking);

• Case 5: casualty estimation for a daytime event, in place of a nighttime one;

• Case 6: like Case 1 with an alternative recovery strategy, termed bridge first, down-up (upgrade priority given to the damaged graph “bridges”, i.e., to those edges whose repair is going to increase

Fig. 13. CDF of resilience for the six analyzed cases.

most the de/d ratio, and then to buildings from smaller to larger).

For each case, an MCS with 1,000 runs has been car- ried out. It is noted that care has been taken to ensure that the same realizations of random variables common to all cases have been used in different simulations, so that the changes in the respective results are only due to differences in the model, and not to pseudo-aleatory uncertainty.

Figure 13 compares the results of the six cases in terms of resilience CDF. Notably, all cases lead to sim- ilar median values (around 0.8). Larger differences are confined to the tails, and especially the left one, ampli- fied in the plot in log-scale. The coincidence of Case 6 with the reference case is strongly related to the spe- cific example and to the RDN topology, that is, to the presence of bridges in the graph. In any case, when a state of complete damage is approached (left tail), the efficiency recovery curves are fundamentally driven by population reallocation, which is the same in the refer- ence and the alternative strategy. As far as Case 3 is con- cerned, neglecting the epistemic uncertainty on building fragility leads to milder damage, a lower number of fa- talities and, hence, increased displaced population; this explains the lower resilience, with respect to Case 1, ob- tained in the left tail. In the opposite tail, the efficiency drop in the aftermath of the event is mainly caused by road damage, which in Case 3 is reduced due to the lower influence of road blockage; this leads to higher resilience, as expected.

Cases 2 and 4 lead to a reduced variance of re- silience values that is apparent in higher resilience in the left tail and lower resilience in the right one. For Case 2, this is expected since correlation ampli- fies extreme loss cases. As far as Case 4 is concerned,

Probabilistic assessment of civil infrastructure resilience to earthquakes 597

Fig. 14. Scatter plot of resilience against normalized displaced population, for the six analyzed cases.

Fig. 15. Relation between resilience and normalized efficiency drop.

results highlight that most of the damage to roads comes from collapsed buildings debris, and hence not con- sidering road blockage leads to an overestimation of resilience.

Finally, Case 5 leads to the lowest resilience values. This is easily explained in terms of the chosen metric and underlying model. As shown in Figure 11, resilience decreases with displaced population, and according to the model, Equation (16), displaced population is com- puted subtracting fatalities. In Case 5, time is set to 11 AM and for the considered example this means that part of the resident population from the central, more vul-

nerable masonry-only district is carrying out other ac- tivities in different districts where less vulnerable RC construction is present. Hence, lower fatalities occur, in- creasing the displaced population.

Figure 14 shows the six cases in terms of resilience versus normalized displaced population plots such as that in Figure 11. This visualization allows appreciating better the reduction in dispersion for Cases 3 and 4, as well as the lower R values for Case 5.

Overall, the sensitivity study confirms the expected influence of each factor and, most importantly, the rel- atively low variability induced in the relationship be- tween initial damage (as expressed by the proxy of dis- placed population) and resilience (as measured by the proposed metric).

4 CONCLUSIONS

The article presented: (1) a novel metric of network- based resilience that, along the lines of the proposal by Asprone et al. (2013), is based on the evolution of efficiency of communication between citizens during the reallocation of displaced population after the event, (2) the extension of a framework recently developed by the authors, based on which this metric can be as- sessed probabilistically, and (3) a study of the influence of different factors on the proposed metric, such as the uncertainty in seismic hazard, physical vulnerability of considered systems and functional consequences. The outcome of the sensitivity study carried out on a syn- thetic city provided the following insights:

� Resilience, as defined herein, is very well predicted based on the initial damage, estimated in terms of displaced population or, better, in terms of the nor- malized efficiency drop [E0 − E(0)]/E0, as shown in Figure 15. The relation is not deterministic, but it is characterized by a low dispersion around a clear, geometrically explainable linear trend given by the equation (black line in Figure 15):

R = 1 − 0.5 E (0) E0

(22)

� The alternative recovery strategy (Case 6), which assigns to network damage highest priority in the rehabilitation works (by choosing to repair all the graph “bridges” in the first step) seems to have a minor influence. Given the structure of the effi- ciency expression at the base of the resilience met- ric, this can only be explained as a nongeneral con- clusion limited to the particular considered case, where the network has been obtained starting from

598 Franchin & Cavalieri

a TD substrate, which is a quite regular and con- nected graph. In this case, graph bridges in an RDN are very few, because they usually corre- spond to real bridges and important crossings, like when a city developed around a major river, which is not the case for the considered synthetic exam- ple. It is expected, however, that this strategy will lead always to the largest resilience values given initial damage.

The presented metric is intended as one component of a vector of resilience metrics that can only jointly give an adequate description of the resilience of a commu- nity. Notably, other metrics should measure economic resilience. The proposed metric cannot presently give account for this aspect. Actually, economic considera- tions are voluntarily avoided by measuring restoration progress through the proxy of reallocated population. This is done to avoid the difficulty of predicting fac- tors such as resources availability and policy-making. The proposed metric can be easily evaluated replac- ing Pd with time, but this will require an extension of the supporting simulation model, to include the time- dimension. This is the object of ongoing research. Fu- ture work will be devoted to investigate the influence on resilience of different recovery strategies, highlight- ing their practical feasibility and meaningfully compar- ing their relative merit accounting for their cost and progress over time. Furthermore, restoration progress will be modeled in a more accurate manner, differen- tiating the restoration processes of each system (trans- portation, power, water, etc.).

ACKNOWLEDGMENT

The authors gratefully acknowledge financial support from the European Commission, through grant agree- ment no. 244061 for the SYNER-G collaborative re- search project, and from the Laboratories University Network of seismic engineering (ReLUIS), within the ReLUIS-DPC 2010-2013 research program.

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